Mathematics in Applied Sciences and Engineering https://ojs.lib.uwo.ca/mase
Volume 3, Number 3, 2022, pp.190-199 https://doi.org/10.5206/mase/15074

THE EXISTENCE AND UNIQUENESS OF SOLUTIONS OF A NONLINEAR

TOXIN-DEPENDENT SIZE-STRUCTURED POPULATION MODEL

YAN LI AND QIHUA HUANG

Abstract. In this paper, we study a toxin-mediated size-structured population model with nonlinear

reproduction, growth, and mortality rates. By using the characteristic method and the contraction

mapping argument, we establish the existence-uniqueness of solutions to the model. We also prove

the continuous dependence of solutions on initial conditions.

1. Introduction

How do anthropogenic and natural environmental toxins affect population dynamics and ecological

integrity? It is an essential question in environmental toxicology [1, 11]. Mathematical models (including

individual-based models, matrix population models, ordinary differential equation models, and so on)

have been widely applied to address this question [4, 5, 6, 7, 9]. In terms of the fact that in a population,

individuals of different sizes may have different sensitivities to toxins, Huang and Wang [8] developed

a size-structured population model for a population living in an aquatic polluted ecosystem, which is

given by the following system of nonlinear first-order hyperbolic equations:


ut + (g(x,P(t))u)x + µ(x,P(t),v(x,t))u = 0, x ∈ (xmin,xmax), t > 0,

vt + g(x,P(t))vx + σ(x,t)v −a(x,t)E(t) = 0, x ∈ (xmin,xmax), t > 0,

g(xmin,P(t))u(xmin, t) =

∫ xmax
xmin

β(x,P(t),v(x,t))u(x,t)dx, t > 0,

v(xmin, t) = 0, t > 0,

u(x, 0) = u0(x), x ∈ (xmin,xmax),

v(x, 0) = v0(x), x ∈ (xmin,xmax)

(1.1)

where u(x,t) represents the density of individuals of size x at time t; P(t) =
∫xmax
xmin

u(x,t)dx is the total

population biomass at time t, where xmin and xmax denote the minimize size and the maximum size of

the population, respectively; v(x,t) denotes the size-dependent body burden — concentration of toxin

per unit population biomass. The function g(x,P(t)) represents the growth rate of an individual of

size x which depends on the total population biomass due to competition for resources. The function

Received by the editors 7 July 2022; accepted 26 September 2022; published online 29 September 2022.

2010 Mathematics Subject Classification. 35L60; 35F15; 92F99.
Key words and phrases. Existence-uniqueness; continuous dependence; size-structured population; characteristic

method; contraction mapping theorem.
Yan Li is supported by the Natural Science Foundation of Shandong Province, China (No.s ZR2021MA028,

ZR2021MA025).

Qihua Huang is supported by the National Natural Science Foundation of China (No. 11871060), the Venture and

Innovation Support Program for Chongqing Overseas Returnees (No. 7820100158).

190

https://ojs.lib.uwo.ca/mase
https://dx.doi.org/https://doi.org/10.5206/mase/15074


THE EXISTENCE AND UNIQUENESS OF SOLUTIONS 191

µ(x,P(t),v(x,t)) denotes the mortality rate of an individual of size x which depend on the total pop-

ulation biomass and the body burden. The function β(x,P(t),v(x,t)) is the reproduction rate of an

individual of size x. The function σ(x,t) is the toxin elimination rate due to the metabolic process

of the population. The function a(x,t) represents the toxin uptake rate by the population from the

environment. The function E(t) is the concentration of toxin in the environment at time t. See [8] for

detailed model derivation.

In [8], an explicit finite difference approximation to partial differential equation problem (1.1) was

developed. The existence and uniqueness of the weak solution — a solution in integral form with two

test functions — were established and convergence of the finite difference approximation to this unique

weak solution was proved.

The main purpose of this paper is to prove the existence-uniqueness of solutions of problem (1.1) by

using the characteristic method and contraction mapping theorem, and the continuous dependence on

initial conditions, which are quite different from the method of numerical approximation used in (1.1).

2. Existence and uniqueness results

Throughout the discussion, we let Ω1 = (xmin,xmax)×(0,∞) and Ω2 = (xmin,xmax)×(0,∞)×(0,∞).
We make the following assumptions on the parameters in problem (1.1):

(H1) g(x,P) is a strictly positive Lipschitz function with respect to x and P in Ω1 with a common

Lipschitz constant Lg.

(H2) µ(x,P,v) is a nonnegative Lipschitz function with respect to x,P, and v in Ω2 with a common

Lipschitz constant Lµ.

(H3) β(x,P,v) is a nonnegative Lipschitz function with respect to x,P,v in Ω2 with a common

Lipschitz constant Lβ. Furthermore, β(x,P,v) is uniformly bounded in Ω2 with 0 ≤ β ≤ βM .
(H4) a(x,t) is a nonnegative Lipschitz function with respect to x in Ω1 with a Lipschitz constant La.

Furthermore, a(x,t) is uniformly bounded in Ω1 with 0 ≤ a ≤ aM .
(H5) σ(x,t) is a nonnegative Lipschitz function with respect to x and t in Ω1 with a Lipschitz constant

Lσ.

(H6) E(t) is a nonnegative continuous function and bounded for 0 < t < ∞ with 0 ≤ E(t) ≤ EM .
(H7) u0(x) ∈ L1(xmin,xmax) and u0(x) ≥ 0.
(H8) v0(x) is a nonnegative Lipschitz function with a Lipschitz constant Lv and bounded for xmin <

x < xmax with 0 ≤ v0(x) ≤ vM .

We begin with the definition of the solutions of problem (1.1).

Definition 2.1. A nonnegative function (u(x,t),v(x,t)) on [xmin,xmax] × [0,T), with u(x,t) and
v(x,t) integrable, is a solution of (1.1) if P(t) =

∫xmax
xmin

u(x,t)dx is a continuous function on [0,T)

and (u(x,t),v(x,t)) satisfies (1.1)3,4,5,6 and the equations

Du(x,t) = −µ̃u(x,t), (2.1)

Dv(x,t) = −[σ(x,t)v −a(x,t)E(t)] (2.2)

with

Du(x,t) = lim
h→0

u(X(t + h; x,t), t + h) −u(x,t)
h

,

Dv(x,t) = lim
h→0

v(X(t + h; x,t), t + h) −v(x,t)
h

,



192 YAN LI AND QIHUA HUANG

where T is a positive constant, µ̃(x,P(t),v(x,t)) = gx(x,P(t)) + µ(x,P(t),v(x,t)) and X(t; x0, t0) is

the solution of the equation for the characteristic curves given by


dx

dt
= g(x,P(t)),

x(t0) = x0.
(2.3)

From (H1), we know that the function X(t; x0, t0) is strictly increasing. Thus a unique inverse

function τ(x; x0, t0) exists. Let Z(t) = X(t; xmin, 0) be the characteristic through the point (xmin, 0).

In what follows, we reduce problem (1.1) to a system of coupled equations for P(t) and B(t) by using

the method of characteristics, where

B(t) =

∫ xmax
xmin

β(x,P(t),v(x,t))u(x,t)dx.

Integrating (2.1) along the characteristics, we have

u(x,t) =



u0(X(0; x,t))e

−
∫
t
0
µ̃(X(s;x,t),P(s),v(X(s;x,t),s))ds, x ≥ Z(t),

B(τ(xmin))

g(xmin; P(τ(xmin)))
e
−

∫
t
τ(xmin)

µ̃(X(s;xmin,τ(xmin)),P(s),v(X(s;xmin,τ(xmin)),s))ds, x < Z(t),

(2.4)

where τ(xmin) = τ(xmin; x,t). Similarly, we have

v(x,t) =



v0(X(0; x,t))e

−
∫
t
0
σ(X(s;x,t),s)ds +

∫ t
0

a(X(s; x,t),s)E(s)e−
∫
t
s
σ(X(τ;x,t),τ)dτ ds, x ≥ Z(t),

0, x < Z(t).

(2.5)

Then

P(t) =

∫ xmax
xmin

u(x,t)dx =

∫ t
0

B(η)e
−

∫
t
η
µ(X(s,xmin,η),P(s),v(X(s;xmin,η),s))dsdη

+

∫ xmax
xmin

u0(ξ)e
−

∫
t
0
µ(X(s;ξ,0),P(s),v(X(s;ξ,0)))dsdξ

(2.6)

and

B(t) =

∫ t
0

β(X(t; xmin,η),P(t),v(X(t; xmin,η), t))B(η)e
−

∫
t
η
µ(X(s,xmin,η),P(s),v(X(s;xmin,η),s))dsdη

+

∫ xmax
xmin

β(X(t; xmin,ξ),P(t),v(X(t; xmin,ξ), t))u0(ξ)e
−

∫
t
0
µ(X(s;ξ,0),P(s),v(X(s;ξ,0)))dsdξ.

(2.7)

If P(t) and B(t) are nonnegative continuous solutions of (2.6) and (2.7), then u(x,t) and v(x,t)

defined by (2.4) and (2.5) respectively are the solutions of (1.1). On the other hand, if u(x,t) and

v(x,t) are the solutions of (1.1), then P(t) and B(t) are nonnegative continuous solutions of (2.6) and

(2.7). Therefore, in order to obtain the existence and uniqueness results for problem (1.1), we only need

to study the solvability of the system consisting of integral equations (2.6) and (2.7).

By using the contraction mapping theorem, we first obtain the local existence and uniqueness results

for problem (1.1). To this end, let

ST;K = {f ∈ C[0,T]|f(0) = ‖u0‖L1, 0 ≤ f(t) ≤ K, where K > ‖u0‖L1},
ST;H = {f ∈ C[0,T]|0 ≤ f(t) ≤ H, where H > βM‖u0‖L1}.

For each P ∈ ST;K, the function X(t; x0, t0) is well-defined by the characteristic curve (2.3). Thus,
there is a unique function v(x,t) determined by (2.5).



THE EXISTENCE AND UNIQUENESS OF SOLUTIONS 193

Define the operator Y : ST;K × ST;H → C[0,T] × C[0,T] by Y (P,B) = (Φ(P,B), Ψ(P,B)) where
Φ(P,B) and Ψ(P,B) are given by the right-hand sides of (2.6) and (2.7) respectively. Then, a fixed point

of the operator Y corresponds to a solution of (2.6) and (2.7). Next lemma establishes the existence

and uniqueness of a fixed point of the operator Y .

Lemma 2.1. Suppose that hypotheses (H1)-(H8) hold. Then there exists a value T > 0 for which Y

has a unique fixed point in ST;K ×ST;H ⊂ C[0,T] ×C[0,T].

Proof. As mentioned above, we just need to show that Y has a unique fixed point in ST;K ×ST;H. For
any P,P̂ ∈ ST,K, B,B̂ ∈ ST,H, let u,û and v, v̂ be given by (2.4) and (2.5) corresponding to B,B̂ and
P,P̂ , respectively. We use the following notations to simplify the expressions:

µ(X
P̂

(s; xmin,η), P̂(s), v̂(s,t)) = µP̂ , µ(Xp(s; xmin,η),P(s),v(s,t)) = µP ;

β(X
P̂

(s; xmin,η), P̂, v̂) = βP̂ , β(Xp(s; xmin,η),P,v) = βP ;

µ(X
P̂

(s; ξ, 0), P̂, v̂) = µ̄
P̂
, µp(X(s; ξ, 0),P,v) = µ̄P ;

β
P̂

(X(s; ξ, 0), P̂, v̂) = β̄
P̂
, βp(X(s; ξ, 0),P,v) = β̄P .

In terms of (2.7), we can conclude that

Ψ(P,B)(t) ≤ βM
∫ t

0

B(η)dη + βM‖u0‖L1 ≤ βMHT + βM‖u0‖L1 ≤ H.

By a series of computations, we have

|Ψ(P,B)(t) − Ψ(P̂,B̂)(t)| ≤ TβM‖B − B̂‖∞ + (βMHLµ + HLβ)T(|XP −XP̂ |

+ |P − P̂ | + |v − v̂|).

Since XP (t; xmin, 0) and XP̂ (t; xmin, 0) are the solutions of


dx

dt
= g(x,P(t))

x(0) = xmin

and 


dx

dt
= g(x,P̂(t))

x(0) = xmin

respectively, we have that

|XP −XP̂ | ≤ Lg
∫ t

0

(|XP −XP̂ | + |P − P̂ |)ds. (2.8)

Gronwall’s inequality tells us that

|XP −XP̂ | ≤ LgTe
LgT‖P − P̂‖∞. (2.9)

Similarly, we get

|βP −βP̂ | ≤ Lβ(|XP −XP̂ | + |P − P̂ | + |v − v̂|)ds, (2.10)

|µP −µP̂ | ≤ Lµ(|XP −XP̂ | + |P − P̂ | + |v − v̂|)ds, (2.11)



194 YAN LI AND QIHUA HUANG

|v − v̂| ≤ vM
∫ t

0

|σ(XP ) −σ(XP̂ )|ds + EM
∫ t

0

a(XP ,s)

∫ t
s

|σ(XP ) −σ(XP̂ )|dsds

+ |v0(XP ) −v0(XP̂ )| + EM
∫ t

0

|a(XP ,s) −a(XP̂ ,s)|ds

≤ (vMTLσ + Lv + EMaMTLσ + EMLaT)|XP −XP̂ |

≤ (vMTLσ + Lv + EMaMTLσ + EMLaT)LgTeLgT‖P − P̂‖∞.

(2.12)

Thus,

|Ψ(P,B)(t) − Ψ(P̂,B̂)(t)| ≤ TβM‖B − B̂‖∞ + h1(T)T‖P − P̂‖∞, (2.13)

where

h1(T) = (βMHLµ + HLβ) T
[
LgTe

LgT + 1

+(vMTLσ + Lv + EMaMTLσ + EMLaT)LgTe
LgT

]
.

For the Φ component, note that

Φ(P,B)(t) − Φ(P̂,B̂)(t)) =
∫ t

0

(B(η) − B̂(η))e−
∫
t
η
µP dsdη

+

∫ t
0

B̂(η)(e
−

∫
t
η
µP ds −e−

∫
t
η
µ
P̂

ds
)dη

+

∫ xmax
xmin

u0(ξ)(e
−

∫
t
0
µ̄P ds −e−

∫
t
0
µ̄
P̂

ds)dξ

≤
∫ t

0

|B(η) − B̂(η)|dη +
∫ t

0

B̂(η)

∫ t
η

|µP −µP̂ |dsdη

+

∫ xmax
xmin

u0(ξ)

∫ t
0

|µ̄P − µ̄P̂ |dsdξ.

(2.14)

Let F(η) = B(η) − B̂(η), by (2.7), we get

|F(t)| ≤βM
∫ t

0

|F(η)|dη +
∫ t

0

B̂(η)|βP −βP̂ |dη + βM
∫ t

0

B̂(η)|µP −µP̂ |dη

+

∫ xmax
xmin

u0(ξ)|β̄Pe−
∫
t
0
µ̄P ds − β̄

P̂
e−

∫
t
0
µ̄
P̂

ds|dξ,
(2.15)

which leads to

|F(t)| ≤ βM
∫ t

0

|F(η)|dη + ψ(t),

where

ψ(t) =

∫ t
0

B̂(η)|βP −βP̂ |dη + βM
∫ t

0

B̂(η)|µP −µP̂ |dη

+

∫ xmax
xmin

u0(ξ)|β̄Pe−
∫
t
0
µ̄P ds − β̄

P̂
e−

∫
t
0
µ̄
P̂

ds|dξ.
(2.16)



THE EXISTENCE AND UNIQUENESS OF SOLUTIONS 195

We also find that∣∣∣β̄Pe−∫ t0 µ̄P ds − β̄P̂e−∫ t0 µ̄P̂ ds∣∣∣ =|(β̄P − β̄p̂)e−∫ t0 µ̄P ds + β̄P̂ (e−∫ t0 µ̄P ds −e−∫ t0 µ̄P̂ ds)|
≤ |β̄P − β̄p̂| + βM

∫ t
0

|µ̄P − µ̄P̂ |ds.
(2.17)

From the above analysis, we can conclude that

ψ(t) ≤ [βM‖u0‖L1eβMT (Lβ + βMLµ) + ‖u0‖L1 (Lβ + βMLµT)]·

[Lge
LgTT + 1 + (vMTLσ + Lv + EMaMTLσ + EMLaT)e

LgTTLg]‖P − P̂‖∞

=: J(T)‖P − P̂‖∞.

Thus,

|F(t)| ≤ βM
∫ t

0

|F(η)|dη + ψ(t) ≤ βM
∫ t

0

|F(η)|dη + J(T)‖P − P̂‖∞.

By Gronwall’s inequality, we have that

|F(t)| ≤ J(T)‖P − P̂‖∞e
∫
t
0
βM dτ = J(T)‖P − P̂‖∞eβMt.

Therefore,

|Φ(P,B)(t) − Φ(P̂,B̂)(t)|

≤ TJ(T)‖P − P̂‖∞eβMT + (βM‖u0‖L1eβMTT + ‖u0‖L1 )T 2Lµ‖P − P̂‖∞
(Lge

LgTT + 1 + (vMTLσ + Lv + EMaMTLσ + EMLaT)TLge
LgT )

=: Th2(T)‖P − P̂‖∞.

(2.18)

Combining (2.13) and (2.18), we obtain

||Y (P,B) −Y (P,B)|| = ||Ψ(P,B) − Ψ(P̂,B̂)|| + ||Φ(P,B) − Φ(P̂,B̂)||

≤ (Th1(T) + Th2(T))||P − P̂ ||∞ + TβM||B − B̂||∞
= r(T)(||P − P̂ ||∞ + ||B − B̂||∞),

(2.19)

where

r(T) = max{(Th1(T) + Th2(T)),TβM}.

Note that r(0) = 0. Therefore, there exists a sufficiently small constant T > 0 such that r(T) ∈ (0, 1).
Hence, for such a small T , the mapping Y is a contractive mapping. By the contracting mapping

theorem, Y has a fixed point. The proof is completed. �

Note that the uniqueness of the solution P(t) and B(t) of system (2.6)-(2.7) implies that the unique-

ness of the solution to problem (1.1) because each u(x; t), v(x,t) given by (2.4) and (2.5) is uniquely

determined by P(t) and B(t). Thus, we have the following result on local existence and uniqueness to

(1.1).

Theorem 2.2. Suppose that hypotheses (H1)-(H8) hold. Then there exists a value T > 0 such that

problem (1.1) has a unique solution up to time T .

In order to establish the global existence-uniqueness result for problem (1.1), we conclude the fol-

lowing upper bound on P(t) for t ∈ [0,T].



196 YAN LI AND QIHUA HUANG

Lemma 2.3. Let u(x,t) and v(x,t) be a solution of (1.1) up to time T . Then for t ∈ [0,T], P(t)
satisfies the following bound

P(t) ≤‖u0‖L1eβMt.

Proof. P(t) is differentiable since P(t) =
∫xmax
xmin

u(x,t)dx and u(x,t) is differentiable by Definition 2.1.

Differentiating (2.6) with respect to t, we get

P ′(t) =

∫ xmax
xmin

(β(x,P(t), ,v(x,t)) −µ(x,P(t), ,v(x,t)))u(x,t)dx ≤ βMP(t),

Gronwall’s inequality tells us that

P(t) ≤‖u0‖L1eβMt.

Using similar arguments as in the proof of Theorem 3 in [2], we are able to derive the following global

existence-uniqueness result. �

Theorem 2.4. Suppose that hypotheses (H1)-(H8) hold. Then problem (1.1) has a unique solution for

t ∈ [0,∞).

3. Continuous dependence on initial conditions

The purpose of this section is to establish the continuous dependence of solutions on initial conditions.

For this purpose, we first show that the fixed point of the operator Φ associated with an initial condition

depends continuously on initial conditions.

Lemma 3.1. Let P1(t) and P2(t) be the fixed points of (2.6) associated with initial conditions (u01,v01)

and (u02,v02), respectively, then

|P1(t) −P2(t)| ≤
eβMt

1 −L
‖u01 −u02‖L1, (3.1)

where L is the contraction constant of the operator Φ.

Proof. It is easy to see that

|P1(t) −P2(t)| ≤ |P1(t) −P3(t)| + |P3(t) −P2(t)|, (3.2)

where

P3(t) =

∫ t
0

B3(η)e
−

∫
t
η
µ(X2(s,xmin,η),P2(s),v2(X2(s;xmin,η),s))dsdη

+

∫ xmax
xmin

u01(ξ)e
−

∫
t
0
µ(X2(s;ξ,0),P2(s),v2(X2(s;ξ,0),s))dsdξ

and

B3(t) =∫ t
0

β(X2(t; xmin,η),P2(t),v2(X2(t; xmin,η), t))B3(η)e
−

∫
t
η
µ(X2(s,xmin,η),P2(s),v2(X2(s;xmin,η),s))dsdη

+

∫ xmax
xmin

β(X2(t; xmin,ξ),P2(t),v2(X2(t; xmin,ξ), t))u01(ξ)e
−

∫
t
0
µ(X2(s;ξ,0),P2(s),v2(X2(s;ξ,0),s))dsdξ.



THE EXISTENCE AND UNIQUENESS OF SOLUTIONS 197

Direct calculations give

|P3(t) −P2(t)| =
∫ t

0

(B3(η) −B2(η))e
−

∫
t
η
µ(X2(s,xmin,η),P2(s),v2(X2(s;xmin,η),s))dsdη

+

∫ xmax
xmin

(u01(ξ) −u02(ξ))e−
∫
t
0
µ(X2(s;ξ,0),P2(s),v2(X2(s;ξ,0)))dsdξ

≤
∫ t

0

|B3(η) −B2(η)|dη + ‖u01 −u02‖L1

and

|B3(t) −B2(t)| ≤ βM
∫ t

0

|B3(η) −B2(η)|dη + βM‖u01 −u02‖L1.

So we can conclude that

|P3(t) −P2(t)| ≤ eβMt‖u01 −u02‖L1.

From (3.2), by the contraction mapping theorem, we have

|P1(t) −P2(t)| ≤ |P1(t) −P3(t)| + |P3(t) −P2(t)| ≤ L|P1(t) −P2(t)| + |P3(t) −P2(t)|

≤ L|P1(t) −P2(t)| + eβMt‖u01 −u02‖L1,

which implies (3.1). �

In the following, in virtue of the above estimates (3.1), we can show the continuous dependence of

solutions on initial conditions.

Theorem 3.2. Let (u1,v1) and (u2,v2) be the solutions of (1.1) with initial conditions (u01,v01) and

(u02,v02), respectively. Then for any ε > 0, there exists δ = δ(ε,t,u0i,v0i) > 0 such that if ‖u01 −
u02‖L1 + ‖v01 −v02‖L1 < δ, then

‖u1 −u2‖L1 + ‖v1 −v2‖L1 ≤ ε.

Proof. Firstly we estimate the difference between the two characteristics. By (2.8) and Gronwall’s

inequality, and combining with (3.1), we find that

|XP1 −XP2| ≤ Lg
∫ t

0

|P1(σ) −P2(σ)|dσeLg(t−s) ≤
Lge

(Lg+βM )(t−s)

βM (1 −L)
‖u01 −u02‖L1,



198 YAN LI AND QIHUA HUANG

which implies that when t ≥ s, |XP1 −XP2|→ 0 as ‖u01 −u02‖L1 → 0. We assume that Z1(t) ≤ Z2(t).
By (2.4), direct calculations show that∫ xmax

xmin

|u1(x,t) −u2(x,t)|dx

≤
∫ Z1(t)
xmin

∣∣∣B(τ1(xmin))
g(xmin,P1)

−
B(τ2(xmin))

g(xmin,P2)

∣∣∣e−∫ tτ1(xmin) µ̃(X1,P1,v1)dsdx
+

∫ Z1(t)
xmin

B2(τ(xmin))

g(xmin,P2)
(e
−

∫
t
τ1(xmin)

µ̃(X1,P1,v1)ds −e
∫
t
τ2(xmin)

µ̃(X2,P2,v2)ds)dx

+

∫ Z2(t)
Z1(t)

(
u01(X1)e

−
∫
t
0
µ̃(X1,P1,v1)ds −

B(τ2(xmin))

g(xmin,P2)
e
∫
t
τ2(xmin)

µ̃(X2,P2,v2)ds
)

dx

+

∫ xmax
Z2(t)

u01(X1)(e
−

∫
t
0
µ̃(X1,P1,v1)ds −e

∫
t
0
µ̃(X2,P2,v2)ds)dx

+
∫xmax
Z2(t)

|u01(X1) −u01(X2)|e−
∫
t
0
µ̃(X2,P2,v2)dsdx

+

∫ xmax
Z2(t)

(u01(X2) −u02(X2))e−
∫
t
0
µ̃(X2,P2,v2)dsdx

and∫ xmax
xmin

|v1(x,t) −v2(x,t)|dx =
∫ Z2(t)
Z1(t)

(
v01(X1)e

−
∫
t
0
σ(X1,s)ds +

∫ t
0

a(X1,s)E(s)e
−

∫
t
s
σ(X1,τ)dτ ds

)
dx

+

∫ xmax
Z2(t)

∫ t
0

[a(X1,s) −a(X2,s)]E(s)e−
∫
t
s
σ(X1,τ)dτ dsdx

+

∫ xmax
Z2(t)

∫ t
0

a(X2,s)E(s)[e
−

∫
t
s
σ(X1,τ)dτ −e−

∫
t
s
σ(X2,τ)dτ ]dsdx

+

∫ xmax
Z2(t)

v01(X1)[e
−

∫
t
0
σ(X1,τ)dτ −e−

∫
t
0
σ(X2,τ)dτ ]dsdx

+

∫ xmax
Z2(t)

(v01(X1) −v01(X2))e−
∫
t
0
σ(X2,τ)dτ dx

+

∫ xmax
Z2(t)

(v01(X2) −v02(X2))e−
∫
t
0
σ(X2,τ)dτ dx,

where Xi = XPi, (i = 1, 2), µ̃ is defined in Definition 2.1. The following proof can be completed by

using similar arguments as in the proof of Theorem 2 in [2]. �

4. Concluding remarks

In this paper, by using the method of characteristic and contracting mapping theorem, we proved the

existence-uniqueness of solutions to problem (1.1). We also derive the continuous dependence on initial

conditions of the solutions. In the future, we plan to study the asymptotic behavior of the population

under the influence of environmental toxins. In addition, problem (1.1) assumes that the population

growth rate g = g(x,P(t)). This mortality rate, however, may depend on the body burden v. Including

the dependence of the growth rate on the body burden (i.e., g = g(x,P(t),v(x,t))) will yield new and

challenging problems.



THE EXISTENCE AND UNIQUENESS OF SOLUTIONS 199

References

[1] S. M. Bartell, R. A. Pastorok, H. R. Akcakaya, H. Regan, S. Ferson and C. Mackay, Realism and relevance of

ecological models used in chemical risk assessment. Hum. Ecol. Risk Assess. 9 (2003), 907–938.

[2] À. Calsina and J. Saldaña, A model of physiologically structured population dynamics with a nonlinear individual

growth rate, J. Math. Biol. 33 (1995), 335-364.

[3] H. I. Freedman, J. Shukla and R. Brown, Models for the effect of toxicant in single-species and predator-Cprey

systems, J. Math. Biol. 9 (1991), 15–30.

[4] T. G. Hallam, C. E. Clark and R. R. Lassiter, Effects of toxicants on populations: aqualitative approach I. Equilib-

rium environmental exposure, Ecol. Model. 18 (1983), 291–304..

[5] T. G. Hallam, C. E. Clark and G. S. Jordan, Effects of toxicants on populations: a qualitative approach II. First

order kinetics, J. Math. Biol. 18 (1983), 25–37.

[6] Q. Huang, G. Seo and C. Shan, Bifurcations and global dynamics in a toxin-dependent aquatic population model,

Math. Bios. 296 (2018), 26–35.

[7] Q. Huang, L. Parshotam, H. Wang, C. Bampfylde and M. A. Lewis, A model for the impact of contaminaants on

fish population dynamics, J. Theor. Biol. 334 (2013), 71–79.

[8] Q. Huang and H. Wang, A toxin-mediated size-structured population model: finite difference approximation and

well-posedness, Math. Biosci. Engin. 13 (2016), 697–722.

[9] Q. Huang, H. Wang and M. A. Lewis, The impact of environmental toxins on predator-Cprey dynamics, J. Theor.

Biol. 378 (2015), 12-30.

[10] R. A. Pastorok, S. M. Bartell, S. Ferson and L. R. Ginzburg, Ecological modeling in risk assessment: chemical effects

on populations, ecosystems, and landscapes, CRC, Boca Raton, FL, USA, 2001.

[11] D. M. Thomas, T. Snell and S. Jaffar, A control problem in a polluted environment, Math. Biosci. 133 (1996),

139-163.

College of Science, China University of Petroleum (East China), Qingdao 266580, P. R. China

Email address: liyan@upc.edu.cn

Corresponding author, School of Mathematics and Statistics, Southwest University, Chongqing 400715,

P. R. China

Email address: qihua@swu.edu.cn


	1. Introduction
	2. Existence and uniqueness results
	3. Continuous dependence on initial conditions
	4. Concluding remarks
	References