Mathematics in Applied Sciences and Engineering https://ojs.lib.uwo.ca/mase
Volume 1, Number 2, June 2020, pp.150-180 https://doi.org/10.5206/mase/10644

SELECTED TOPICS ON REACTION-DIFFUSION-ADVECTION MODELS FROM

SPATIAL ECOLOGY

KING-YEUNG LAM, SHUANG LIU, AND YUAN LOU

Abstract. We discuss the effects of movement and spatial heterogeneity on population dynamics

via reaction-diffusion-advection models, focusing on the persistence, competition, and evolution of

organisms in spatially heterogeneous environments. Topics include Lokta-Volterra competition models,

river models, evolution of biased movement, phytoplankton growth, and spatial spread of epidemic

disease. Open problems and conjectures are presented. Parts of this survey are adopted from the

materials in [89, 108, 109], and some very recent progress are also included.

1. Introduction

Recent years have witnessed unprecedented progress of experimental technologies in the life sci-

ences. The explosion of empirical results have rapidly generated massive sets of loosely structured data.

The analytical methods from mathematics and statistics are required to synthesize the large data sets

and extract insightful information from them. This trend continues to accelerate the development in

mathematical biology. The study of mathematical biology usually includes two aspects. On the one

hand, by introducing and analyzing mathematical models, researchers can elucidate and predict the

basic mechanisms of underlying biological processes. On the other hand, deeper understanding of these

mechanisms may drive the discovery of new mathematical problems, new analytical techniques, and

even launch new research directions. Most, if not all, areas of mathematics have found applications in

mathematical biology.

As an important mathematical area, nonlinear partial differential equations have been one of the

most active research fields in the 21st century. It has also found new opportunities in mathematical

biology in recent years. In [47] Dr. Avner Friedman discussed some challenging problems arising from

mathematical biology, and one of the major mathematical tools he used is nonlinear partial differential

equations. The subject of partial differential equations concerns the changes of quantities in space and

time. For biology, the importance of space is hardly a question, and a practical question is to quantify

the effect of space in different biological scenarios. In this survey, we consider some issues in spatial

ecology via reaction-diffusion models, focusing upon the effect of movement of species on population

dynamics in spatially heterogeneous environments. An important ongoing trend in population dynamics

is the integration with other directions of biosciences, such as evolution, epidemiology, cell biology,

cancer research, etc. Our main goal is to showcase the role of reaction-diffusion models by integrating

across different research directions in mathematical biology, including spatial ecology, evolution and

Received by the editors 15 April, 2020; revised 15 May 2020; accepted 15 May 2020; published online 17 May 2020.

2010 Mathematics Subject Classification. 35K57, 92D25, 92D40, 92D30, 37N25.

Key words and phrases. reaction-diffusion; advection; competition; evolution; dispersal; population dynamics.
KYL and YL were supported in part by NSF Grant DMS-1853561.

SL was partially supported by the Outstanding Innovative Talents Cultivation Funded Programs 2018 of Renmin

Univertity of China and the NSFC grant No. 11571364.

150

https://ojs.lib.uwo.ca/mase
https://dx.doi.org/https://doi.org/10.5206/mase/10644


SELECTED TOPICS ON REACTION-DIFFUSION-ADVECTION MODELS 151

disease transmissions. We will illustrate that the quantitative analysis of some important questions

in biosciences does bring interesting and novel mathematical problems, which in turn calls for the

development of new mathematical tools. We will also formulate a number of potential research questions

along the way.

This paper is organized as follows. In Section 2 we discuss several single species models, for which the

dynamics have important implications for the invasion of exotic species. Section 3 is devoted to the study

of two-species competition models, with emphasis on the effects of dispersal and spatial heterogeneity

on the outcome of competition for two similar species. In Section 4 we investigate the competition

models with directed movement and show how different dispersal (either random or biased) can bring

dramatic changes to population dynamics. Section 5 concerns the continuous trait models and evolution

of dispersal. Finally, in Sections 6 and 7 we discuss some progress on dynamics of phytoplankton growth

and disease transmissions, respectively.

2. Single species models

The study of mathematical models for single species has played a primary role in population dynam-

ics. It is also crucial in analyzing the dynamics of multiple interacting species, e.g. on issues concerning

the invasions of exotic species. In this section we will focus on two types of single species models and

study the effect of varying the diffusion rate.

2.1. Logistic model. In this subsection, we consider the logistic model

∂tu = d∆u + u[m(x) −u] (x,t) ∈ Ω × (0,∞),
∂u

∂ν
= 0 (x,t) ∈ ∂Ω × (0,∞),

u(x, 0) = u0(x) x ∈ Ω,

(2.1)

where Ω is a bounded domain in RN with smooth boundary ∂Ω and ν(x) denotes the unit outward
normal vector at x ∈ ∂Ω. Here u(x,t) represents the density of the single species at location x and time
t. Parameter d > 0 is the diffusion rate and ∆ =

∑N
i=1

∂2

∂x2
i

is the usual Laplace operator. The function

m ∈ C2(Ω) accounts for the local carrying capacity or the intrinsic growth rate of the species, which is
assumed to be strictly positive in Ω for the sake of clarity. The Neumann boundary condition indicates

that no individuals can move across ∂Ω, i.e. the habitat is closed. We assume that the initial data u0
is non-negative and not identically zero.

It is well known [12] that

lim
t→∞

u(x,t) = u∗(x,d) uniformly for x ∈ Ω,

where for each d > 0, the function u∗(·,d) is the unique positive solution of

d∆u + u(m−u) = 0 x ∈ Ω,
∂u

∂ν
= 0 x ∈ ∂Ω.

(2.2)

If the spatial environment is homogeneous, i.e. m is a positive constant, then u∗(x,d) ≡ m is the unique
positive solution of (2.2), which is independent of the diffusion rate d. If the spatial environment is

heterogeneous, i.e. m = m(x) is a non-constant function, then u∗(x,d) depends non-trivially on d. In

this case, a natural question is how the total biomass of the species at equilibrium, i.e.
∫

Ω
u∗(x,d) dx,

depends on the diffusion rate d.

The following property was observed in [105]:



152 K.-Y. LAM, S. LIU, AND Y. LOU

Lemma 2.1. If m is non-constant, then for any diffusion rate d > 0,∫
Ω

u∗ >

∫
Ω

m = lim
d→0

∫
Ω

u∗ = lim
d→∞

∫
Ω

u∗.

Lemma 2.1 means that (i) the heterogeneous environment can support a total biomass greater than

the total carrying capacity of the environment, which is quite different from the homogeneous case; (ii)

the total biomass, as a function of d, is non-monotone; it is maximized at some intermediate diffusion

rate, and is minimized at d = 0 and d = ∞, respectively. Examples were constructed in [98] to illustrate
that this function can have two or more local maxima.

Lemma 2.1 implies that
∫

Ω
u∗/

∫
Ω
m > 1 for all d > 0, and the lower bound 1 is optimal. For the

upper bound of
∫

Ω
u∗/

∫
Ω
m, Ni raised the following question:

Conjecture 1 ([130]). There exists some constant C = C(N) depending only on N such that∫
Ω

u∗
/∫

Ω

m ≤ C(N), and C(1) = 3.

Conjecture 1 suggests that the supremum of the ratio of the biomass to the total carrying capacity

depends only on the spatial dimension. For one-dimensional case, C(1) = 3 was proved in [6] and it

turns out to be optimal. However, it is recently shown by Inoue and Kuto [74] that Conjecture 1 fails for

the higher-dimensional case even when Ω is a multi-dimensional ball. That is, the nonlinear operator

which maps m to u∗ is not bounded as a map from the positive cone of L1(Ω) to itself.

A related question is the optimal control of total biomass [38]:

Optimization problem. Fix any δ ∈ (0, 1), and define the control set

U :=
{
m ∈ L∞(Ω) : 0 ≤ m ≤ 1,

∫
Ω
m = δ|Ω|

}
.

Determine those m∗ ∈ U which can maximize the total biomass
∫

Ω
u∗ in U.

Nagahara and Yanagida [128] showed that, under a regularity assumption, the optimal m∗ is of

“bang-bang” type, i.e. m∗ = χE for some measurable set E ⊂ Ω, where χE denotes the characteristic
function of E. More recently, Mazari et al. [121] proved that the bang-bang property holds for all large

enough diffusion rates. Biologically, the set E can be viewed as the protected area, and the condition∫
Ω
m = δ|Ω| means that the total resources are limited. The results in [121, 128] imply that under the

limited resources, the way to maximize the total population size of species is to place all the resources

evenly in some suitable subset E of Ω. However, the characterization of the set E is a challenging

problem. It seems that the problem has not been completely solved even in the one-dimensional case;

see [79] for some progress on cylindrical domains. Recent work [127] for the discrete patch model suggests

that the set E is periodically fragmented; see also [123] for general fragmentation phenomenon. For the

biased movement model with dispersal term ∇· ((1 + m)∇u), the optimal distribution of resources for
maximizing the survival ability of a species was considered in [122]. They showed that the problem has

“regular” solutions only when the domain is a ball and the optimal distribution can be characterized

in this case; see [11] for the one-dimensional case.

Compared to (2.2), a more general model is

d∆u + r(x)u

(
1 −

u

K(x)

)
= 0 x ∈ Ω,

∂u

∂ν
= 0 x ∈ ∂Ω,

(2.3)

where r is the intrinsic growth rate and K is the carrying capacity of environment. Is it possible to find

general sufficient conditions for r,K such that the maximal biomass size of species can be reached at



SELECTED TOPICS ON REACTION-DIFFUSION-ADVECTION MODELS 153

some intermediate diffusion rate? This question for (2.3) was proposed and studied by DeAngelis et al.

[35]. We refer to [59] for some relevant developments in this regard, which shows that the total biomass

can be a monotone increasing function of the diffusion rate for some choices of r and K.

In the study of a predator-prey model [113], the following problem arises: Is the maximum of the

density in (2.2), i.e. maxx∈Ω u
∗(x,d), monotone decreasing with respect to the diffusion rate d? Some

recent progress is obtained in [97], where the monotonicity was proved for several classes of function m.

However, it remains open to show the monotonicity for general function m. In contrast, the minimum

of the density, i.e. minx∈Ω u
∗(x,d), is not necessarily monotone increasing in d; see [63]. We conjecture

that the difference between the maximum and minimum values of the density, which measures the

spatial variations of the density, is a monotone decreasing function of d. It is known that
∫

Ω
|∇u∗|2,

which also measures the spatial variations of the density, is monotone decreasing with respect to d; see

[98] for more details.

2.2. Single species models in rivers. How do populations persist in streams when they are con-

stantly subject to downstream drift? This problem is intensified when the habitat quality at the

downstream end is very poor. Once the species are washed downstream, the chances of survival will

be greatly reduced. This problem, termed as the “drift paradox, has received considerable attention

[65, 126]. Speirs and Gurney [144] pointed out that the action of dispersal can permit persistence in an

advective environment, and they proposed the following reaction-diffusion-advection model:

∂tu = d∂xxu−α∂xu + u(r −u) x ∈ (0,L), t > 0,
d∂xu(0, t) −αu(0, t) = u(L,t) = 0 t > 0,
u(x, 0) = u0(x) x ∈ (0,L),

(2.4)

where positive constants α and r are the advection rate and the growth rate of species, respectively.

The interval [0,L] represents an idealized river, assuming that the upstream end x = 0 is a no-flux

or closed boundary, while at the downstream end x = L the zero Dirichlet boundary condition (also

called lethal boundary condition) is imposed. Biologically interpreted, the downstream end describes

the junction of rivers and seas, as freshwater fishes cannot survive in the seawater. The persistence of

species is equivalent to the instability of the trivial equilibrium, which is in turn characterized by the

negativity of the principal eigenvalue of the following problem:{
d∂xxϕ−α∂xϕ + (r + λ)ϕ = 0 x ∈ (0,L),
d∂xϕ(0) −αϕ(0) = ϕ(L) = 0.

(2.5)

By an explicit calculation, Speirs and Gurney [144] obtained the following necessary and sufficient

condition for the persistence of species:

d >
α2

4r
and L > L∗(d) :=

π − arctan
(√

4dr −α2/α
)

√
4dr −α2/2d

. (2.6)

Set L := infd>0 L
∗(d). As illustrated in Figure 1, condition (2.6) implies that

(i) if L ≤ L, then the species cannot persist for any diffusion rate d. In this case, the equilibrium
u = 0 is globally stable among all non-negative initial values;

(ii) if L > L, then there exist 0 < d < d such that the species persists if and only if d ∈ (d,d).
Furthermore, if d 6∈ (d,d), then u = 0 is globally stable; if d ∈ (d,d) then (2.4) admits a unique
positive steady state which is globally stable.

Biologically, the conditions for species survival are that the river has a minimal length and the

diffusion rate of the species lies within a certain range: when the diffusion rate is small, the drift is so

dominant that the species is washed downstream and cannot survive; when the diffusion rate is large,



154 K.-Y. LAM, S. LIU, AND Y. LOU

 

 

 

 

 

 

 

 

Figure 1. The diagram of L∗(d) illustrated by the black colored solid curve. The

parameter region where the species persists is shaded in blue.

the species is exposed too much to the lethal boundary at the downstream end and will also go to

extinction. Therefore, it was conjectured in [89, 110] that there is an intermediate diffusion rate d∗

which is the “best” strategy, i.e. once the diffusion rate of a species is d∗, no other species can replace

it. This conjecture implies that if the majority of individuals in a population adopt the strategy d∗,

then any invasion fails so long as the invading species is adopting a strategy different from d∗. In the

evolutionary game theory, such d∗ is referred to as an evolutionary stable strategy [119]. See Subsection

3.2 and Conjecture 3 for further details.

In (2.4), the downstream end is assumed to be the lethal boundary. Recent work considering general

boundary conditions [115, 148] showed that the species persists if and only if the velocity of flow drift

is less than some threshold value. Under some other boundary conditions, this threshold value turns

out to be a monotone decreasing function of the diffusion rate, while this is not the case for (2.4). This

leads to a natural evolutionary question: Is the fast diffusion or slow diffusion more beneficial to the

species in streams? We will discuss this question in Subsection 3.2. Some new persistence criteria has

been found for general boundary conditions at the downstream in a most recent work [57], where the

critical river length can be either monotone decreasing in the diffusion rate d or it can first decrease and

then increase as d increases, depending on the loss rate at the downstream; see also [156] for related

discussions. The diagram in Figure 1 can be regarded as an extreme case of the new persistence criteria

found in [57].

Finally, we refer to [43, 78, 116, 117, 147, 151, 152] for related work on the persistence problem in

rivers and river networks.

3. Competing species models

One basic question in spatial ecology is: How does the spatial heterogeneity of environment affect the

invasion of exotic species and the competition outcome of native and exotic species? The term hetero-

geneity refers to the non-uniform distribution of various environmental conditions. For example, in the

oceans, light intensity decreases with respect to the depth due to light absorption by phytoplankton and



SELECTED TOPICS ON REACTION-DIFFUSION-ADVECTION MODELS 155

water [80], and thus is unevenly distributed. An exotic species can successfully invade often because it

has some competitive advantage over the resident species. However, in the heterogeneous environment,

with the help of diffusion alone, invasion is still possible even though the exotic species does not have

any apparent competitive advantage. In this section, we consider some classical competition models to

illustrate this phenomenon.

3.1. Lotka-Volterra competition models. We first consider the following Lotka-Volterra competi-

tion model in the homogeneous environment:




∂tu = d∆u + u(a1 − b1u− c1v) (x,t) ∈ Ω × (0,∞),
∂tu = D∆v + v(a2 − b2u− c2v) (x,t) ∈ Ω × (0,∞),
∂u

∂ν
=
∂v

∂ν
= 0 (x,t) ∈ ∂Ω × (0,∞),

u(x, 0) = u0(x), v(x, 0) = v0(x) x ∈ Ω,

(3.1)

where u(x,t) and v(x,t) are the population densities of two competing species with diffusion rates d

and D > 0, parameters a1,a2 are their intrinsic growth rates, b1,c2 are the intra-specific competition

coefficients, and b2,c1 are the inter-specific competition coefficients. We assume for the moment that

ai,bi,ci, i = 1, 2, are positive constants, i.e. the spatial environment is homogeneous. If the coefficients

satisfy the weak competition condition

b1
b2
>
a1
a2

>
c1
c2
, (3.2)

then (2.4) admits a unique positive steady state (u∗,v∗) given by

u∗ =
a1c2 −a2c1
b1c2 − b2c1

, v∗ =
a2b1 −a1b2
b1c2 − b2c1

. (3.3)

Theorem 3.1 ([10]). Suppose that the environment is homogeneous and (3.2) holds. Then the posi-

tive steady state (u∗,v∗) is globally asymptotically stable among all non-negative and non-trivial initial

conditions.

Theorem 3.1 shows that when the environment is homogeneous and the competition is weak, two

competing species can always coexist regardless of the initial conditions. A natural question arises:

Does Theorem 3.1 remain valid in spatially heterogeneous environment?

Obviously, when ai,bi,ci are non-constant functions in Ω, (u
∗,v∗) given by (3.3) is generally non-

constant, and thus not necessarily a positive steady state of (3.1). However we may still ask: If

condition (3.2) holds pointwisely on Ω, does (3.1) admit a unique positive steady state which is globally

asymptotically stable? Surprisingly, even if (3.2) holds pointwisely in Ω, it is possible that one of the

two competing species eventually becomes extinct, regardless of initial conditions, provided that two

competitors have proper diffusion rates. Mathematically, for the homogeneous case, (3.1) can be viewed

as an ordinary differential equation, which has a globally stable positive steady state for each x ∈ Ω.
In contrast, for the heterogeneous case, the global attractor of the partial differential equation model

(3.1) could simply be a semi-trivial steady state, so that the density of one competitor stays bounded

from below while the other tends to zero, as t → ∞. Thus, the spatial heterogeneity of environment
also plays a crucial role here.



156 K.-Y. LAM, S. LIU, AND Y. LOU

For further discussions on the competitive exclusion in spatially heterogeneous environment, we

consider the following simplified competition model:


∂tu = d∆u + u[m(x) −u− bv] (x,t) ∈ Ω × (0,∞),
∂tv = D∆v + v[m(x) − cu−v] (x,t) ∈ Ω × (0,∞),
∂u

∂ν
=
∂v

∂ν
= 0 (x,t) ∈ ∂Ω × (0,∞),

u(x, 0) = u0(x), v(x, 0) = v0(x) x ∈ Ω,

(3.4)

where m(x) represents the common resource of two competitors and the weak competition condition

(3.2) is reduced to 0 < b,c < 1.

We assume that m(x) is positive, smooth and non-constant to reflect the inhomogeneous distribution

of resources across the habitat. Then for d,D > 0, (3.4) has two semi-trivial states, (u∗, 0) and (0,v∗),

where u∗ is the unique positive solution of (2.2), while v∗ is the unique positive solution of (2.2) with

d replaced by D. For the spatially heterogeneous environment, the stability of semi-trivial states is a

subtle question. For example, the stability of (u∗, 0) is determined by the sign of the principal eigenvalue

λ1 of the problem 

D∆ϕ + (m− cu∗)ϕ + λϕ = 0 x ∈ Ω,
∂ϕ

∂ν
= 0 x ∈ ∂Ω,

(3.5)

where u∗(x,d) depends implicitly on d. It is well-known [12] that if λ1 > 0, then (u
∗, 0) is stable; if

λ1 < 0, then (u
∗, 0) is unstable. When m is constant, under weak competition condition 0 < b,c < 1,

direct calculations give λ1 < 0, i.e. (u
∗, 0) is always unstable for any d,D > 0. In contrast, for non-

constant m, the situation becomes complicated and interesting: First, the variational characterization

of the principal eigenvalue [12] implies that λ1 is monotone increasing with respect to D, and it tends

to minΩ(cu
∗ −m) as D approaches 0. Notice from (2.2) that m−u∗ must change sign in Ω. If c < 1

(in fact c ≤ 1 is sufficient), then minΩ(cu
∗ −m) < 0, which implies that λ1 < 0 for small D. For large

D we have

lim
D→∞

λ1 =
1

|Ω|

(
c

∫
Ω

u∗ −
∫

Ω

m

)
,

which highlights the role played by the total biomass
∫

Ω
u∗ in the stability of (u∗, 0). To be more precise,

define

c∗ := inf
d>0

∫
Ω
m∫

Ω
u∗
.

From Subsection 2.1 we see that c∗ ∈ (0, 1), and if c < c∗, then (u∗, 0) is unstable for all d,D > 0. It
is interesting that when c ∈ (c∗, 1), we can find some non-empty open set Σ such that the semi-trivial
solution (u∗, 0) is stable if and only of (d,D) ∈ Σ, so that we can define

Σ := {(d,D) ∈ (0,∞) × (0,∞) : (u∗, 0) is stable} . (3.6)

Observe that for any c ∈ (c∗, 1), the set Σ is always a subset of {(d,D) : 0 < d < D}.
Recently, He and Ni [62] proved a surprising and insightful result: for a class of competition models,

if the semi-trivial or positive solution is locally stable, then it must be globally stable. The following

result, as a corollary of their general result, completely solves a conjecture proposed in [105, 107]:

Theorem 3.2. Suppose that 0 < b ≤ 1, c ∈ (c∗, 1) and m is non-constant. If (d,D) ∈ Σ, then (u∗, 0)
is globally stable among all non-negative and non-trivial initial conditions; if (d,D) 6∈ Σ and d < D,
then (3.4) admits a unique positive steady state which is globally stable.



SELECTED TOPICS ON REACTION-DIFFUSION-ADVECTION MODELS 157

Theorem 3.2 shows that for some diffusion rates, competition exclusion takes place even if the weak

competition condition is satisfied everywhere in Ω. The case b = 1 and c∗ < c < 1 should be pointed

out particularly. In this case, when the environment is homogeneous, i.e. m is constant, the equilibrium

solution (0,m) is globally stable, so that the faster diffusing species excludes the slower one. In contrast,

for the spatially heterogeneous environment, Theorem 3.2 implies that for proper diffusion coefficients,

the final outcome of the competition is reversed: the slower diffusing species excludes the faster one!

Furthermore, when b = 1 and c approaches 1 from below, the following result holds:

Theorem 3.3. Assume that b = c = 1. If d < D, then (0,v∗) is unstable and (u∗, 0) is globally stable.

Conversely, if d > D, then (0,v∗) is globally stable.

In fact, as c tends to 1 from below, the set Σ increases monotonically and converges to the set

{(d,D) : 0 < d < D} in the Hausdorff sense. Biologically, this implies that the slower diffusing species
will replace the faster one in the spatially heterogeneous environment. This is the so-called evolution of

slow dispersal. More precisely, if we consider the slower diffuser as a mutant among the faster diffusing

population, then mutations for slower diffusion rate are selected, i.e. the diffusion rate of the “new

resident” is smaller than the “previous resident”. Through constant mutation and competition, the

diffusion of new species thus becomes slower and slower. This phenomenon was first discovered by

Hastings [58], and Theorem 3.3 was proved by Dockery et al. [39]. Moreover, it is shown in [71] that

this result may fail for spatio-temporally varying environments; see also [104].

The above discussion illustrates that diffusion in a spatially heterogeneous environment has a dra-

matic effect on the dynamics of competition models, which is rather different from the case of a ho-

mogeneous environment. It should be noted that there are still many questions which remain open

even for two competing species. We refer the interested readers to [23, 60, 61, 92, 129] for some related

discussions. For general resource distributions, it is challenging to fully determine the stability of the

semi-trivial state of (3.1); see [14, 70, 106] for examples of multiple reversals of the competition outcomes

for two-species Lotka-Volterra competition models. A remaining problem is to find some conditions on

the resource distribution, so that the local stability of two semi-trivial steady states of (3.1) can be

accurately characterized. The answer may depend upon how to give some sufficient conditions ensuring

that the total biomass of a single species as a function of diffusion rate has a unique local maximum

(and thus also a global maximum). Another interesting question is what happens if all the coefficients

ai,bi,ci in (3.1) are non-constant, and the weak competition condition (3.2) still holds for all x ∈ Ω.
See the recent progress in [131] via construction of special Lyapunov functions.

For other population models, such as the predator-prey models, we can enquire similar issues: How

do diffusion and spatial heterogeneity affect the dynamics of predator-prey models? For the competition

models, under the weak competition condition, if the diffusion rate of a species lies within some proper

ranges, the remaining average resources can be negative, which is closely related to the phenomenon

that the total biomass of a single species can be greater than the total amount of resources, as discussed

in Subsection 2.1. Therefore, the fast diffusing invader cannot invade successfully in this case as their

available resources can be negative. Similar consideration is applicable for the predator-prey models.

However, the study of predator-prey models turns out to be more complicated [113]: Understanding the

relationship between the total biomass of species and the total amount of resources is far from being

sufficient, and other issues are also involved, such as the dependence of the maximum density of species

on the diffusion rate [97].

The study of reaction-diffusion models not only uncover some interesting biological phenomena,

but also bring new and interesting mathematical problems. In the next two subsections, we continue

on the theme of competition, but focus on the effect of biased movement of species in the spatially

heterogeneous environment.



158 K.-Y. LAM, S. LIU, AND Y. LOU

3.2. Competition in rivers. In this subsection we consider the following two-species competition

model in a stream:

∂tu = d∂xxu−α1∂xu + u(r −u−v) (x,t) ∈ (0,L) × (0,∞),
∂tv = D∂xxv −α2∂xv + v(r −u−v) (x,t) ∈ (0,L) × (0,∞),
d∂xu−α1u = D∂xv −α2v = 0 (x,t) ∈{0,L}× (0,∞),
u(x, 0) = u0(x), v(x, 0) = v0(x) x ∈ (0,L),

(3.7)

where α1 and α2 represent the advection rates of species u and v, respectively. When α1 = α2 = 0, it

is shown in Subsection 3.1 that the species with the smaller diffusion rate will drive the other species

to extinction, provided that r is non-constant. What happens if α1,α2 6= 0?
For the case when r is a positive constant (i.e. the environment is homogeneous) and α1 = α2 = α >

0, the following result was established in [115]:

Theorem 3.4. Suppose that r > 0 is constant and α > 0, D > d. Then the semi-trivial state (0,v∗) is

globally stable, where v∗ is the unique positive solution of{
D∂xxv −α∂xv + v(r −v) = 0 x ∈ (0,L),
D∂xv −αv = 0 x = 0,L.

(3.8)

Theorem 3.4 shows that under the influence of passive drift, the species with faster diffusion will

be favored, which is contrary to Theorem 3.3 for model (3.4) in the non-advective environment. In

the river, the “fate” of the two competitors is reversed: the slower diffusing species will be eventually

wiped out! An intuitive reasoning is that the passive drift will push more individuals downstream to

force a concentration of population at the downstream end, which causes mismatch of the population

density with the resource distribution. Faster diffusion can counterbalance such mismatch by sending

more individuals upstream and therefore, it is competitively advantageous for species to adopt faster

diffusion.

When r is constant, the case α1 6= α2 and D = d was considered in [112], where it was proved that
the smaller advection is always beneficial. A similar explanation is that under the drift, the species

will move towards the downstream end, leading to the overcrowding and the overmatching of resources

there, which in turn results in the extinction of the species with the stronger advection that can send

more individuals to the downstream. The general case D 6= d and α1 6= α2 was addressed in [158],
which showed that the fast diffusion and weak advection will be favored, consistent with the findings

in [112, 115].

When r is non-constant, i.e. the spatial environment is heterogeneous, the dynamics of (3.7) turns

out to be more complicated. The simplest situation might be when r is decreasing, for which we have

the following conjecture:

Conjecture 2. Suppose that r is positive and decreasing in (0,L). Then there exists some α∗ > 0,

independent of d and D, such that

(i) for α ∈ (0,α∗), there exists D∗ = D∗(α) > 0 such that if D = D∗ and d 6= D∗, then the
semi-trivial steady state (0,v∗) of (3.7) is globally stable;

(ii) for α ≥ α∗, if D > d > 0, then (0,v∗) of (3.7) is globally stable.

We observe that if r is non-constant and α = 0, the slower diffusion rate is always favored. This

suggests that D∗(α) → 0 as α → 0. Furthermore, we expect D∗(α) → +∞ as α → α∗−. We refer
to [114, 157] for some recent developments for the case of decreasing r. The general case of r(x) is

studied in [90], where sufficient conditions for the existence of evolutionarily stable strategies, or that



SELECTED TOPICS ON REACTION-DIFFUSION-ADVECTION MODELS 159

of branching points, are found. We refer the readers to [36] for the definition of a branching point, and

to Subsection 4.2 for further discussions.

For the Speirs and Gurney model (2.4), it was conjectured that there is an intermediate diffusion

rate which is optimal for the evolution of dispersal. This seems to be in contrast to Theorem 3.4,

where the faster diffusion is favored. Such discrepancy is due to the different boundary conditions at

the downstream end x = L; see [110, 115, 148] for further results on the effect of boundary conditions.

Next, we impose the zero Dirichlet boundary conditions at the downstream end and consider the

following competition model:


∂tu = d∂xxu−α∂xu + u(r −u−v) (x,t) ∈ (0,L) × (0,∞),
∂tv = D∂xxv −α∂xv + v(r −u−v) (x,t) ∈ (0,L) × (0,∞),
d∂xu−αu = D∂xv −αv = 0 (x,t) ∈{0}× (0,∞),
u = v = 0 (x,t) ∈{L}× (0,∞),
u(x, 0) = u0(x), v(x, 0) = v0(x) x ∈ (0,L).

(3.9)

As shown in Subsection 2.2, neither large nor small diffusion rate is beneficial to each species in this

case. A natural conjecture is:

Conjecture 3. There exists some D∗(α) > 0 such that if D = D∗ and d 6= D∗, the semi-trivial steady
state (0,v∗) of (3.9), whenever it exists, is globally stable.

For the case of weak advection (α small), we conjecture that the approaches developed in [87, 88]

may be applicable to yield the existence of D∗ for small α, satisfying D∗(α) → 0 as α → 0. However,
the proof remains quite challenging for general α. Recently it was shown in [155] that such diffusion

rate D∗ exists and is unique for two-patch models: The semi-trivial steady state (0,v∗) is locally stable

when D = D∗ and d 6= D∗. However, the global stability of (0,v∗) remains unclear. For multi-patch
models, it remains open to determine the existence of D∗.

4. Competition models with directed movement

Belgacem and Cosner [7] argued that in the spatially heterogeneous environment, the species may

have a tendency to move upward along the resource gradient, in addition to random diffusion. In

this connection, they added an advection term to (2.1) and considered the reaction-diffusion-advection

model 

∂tu = ∇· [d∇u−αu∇m] + u(m−u) (x,t) ∈ Ω × (0,∞),
∂u

∂ν
−αu

∂m

∂ν
= 0 (x,t) ∈ ∂Ω × (0,∞),

u(x, 0) = u0(x) x ∈ Ω.

(4.1)

Here α ≥ 0 is a parameter which measures the tendency of the population to move upward along the
gradient of the resource m(x). Belgacem and Cosner [7] and subsequent work [30] indicate that for

a single species, the advection along the resource gradient can be beneficial to the persistence of the

single species in convex habitats, but not necessary for non-convex Ω. In particular, when m is positive

somewhere, the species persists whenever the advection rate α is large. The biological explanation is

that the species can utilize the large advection to concentrate at places with locally most abundant

resources.



160 K.-Y. LAM, S. LIU, AND Y. LOU

It is natural to enquire whether the advection along the resource gradient can confer competition

advantage. To this end, Cantrell et al. [15] proposed the following model:


∂tu = ∇· [d∇u−αu∇m] + u(m−u−v) (x,t) ∈ Ω × (0,∞),
∂tv = D∆v + v(m−u−v) (x,t) ∈ Ω × (0,∞),

d
∂u

∂ν
−αu

∂m

∂ν
=
∂v

∂ν
= 0 (x,t) ∈ ∂Ω × (0,∞),

u(x, 0) = u0(x), v(x, 0) = v0(x) x ∈ Ω.

(4.2)

When m is non-constant and
∫

Ω
m ≥ 0, (4.2) has two semi-trivial states, denoted by (u∗, 0) and

(0,v∗), for every d,D > 0 and α ≥ 0 (see [30]), where u∗ is the unique positive steady state of (4.1).
It is shown in [16] that if d = D and Ω is convex, then the semi-trivial state (u∗, 0) is globally stable

for small α > 0. This implies that for two species which differ only in their dispersal mechanisms,

the weak advection can confer some competition advantages, provided that the underlying habitat is

convex. However, for some non-convex Ω, it is also proved in [16] that the opposite case may happen

for proper m(x): the weak advection can actually put the species at a disadvantage.

Does the species have a competitive advantage if the advection is strong? The following result of [5]

might seem surprising at the first look. It was first proved in [16] under the additional assumption that

the set of critical points is of measure zero and has at least one strict local maximum point.

Theorem 4.1. Suppose that m ∈ C2(Ω) is positive and non-constant. Then for α
d
≥ 1

minΩ m
, both

semi-trivial steady states (u∗, 0) and (0,v∗) are unstable, and (4.2) has at least one stable coexistence

state.

Theorem 4.1 implies that the two competing species are able to coexist when the advection is strong,

which is in stark contrast with the case of weak advection. What is preventing the species u to exclude

species v by strong advection? A series of work suggests that as α increases, species u resides only at

the set M of local maximum points of m(x), and retreats from Ω\M. If m is positive, then that leaves
ample room for species v to survive in Ω \ M, i.e. places with less resources [16, 26, 83, 84, 91]. This
is proved in Theorem 4.1. When m(x) changes sign, however, the overall leftover resources in Ω \ M
might be poor and u may competitively exclude v by strong advection.

Theorem 4.2 ([93]). Suppose that m ∈ C2(Ω) and it is sign-changing.
(a) If

∫
Ω\M m > 0, then for any d,D > 0, u and v coexist for large α.

(b) If
∫

Ω\M m < 0, then for some d,D > 0, u competitively excludes v for large α.

Theorem 4.1 also reveals that the two competitors modeled by (4.2) can achieve coexistence by

different dispersal strategies including random diffusion and directed advection towards resources, which

suggests a new mechanism for coexistence of two or more competing species. In [111] we apply the

similar ideas to discuss the possibility of coexistence for three-species competition systems with the

random diffusion and directed advection. The analysis of three-species models turns out to be much

more difficult. An obvious challenge is that systems of three competing species, unlike two-species

competition model, are not order-preserving [142].

As mentioned earlier, if the advection rate α is large, then the species u will concentrate near

some local maximum points of m. Further studies [84, 91] showed that the concentration positions

are precisely those local maximum points of m where the function m − v∗ is also strictly positive,
by establishing the uniform L∞ estimate and a new Liouville-type result. We note that m is the

common resource for both species u and v, and m − v∗ is the resource available for species u when
the species v reaches its equilibrium. It is observed in [91, Rmk. 1.4] that for some positive local



SELECTED TOPICS ON REACTION-DIFFUSION-ADVECTION MODELS 161

Figure 2. Decomposition of the d-D parameter space into three regions: R1 and R2
are separated by the curve D = f(d). The regions R2 and R3 are separated by the line

D = d. The bifurcation diagrams for each of these regions, with α as the bifurcation

parameter, are given in Figure 3. (First published in Mem. Amer. Math. Soc. 245

(Spring 2017), published by the American Mathematical Society. @ 2017 American

Mathematical Society.)

maximum points of m, function m − v∗ can be negative, i.e. some apparently favorable places are
actually dangerous ecological traps for the species u. This suggests that both distribution of resources

and competitors should be taken into account when studying the directed movement of species along

the resource gradient.

4.1. Global bifurcation diagrams. The above results show that when the advection is weak, the two

species cannot coexist, and the advection confers some competitive advantage for species u; when the

advection is strong, the two species coexist for any diffusion rates. What happens for general advection

rates? For instance, assuming d > D, when we increase the advection from weak to strong, how does

that affect the dynamics of (4.2)? If α = 0, the result of Dockery et al. [39] says that the semi-trivial

state (0,v∗) is globally stable; if α is sufficiently large, Theorem 4.1 shows that both (u∗, 0) and (0,v∗)

are unstable and the coexistence happens. A possible conjecture is there exists some critical advection

rate α∗ > 0 such that for α < α∗, (0,v∗) is globally stable, while for α > α∗, (4.2) has a unique positive

steady state which is globally stable. However this conjecture fails in general and a more complicated

bifurcation diagram of positive steady states occurs. To be more precise, Averill et al. [5] showed that

under some conditions on m(x), there exists a function f(x) : [0,∞) → [0,∞) satisfying f(0) = 0 and
0 < f(x) < x such that the d-D parameter space is divided into three regions R1,R2,R3, as shown in

Figure 2. Using α as a bifurcation parameter, we see that the following three situations occur:

(1) for D < f(d), there exists some critical value α1 > 0 such that as α < α1, the semi-trivial state

(0,v∗) is stable but (u∗, 0) is unstable, while as α > α1, both (u
∗, 0) and (0,v∗) are unstable so

that (4.2) has at least one positive steady state. By the bifurcation theory, there is a branch

of positive steady states, denoted by Γ1, which bifurcates from (0,v
∗) at α = α1 and can be

extended to infinity in α. This is illustrated in Figure 3(a). It is unclear whether (0,v∗) is



162 K.-Y. LAM, S. LIU, AND Y. LOU

globally stable when α < α1, and whether the positive steady state is unique and globally

stable when α > α1;

(2) for f(d) < D < d, there exist three critical values 0 < α1 < α2 < α3 such that (i) as α < α1,

(0,v∗) is stable but (u∗, 0) is unstable; (ii) as α2 < α < α3, (u
∗, 0) is stable but (0,v∗) is

unstable; (iii) as α ∈ (α1,α2) ∪ (α3,∞), both (u∗, 0) and (0,v∗) are unstable so that (4.2) has
at least one positive steady state. Similarly, by the bifurcation theory, there is still a branch of

positive steady states bifurcating from (0,v∗) at α = α1, which is denoted as Γ2. The difference

is that the branch Γ2 cannot be extended to α = ∞, but is connected to (u∗, 0) at α = α2.
Interestingly, when α ∈ (α2,α3), the semi-trivial state (u∗, 0) is stable, which indicates that the
advection can indeed be beneficial to the species in this case! When α > α3, the branch of a

positive steady state, denoted as Γ3, bifurcates from (u
∗, 0) at α = α3, and can be extended

to α = ∞. We conjecture that as α < α1, (0,v∗) is globally stable; as α2 < α < α3, (u∗, 0) is
globally stable; as α ∈ (α1,α2) ∪ (α3,∞), (4.2) admits a unique positive steady state which is
globally stable. This is illustrated in Figure 3(b).

(3) for D > d, there exists some critical value α1 > 0 such that as α < α1, (u
∗, 0) is stable, while

as α > α1, both (u
∗, 0) and (0,v∗) are unstable so that (4.2) has at least one positive steady

state. Similarly, a branch of positive steady states bifurcates from (u∗, 0) at α = α1, denoted

as Γ4, which can be extended to α = ∞. It is conjectured that (u∗, 0) is globally stable when
α < α1, and the positive steady state is unique when α > α1. This is illustrated in Figure 3(c).

How, then, is the branch Γ1 transformed to Γ2 and Γ3, and finally to Γ4, as the diffusion rate D

varies, with d being fixed? The result in [5] shows that in the critical situation D = f(d), the branch

Γ1 is connected to the semi-trivial state (u
∗, 0) at α = α∗ for some α∗ > 0. Once D > f(d), α∗ is

split into two different points α2 and α3, so that Γ1 is split into two disjoint branches Γ2 and Γ3. As

D approaches d from below, the bounded branch Γ2 will approach α = 0 and vanishes; meanwhile Γ3
approaches Γ4.

By these discussions, we explained that for the models with multiple parameters, it is possible to

apply the bifurcation theory to describe the structure of positive steady states. Of course, it remains

challenging to completely characterize the global dynamics of (4.2), and many interesting mathematical

questions, e.g. uniqueness and global stability, are left open [5]. One particular conjecture is that, for α

sufficiently large, the positive steady state of the competition system (4.2) is unique and thus globally

asymptotically stable. One way of proving it is to determine the precise limiting profile and show that

each positive steady state is linearly stable, whenever it exists. One can then conclude the uniqueness

and global asymptotic stability by the theory of monotone dynamical system [64]. This was carried out

for the case when all critical points of m are non-degenerate, and that the value of m is constant on

the set of all local maximum points [82, Thm. 8.1]. The general case remains open.

It is illustrated here that proper advection can confer some competition advantages. In next subsec-

tion, we will use evolutionary game theory to further show that proper advection can be an evolutionary

stable strategy.

4.2. Evolution of biased movement. In this subsection, we adopt the viewpoint of adaptive dy-

namics [36, 54] to consider the evolution of dispersal and the ecological impact of dispersal in spatially

heterogeneous environments. An important concept in adaptive dynamics is Evolutionarily Stable Strat-

egy (ESS), known also as Evolutionarily Steady Strategy, which was introduced by Maynard Smith and

Price in the seminal paper [119]. A strategy is said to be evolutionarily stable if a population using it

cannot be invaded by any small population using a different strategy. ESS is a very general concept. In



SELECTED TOPICS ON REACTION-DIFFUSION-ADVECTION MODELS 163

Figure 3. Bifurcation results from [5], where d,D are fixed and α is used as the

bifurcation parameter. The three figures, from top to bottom, are the bifurcation

diagrams concerning the parameter regions R1,R2, and R3 (respectively, D < f(d),

f(d) < D < d and D > d) in Figure 2. (First published in Mem. Amer. Math. Soc.

245 (Spring 2017), published by the American Mathematical Society. @ 2017 American

Mathematical Society.)

this subsection we aim to connect it with the local stability of the semi-trivial steady states discussed

earlier.

An interesting question is what kind of advection rate will be evolutionarily stable, by regarding the

rates of advection along the environmental gradient as movement strategies of the species. For this

purpose, we consider the following model:




∂tu = ∇· [d∇u−αu∇m] + u(m−u−v) (x,t) ∈ Ω × (0,∞),
∂tv = ∇· [d∇v −βv∇m] + v(m−u−v) (x,t) ∈ Ω × (0,∞),

d
∂u

∂ν
−αu

∂m

∂ν
= d

∂v

∂ν
−βv

∂m

∂ν
= 0 (x,t) ∈ ∂Ω × (0,∞),

u(x, 0) = u0(x), v(x, 0) = v0(x) x ∈ Ω,

(4.3)

where u and v are the densities of resident and mutant populations, and the two species differ only in

their advection rates, given by the non-negative parameters α and β, respectively.



164 K.-Y. LAM, S. LIU, AND Y. LOU

The mutant population v, which is assumed to be small initially, can invade successfully if and only

if the semi-trivial steady state (u∗, 0) is unstable. Here u∗ denotes the unique positive solution of

∇· [d∇u−αu∇m] + u(m−u) = 0 x ∈ Ω,

d
∂u

∂ν
−αu

∂m

∂ν
= 0 x ∈ ∂Ω.

(4.4)

A natural question is whether there is a critical value α∗ > 0 such that (u∗, 0) is a stable steady state

of (4.3) whenever α = α∗ and β 6= α∗. We call such α∗, if it exists, an ESS. To find such a magical
strategy, we note that the stability of (u∗, 0) is determined by the principal eigenvalue, denoted by Λ1,

of the problem 

∇· [d∇ϕ−βϕ∇m] + ϕ(m−u∗) + λϕ = 0 x ∈ Ω,

d
∂ϕ

∂ν
−βϕ

∂m

∂ν
= 0 x ∈ ∂Ω.

(4.5)

It is classical that the set of eigenvalues of (4.5) is real and bounded from below. The principal eigenvalue

Λ1 refers to the smallest eigenvalue. It is simple and corresponds to a strictly positive eigenfunction in

Ω; see [12, 26]. Since u∗ is an implicit function of α, we regard Λ1 = Λ1(α,β). Observe that Λ1 = 0 for

α = β, since the principal eigenfunction ϕ is simply a constant multiple of u∗ in such a case.

It is well known that if Λ1 > 0, then (u
∗, 0) is stable; otherwise, (u∗, 0) is unstable. Hence if α∗ is

an ESS, then Λ1(α
∗,β) ≥ 0 for all β, which together with Λ1(α∗,α∗) = 0 implies that

∂Λ1
∂β

∣∣∣
α=β=α∗

= 0. (4.6)

We call any strategy α∗ satisfying (4.6) an evolutionarily singular strategy, hence any ESS is automati-

cally an evolutionarily singular strategy. For the existence of evolutionarily singular strategies, we have

the following result:

Theorem 4.3 ([87, Thm. 2.2]). Suppose that m ∈ C2(Ω) is strictly positive in Ω and maxΩ m/ minΩ m ≤
3 + 2

√
2. Given any A > 1/ minΩ m, for all small positive d, there is exactly one evolutionarily singular

strategy α∗ ∈ (0,dA).

The condition maxΩ m/ minΩ m ≤ 3 + 2
√

2 is sharp, and Theorem 4.3 may fail for general m [87,

Thm. 2.6]. When Ω is one-dimensional, the evolutionarily singular strategy found in Theorem 4.3 is

also unique in the whole interval [0,∞), provided d > 0 is small enough [87, Cor. 2.3]. We conjecture
that the uniqueness holds for Ω of higher dimensions as well. For intermediate or large d, however, the

uniqueness of singular strategies remains open; see [20] for some recent progress. We conjecture that

for any evolutionarily singular strategy α∗ = α∗(d), it holds that, for all d,

min
Ω
m <

d

α∗
< max

Ω
m. (4.7)

If this estimate were true, it suggests that a balanced combination of diffusion and advection might

be a better movement strategy for populations. The following result shows that this is indeed the case:

Theorem 4.4 ([87, Thm. 2.5]). Suppose that Ω is convex and m ∈ C2(Ω) is strictly positive in Ω.
Assume that ‖∇(ln m)‖L∞(Ω) ≤ α0/diam(Ω), where α0 ≈ 0.8 and diam(Ω) is the diameter of Ω. Then
for small d, the strategy α∗ given by Theorem 4.3 is a local ESS, i.e. there exists some small δ > 0 such

that Λ1(α
∗,β) > 0 for all β ∈ (α∗ −δ,α) ∪ (α,α∗ + δ).

Some other work also considered model (4.3) and related models. For instance, in [88] we discussed

the evolution of diffusion rate in an advective environment. We refer to [20, 24, 25, 28, 29, 53, 55] for

further discussions.



SELECTED TOPICS ON REACTION-DIFFUSION-ADVECTION MODELS 165

Can we decide whether the local ESS α∗ given in Theorem 4.3 is actually a global ESS? Or more

generally, what can we say about the global structure of nodal set for the function Λ1 = Λ1(α,β)? It

is clear that the nodal set of Λ1(α,β) must include the diagonal line α = β, but otherwise little is

known. For the reaction-diffusion models discussed in this paper, determining the complete structure

of the nodal set for Λ1(α,β) poses a challenging problem, both analytically and numerically [52]. To

this end, some asymptotic behaviors of Λ1 were studied in [26, 27, 99, 139] in order to have a better

understanding for this problem. Recently, the asymptotic behaviors for the principal eigenvalues of

time-periodic parabolic operators were also studied, and we refer to [64, 69, 71, 101, 102, 103, 137].

These new developments of the related linear PDE theory may open up a new venue to study the

evolution of dispersal in spatio-temporally varying environments.

The evolutionary dynamics, e.g. existence of ESS, have various consequences on the two-species com-

petition dynamics. This connection has been investigated in [13] for a broad class of model including

reaction-diffusion equations and nonlocal diffusion equations. It was shown that frequently the ESS

playing species can competitively exclude the species playing a different but nearby strategy, regardless

of the initial condition. In the jargon of adaptive dynamics, this result says that ESS implies Neigh-

borhood Invader Strategy. This is quite unexpected, since an ESS by its very definition only concerns

the local stability of the semi-trivial steady state. The main tool in [13] is Hadamard’s graph transform

method.

5. Continuous trait models

Regarding the advection rates as strategies of the species, in Subsection 4.2 we investigated the

course of evolution for two-species competition model (4.3). Therein the strategies {α,β} form a
discrete set, and the competition and mutation occur independently of each other. In this section, we

consider population models with continuous traits, in which the competition and mutation can occur

simultaneously.

5.1. Mutation-selection model. To study the selection of random dispersal rate in multi-species

competition, Dockery et al. [39] considered the following model:

∂tui = ξi∆ui + ui


m(x) − K∑

j=1

uj


 + K∑

j=1

Mijuj, (5.1)

where K interacting species, with densities ui(x,t), 1 ≤ i ≤ K, compete for a common resource m(x)
and differ only in their diffusion rates ξi. These species are subject to mutation with switching rate

Mij from phenotype j to i. For K = 2 and Mij = 0, a well-known result is that the slower diffusing

species always win in the competition, but it remains open to determine the global dynamics for the

case K ≥ 3.
Next, we extend the discrete set of strategies {ξi}Ki=1 in (5.1) to a continuum set (ξ,ξ). By viewing

the population u(x,ξ,t) as being structured by space, trait and time (i.e. u(·,ξ,t) gives the spatial
distribution of the individuals playing strategy ξ at time t), we obtain the following mutation-selection

model: 


∂tu = ξ∆xu + u [m(x) − û(x,t)] + �2∂ξξu for x ∈ Ω,ξ ∈ (ξ,ξ), t > 0,

ν(x) ·∇xu = 0 for x ∈ ∂Ω,ξ ∈ (ξ,ξ), t > 0,

∂ξu = 0 for x ∈ Ω,ξ ∈{ξ,ξ}, t > 0,

u(x,ξ, 0) = u0(x,ξ) for x ∈ Ω,ξ ∈ (ξ,ξ).

(5.2)



166 K.-Y. LAM, S. LIU, AND Y. LOU

Here the parameters ξ,ξ, which are assumed to be positive, determine the range of continuous strategy

ξ, and

û(x,t) :=

∫ ξ
ξ

u(x,ξ′, t) dξ′

represents the total population density at location x and time t, and the term �2∂ξξu accounts for the

rare mutation, which is modeled by a diffusion with covariance
√

2�. We refer to [22] for a derivation

of this type of equations from individual based on stochastic models. Model (5.2) is an integro-PDE

model describing the diffusion, competition and mutation of a population with continuous traits.

If
∫

Ω
m(x)dx > 0, then it is shown in [85] that (5.2) admits a unique positive steady state u�(x,ξ)

for small �, which is locally asymptotically stable and satisfies

ξ∆xu + �

2∂ξξu + u [m(x) − û(x)] = 0 (x,ξ) ∈ Ω × (ξ,ξ),

ν(x) ·∇xu = 0 (x,ξ) ∈ ∂Ω × (ξ,ξ),

∂ξu = 0 (x,ξ) ∈ Ω ×{ξ,ξ},

(5.3)

where û(x) =
∫ ξ
ξ
u(x,ξ′)dξ′. Observe that when m is constant, i.e. m ≡ m0 for some positive constant

m0, we have u� ≡ m0/(ξ − ξ) which is globally stable for any � > 0 [85]. This suggests that selection
does not favor any particular strategy ξ ∈ [ξ,ξ], and the mutation �2∂ξξu has no effect on the population
dynamics in the homogeneous environment. What happens in the spatially heterogeneous environment?

The following result gives the answer:

Theorem 5.1 ([86]). Suppose that
∫

Ω
m(x)dx > 0 and m is non-constant. Then as � → 0,∥∥∥∥�2/3u�(x,ξ) −u∗(x,ξ)η∗

(
ξ − ξ
�2/3

)∥∥∥∥
L∞(Ω×(ξ,ξ))

→ 0,

where η∗(s) satisfying the following ordinary differential equation:{
η′′ + (a0 −a1s)η = 0 s > 0,

η′(0) = η(+∞) = 0,
∫∞

0
η(s) ds = 1,

with some positive constants a0 and a1. In particular, we have as � → 0,

u�(x,ξ) → u∗(x,ξ) ·δ(ξ − ξ) in distribution,

where u∗(x,ξ) denotes the unique positive solution of (2.2) with d = ξ, and δ(·) is a Dirac mass
supported at 0.

It can be verified that for any ξ0 ∈ [ξ,ξ], the limit profile u∗(x,ξ0) · δ(ξ − ξ0) is a solution of (5.3)
with � = 0. Theorem 5.1 says that only the steady state corresponding to the smallest diffusion rate is

preserved by the perturbation of � = 0 to 0 < � � 1. Theorem 5.1 implies that when the mutation rate
is small, the species with the smallest diffusion rate ξ will eventually become the dominant phenotype by

driving other phenotypes to extinction. This result connects with the case of two-species competition:

The smallest available diffusion rate is selected. Note from [85] that the steady state u� is the unique

solution of (5.3) for small � > 0, and it is locally asymptotically stable. An unsolved problem here is

the global stability of the unique positive steady state u�(x,ξ).



SELECTED TOPICS ON REACTION-DIFFUSION-ADVECTION MODELS 167

ξ

lo
g
(t

im
e
)

0.5 1 1.5
0

1

2

3

4

5

2

4

6

8

10

12

ξ

lo
g
(t

im
e
)

0.5 1 1.5
0

1

2

3

4

5

2

4

6

8

10

12

Figure 4. Contour plot of
∫
u(x,ξ,t)dx as a function of ξ and time (log(time) for

vertical axis) for a = 1
4

(left) and a = −1
4

(right). (First published in [W. Hao et al.,

Indiana Univ. Math. J. 68 (2019), 881-923.].)

A closely related work is due to Perthame and Souganidis [140], where the following mutation-

selection model was considered:

ξ(z)∆xu + �

2∂zzu + u(m(x) − û) = 0 (x,z) ∈ Ω × [0, 1],
ν(x) ·∇xu = 0 (x,z) ∈ ∂Ω × [0, 1],
u(x, 0) = u(x, 1) x ∈ Ω,

(5.4)

where Ω is assumed to be convex and ξ(z) is assumed to be positive and periodic with unit period.

For the solution u�(x,z) of (5.4), it is proved in [140] that u�(x,z) → u∗(x,ξ(z∗)) · δ(z − ξ(z∗)) in
distribution as � → 0, where ξ(z∗) = min0≤z≤1 ξ(z). This implies that for rare mutations, the population
concentrates on a single trait associated to the smallest diffusion rate, in agreement with the result for

(5.3). See also [8, 76, 125] for some progress in this direction.

5.2. Mutation-selection model with drift. Similar to (5.2), we introduce the following integro-PDE

model in a stream:


∂tu = ξ∂xxu−α∂xu + u [r(x) − û(x,t)] + �2∂ξξu for x ∈ (0,L),ξ ∈ (ξ,ξ), t > 0,

(ξ∂xu−αu)(0,ξ,t) = (ξ∂xu−αu)(L,ξ,t) = 0 for ξ ∈ (ξ,ξ), t > 0,

u(x,ξ,t) = u(x,ξ,t) = 0 for x ∈ (0,L) t > 0,

u(x,ξ, 0) = u0(x,ξ) for x ∈ (0,L),ξ ∈ (ξ,ξ),

(5.5)

where α > 0 is the advection rate and û(x,t) :=
∫ ξ
ξ
u(x,ξ′, t) dξ′ denotes the total population density

at location x and time t.

When r is a positive constant, from Theorem 3.4 we conjecture that (i) the integro-PDE (5.5) admits

at most one positive steady state, denoted by u�(x,ξ), which is globally stable whenever it exists; (ii) as

� → 0, u�(x,ξ) → u∗(x,ξ) · δ(ξ − ξ) in distribution sense, where u∗(x,ξ) is the unique positive solution
of (3.8) with D = ξ. Again, this predicts that the larger diffusion is selected as in Theorem 3.4.

To study the dynamics of (5.5) for non-constant r, we carried out some numerical simulations in

[56], where the corresponding parameters were selected as follows:

L = 1, α = 1, ξ = 0.5, ξ = 1.5, r(x) = e(1−a)x+ax
2

, � = 10−3. (5.6)

We take initial conditions in the form of one Dirac mass on the phenotypic space, and investigate their

evolution for a = ±1
4
. The numerical results are presented in Figure 4.

(i) The numerical result in Figure 4 (right) shows that when a = 1
4
, the steady state of (5.5)

concentrates on the trait ξ∗ ≈ 1.1 in the limit of rare mutation, which suggests that the species



168 K.-Y. LAM, S. LIU, AND Y. LOU

adopting the strategy ξ∗ persists and other species will disappear in the competition. This

suggests that ξ∗ is an ESS, which is in contrast with the assertion that “faster diffusion is more

favorable” for the homogeneous environment.

(ii) More interestingly, when a = −1
4
, Figure 4 (left) shows that the steady state of (5.5) concen-

trates on two different traits, so that two species adopting different strategies form a coalition

that is able to resist invasion by any of the remaining species! This phenomenon is called “evolu-

tionary branching” in adaptive dynamics, and it is also clearly different from the homogeneous

case.

To explain these phenomena mathematically, we investigate the asymptotic behaviors of steady state

u�(x,ξ) for (5.5) as � → 0. To this end, we first consider the two-species competition model (3.7) and
denote λ = λ(ξ1,ξ2) as the principal eigenvalue of the problem{

ξ2∂xxϕ−α∂xϕ + [r −u∗(·,ξ1)] ϕ + λϕ = 0 x ∈ (0,L),
ξ2∂xϕ−αϕ = 0 x = 0,L,

where x 7→ u∗(x,ξ1) is the unique positive solution of (3.8) with D = ξ1. In the adaptive dynamics
framework, λ(ξ1,ξ2) is called the invasion fitness, and an invader with trait ξ2 can (resp. cannot)

invade an established trait ξ1 at equilibrium when rare if λ(ξ1,ξ2) < 0 (resp. λ(ξ1,ξ2) > 0). The

limiting profiles of u�(x,ξ) depends critically on the behavior of λ(ξ1,ξ2) near ξ2 = ξ1. Note that

λ(ξ,ξ) ≡ 0 for all ξ > 0. We first consider the case of ∂ξ2λ(ξ,ξ) 6= 0, where the following result holds:

Theorem 5.2 ([56]). Suppose that ∂ξ2λ(ξ,ξ) > 0 for all ξ ∈ (ξ,ξ), and ξ−ξ is sufficiently small. Then
any positive steady state u� of (5.5) satisfies as � → 0,

u�(x,ξ) → δ(ξ − ξ) ·u∗(x,ξ) in distribution,

where x 7→ u∗(x,ξ) is the unique positive solution of (3.8) with D = ξ.

Theorem 5.2 implies that for the case ∂ξ2λ(ξ,ξ) > 0, the slowest diffusion will be selected in the

competition. Similarly, we may conclude that if ∂ξ2λ(ξ,ξ) < 0 for all ξ ∈ (ξ,ξ), then the steady
state u� concentrates on the trait ξ = ξ, so that the fastest diffusion will be selected, coinciding with

the observation in Theorem 3.4. These results show that if ∂ξ2λ 6= 0, it gives rise to a single Dirac-
concentration at one of the two extreme traits, depending upon the sign of ∂ξ2λ.

To study the case when ∂ξ2λ vanishes somewhere in (ξ,ξ), we introduce the convergence stable

strategy ξ̂ (see [45]), which is characterized by the following relations:

∂ξ2λ(ξ̂, ξ̂) = 0 and
d
ds

[∂ξ2λ(s,s)]
∣∣∣
s=ξ̂

> 0. (5.7)

This leads to two generic cases: The convergence stable strategy ξ̂ is an ESS if ∂2ξ2λ(ξ̂, ξ̂) > 0, and is a

branching point (BP) if ∂2ξ2λ(ξ̂, ξ̂) < 0.

In the case when the ESS exists, there is concentration in the mutation-selection model.

Theorem 5.3. [56] Suppose that ξ̂ is an ESS and (5.7) holds. If ξ̂ ∈ (ξ,ξ) and ξ − ξ is sufficiently
small, then as � → 0, any positive steady state u� of (5.5) satisfies

u�(x,ξ) → u∗(x, ξ̂) ·δ(ξ − ξ̂) in distribution.

Theorem 5.3 indicates that if there is an ESS in (ξ,ξ), then the phenotype adopting the ESS dominates

the competition in the limit of rare mutation. This corresponds to the numerical result in Figure 4

(right) with ξ̂ ≈ 1.1. To further understand Theorem 5.3, we consider a special case of (3.7) where
r(x) = becx for some positive constants b and c. In this case, it is shown in [4] that the semi-trivial state

(u∗(x,ξ1), 0) is globally stable when ξ1 = α/c and ξ2 6= α/c, i.e. the intermediate trait ξ̂ = α/c is an ESS.



SELECTED TOPICS ON REACTION-DIFFUSION-ADVECTION MODELS 169

We can offer some intuitive reasoning for this result by employing the theory of ideal free distribution

[46]: When r(x) = becx, if ξ1 = α/c, it can be verified from (3.8) that u
∗(x,ξ1) ≡ becx ≡ r(x), so that

species u attains the ideal free distribution, and thus α/c being an ESS is natural. As c → 0, i.e. the
environment tends to be homogeneous, we find ξ̂ = α/c approaches +∞, which is in agreement with
the prediction of Theorem 3.4.

Finally, in the neighborhood of a BP, we have the following result:

Theorem 5.4 ([56]). Suppose that ξ̂ ∈ (ξ,ξ) is a BP and (5.7) holds. If ξ−ξ is sufficiently small, then
as � → 0, any positive steady state u� of (5.5) satisfies

u�(x,ξ) → δ(ξ − ξ) ·u1(x) + δ(ξ − ξ) ·u2(x) in distribution,

where (u1,u2) is a positive solution of

ξ∂xxu1 −α∂xu1 + u1(r −u1 −u2) = 0 x ∈ (0,L),

ξ∂xxu2 −α∂xu2 + u2(r −u1 −u2) = 0 x ∈ (0,L),

ξ∂xu1 −αu1 = ξ∂xu2 −αu2 = 0 x = 0,L.
(5.8)

Theorem 5.4 reveals a new phenomenon: In the neighborhood of a BP, no single trait can dominate

the competition; instead, the two extreme traits, ξ and ξ, form a coalition that together dominates

the competition such that any species with other traits cannot invade. To illustrate Theorem 5.4, we

consider the case r(x) = b1e
c1x + b2e

c2x in (3.7) with positive constants bi and ci, i = 1, 2. We observe

that if ξ = α/c1 and ξ = α/c2, then (5.8) has the unique positive solution ui(x) = bie
cix, i = 1, 2. Again,

in the framework of ideal free distribution, it is not difficult to understand the biological reasoning: In

view of u1(x) + u2(x) = b1e
c1x + b2e

c2x = r(x), the total density of the two species can jointly reach

an ideal free distribution, so that ξ = α/c1 and ξ = α/c2 potentially become a pair of ESS. We refer to

[17, 18, 19, 21, 53] for related work on the evolution of dispersal and ideal free distribution.

An open problem is whether (5.5) or (5.8) has at most one positive solution, and if it exists, whether

it is globally asymptotically stable. Another challenging question is to determine the limit of the time-

dependent solutions uε(x,ξ,t) of (5.3) and (5.5) as � → 0. See [94, 132] and references therein for
recent progress on the rigorous derivations of the canonical equations for the trait evolution in spatially

structured mutation-selection models.

6. Dynamics of phytoplankton

Besides the models discussed above, there are many other types of reaction-diffusion models in

population dynamics. In this section, we briefly discuss some problems arising from the phytoplankton

growth.

Phytoplankton are microscopic plant-like photosynthetic organisms drifting in lakes and oceans and

are the foundation of the marine food chain. Since they transport significant amounts of atmospheric

carbon dioxide into the deep oceans, they play a crucial role in climate dynamics and have been one of

the central topics in marine ecology. Nutrients and light are the essential resources for the growth of

phytoplankton. Since most phytoplankton are heavier than water, they will sink into the bottom of the

lakes or oceans where the light intensity is too weak for their growth. So how do these phytoplankton

persist in the water columns? Some biologists propose that biased movement, combining with water

turbulence, can help phytoplankton diffuse to a position closer to the top of lakes or oceans, so that

they can get access to sunlight. In this connection, a series of reaction-diffusion models including single

and multiple phytoplankton species are introduced in [66, 67, 68] and the references therein to model

the spatio-temporal dynamics of phytoplankton growth.



170 K.-Y. LAM, S. LIU, AND Y. LOU

The following system of reaction-diffusion-advection equations was used by Huisman et al. [68] to

describe the population dynamics of two phytoplankton species:


∂tu = D1∂xxu−α1ux + [g1(I(x,t)) −d1]u 0 < x < L, t > 0,
∂tv = D2∂xxv −α2vx + [g2(I(x,t)) −d2]v 0 < x < L, t > 0,
D1∂xu(x,t) −α1u(x,t) = 0 x = 0,L, t > 0,
D2∂xv(x,t) −α2v(x,t) = 0 x = 0,L, t > 0,
u(x, 0) = u0(x) ≥, 6≡ 0, v(x, 0) = v0(x) ≥, 6≡ 0 0 ≤ x ≤ L.

(6.1)

Here u(x,t),v(x,t) denote the population density of the phytoplankton species at depth x and time t,

and D1,D2 > 0 are their diffusion rates. For i = 1, 2, αi ∈ R is the sinking (αi > 0) or buoyant (αi < 0)
velocity, di > 0 is the death rate, and gi(I) represents the specific growth rate of phytoplankton species

as a function of light intensity I, given by the Lambert-Beer law

I(x,t) = I0 exp

[
−k0x−

∫ x
0

(k1u(s,t) + k2v(s,t))ds

]
, (6.2)

where I0 > 0 is the incident light intensity, k0 > 0 is the background turbidity that summarizes

light absorption by all non-phytoplankton components, and ki is the absorption coefficient of the i-th

phytoplankton species. Function gi(I) is smooth and satisfies

gi(0) = 0 and g
′
i(I) > 0 for I ≥ 0.

6.1. Single phytoplankton species. The single species model, i.e. v = 0 in (6.1), was first considered

in [141] for the self-shading case and infinite long water column; see also [75, 81]. The authors [44]

considered the global dynamics of the single species model when diffusion and drift rates are spatially

dependent. In [42] the authors studied the effect of photoinhibition on the single phytoplankton species,

and they found that, in contrast to the case of no photoinhibition, the model with photoinhibition

possesses at least two positive steady states in certain parameter ranges. Hsu et al. [73] studied the

single species growth consuming inorganic carbon with internal storage in a poorly mixed habitat,

building upon some interesting recent findings of the principal eigenvalue of a 1-homogeneous positive

compact operator. In [137, 138], the authors considered the effect of time-periodic light intensity at the

surface. Ma and Ou [118] recently made an important finding that the biomass of the single species

satisfies a comparison principle, even though the density itself does not obey such orders.

In [72], we investigated a single phytoplankton model and obtained some necessary and sufficient

conditions for the growth of phytoplankton, and the critical death rate, critical water column depth,

critical sinking or buoyant coefficient and critical turbulent diffusion rate were studied respectively.

One of the results is that the phytoplankton population persists if and only if the sinking velocity of

phytoplankton is less than a critical value. There are many simplified assumptions in the model, e.g. the

death rate is assumed to be a constant. An interesting issue is to consider the case when the death rate

is non-constant in space and time. Some preliminary results show that this situation is quite different

from that of constant death rate. For instance, under some conditions phytoplankton can persist if and

only if the sinking velocity stays within an intermediate range, rather than below a single critical value.

6.2. Two phytoplankton species. The coexistence of two or multiple phytoplankton is an important

issue. In contrast to single phytoplakton species, very few results exist for two or more phytoplankton

species; see [40, 41, 124]. On the one hand, the coexistence of many species can often be observed

in reality. On the other hand, the classical competition theory shows that only the most dominant

phytoplankton persists. They seem to contradict each other. The reason is that the classical competition

theory is generally established for ordinary differential equation models, i.e. it is assumed that the



SELECTED TOPICS ON REACTION-DIFFUSION-ADVECTION MODELS 171

diffusion rates of phytoplankton are sufficiently large, so that only their average densities are considered;

i.e. the water column is well mixed. A natural question is whether small diffusion can increase the

possibility of coexistence. In this connection, we recently investigated the outcome of model (6.1)-(6.2)

in [77] and established the following results:

(i) If two phytoplankton differ only in their sinking velocities (D1 = D2,α1 6= α2,g1 = g2, d1 = d2),
in [77] we showed that the phytoplankton with smaller sinking velocity has the competitive ad-

vantage. In terms of evolution, the mutation of phytoplankton can continually reduce their

sinking velocities, so that the density of phytoplankton and water will get closer as the popu-

lation evolves. In other words, as the environment permits, the natural selection may favor the

phytoplankton whose density is lighter that of water in some circumstances.

(ii) If two phytoplankton differ only in their diffusion rates (D1 6= D2,α1 = α2,g1 = g2, d1 = d2), it
is shown in [77] that slower diffusion rate will be selected when buoyant, and in contrast, faster

diffuser wins when they are sinking with large velocity. An underlying biological reasoning is

that when the phytoplankton are buoyant, the slower diffuser are more likely to reach the top

of the water column (i.e. without water turbulence), where the light intensity is the strongest,

while when sinking with large velocity, faster diffusion can counterbalance the tendency to sink

and provide individuals with better access to light.

An important tool is the extension of the comparison principle in [118] for single-species models to

two-species phytoplakton models as follows:

Theorem 6.1. Suppose {(ui,vi)}i=1,2 are non-negative solutions of (6.1)-(6.2) such that∫ x
0

u1(s, 0) ds ≤, 6≡
∫ x

0

u2(s, 0) ds and

∫ x
0

v1(s, 0) ds ≥, 6≡
∫ x

0

v2(s, 0) ds

hold for all x ∈ (0,L]. Then for all x ∈ (0,L] and t > 0,∫ x
0

u1(s,t) ds <

∫ x
0

u2(s,t) ds and

∫ x
0

v1(s,t) ds >

∫ x
0

v2(s,t) ds.

By Theorem 6.1, system (6.1)-(6.2) is a strongly monotone dynamical system with respect to the

order generated by the cone K := K1 × (−K1), where

K1 :=
{
φ ∈ C([0,L]; R) :

∫ x
0

φ(s) ds ≥ 0, ∀x ∈ (0,L]
}
.

This in turn enables the application of the theory of strongly monotone dynamical system, which

provides a powerful tool to investigate the global dynamics of two-species system (6.1)-(6.2).

The above findings in part (ii) naturally lead to the following conjecture:

Conjecture 4. Suppose that α1 = α2 := α, g1 = g2 and d1 = d2. There exist two critical sinking

velocities αmin and αmax such that the slower diffuser wins if α < αmin, the faster diffuser wins if

α > αmax, and two species coexist if α ∈ (αmin,αmax).

7. Spatial dynamics of epidemic diseases

The COVID-19 pandemic has impacted or changed almost everyone’s life. There are many mysteries

about the novel coronavirus, among which the effect and possible control strategies of the movement

of individuals is an emerging question. The COVID-19 pandemic has been spreading so fast, at least

partially due to the movement of those infected individuals who are showing little or no symptoms.

What is the general impact of movement on the persistence and extinction of a disease in spatially

heterogeneous environment? We studied various susceptible-infected-susceptible (SIS) models in the



172 K.-Y. LAM, S. LIU, AND Y. LOU

heterogeneous environment [1, 2, 3], including the following reaction-diffusion model [2], where the

susceptible and infected individuals move randomly and the disease transmission and recovery rates

could be uneven across the space:


∂tS −dS∆S = −
βSI

S + I
+ γI (x,t) ∈ Ω × (0,∞),

∂tI −dI∆I =
βSI

S + I
−γI (x,t) ∈ Ω × (0,∞),

∂S

∂ν
=
∂I

∂ν
= 0 (x,t) ∈ ∂Ω × (0,∞),

S(x, 0) = S0(x), I(x, 0) = I0(x) x ∈ Ω,

(7.1)

where Ω is a bounded domain in RN with smooth boundary ∂Ω and ν denotes the unit outward normal
vector on ∂Ω. Here S(x,t) and I(x,t) represent the density of susceptible and infected populations at

location x and time t, respectively. Parameters dS,dI > 0 denote their diffusion rates. The functions

β(x),γ(x) account for the rates of disease transmission and recovery, respectively. They are assumed to

be positive and at least one of them is non-constant to reflect the spatial heterogeneity of the residing

habitat.

In [2], we defined the basic reproduction number R0 for the spatial SIS model (7.1), which is consistent
with the next generation approach for heterogeneous populations [37, 146, 150]. It is of practical

significance to consider the effect of spatial heterogeneity of disease transmission and recovery rates on

the basic reproduction number. For instance, in the study of dengue fever, it is shown in [149] that

the infection risk may be underestimated if the spatially averaged parameters are used to compute

the basic reproduction number for spatially heterogeneous infections. We proved that R0 is monotone
decreasing with respect to the diffusion rate of the infected individuals. The basic reproduction number

R0 characterizes the infection risk of disease and serves as the threshold value for the extinction and
persistence of disease: Namely, if R0 < 1, the unique disease-free equilibrium is globally stable, and if
R0 > 1, the disease-free equilibrium is unstable and there is a unique endemic equilibrium. A standing
open question is whether this unique endemic equilibrium is globally stable.

Interestingly, we also found in [2] that a disease may persist but it can be controlled by limiting

the movement of the susceptible populations. To be more specific, if the spatial environment can

be transformed to include low-risk sites where the recovery rates are greater than transmission rates

(such as through vaccination and treatment), and the movement of the susceptible individuals can be

restricted (such as through isolation or lock down), then it may be possible to control the disease by

flattening the disease outbreak curve, i.e. decreasing the daily number of infected individuals.

In recent years there have been many studies on SIS and other types of disease transmission models

in spatially and/or temporally varying environments; see [31, 32, 33, 34, 48, 49, 50, 51, 95, 96, 120,

133, 134, 135, 136, 154]. For instance, to study the effect of the movement of exposed individuals on

disease outbreaks, the following SEIRS (susceptible-exposed-infected-recovered-susceptible) epidemic

reaction-diffusion model was considered in [143]:


∂tS = dS∆S −
β(x)SI

S + I + E + R
+ αR (x,t) ∈ Ω × (0,∞),

∂tE = dE∆E +
β(x)SI

S + I + E + R
−σE (x,t) ∈ Ω × (0,∞),

∂tI = dI∆I + σE −γ(x)I (x,t) ∈ Ω × (0,∞),
∂tR = dR∆R + γ(x)I −αR (x,t) ∈ Ω × (0,∞),
∂S

∂ν
=
∂E

∂ν
=
∂I

∂ν
=
∂R

∂ν
= 0 (x,t) ∈ ∂Ω × (0,∞).

(7.2)



SELECTED TOPICS ON REACTION-DIFFUSION-ADVECTION MODELS 173

Here we divide the individuals into four different compartments: susceptible (S), exposed (E), infectious

(I), recovered (immune by vaccination, R). The susceptible individuals are infected by infectious

individuals with a rate of β, and become exposed; exposed individuals become infectious with a rate σ;

infected individuals are recovered with a rate γ; recovered individuals lose immunity and go back into

the susceptible class with a rate of α. Here S(x,t), E(x,t), I(x,t) and R(x,t) denote the density of

susceptible, exposed, infected and recovered individuals at location x and time t, and dS, dE, dI, dR
represent the diffusion rates for susceptible, exposed, infected and recovered populations, respectively.

We assume that the disease transmission rate β(x) and recovery rate γ(x) are environmentally dependent

and could be spatially heterogeneous. Our results in [143] reveal that the travel of exposed individuals

could have an important impact on the persistence of disease and the movement of recovered individuals

may enhance the endemic. Hence, further understanding of the behaviors of the exposed and recovered

individuals could be important in designing effective disease control measures.

In [100] we investigated the spatial SIS model (7.1) with spatially heterogeneous and time-periodic

coefficients and proved that the basic reproduction number R0 is non-decreasing with respect to the
time-period T . In some scenario, this would imply that there is a critical period T∗ > 0 such that R0 < 1
for T < T∗ and R0 > 1 for T > T∗; i.e. increasing the period of disease transmissions and recovery
will increase the chance of disease outbreak. This might suggest why the outbreak of some epidemic

diseases are less frequent than others, e.g. annually vs bi-annually, as different epidemic diseases could

have different transmission and recovery rates so that the threshold values T∗ are different.

How do movement and spatial heterogeneity affect the competition among multiple strains? Many

diseases are present in multiple phenotypes or strains, e.g. the novel coronavirus has found three major

strains which are prevalent in different continents. It is an important evolutionary problem to deter-

mine the environmental conditions under which the adaptability of a certain strain becomes dominant.

Bremermann and Thieme [9] showed that for an SIR model with one host and multiple pathogens,

different pathogens will die out eventually except those that optimize the basic reproduction number.

Recent studies showed that for some non-autonomous multi-strain models, even if the transmission and

recovery rates change time-periodically, these strains cannot coexist; i.e. the temporal heterogeneity

may not increase the chance of the coexistence for multiple pathogens. Tuncer and Martcheva [145] and

Wu et al. [153] considered the following two-strain SIS spatial model to address the question whether

the presence of spatial structure would allow two strains to coexist, as the corresponding spatially

homogeneous model generally leads to competitive exclusion:




∂tS −dS∆S = −
S(β1I1 + β2I2)

S + I1 + I2
+ γ1I1 + γ2I2 (x,t) ∈ Ω × (0,∞),

∂tIi −dI,i∆Ii =
βiSIi

S + I1 + I2
−γiIi (x,t) ∈ Ω × (0,∞),

∂S

∂ν
=
∂Ii
∂ν

= 0 (x,t) ∈ ∂Ω × (0,∞),

S(x, 0) = S0(x), Ii(x, 0) = I0,i(x) x ∈ Ω,

(7.3)

where i = 1, 2. These two strains could be different in dispersal rates, transmission rates and recovery

rates. For the two pathogens differing only in their diffusion rates, a conjecture is that the pathogen

with the slower diffusion will drive the other one to extinction for proper transmission and recovery

rates; see [153] for some partial results. For the two pathogens differing only in their recovery rates, we

conjecture that the strain with the recovery rate of greater spatial variation will be selected eventually.

The coexistence of multiple pathogens in spatially heterogeneous and temporally varying environment

remains a promising open research direction.



174 K.-Y. LAM, S. LIU, AND Y. LOU

Acknowledgment. We thank the anonymous referee and Grégoire Nadin and Lei Zhang for their

helpful comments.

References

[1] L. Allen, B. Bolker, Y. Lou, A. Nevai, Asymptotic profile of the steady states for an SIS epidemic patch model, SIAM

J. Appl. Math. 67 (2007), 1283-1309.

[2] L. Allen, B. Bolker, Y. Lou, A. Nevai, Asymptotic profile of the steady states for a spatial SIS epidemic disease

reaction-diffusion model, Discrete Contin. Dyn. Syst. 21 (2008), 145-164.

[3] L. Allen, Y. Lou, A. Nevai, Spatial patterns in a discrete-time SIS patch model, J. Math Biol. 58 (2009), 339-375.

[4] I. Averill, Y. Lou, D. Munther, On several conjectures from evolution of dispersal, J. Biol. Dyn. 6 (2012), 117-130.

[5] I. Averill, K.-Y. Lam, Y. Lou, The role of advection in a two-species competition model: a bifurcation approach,

Mem. Amer. Math. Soc. 245 (2017): 1161.

[6] X. Bai, X. He, F. Li, An optimization problem and its application in population dynamics, Proc. Amer. Math. Soc.

144 (2016), 2161-2170.

[7] F. Belgacem, C. Cosner, The effects of dispersal along environmental gradients on the dynamics of populations in

heterogeneous environment, Canadian Appl. Math. Quarterly 3 (1995), 379-397.

[8] E. Bouin, S. Mirrahimi, A Hamilton-Jacobi limit for a model of population structured by space and trait, Comm.

Math Sci. 13 (2015), 1431-1452.

[9] H. Bremermann, H.Thieme, A competitive exclusion principle for pathogen virulence, J. Math. Biol. 27 (1989),

179-190.

[10] P.N. Brown, Decay to uniform states in ecological interactions, SIAM J. Appl. Math. 38 (1980), 22-37.

[11] F. Caubet, T. Deheuvels, Y. Privat, Optimal location of resources for biased movement of species: The 1D Case,

SIAM J. Appl. Math. 77 (2017), 1876-1903.

[12] R.S. Cantrell, C. Cosner, Spatial Ecology via reaction-diffusion Equations, Series in Mathematical and Computational

Biology, John Wiley and Sons, Chichester, UK, 2003.

[13] R.S. Cantrell, C. Cosner, K-Y. Lam, On resident-invader dynamics in infinite dimensional dynamical systems, J.

Differential Equations 263 (2017), 4565-4616.

[14] R.S. Cantrell, C. Cosner, Y. Lou, Multiple reversals of competitive dominance in ecological reserves via external

habitat degradation, J. Dyn. Diff. Eqs. 16 (2004), 973-1010.

[15] R.S. Cantrell, C. Cosner, Y. Lou, Movement towards better environments and the evolution of rapid diffusion, Math.

Biosci. 240 (2006), 199-214.

[16] R.S. Cantrell, C. Cosner, Y. Lou, Advection-mediated coexistence of competing species, Proc. Roy. Soc. Edinburgh

Sect. A 137 (2007), 497-518.

[17] R.S. Cantrell, C. Cosner, Y. Lou, Evolution of dispersal and the ideal free distribution, Math. Biosci. Eng. 7 (2010),

17-36.

[18] R.S. Cantrell, C. Cosner, Y. Lou, Evolutionary stability of ideal free dispersal strategies in patchy environments, J.

Math. Biol. 65 (2012), 943-965.

[19] R.S. Cantrell, C. Cosner, Y. Lou, D. Ryan, Evolutionary stability of ideal free dispersal strategies: a nonlocal dispersal

model, Can. Appl. Math. Q. 20 (2012), 15-38.

[20] R.S. Cantrell, C. Cosner, M.A. Lewis, Y. Lou, Evolution of dispersal in spatial population models with multiple

timescales, J. Math. Biol. 80 (2020), 3-37.

[21] R.S. Cantrell, C. Cosner, Y. Lou, S.J. Schreiber, Evolution of natal dispersal in spatially heterogenous environments,

Math. Biosci. 283 (2017), 136-144.

[22] N. Champagnat, R. Ferrière, S. Méléard, Unifying evolutionary dynamics: From individual stochastic processes to

macroscopic models, Theor. Pop. Biol. 69 (2006), 297-321.

[23] S.S. Chen, J.P. Shi, Global dynamics of the diffusive Lotka-Volterra competition model with stage structure, Calc.

Var. Partial Differential Equations 59 (2020): 33.

[24] X.F. Chen, R. Hambrock, Y. Lou, Evolution of conditional dispersal, a reaction-diffusion-advection model, J. Math.

Biol. 57 (2008), 361-386.



SELECTED TOPICS ON REACTION-DIFFUSION-ADVECTION MODELS 175

[25] X.F. Chen, K.-Y. Lam, Y. Lou, Dynamics of a reaction-diffusion-advection model for two competing species, Discrete

Contin. Dyn. Syst. 32 (2012), 3841-3859.

[26] X.F. Chen, Y. Lou, Principal eigenvalue and eigenfunctions of an elliptic operator with large advection and its

application to a competition model, Indiana Univ. Math. J. 57 (2008), 627-657.

[27] X.F. Chen, Y. Lou, Effects of diffusion and advection on the smallest eigenvalue of an elliptic operators and their

applications, Indiana Univ. Math J. 60 (2012), 45-80.

[28] C. Cosner, Beyond Diffusion: Conditional Dispersal in Ecological Models, Infinite dimensional dynamical systems,

305-317, Fields Inst. Commun. 64, Springer, New York, 2013.

[29] C. Cosner, Reaction-diffusion-advection models for the effects and evolution of dispersal, Discrete Contin. Dyn. Syst.

34 (2014), 1701-1745.

[30] C. Cosner, Y. Lou, Does movement toward better environments always benefit a population?, J. Math. Anal. Appl.

277 (2003), 489-503.

[31] R. Cui, K.-Y. Lam, Y. Lou, Dynamics and asymptotic profiles of steady states of an epidemic model in advective

environments, J. Differential Equations 263 (2017), 2343-2373.

[32] R. Cui, Y. Lou, A spatial SIS model in advective heterogeneous environments, J. Differential Equations 261 (2016),

3305-3343.

[33] K. Deng, Asymptotic behavior of an SIR reaction-diffusion model with a linear source, Discrete Contin. Dyn. Syst.

Ser. B 24 (2019), 5945-5957.

[34] K. Deng, Y. Wu, Dynamics of a susceptible-infected-susceptible epidemic reaction-diffusion model, Proc. Roy. Soc.

Edinburgh Sect. A 146 (2016), 929-946.

[35] D. DeAngelis, W.-M. Ni, B. Zhang, Dispersal and spatial heterogeneity: single species, J. Math. Biol. 72 (2016),

239-254.

[36] O. Diekmann, A beginners guide to adaptive dynamics, in: Mathematical modelling of population dynamics, Banach

Center Publications, Vol. 63, Institute of Mathematics Polish Academy of Sciences, Warszawa, 2004, 47-86.

[37] O. Diekmann, J.A.P. Heesterbeek, J.A.J. Metz, On the definition and the computation of the basic reproduction ratio

R0 in models for infectious diseases in heterogeneous populations, J. Math. Biol. 28 (1990), 365-382.

[38] W. Ding, H. Finotti, S. Lenhart, Y. Lou, Q. Ye, Optimal control of growth coefficient on a steady-state population

model, Nonlinear Anal. Real World Appl. 11 (2010), 688-704.

[39] J. Dockery, V. Hutson, K. Mischaikow, M. Pernarowski, The evolution of slow dispersal rates: a reaction-diffusion

model, J. Math. Biol. 37 (1998), 61-83.

[40] Y. Du, S.-B. Hsu, Concentration phenomena in a nonlocal quasi-linear problem modeling phytoplankton I: existence,

SIAM J. Math. Anal. 40 (2008), 1419-1440.

[41] Y. Du, S.-B. Hsu, Concentration phenomena in a nonlocal quasi-linear problem modeling phytoplankton II: limiting

profile, SIAM J. Math. Anal. 40 (2008), 1441-1470.

[42] Y. Du, S.-B. Hsu, Y. Lou, Multiple steady-states in phytoplankton population induced by photoinhibition, J. Differ-

ential Equations 258 (2015), 2408-2434.

[43] Y. Du, B. Lou, R. Peng, M. Zhou, The Fisher-KPP equation over simple graphs: Varied persistence states in river

networks, J. Math. Biol. 80 (2020), 1559-1616.

[44] Y. Du, L. Mei, On a nonlocal reaction-diffusion-advection equation modeling phytoplankton dynamics, Nonlinearity

24 (2011), 319-349.

[45] I. Eshel, U. Motro, Kin selection and strong evolutionary stability of mutual help, Theor. Pop. Biol. 19 (1981),

420-433.

[46] S.D. Fretwell, H.L. Lucas, On territorial behavior and other factors influencing habitat selection in birds, Acta

Biotheretica 19 (1970), 16-36.

[47] A. Friedman, What is mathematical biology and how useful is it ? Notices Amer. Math. Soc. 57 (2010), 851-857.

[48] D. Gao, Travel frequency and infectious disease, SIAM J. Appl. Math. 79 (2019), 1581-1606.

[49] D. Gao, C. Dong, Fast diffusion inhibits disease outbreaks, Proc. Amer. Math. Soc. 148 (2020), 1709-1722.

[50] D. Gao, S. Ruan, An SIS patch model with variable transmission coefficients, Math. Biosci. 232 (2011), 110-115.

[51] J. Ge, K.I. Kim, Z. Lin, H. Zhu, An SIS reaction-diffusion-advection model in a low-risk and high-risk domain, J.

Differential Equations 259 (2015), 5486-5509.



176 K.-Y. LAM, S. LIU, AND Y. LOU

[52] M. Golubitsky, W. Hao, K.-Y. Lam, Y. Lou, Dimorphism by singularity theory in a model for river ecology, Bull.

Math. Biol. 79 (2017), 1051-1069.

[53] R. Gejji, Y. Lou, D. Munther, J. Peyton, Evolutionary convergence to ideal free dispersal strategies and coexistence,

Bull. Math Biol. 74 (2012), 257-299.

[54] S. A. H. Geritz, E. Kisdi, G. Meszéna and J. A. J. Metz, Evolutionarily singular strategies and the adaptive growth

and branching of the evolutionary tree, Evol. Ecol. 12 (1998), 35-57.

[55] R. Hambrock, Y. Lou, The evolution of conditional dispersal strategy in spatially heterogeneous habitats, Bull. Math.

Biol. 71 (2009), 1793-1817.

[56] W. Hao, K.-Y. Lam, Y. Lou, Concentration phenomena in an integro-PDE model for evolution of conditional

dispersal, Indiana Univ. Math. J. 68 (2019), 881-923.

[57] W. Hao, K.-Y. Lam, Y. Lou, Ecological and evolutionary dynamics in advective environments: Critical domain size

and boundary conditions, submitted, 2020.

[58] A. Hastings, Can spatial variation alone lead to selection for dispersal?, Theor. Pop. Biol. 24 (1983), 244-251.

[59] X. He, K.-Y. Lam, Y. Lou, W.-M. Ni, Dynamics of a consumer-resource reaction-diffusion model: Homogeneous

versus heterogeneous environments, J. Math Biol. 78 (2019), 1605-1636.

[60] X. He, W.-M. Ni, The effects of diffusion and spatial variation in Lotka-Volterra competition-diffusion system I:

Heterogeneity vs. homogeneity, J. Differential Equations 254 (2013), 528-546.

[61] X. He, W.-M. Ni, The effects of diffusion and spatial variation in Lotka-Volterra competition-diffusion system II:

The general case, J. Differential Equations 254 (2013), 4088-4108.

[62] X. He, W.-M. Ni, Global dynamics of the Lotka-Volterra competition-diffusion system: Diffusion and spatial hetero-

geneity I, Comm. Pure. Appl. Math. 69 (2016), 981-1014.

[63] X. He, W.-M. Ni, Global dynamics of the Lotka-Volterra competition-diffusion system with equal amount of total

resources II, Calc. Var. Partial Differential Equations 55 (2016): 25.

[64] P. Hess, Periodic-Parabolic Boundary Value Problems and Positivity, Pitman Research Notes in Mathematics Series

247, Longman Scientific & Technical, Harlow, 1991.

[65] A.E. Hershey, J. Pastor, B.J. Peterson, G.W. Kling, Stable isotopes resolve the drift paradox for Baetis mayflies in

an arctic river, Ecology 74 (1993), 2315-2325.

[66] J. Huisman, M. Arrayas, U. Ebert, B. Sommeijer, How do sinking phytoplankton species manage to persist ?, Amer.

Naturalist 159 (2002), 245-254.

[67] J. Huisman, P. van Oostveen, F.J. Weissing, Species dynamics in phytoplankton blooms: incomplete mixing and

competition for light, Amer. Naturalist 154 (1999), 46-67.

[68] J. Huisman, N.N. Pham Thi, D.K. Karl, B. Sommeijer, Reduced mixing generates oscillations and chaos in oceanic

deep chlorophyll, Nature 439 (2006), 322-325.

[69] V. Hutson, W. Shen, G.T. Vickers, Estimates for the principal spectrum point for certain time-dependent parabolic

operators, Proc. Amer. Math. Soc. 129 (2000), 1669-1679.

[70] V. Hutson, Y. Lou, K. Mischaikow, P. Poláčik, Competing species near the degenerate limit, SIAM J. Math. Anal.

35 (2003), 453-491.

[71] V. Hutson, K. Mischaikow, P. Poláčik, The evolution of dispersal rates in a heterogeneous time-periodic environment,

J. Math. Biol. 43 (2001), 501-533.

[72] S.-B. Hsu, Y. Lou, Single phytoplankton species growth with light and advection in a water column, SIAM. J. Appl.

Math. 70 (2010), 2942-2974.

[73] S.-B. Hsu, K.-Y. Lam, F.-B. Wang, Single species growth consuming inorganic carbon with internal storage in a

poorly mixed habitat, J. Math. Biol. 75 (2017), 1775-1825.

[74] J. Inoue, K. Kuto, On the unboundedness of the ratio of species and resources for the diffusive logistic equation,

ArXiv preprint, 2020. https://arxiv.org/abs/2001.06308.

[75] H. Ishii, I. Takagi, Global stability of stationary solutions to a nonlinear diffusion equation in phytoplankton dynam-

ics, J. Math. Biol. 16 (1982), 1-24.

[76] P.E. Jabin, R. Schram, Selection-Mutation dynamics with spatial dependence, ArXiv preprint, 2020.

http://arxiv.org/abs/1601.04553.

[77] D. Jiang, K.-Y. Lam, Y. Lou, Z.-C. Wang, Monotonicity and global dynamics of a nonlocal two-species phytoplankton

model, SIAM J. Appl. Math. 79 (2019), 716-742.

https://arxiv.org/abs/2001.06308
http://arxiv.org/abs/1601.04553


SELECTED TOPICS ON REACTION-DIFFUSION-ADVECTION MODELS 177

[78] Y. Jin, R. Peng, J.P. Shi, Population dynamics in river networks, J. Nonlinear Sci. 29 (2019), 2501-2545.

[79] C.Y Kao, Y. Lou, E. Yanagida, Principal eigenvalue for an elliptic problem with indefinite weight on cylindrical

domains, Math. Biosci. Eng. 5 (2008), 315-335.

[80] J.T. Kirk, Light and Photosynthesis in Aquatic Ecosystems, Cambridge University Press, Cambridge, UK, 1994.

[81] T. Kolokolnikov, C.H. Ou, Y. Yuan, Phytoplankton depth profiles and their transitions near the critical sinking

velocity, J. Math. Biol. 59 (2009), 105-122.

[82] K.-Y. Lam, A semilinear equation with large advection in population dynamics, Thesis (Ph.D.)-University of Min-

nesota. 2011. 91 pp. ISBN: 978-1124-67112-3, ProQuest LLC

[83] K.-Y. Lam, Concentration phenomena of a semilinear elliptic equation with large advection in an ecological model,

J. Differential Equations 250 (2011), 161-181.

[84] K.-Y. Lam, Limiting profiles of semilinear elliptic equations with large advection in population dynamics II, SIAM

J. Math. Anal. 44 (2012), 1808-1830.

[85] K.-Y. Lam, Stability of Dirac concentrations in an integro-PDE model for evolution of dispersal, Calc. Var. Partial

Differential Equations 56 (2017), no. 3, Paper No. 79, 32 pp.

[86] K.-Y. Lam, Y. Lou, An integro-PDE from evolution of random dispersal, J. Func. Anal. 272 (2017), 1755-1790.

[87] K.-Y. Lam, Y. Lou, Evolution of dispersal: ESS in spatial models, J. Math. Biol. 68 (2014), 851-877.

[88] K.-Y. Lam, Y. Lou, Evolutionarily stable and convergent stable strategies in reaction-diffusion models for conditional

dispersal, Bull. Math. Biol. 76 (2014), 261-291.

[89] K.-Y. Lam, Y. Lou, Persistence, Competition, and Evolution. In: Bianchi A., Hillen T., Lewis M., Yi Y. (eds) The

Dynamics of Biological Systems. Mathematics of Planet Earth, vol 4. Springer, Cham, 2019.

[90] K.-Y. Lam, Y. Lou, F. Lutscher, Evolution of dispersal in closed advective environments, J. Biol. Dyn. 9 (2015),

188-212.

[91] K.-Y. Lam, W.-M. Ni, Limiting profiles of semilinear elliptic equations with large advection in population dynamics,

Discrete Contin. Dyn. Syst. 28 (2010), 1051-1067.

[92] K.-Y. Lam, W.-M. Ni, Uniqueness and complete dynamics of the Lotka-Volterra competition diffusion system, SIAM

J. Appl. Math. 72 (2012), 1695-1712.

[93] K.-Y. Lam, W.-M. Ni, Advection-mediated competition in general environments, J. Differential Equations 257 (2014),

3466-3500.

[94] K.-Y. Lam, Y. Lou, B. Perthame, A Hamilton-Jacobi approach to evolution of dispersal, in preparation.

[95] H. Li, R. Peng, F.-B. Wang, Varying total population enhances disease persistence: qualitative analysis on a diffusive

SIS epidemic model, J. Differential Equations 262 (2017), 885-913.

[96] H. Li, R. Peng, T. Xiang, Dynamics and asymptotic profiles of endemic equilibrium for two frequency-dependent SIS

epidemic models with cross-diffusion, Eur. J. Appl. Math. 31 (2020), 26-56.

[97] R. Li, Y. Lou, Some monotone properties for solutions to a reaction-diffusion model, Discrete Contin. Dyn. Syst.

Ser. B 24 (2019), 4445-4455.

[98] S. Liang, Y. Lou, On the dependence of the population size on the dispersal rate, Discrete Contin. Dyn. Syst. Ser. B

17 (2012), 2771-2788.

[99] S. Liu, Y. Lou, A functional approach towards eigenvalue problems associated with incompressible flow, Discrete

Contin. Dyn. Syst. 40 (2020), 3715-3736.

[100] S. Liu, Y. Lou, The roles of frequency and movement in a spatial SIS model with time-periodic environment,

submitted, 2020.

[101] S. Liu, Y. Lou, R. Peng, M. Zhou, Monotonicity of the principal eigenvalue for a linear time-periodic parabolic

operator, Proc. Amer. Math. Soc. 47 (2019), 5291-5302.

[102] S. Liu, Y. Lou, R. Peng, M. Zhou, Asymptotics of the principal eigenvalue for a linear time-periodic parabolic

operator I: Large advection, ArXiv preprint, 2020. https://arxiv.org/abs/2002.01330.

[103] S. Liu, Y. Lou, R. Peng, M. Zhou, Asymptotics of the principal eigenvalue for a linear time-periodic parabolic

operator II: Small diffusion, ArXiv preprint, 2020. https://arxiv.org/abs/2002.01357.

[104] S. Liu, Y. Lou, P. Song, A new monotonicity for principal eigenvalues with applications to time-periodic patch

models, submitted, 2020.

[105] Y. Lou, On the effects of migration and spatial heterogeneity on single and multiple species, J. Differential Equations

223 (2006), 400-426.

https://arxiv.org/abs/2002.01330
https://arxiv.org/abs/2002.01357


178 K.-Y. LAM, S. LIU, AND Y. LOU

[106] Y. Lou, S. Martinez, P. Poláčik, Loops and branches of coexistence states in a Lotka-Volterra competition model,

J. Differential Equations 230 (2006), 720-742.

[107] Y. Lou, Some challenging mathematical problems in evolution of dispersal and population dynamics, Tutorials in

mathematical biosciences. IV, 171-205, Lecture Notes in Math., 1922, Math. Biosci. Subser., Springer, Berlin, 2008.

[108] Y. Lou, Some reaction-diffusion models in spatial ecology (in Chinese), Sci. Sin. Math 45 (2015), 1619-1634.

[109] Y. Lou, On several partial differential equation models in population dynamics (in Chinese), to appears in the

Lecture Series of Institute for Mathematics, Chinese Academy of Sciences.

[110] Y. Lou, F. Lutscher, Evolution of dispersal in advective environments, J. Math Biol. 69 (2014), 1319-1342.

[111] Y. Lou, D. Munther, Dynamics of a three species competition model, Discrete Contin. Dyn. Syst. 32 (2012), 3099-

3131.

[112] Y. Lou, D. Xiao, P. Zhou, Qualitative analysis for a Lotka-Volterra competition system in advective homogeneous

environment, Discrete Contin. Dyn. Syst. 36 (2016), 953-969.

[113] Y. Lou, B. Wang, Local dynamics of a diffusive predator-prey model in spatially heterogeneous environment, J.

Fixed Point Theory Appl. 19 (2017), 755-772.

[114] Y. Lou, X.-Q. Zhao, P. Zhou, Global dynamics of a Lotka-Volterre competition-diffusion-advection system in het-

erogeneous environments, J. Math. Pure. Appl. 121 (2019), 47-82.

[115] Y. Lou, P. Zhou, Evolution of dispersal in advective homogeneous environments: The effect of boundary conditions,

J. Differential Equations 259 (2015), 141-171.

[116] F. Lutscher, E. Pachepsky, M.A. Lewis, The effect of dispersal patterns on stream populations, SIAM Rev. 47

(2005), 749-772.

[117] F. Lutscher, E. McCauley, M.A. Lewis, Effects of heterogeneity on spread and persistence in rivers, Bull. Math.

Biol. 68 (2006), 2129-2160.

[118] M.J. Ma, C.H. Ou, Existence, uniqueness, stability and bifurcation of periodic patterns for a seasonal single phyto-

plankton model with self-shading effect, J. Differential Equations 263 (2017), 5630-5655.

[119] J. Maynard Smith, G. Price, The logic of animal conflict, Nature 246 (1973), 15-18.

[120] P. Magal, G. F. Webb, Y. Wu, On the basic reproduction number of reaction-diffusion epidemic models, SIAM J.

Appl. Math. 79 (2019), 284-304.

[121] I. Mazari, G. Nadin, Y. Privat, Optimal location of resources maximizing the total population size in logistic models,

J. Math. Pure. Appl. 134 (2020), 1-35.

[122] I. Mazari, G. Nadin, Y. Privat, Shape optimization of a weighted two-phase Dirichlet eigenvalue, ArXiv preprint,

2020. https://arxiv.org/abs/2001.02958.

[123] I. Mazari, D. Ruiz-Balet, A fragmentation phenomenon for a non-energetic optimal control problem: optimisation

of the total population size in logistic diffusive models, preprint, 2020. https://hal.archives-ouvertes.fr/hal-02523753.

[124] L. Mei, X.Y. Zhang, Existence and nonexistence of positive steady states in multi-species phytoplankton dynamics,

J. Differential Equations 253 (2012), 2025-2063.

[125] S. Mirrahimi, B. Perthame, Asymptotic analysis of a selection model with space, J. Math. Pures Appl. 104 (2015),

1108-1118.

[126] K. Müller, Investigations on the organic drift in North Swedish streams, Rep. Inst. Freshw. Res. Drottningholm 35

(1954), 133-148.

[127] K. Nagahara, Y. Lou, E. Yanagida, Maximizing the total population in a patchy environment, submitted, 2020.

[128] K. Nagahara, E. Yanagida, Maximization of the total population in a reaction-diffusion model with logistic growth,

Calc. Var. Partial Differential Equations 57 (2018): 80.

[129] W.-M. Ni, The Mathematics of Diffusion, CBMS Reg. Conf. Ser. Appl. Math. 82, SIAM, Philadelphia, 2011.

[130] W.-M. Ni, private communication.

[131] W. Ni, J. Shi, M. Wang, Global stability of nonhomogeneous equilibrium solution for the diffusive Lotka-Volterra

competition model, ArXiv preprint, 2020. http://arxiv.org/abs/1909.03376.

[132] S. Nordmann, B. Perthame, Dynamics of concentration in a population structured by age and a phenotypic trait

with mutations. Convergence of the corrector, ArXiv preprint, 2020. http://arxiv.org/abs/2001.04323v1.

[133] R. Peng, Asymptotic profiles of the positive steady state for an SIS epidemic reaction-diffusion model I, J. Differ-

ential Equations 247 (2009), 1096-1119.

https://arxiv.org/abs/2001.02958
https://hal.archives-ouvertes.fr/hal-02523753
http://arxiv.org/abs/1909.03376
http://arxiv.org/abs/2001.04323v1


SELECTED TOPICS ON REACTION-DIFFUSION-ADVECTION MODELS 179

[134] R. Peng S. Liu, Global stability of the steady states of an SIS epidemic reaction-diffusion model, Nonlinear Anal.

71 (2009), 239-247.

[135] R. Peng, F. Yi, Asymptotic profile of the positive steady state for an SIS epidemic reaction-diffusion model: effects

of epidemic risk and population movement, Phys. D 259 (2013), 8-25.

[136] R. Peng, X.Q. Zhao, A reaction-diffusion SIS epidemic model in a time-periodic environment, Nonlinearity 25

(2012), 1451-1471.

[137] R. Peng, X.-Q. Zhao, Effects of diffusion and advection on the principal eigenvalue of a periodic parabolic problem

with applications, Calc. Var. Partial Differential Equations 54 (2015), 1611-1642.

[138] R. Peng, X.-Q. Zhao, A nonlocal and periodic reaction-diffusion-advection model of a single phytoplankton species,

J. Math. Biol. 72 (2016), 755-791.

[139] R. Peng, M. Zhou, Effects of large degenerate advection and boundary conditions on the principal eigenvalue and

its eigenfunction of a linear second order elliptic operator, Indiana Univ. Math J. 67 (2018), 2523-2568.

[140] B. Perthame, P.E. Souganidis, Rare mutations limit of a steady state dispersion trait model, Math. Model. Nat.

Phenom. 11 (2016), 154-166.

[141] N. Shigesada, A. Okubo, Analysis of the self-shading effect on algal vertical distribution in natural waters, J. Math.

Biol. 12 (1981), 311-326.

[142] H. Smith, Monotone Dynamical Systems, Mathematical Surveys and Monographs 41, American Mathematical

Society, Providence, Rhode Island, U.S.A. 1995

[143] P. Song, Y. Lou, Y. Xiao, A spatial SEIRS reaction-diffusion model in heterogeneous environment, J. Differential

Equations 267 (2018), 5084-5114.

[144] D.C. Speirs, W.S.C. Gurney, Population persistence in rivers and estuaries, Ecology 82 (2001), 1219-1237.

[145] N. Tuncer, M. Martcheva, Analytical and numerical approaches coexistence of strains in a two-strain SIS model

with diffusion, J Biol. Dyn. 6 (2012), 406-439.

[146] P. van den Driessche, J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental

models of disease transmission, John A. Jacquez memorial volume, Math. Biosci. 180 (2002), 29-48.

[147] O. Vasilyeva, Population dynamics in river networks: analysis of steady states, J. Math. Biol. 79 (2019), 63-100.

[148] O. Vasilyeva, F. Lutscher, Population dynamics in rivers: analysis of steady states, Can. Appl. Math. Quart. 18

(2011), 439-469.

[149] W. Wang, X.-Q. Zhao, A nonlocal and time-delayed reaction-diffusion model of dengue transmission, SIAM J. Appl.

Math. 71 (2011), 147-168.

[150] W.-D. Wang, X.-Q. Zhao, Basic reproduction numbers for reaction-diffusion epidemic models, SIAM J. Appl. Dyn.

Syst. 11 (2012), 1652-1673.

[151] Y. Wang, J.P. Shi, Persistence and extinction of population in reaction-diffusion-advection model with weak Allee

effect growth, SIAM J. Appl. Math. 79 (2019), 1293-1313.

[152] Y. Wang, J.P. Shi, J.F. Wang, Persistence and extinction of population in reaction-diffusion-advection model with

strong Allee effect growth, J. Math. Biol. 78 (2019), 2093-2140.

[153] Y. Wu, N. Tuncer, M. Martcheva, Coexistence and competitive exclusion in an SIS model with standard incidence

and diffusion, Discrete Contin. Dyn. Syst. Ser. B 22 (2017), 1167-1187.

[154] Y. Wu, X. Zou, Asymptotic profiles of steady states for a diffusive SIS epidemic model with mass action infection

mechanism, J. Differential Equations 261 (2016), 4424-4447.

[155] J.-J. Xiang, Y. Fang, Evolutionarily stable dispersal strategies in a two-patch advective environment, Discrete

Contin. Dyn. Syst. Ser. B 24 (2019), 1875-1887.

[156] F. Xu, W. Gan, D. Tang, Population dynamics and evolution in river ecosystems, Nonlinear Anal. Real World

Appl. 51 (2020): 102983.

[157] X.-Q. Zhao, P. Zhou, On a Lotka-Volterra competition model: the effects of advection and spatial variation, Calc.

Var. Partial Differential Equations 55 (2016): 73.

[158] P. Zhou, On a Lotka-Volterra competition system: diffusion vs advection, Calc. Var. Partial Differential Equations

55 (2016): 137.



180 K.-Y. LAM, S. LIU, AND Y. LOU

Corresponding author. Department of Mathematics, The Ohio State University, Columbus, OH 43210,

USA

E-mail address: lam.184@math.ohio-state.edu

Institute for Mathematical Sciences, Renmin University of China, Beijing 100872, PRC

E-mail address: liushuangnqkg@ruc.edu.cn

Department of Mathematics, The Ohio State University, Columbus, OH 43210, USA

E-mail address: lou@math.ohio-state.edu


	1. Introduction
	2. Single species models
	2.1. Logistic model
	2.2. Single species models in rivers

	3. Competing species models
	3.1. Lotka-Volterra competition models
	3.2. Competition in rivers

	4. Competition models with directed movement
	4.1. Global bifurcation diagrams
	4.2. Evolution of biased movement

	5. Continuous trait models
	5.1. Mutation-selection model
	5.2. Mutation-selection model with drift

	6. Dynamics of phytoplankton
	6.1. Single phytoplankton species
	6.2. Two phytoplankton species

	7. Spatial dynamics of epidemic diseases
	References