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ORIGINAL ARTICLE

A class of individual–based models
Mirosław Lachowicz

Institute of Applied Mathematics and Mechanics
University of Warsaw

Warsaw, Poland
M.Lachowicz@mimuw.edu.pl

Received: 9 February 2018, accepted: 12 April 2018, published: 15 April 2018

Abstract—We discuss a class of mathematical
models of biological systems at microscopic level
— i.e. at the level of interacting individuals of
a population. The class leads to partially integral
stochastic semigroups — [25]. We state general
conditions guaranteeing the asymptotic stability. In
particular under some rather restrictive assump-
tions we observe that any, even non–factorized,
initial probability density tends in the evolution
to a factorized equilibrium probability density —
[16]. We discuss possible applications of the gen-
eral theory such as redistribution of individuals —
[10], thermal denaturation of DNA [7], and tendon
healing process — [11].

Keywords-Individual–based models; Markov
jump processes; Integro–differential equations,
Stochastic semigroups, Stability.

I. MICROSCOPIC SCALE

In the present paper we review the general class
of individual–based models in Biology developed
in Ref. [15] — see also [3], [14], [16] and refer-
ences therein. We show that the class corresponds
to the partially integral stochastic semigroups and
under some more restrictive assumptions leads to
the stability result. We discuss possible applica-

tions of the general theory such as redistribution of
individuals — [10], thermal denaturation of DNA
[7], and tendon healing process — [11].

We consider the general equation that defines
the evolution of a number N of individuals of
biological populations — cf. Refs. [3], [14], [15]
and references therein. Each individual n (n ∈
{1, . . . ,N}) is characterized by its inner (micro-
scopic ) state

un ∈ U ,

where U is a Borel set in Rd, d ∈ {1, 2, 3, . . .}.
The variable un related to the individual n may
have various meanings: it may be any vector pa-
rameter that characterize an individual biological
state of any of the individuals. In particular it may
also contain an information of a subpopulation to
which the individuals belongs (a discrete compo-
nent) — see [3], [14], [15].

In the general setting (U,B,µ) is a σ–finite
measure space. In some applications U is a product
of a discrete set and a Lebesgue–measurable subset
(e.g. a closed bounded interval) in the discrete–
continuous picture or a discrete set in the discrete–
discrete case and the measure µ is a product of the

Copyright: c© 2018 Lachowicz et al. This article is distributed under the terms of the Creative Commons Attribution License
(CC BY 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and
source are credited.

Citation: Mirosław Lachowicz, A class of individual–based models, Biomath 7 (2018), 1804127,
http://dx.doi.org/10.11145/j.biomath.2018.04.127

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Mirosław Lachowicz, A class of individual–based models

counting measure and the counting measure (in the
discrete–discrete picture) or the Lebesgue measure
(in the discrete–continuous picture).

The n1–individual changes its state at random
times. We consider the possibility of the following
stochastic changes
• without interactions,
• due to interaction with the n2–individual,
n2 ∈{1, . . . ,N}, n2 6= n1 ,

• due to interaction with the n2 and n3 individ-
uals,

• ...
• due to interaction with n2, n3, ..., nM indi-

viduals, where M is an integer 2 ≤ M ≤ N.
Consider interactions of the given individual

with m− 1 individuals, m = 1, . . . ,M.

Assumption 1. The rate of interaction between
the individual with state un1 and the individuals
with states un2 , ..., unm is given by the measurable
function a[m] = a[m](un1, . . . ,unm), such that

0 ≤ a[m](un1,un2, . . . ,unm) ≤ a
[m]
+ , (1)

for all un1,un2, . . . ,unm ∈ U , where a
[m]
+ < ∞

is a constant.

Assumption 2. The transition into state v of an
n1–individual with state un1 , due to the inter-
action with individuals of n2,...,nm with states
un2 ,...,unm, respectively, is described by the mea-
surable function A[m] = A[m] (v; un1, . . . ,unm) ≥
0 , where∫

U

A[m]
(
v ; un1,un2, . . . ,unm

)
dµ(v) = 1 ,

(2)
for all un1,un2, . . . ,unm ∈ U .

The stochastic model (at the microscopic level)
is determined by the functions a[m] and A[m].
L
(N)
1 is the space equipped

with the norm ‖f‖
L

(N)
1

=∫
UN

∣∣∣f(u1, . . . ,uN)∣∣∣dµ(u1) . . . dµ(uN ) . If N = 1
we simply write L1.

Given N, M, and a[m], A[m], for m =
1, . . . ,M, we consider the stochastic system that

is defined by the Markov jump process of N indi-
viduals through the following generator Λ acting
on densities

Λ = Λ+ − Λ− =
M∑
m=1

(
Λ[m] + − Λ[m]−

)
,

Λ[m] +f
(
t,u1,u2, ...,uN

)
=

cN,m
∑

1≤n1,...,nm≤N
ni 6=nj ∀i6=j

∫
U
A[m]

(
un1 ; v,un2, . . . ,unm

)
×

a[m]
(
v,un2, . . . ,unm

)
×

f
(
t,u1, . . . ,un1−1,v,un1+1, . . . ,uN

)
dµ(v) ,

Λ[m]−f
(
t,u1,u2, ...,uN

)
=

cN,m
∑

1≤n1,...,nm≤N
ni 6=nj ∀i6=j

a[m]
(
un1,un2, . . . ,unm

)
×

f
(
t,u1, . . . ,uN

)
,

on UN , where cN,m = 1
(m−1)! ( N

m−1)
are normaliz-

ing constants.
Assume that the system is initially distributed

according to F ∈ L(N)1 and time evolution is
described by the following (linear) equation — the
modified Liouville equation ,

∂

∂t
f = Λf , ; f

∣∣∣
t=0

= F . (3)

with the initial data

f
∣∣∣
t=0

= F . (4)

We refer here to the Liouville equation in the
sense of particle dynamics: Eq. (3) plays a similar
role as the Liouville equation in kinetic theory —
see Ref. [6] (cf. also [18]). The generator Λ is the
difference between the gain term and loss terms
Λ = Λ+ − Λ−, where
• the gain term Λ+ is a sum of terms describing

the changes from state v of n1–individual into
un1 due to the interaction with n2, ..., nm
individuals with states un2 , ..., unm, respec-
tively for 2 ≤ m ≤ M and the term (m = 1)
describing the direct changes of state v of n1–
individual into un1 without interactions;

• the loss term Λ− is a sum of terms describing
the changes from state un1 of n1–individual
into another state due to the interaction with

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Mirosław Lachowicz, A class of individual–based models

n2, ..., nm individuals with states u2, ..., um,
respectively for 2 ≤ m ≤ M or without
interactions for m = 1.

It is easy to see that under Assumptions 1—2
the operator Λ is a bounded in L(N)1 . Thus the
Cauchy Problem has the unique solution

f(t) = exp
(
tΛ
)
F (5)

in L(N)1 for all t ≥ 0. The solution is nonnegative
for nonnegative initial data and the L(N)1 –norm is
preserved for any t > 0. Therefore

{
exp

(
tΛ
)}

t≥0
defines a continuous (linear) semigroup of Markov
operators that is a stochastic semigroup in the
sense of Ref. [21]. Actually we may note that here
we have even a group.

In Ref. [15] (see also [14], [3] and the references
therein) the limit N → ∞ was studied. Under
suitable assumption Eq. (3) results in a nonlin-
ear kinetic equation referred to the correspond-
ing mesoscopic level. Moreover in various cases
macroscopic limits can be obtain.

In the next section (Section II) we show that{
exp

(
tΛ
)}

t≥0 is a partially integral stochastic
semigroup (see Refs. [26], [25]) and, under some
additional assumptions, leads to a stability result.
In Section III we review some possible applica-
tions.

II. ASYMPTOTIC BEHAVIOUR

In order to formulate the time asymptotic result
we can refer to the notion of the partially integral
stochastic semigroups — see Refs. [26], [25]) —
and the Lower Function Theorem by Lasota and
Yorke — see [22] (Theorem 2; cf. also Corollary
IV.16 in Ref. [28]).

Using similar strategy as in Refs. [16] we prove
a more general result that may be related to a gen-
eral class of microscopic systems in the form given
by Eq. (5) under reasonably general Assumptions
1 and 2.

Lemma II.1. Let Assumptions 1 and 2 be satis-
fied. Assume moreover that a[m] is non–zero, for
some m ∈{1, . . . ,M}. Then

{
exp

(
t Λ
)}

t≥0 is a
partially integral stochastic semigroup.

Proof: Let Γ be the operator given by

ΓF = ΛF + a+F ,

for F ∈ L(N)1 , where a
+ = max

m=1,...,M
a[m]. Then

ΓF ≥ Λ+F ≥ max
{

0, ΛF
}

(6)

and

exp
(
t Λ
)
F = exp

(
−a+ t

)
exp

(
t Γ
)
F , (7)

for any probability density F on UN .
By Eqs. (6) and (7), for any probability density

F on UN , we have
exp

(
t Λ
)
F
(
u1, . . . ,uN

)
≥ t

N

N!
exp

(
−a+ t

)(
Λ+
)N
F
(
u1, . . . ,uN

)
≥ cN,m(t)

∫
UN
k[m]
(
u1, . . . ,uN,v1, . . . ,vN

)
×F
(
v1, . . . ,vN

)
dµ(v1) . . . dµ(vN ) ,

(8)

where cN,m(t) is a constant that depends on m,n
and t > 0 and k[m] is a complicated function that
depends on A[m] and a[m],

k[m]
(
u1, . . . ,uN,v1, . . . ,vN

)
=

= A[m]
(
u1; v1,u2, . . .um

)
a[m]

(
v1,u2, . . .um

)
×A[m]

(
u2; v2,u3, . . .um+1

)
a[m]
(
v2,u3, . . .um+1

)
× . . . A[m]

(
uN ; vN,v1, . . .vm−1

)
×a[m]

(
vN,v1, . . .vm−1

)
.

Thus (see [26])
{

exp
(
t Λ
)}

t≥0 is a partially
integral stochastic semigroup.

We note that the above result does not need
any additional assumption. The class of partially
integral stochastic semigroups is particularly im-
portant (see [25], [26], [28]) in the analysis of
asymptotic behaviour of stochastic semigroups. To
state the asymptotic stability we need however a
stronger assumption

Theorem II.2. Let Assumptions 1 and 2 be sat-
isfied. Additionally, for some m ∈{1, . . . ,M} we
assume that a[m] is nonzero, and there exists a
measurable nonnegative function h on U such that∫

U

h(u) dµ(u) > 0 , (9)

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Mirosław Lachowicz, A class of individual–based models

and

A[m]
(
v,u1, . . . ,um

)
a[m]

(
u1, . . . ,um

)
≥ h(v) ,

(10)
for any v,u1,u2, . . . ,um ∈ U. Then the stochastic
semigroup

{
exp

(
tΛ
)}
t≥0 is asymptotically stable.

Proof: As in the proof of Lemma II.1, for any
probability density F on UN , we have

exp
(
t Λ
)
F
(
u1, . . . ,uN

)
≥ t

N

N!
exp

(
−a+t

)(
Λ+
)N
F
(
u1, . . . ,uN

)
≥ cN,m(t)

N∏
i=1

h(ui)

×
∫
UN

F
(
v1, . . . ,vN

)
dµ(v1) . . . , dµ(vN )

= cN,m(t)
N∏
i=1

h(ui) ,

(11)

where cN,m(t) is a constant that depends on m,n
and t > 0. Let t0 > 0 be fixed, e.g. t0 = 1. We
have

exp
(
t Λ
)
F = exp

(
t0 Λ

)
exp

(
(t− t0) Λ

)
F .

By Eq. (11) we obtain

exp
(
t Λ
)
F
(
u1, . . . ,uN

)
≥ cN,m(t0)

N∏
i=1

h(ui)
∥∥∥ exp ((t− t0) Λ)F∥∥∥

L
(N)
1

,

(12)
for t > t0. Keeping in mind that exp

(
(t−t0) Λ

)
F

is a probability density, for each t > t0 we
conclude

exp
(
t Λ
)
F
(
u1, . . . ,uN

)
≥ `(u1, . . . ,uN ) ,

(13)

where `(u1, . . . ,uN ) = cN,m(t0)
N∏
i=1

h(ui) de-

pends on N but does not depend on t and F . By
Assumptions 1 and 2 it follows that ` ∈ L(N)1 .
Moreover by Eq. (9)∫
UN

`(u1, . . . ,uN ) dµ(u1) . . . dµ(uN ) > 0 , (14)

and ` is a lower function in the sense of Lasota and
Yorke [22]. In fact the following condition holds

lim
t→∞

∥∥∥( exp(t Λ) F − `)−∥∥∥
L

(N)
1

= 0 ,

for every probability density F , where X− = 0
if X ≥ 0 and X− = −X if X < 0. Thus by
the lower function theorem of Lasota and Yorke
the semigroup

{
exp(t Λ)

}
t≥0 is asymptotically

stable.
As a by–product of Theorem II.2 we obtain the

uniqueness of an equilibrium (stationary) solution
corresponding to Eq. (3). The identification of
possible equilibrium solutions is an essential step
in studying macroscopic limits corresponding to
the microscopic models — see [3].

In a particular case referred to a microscopic
system in Ref. [16], under some (rather strong)
assumption, it is shown that any, even non–
factorized, initial probability density tends in the
evolution to a factorized equilibrium probability
density. Such a situation one can refer to as
asymptotic annihilation of initial correlations in
the system. On the other hand it was also shown
that if the mentioned assumptions are not satisfied
— a number of equilibrium states could be large
and no annihilation is observed.

The possible relationships between micro-,
meso- and macro- scales were discussed in Ref.
[3] (see also references therein) — Chapter 8 and
in particular Subsection 8.3.3.

III. APPLICATIONS

There are many possible applications of the gen-
eral theory presented in section I. The stochastic
systems that corresponds at the macroscopic level
to standard logistic growth were considered in
Ref. [16]. The parameter un, n ∈ {1, 2, . . . ,N},
describing the microscopic (individual) state of
n–individual, may be related to its activity (cf.
Refs. [4], [3]). The parameter may also describe
dominance [13] or social state (c.f. Refs. [1], [5],
[9]). The references mentioned above refer to the
mesoscopic (kinetic) description whereas Ref. [16]
- to microscopic (individual–based) one.

Usually the importance of the microscopic ap-
proach may be particularly visible in a case when
the number of interacting entities of the system is
not huge which is typical for biological systems.
In such cases the kinetic (mesoscopic) description
not always may be properly justified.

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Mirosław Lachowicz, A class of individual–based models

A. Redistribution of individuals

An application of the general model at the
microscopic level of the redistribution of individ-
uals in a closed domain featuring, as an example,
an elevator — see [10]. The modeling bases on
experiments performed in order to elucidate the
interactions between pedestrians — see [12] and
the references in [10]. In these experiments the
inflow of persons into a spatially restricted area,
e.g. an elevator, was studied, featuring the process
inverse to evacuation. In Ref. [10] an n–th individ-
ual, n = 1, . . . ,N, is characterized by its position
un. The model and analysis presented in [10] have
a preliminary nature and still have to be developed.

B. Thermal denaturation of DNA

In Ref. [7] some aspects of deoxyribonucleic
acid DNA thermal denaturation process (cf. [23],
[24], [2]) were considered. Ref. [7] is a continuing
the idea of [8] where a preliminary mesoscopic
(kinetic) model was discussed. In Ref. [7] a new,
more adequate, model that takes into account at the
individual (microscopic) level the time evolution
of the probability distribution of the state of all
hydrogen bonds. Two types of bonds (with two and
three hydrogen bonds) and the direct dependence
on the temperature are included in the model. The
base pairs A–T, C–G are numbered by the discrete
variable n ∈ {1, 2, . . . ,N}, and the continuous
variable u ∈ [0,∞[ representing the stretching of
the distance between the two connected base is
used. The variable u is called stretching parameter.
Every base pair (’bond’) n is then characterized by
the variables un. The discrete variable belongs to
one of two subsets of bonds: J2 — two hydro-
gen bonds connecting A and T and J3 — three
hydrogen bonds connecting C and G

J2 ∪J3 = J, J2 ∩J3 = ∅ .

According to the biological knowledge it is as-
sumed that the three hydrogen bonds are more
resistant to heating than the two hydrogen bonds.
The probability densities f = f(t,u1, . . . ,uN )
that describe the distribution of the variable
u1, . . . ,uN at all bonds is considered.

C. Tendon healing process

In Ref. [11] a kinetic model of collagen remod-
eling occurring in latter stage of tendon healing
process was proposed and studied. The model
is an integro–differential equation describing the
alignment of collagen fibers in a finite time. An
important feature of tendon structure is the colla-
gen fibers orientation. In the healthy tendon they
are aligned. The result of the tendon injury is a
disturbance of the parallel structure. The healing
process consists in the reconstruction of parallel
structure. Scars that may be formed during the
healing process cause no proper alignment of col-
lagen fibers. One of the most important indicators
of the success of the treatment of tendon injury is
the degree of alignment of collagen fibers.

The model in [11] refers to the function
g(t,x,v) that describes a statistical state of col-
lagen, i.e. the probability density g = g(t,x,v) to
find a collagen fiber at the instant of time t > 0
at point x ∈ D and with orientation v ∈ V, where
D, V are domains in Rd. Thus the model has a
mesoscopic nature.

We consider the following equation

∂
∂t
g(t,x,v) =

∫
V

∫
D

(
Tg(y,v; x,w)g(t,x,w)

−g(t,x,v)Tg(y,w; x,v)
)

dydw,

(15)

where Tg(y,v; x,w) describes the transition prob-
ability from the orientation w ∈ V at x ∈ D to the
orientation v ∈ V at x caused by an adaptation
to the orientation at y ∈ D. The model bases
on a proper choice of the function Tf that in
general may depend on both collagen distribution
g and tenocytes density c. In Ref. [11] a simplified
case of constant (uniform) tenocytes density was
considered.

A realistic definition is

Tg(y,v; x,w) = β(y,v; x,w)g
γ(t,y,v) ,

where γ > 0 describes the strength of influence
of collagen fibers from neighborhood on collagen
fiber in considered point. The bigger is the γ the
stronger the influence is. That choice leads to the

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Mirosław Lachowicz, A class of individual–based models

following general class of equations
∂
∂ t
g(t,x,v) =∫

D

∫
V

(
β(y,v; x,w) gγ(t,y,v) g(t,x,w)

−g(t,x,v)β(y,w; x,v)gγ(t,y,w)
)

dydw.

(16)

The function β(y,v,x,w) is related to the inter-
action between the collagen fiber with orientation
w at point x with collagen fiber with orientation
v located at point y and describes the transition
from orientation w to orientation v.

In Ref. [11] we show that the solutions may
exist globally in time or may blow–up in a finite
time depending on initial data. The latter behavior
is related to the healing of injury without the scar
formation in a finite time: a full alignment of
collagen fibers occurs. The approach of [11] may
be related to the mesoscopic scale.

One may however propose a stochastic individu-
ally based (microscopic) model following the idea
stated in Section I. The number N may be then
related to the number of collagen fibers that are
taken into account in the modeling process. The
system is described in terms of a Markov jump
process and the related linear evolution equations
as in Section I. The equation describes the evolu-
tion of probability densities, with microscopic rep-
resentation of the system of N interacting agents.

We consider the interactions of a given agent
with γ agents. The system is initially distributed
according to the probability density F ∈ L(N)1 .

The time evolution is described by Eq. (3) where
Λ is the generator that takes the form

Λf
(
t,x1,v1, . . . ,xN,uN

)
=

cN,γ
∑

1≤n1,...,nγ+1≤N
ni 6=nj ∀i6=j

( ∫
D×V

A
(
xn1,vn1 ; y,w,xn2,vn2, . . . ,vnγ+1,vnγ+1

)
×a
(
y,w,xn2,vn2, . . . ,xnγ+1,vnγ+1

)
×f
(
t,x1,v1, . . . ,xn1−1,vn1−1,y,w,

xn1+1,vn1+1, . . . ,xN,vN
)

dy dw
−a
(
xn1,vn1, . . . ,xnγ+1,vnγ+1

)
f
(
t,x1,v1, . . . ,xN,vN

))
.

In the limit N → ∞, the (linear) modified
Liouville equation (3) yields, [15], [3], a nonlinear
integro–differential equation that can be related to
the mesoscopic description

∂
∂t
f(t,u) = G[f](t,u) −f(t,u)L[f](t,u) ,

u = (x,v) ∈ D×V ,
(17)

where G[f] is the gain term , given by

G[f](t,x,v) =
∑
{}

∫(
D×V

)γ+1
A
(
x,v; y,w,{x2,v2, . . . ,xγ+1,uγ+1}

)
×a
(
y,w,{x2,u2, . . . ,xγ+1,uγ+1}

)
×f(t,y,w)f(t,x2,v2) . . .f(t,xγ+1,vγ+1)
dy dw dx2, dv2, . . . dxγ+1, dvγ+1 ,

and fL[f] is the loss term , defined as

L[f](t,x,v) =
∑
{}

∫(
D×V

)γ
a
(
x,v,{x2,v2, . . . ,xγ+1,vγ+1})
×f(t,x2,v2) . . .f(t,xγ+1,vγ+1)
×dx2 dv2 . . . dxγ+1 dvγ+1 ,

and
∑
{}

means the sum over all permutation of

variables within {}.
It is easy to see that the global (in time) exis-

tence and uniqueness of solutions f = f(t) to Eq.
(17) in L(1)1 follows.

One may now state the theorem (cf. [15], [3])
that defines the links between the solutions to Eq.
(3) and to Eq. (17) or, in other words, that defines
the transition from the microscopic level to the
mesoscopic level.

The mathematical properties of Eq. (17) are
different than those of Eq. (16). The possible
rich behavior of solutions of Eq. (16), see [11],
[17], leading to blow–ups in a finite of time are
not possible in the case of solutions of Eq. (17).
On the other hand in some limit (approximating
”delta –function”) the solutions of Eq. (16) may
be approximated by the solutions of Eq. (17).
This defines the relationship between a stochastic,
individually–based (microscopic) description and
Eq. (16).

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Mirosław Lachowicz, A class of individual–based models

A formal series with respect to γ = 1, 2, . . . ,
results in the following nonlinear kinetic (meso-
scopic) equation alternative to Eq. (16)

∂
∂ t
g(t,x,v) =∫

D

∫
V

(
β(y,v; x,w)

g(t,y,v)
1−g(t,y,v) g(t,x,w)

−g(t,x,v) β(y,w; x,v) g(t,y,w)
1−g(t,y,w)

)
dy dw.

(18)
At present, the mathematical theory of Eq. (18)

is missing.

IV. CONCLUSIONS

In Section I we review the general class of
microscopic models that are able to describe inter-
actions between individuals of a biological popula-
tion. The class refers to the stochastic semigroups.
In Section II we show the methods that leads to the
asymptotic stability under some rather restrictive
assumptions. On the other hand the asymptotic
behaviour in the general case is still an open prob-
lem. The important technical tool could be the fact
that the semigroup is partially integral (without
any particular additional assumption). This may
be treated as a preliminary step towards the de-
scription of macroscopic (”hydrodynamic ”) limits
that seems to be essential part of the program of
giving full description on various scales starting
from microscopic, then mesoscopic and finally —
macroscopic.

In a very simple case considered in [16] —
the macroscopic equation was obtain from the
mesoscopic equation by the averaging with respect
to microscopic variable. In the general case it is
far from being solved. Therefore we may believe
that the methods of the present paper can indicate
the possible further research.

In Section III we review some important appli-
cations. They show that the general framework is
suitable to describe various systems in which the
interactions between individuals are essential.

We point out some new equations that result in
various limits that can be interesting for further
mathematical studies.

ACKNOWLEDGMENT

The author was supported by the National Sci-
ence Centre Poland Grant 2017/25/B/ST1/00051.

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	Microscopic scale
	Asymptotic behaviour
	Applications
	Redistribution of individuals
	Thermal denaturation of DNA
	Tendon healing process

	Conclusions
	References