Trajectory analysis of nonlinear kinetic models of population dynamics of several species

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Mathematical and Computer Modelling

journal homepage:www.elsevier.com/locate/mcm

Trajectory analysis of nonlinear kinetic models of population dynamics of several species

B. Aylaj

^∗

, A. Noussair

INRIA, IMB Université Bordeaux 1, 351 cours de la Libération 33405, Talence cedex, Bordeaux, France

a r t i c l e i n f o

Article history:

Received 14 February 2008 Accepted 24 July 2008 Keywords:

Population dynamics Kinetic models Equilibrium solutions Stability

Dissipativity PositiveC0-semigroups Nonlocal boundary conditions

a b s t r a c t

The existence and uniqueness of global solutions are proved for kinetic models of coupled population dynamics. Moreover, under consideration, the uniqueness and the stability analysis of this steady-state solution are discussed. This analysis uses, essentially, the dissipativity, a subtangential condition and the positivity of the relatedC0-semigroup.

Crown Copyright©2008 Published by Elsevier Ltd. All rights reserved.

1. Introduction

In this paper, we study a mathematical model that can be regarded as a generalization of the model proposed by Jager and Segel [10,1,8] to several interacting populations with coupled interactions and nonlocal boundary conditions. The kinetic modeling approach is based upon a description of the statistical distribution over the state of the cells for each population, and on the derivation of suitable evolution equations. For each population, it is necessary to identify the most important set of activities. Each activity differs from population to population, and is tagged by a suitable dimensionless variablex. In this paper, the variablex

∈ [

0

,

¹

]

denotes the ability of the cell to express its main activity, which differs from cell type to cell type. The valuex

=

1 corresponds to the highest value of the peculiar activities of the cell, whilex

=

0 corresponds to complete functional inhibition (no activities). On the other hand, ifx

∈ (

⁰

,

¹

]

, cells possess a certain progression velocity.

The governing equations for this model are defined by a system of integro-differential equations, with quadratic nonlinearity

∂

^fi

(

^t

,

^x

)

∂

^t

+ ∂(v

i

(

^x

)

^fi

(

^t

,

^x

))

∂

^x

+ µ

i

(

^x

)

^fi

(

^t

,

^x

) =

n

X

j=1

Q_ij

(

^fi

,

^fj

)(

^t

,

^x

),

^(1.1)

v

i

(

⁰

)

^fi

(

^t

,

⁰

) = Z

1

0

β

i

(

^x

)

^fi

(

^t

,

^x

)

^dx

,

^(1.2)

f_i

(

^t

,

⁰

) =

f_i0

(

^x

),

^(1.3)

∗Corresponding author.

E-mail addresses:Bouchra.Aylaj@math.u-bordeaux.fr(B. Aylaj),Ahmed.Noussair@math.u-bordeaux.fr(A. Noussair).

0895-7177/$ – see front matter Crown Copyright©2008 Published by Elsevier Ltd. All rights reserved.

doi:10.1016/j.mcm.2008.07.022

where the operatorQ_ij

(

^fi

,

^fj

) =

Q_ij⁺

(

^fi

,

^fj

) −

Q_ij⁻

(

^fi

,

^fj

)

^with Q_ij⁺

(

^fi

,

^fj

)(

^t

,

^x

) =

Z

1

0

Z

1

0

η

ij

(

^y

,

^z

)ψ

ij

(

^y

,

^z

,

^x

)

^fi

(

^t

,

^y

)

^fj

(

^t

,

^z

)

^dydz

,

^(1.4)

Q_ij⁻

(

^fi

,

^fj

)(

^t

,

^x

) =

f_i

(

^t

,

^x

) Z

1

0

η

ij

(

^x

,

^y

)

^fj

(

^t

,

^y

)

^dy

.

^(1.5)

In the above equations, the functions

v

i

(

^x

), µ

i

(

^x

)

^and

β

i

(

^x

)

denote the natural vital (growth, death and birth) rates of theith species. All quantities

v

i

(

^x

), µ

i

(

^x

)

^and

β

i

(

^x

)

are nonnegative for alli, for further discussion of the vital rates, we refer to [15].

For eachi

=

1

, . . . ,

n, the following assumptions will be used throughout this paper:

(

^Hv

) v

isatisfies a Lipschitz condition on

[

0

,

¹

] , v

i

(

^x

) >

^{0 for 0}

≤

x

<

^{1 and}

v

i

(

¹

) =

0

. (

^Hµ

) µ

i

∈

L^∞_loc

( [

0

,

¹

] ), µ

i

≥

0 a.e. on

[

0

,

¹

] .

(

^Hβ

) β

i

∈

L^∞

(

⁰

,

¹

), β

i

≥

0 a.e. on

[

0

,

¹

] .

The encounter rate between pairs of individuals, of theith population in the statexand of thejth population in the state yis identified by the term

η

ij

(

^x

,

^y

)

^{. We have}

η

ij

(

^x

,

^y

) = η

ij

(

^y

,

^x

)

and we assume

(

^Hη

)

⁰

≤ η

ij

≤ η, η

ij

∈

L^∞

∀

i

,

^j

=

1

, . . . ,

ⁿ

,

where

η

is positive constant. The probability that an individual of theith population in the stateyends up in the statex, conditionally to an encounter with an individual of thejth population in the statez, has a density denoted by

ψ

ij

(

^y

,

^z

,

^x

),

we have

(

^Hψ

) ψ

ij

≥

0

, ψ

ij

∈

L^∞

,

^and

R

1

0

ψ

ij

(

^y

,

^z

,

^x

)

^dx

=

1.

The nonlinear models considered in this paper have been studied in a qualitative manner by several authors.

In the casen

=

1, when no linear operator is considered, Segel [10] analyzed the evolution of a physical state, called dominance, which characterizes a certain population of insects. This latter result has been generalized by [2], for a kinetic model of population dynamics with character dependence and spatial structure, which established the existence and uniqueness of the solution. In the case when no gain term exists (i.e.Q_ij⁺

=

0), the model is a particular case of [17]. Prüß [17]

established the global existence as well as the stability of equilibrium solutions of age specific population dynamics of several species (see also [18,19]). When no diffusion is considered, Bellomo and coworkers in a series of papers analyzed, analytically and/or numerically, the previous model, where various examples and applications are proposed and critically analyzed [3, 8,4,7]. In particular, recently Bellouquid and Delitala [9] proposed a new biological model for the analysis of competition between cells of an aggressive host and cells of a corresponding immune system. In the case when the quantity

η

ijis a constant, the real valued functions

ψ

ijare continuous and

v

i

= µ

i

=

0, the existence of steady-state positive solutions has been proved in [3]. Moreover, simulation analysis of the stability of stationary solutions of these populations was developed by Preziosi in [16].

This paper is organized as follows. In Section2, we will briefly recall some basic results and preliminary facts from semilinear evolution equations which will be used throughout Section4. In Section3, the problem(1.1)–(1.3)is converted through some transformations to a homogeneous form, where semigroup theory applies. In Section4, we establish the main global existence result for system(1.1)–(1.3). We report the existence, uniqueness and stability of equilibrium profiles results in Section5. The background of our approach is to be found in [12,14,6,17,11,5]. In Section6, we report some numerical results. Finally, the main conclusions are outlined in Section7.

2. Preliminaries

Let

(

^X

, k k )

be a real Banach space,

(

^T

(

^t

))

t≥0aC₀-semigroup of linear operators such that

k

_T

(

^t

) k ≤

e^wt, for allt

≥

0, for some

w ∈

_R.Ais the infinitesimal generator of

(

^T

(

^t

))

t≥0

,

^Qbe a continuous function from a closed subsetDofXinto XandIbe the identity operator ofX. Recall that

d

(

^f

;

_D

) =

inf

{k

f

−

g

k ,

^g

∈

_D

} .

By

h ., . i

₋we denote the semi-inner products onXgiven by

h

f

,

^g

i

₋

=

min

{h

f

,

^g∗

i :

g^∗

∈

X^∗

, k

g^∗

k = k

g

k , h

g

,

^g∗

i = k

g

k

²

}

for allf

,

^g

∈

X

,

whereX^∗stands for the normed dual ofXand

h ., . i

indicates the natural pairing betweenXandX^∗

.

We are interested in the following abstract initial value problem

˙

f

(

^t

) =

_Af

(

^t

) +

_Q

(

^f

(

^t

)),

f

(

⁰

) =

f₀

∈

_D

.

^(2.1)

A functionfofC¹with values inD

∩

D

(

^A

)

^{satisfying(2.1)}is called a strict solution of(2.1). It is convenient to also consider f

(

^t

) =

_T

(

^t

)

^f0

+

Z

t

0

T

(

^t

−

s

)

^Q

(

^f

(

^s

))

^ds

.

^(2.2)

Continuous solutions of(2.2)are called mild solutions of(2.1). Clearly, each strict solution of(2.1)is a mild solution, but the converse need not be true (see [12,14]).

For the abstract Cauchy problem(2.1), we have

Theorem 2.1 ([12, p. 355]).If the following conditions are satisfied:

(i) DisT

(

^t

)

-invariant, .i.e.T

(

^t

)

^D

⊂

_D

,

^{for all t}

≥

0

;

(ii) for all f

∈

_D

;

lim_h→0+ 1

hd

(

^f

+

hQ

(

^f

) ;

_D

) =

0

;

(iii) Qis continuous onDand there exists l_Q

∈

_Rsuch that

h

_Qf

−

_Qg

,

^f

−

g

i

₋

≤

l_Q

k

f

−

g

k

for all f

,

^g

∈

_D

.

Then the Eq.(2.1)has a unique mild solution f

(

^t

,

^f0

)

^on

[

0

, +∞[

, for all f₀

∈

_D

.

Furthermore, if

(

^S

(

^t

))

t≥0is defined on D by S

(

^t

)

^f0

=

f

(

^t

,

^f0

)

, for all t

≥

0and f₀

∈

_D, it is a nonlinear semigroup onD, with

(

^A

+

_Q

)

as its generator and D

(

^A

+

_Q

) =

D

(

^A

) ∩

_D

.

Remark 2.1. IfQisC¹inD, solutionsf

(

^t

)

^withf0

∈

D

(

^A

)

are strict solutions of(2.1)(see [17, Theorem B]).

At the end of this section, assume thatΛis a closed interval ofR,X

=

L¹

[

0

,

¹

]

and define K

(

Λ

,

^X

) = { φ ∈

X

, φ(

^z

) ∈

_Λfor almost allz

∈ [

0

,

¹

]} .

As a convenient criterion for the subtangential condition given by (ii) ofTheorem 2.1, we have the following result:

Lemma 2.1 ([11]).Assume that X

=

L¹

[

0

,

¹

] ,

Λ

= [

a

, ∞ ),

^Fc

:

_Λ

→

_Ris a continuous function, and F^p

: [

0

,

¹

] →

_Ris a nonnegative bounded measurable function. If F^c

(

^a

) ≥

_{0, then}

lim

h→0+

1

hd

(φ +

hB

(φ),

^K

(

Λ

,

^X

)) =

0

,

where the substitution operator B is defined on K

(

Λ

,

^X

)

^by

[

_B

(φ) ] (

^z

) =

F^p

(

^z

).

^Fc

(φ(

^z

))

, for all z

∈ [

0

,

¹

]

and for all

φ ∈

_K

(

Λ

,

^X

).

3. Abstract semigroup formulation

Sincef_i

(

^t

,

^x

)

represents a density,f_i

(

^t

,

^x

)

should be nonnegative for allx

,

^t. Furthermore, sinceP_i

(

^t

) = R

1

0 f_i

(

^t

,

^x

)

^{dxis the} total number of individuals of theith species in the population, we should havef_i

(

^t

, .) ∈

L¹

( [

0

,

¹

] )

^{for allt}

,

i. It is natural to considerX

= (

^L1

[

0

,

¹

] )

ⁿ, with the usual inner product

h

f

,

^g

i =

n

X

i=1

h

f_i

,

^gi

i

_L1

and normed by

k

_f

k =

n

X

i=1

k

_fi

k

_L1 (3.1)

for allf andginX

.

Clearly, the Hilbert spaceXis a real Banach lattice (for more details, see e.g. [13]) where, for all given f

∈

X

,

^g

∈

X

,

f

≤

g if and only if f_i

(

^x

) ≤

g_i

(

^x

)

^{for a.e.x}

∈ [

0

,

¹

] ,

ⁱ

=

1

, . . . ,

ⁿ

.

The standard cone inX, is given by

D

=

n

Y

i=1

D_i

= {

f

∈

X

:

f_i

(

^x

) ≥

0 a.e.

,

^x

∈ [

0

,

¹

] ,

ⁱ

=

1

, . . . ,

ⁿ

} .

One recalls that a bounded linear operatorT onXis said to be positive if 0

≤

_Tf, for all 0

≤

f. Similarly, a family of bounded linear operators

(

^T

(

^t

))

t≥0ofXis said to be a positiveC₀-semigroup onXifT

(

^t

)

^{is aC}0-semigroup onXandT

(

^t

)

is a positive operator for allt

≥

0

.

The PDEs(1.1)–(1.3)can be written in the compact form as:

˙

f

(

^t

) =

Af

(

^t

) +

Q

(

^f

(

^t

)),

^(3.2)

f

(

⁰

) =

f₀

∈

X

.

^(3.3)

The linear operatorAdefined byA

=

diag

(

^A1

, . . . ,

^An

)

^whereAifori

=

1

, . . . ,

n, denote the following operators:

(

^Aif_i

)(

^x

) = −

^d

(v

i

(

^x

)

^fi

(

^x

))

dx

− µ

i

(

^x

)

^fi

(

^x

),

^forfi

∈

D

(

^Ai

),

^(3.4)

with domain D

(

^Ai

) =

f_i

∈

L¹

( [

0

,

¹

] ) : v

if_iabsolutely continuous on

[

0

,

¹

] ,

^Aif_i

∈

L¹

( [

0

,

¹

] ),

lim

x↑₁

v

i

(

^x

)

^fi

(

^x

) =

0

, v

i

(

⁰

)

^fi

(

⁰

) = Z

1

0

β

i

(

^x

)

^fi

(

^x

)

^dx

.

^(3.5)

Under assumptions

(

^Hv

)

^–

(

^Hβ

)

, it is shown, in [6], that the linear operator A_i is the infinitesimal generator of a C₀-semigroup

, (

^Ti

(

^t

))

t≥0, of bounded linear operators onL¹

( [

0

,

¹

] )

^{such that}

k

T_i

(

^t

) k ≤

exp

(w

it

)

^{, for alltandi}

=

1

, . . . ,

ⁿ with

w

i

= k β

i

k

_L∞. Then, if we put

w =

max_i

w

i, we haveAgenerates aC₀-semigroup of bounded linear operators onX

(

^T

(

^t

))

t≥0

=

diag

((

^T1

(

^t

))

t≥0

, . . . , (

^Tn

(

^t

))

t≥0

),

such that

k

T

(

^t

) k ≤

exp

(w

^t

)

^{, for allt}

.

The nonlinear operatorQ is definedQ

=

Q⁺

−

Q⁻with the gain termQ⁺

= {

Q_i⁺

}

ⁿ_i

= { P

n

jQ_i⁺_,j

}

ⁿ_i and loss term Q⁻

= {

Q_i⁻

}

ⁿ_i

= { P

n

jQ_i⁻_,j

}

ⁿ_i

.

Now, we shall show that the resolvent ofA_i

, (λ

^I

−

A_i

)

⁻¹is positive for eachiand for

λ

sufficiently large. Therefore, we solve

λ

^fi

(

^x

) +

^d

(v

i

(

^x

)

^fi

(

^x

))

dx

+ µ

i

(

^x

)

^fi

(

^x

) =

u_i

(

^x

),

^(3.6)

v

i

(

⁰

)

^fi

(

⁰

) = Z

1

0

β

i

(

^x

)

^fi

(

^x

)

^{dx (3.7)}

whereu

= {

u_i

}

ⁿ_i=1

∈

Xand

λ ∈

_C. The solutions of(3.6)are given by f_i

(

^x

) =

f_i

(

⁰

) v

i

(

⁰

)

v

i

(

^x

)

^exp

− Z

x

0

λ + µ

i

(σ) v

i

(σ)

^d

σ

+

Z

x

0

u_i

(τ) v

i

(

^x

)

^exp

− Z

x

τ

λ + µ

i

(σ ) v

i

(σ )

^d

σ

d

τ.

^(3.8)

Inserting into(3.7)we find

v

i

(

⁰

)

^fi

(

⁰

)

1

−

Z

1

0

β

i

(

^x

) v

i

(

^x

)

^exp

− Z

x

0

λ + µ

i

(σ) v

i

(σ)

^d

σ

dx

= Z

1

0

β

i

(

^x

) v

i

(

^x

)

Z

x

0

u_i

(τ)

^exp

− Z

x

τ

λ + µ

i

(σ ) v

i

(σ)

^d

σ

d

τ

^dx

.

^(3.9)

By [6], taking into account the assumption

(

^Hv

)

, we can solve Eq.(3.9), for

λ

sufficiently large independent ofuand we see that

Z

1

0

β

i

(

^x

) v

i

(

^x

)

^exp

− Z

x

0

λ + µ

i

(σ ) v

i

(σ)

^d

σ

dx

<

¹

.

Moreover,(3.8)and(3.9)imply

(λ

^I

−

A_i

)

^−1Di

⊂

D_ifor

λ

sufficiently large, thus from the exponential formula easily follows invariance ofD_iunder

(

^Ti

(

^t

))

t≥0, i.e.T_i

(

^t

)

^Di

⊂

D_i, for allt

≥

0 andi. which is leavingDinvariant. Then we have

Lemma 3.1. The C₀-semigroup

(

^T

(

^t

))

t≥0is positive.

4. Global existence of state trajectories

For instance, the problem(1.1)–(1.3), has a unique local mild solution on some interval

[

0

,

^tmax

),

^tmax

∈ [

0

, +∞]

given by

f

(

^t

) =

T

(

^t

)

^f0

+ Z

t

0

T

(

^t

−

s

)

^Q

(

^f

(

^s

))

^ds

,

⁰

≤

t

≤

t_max

.

Indeed,

Lemma 4.1. For all f

,

^g

∈

X , we have

k

Q

(

^f

) −

Q

(

^g

) k ≤

2

η( k

f

k + k

g

k ) k

f

−

g

k .

SinceAis the infinitesimal generator of aC₀-semigroup

(

^T

(

^t

))

t≥0onX. Then, by [14, Theorem 1.4], for everyf₀

∈

Xthere is at_max

≤ ∞

, such that(1.1)–(1.3)has a unique mild solutionf on

[

0

,

^tmax

)

. Yet for the stability analysis, it is necessary to havet_max

= ∞

. A sufficient condition for the mild solution of(1.1)–(1.3)to be a global solution is given next, by applying Theorem 2.1.

In order to apply the result given inTheorem 2.1, we need the following lemmas concerning the nonlinear operatorQ, involved in the dynamics(1.1)–(1.3).

Lemma 4.2. For each f

∈

D, the following subtangential condition holds:

lim

h→0+

1

hd

(

^f

+

hQ

(

^f

) ;

D

) =

0

.

^(4.1)

Proof. First of all, observe thatDis given byD

= Q

n

i=1D_i

,

^where D_i

= {

f_i

∈

X_i

=

L²

[

0

,

¹

] :

f_i

(

^x

) ∈

_Λ

= [

0

, ∞ )

^{a.e. on}

[

0

,

¹

]} .

Letf

= (

^fi

)

ⁿi=1be arbitrarily fixed onD.

Now, we applyLemma 2.1where K

(

Λ

,

^Xi

) =

D_i

,

^Fi^p

(

^x

) =

n

X

j=₁

Z

1

0

η

ij

(

^x

,

^y

)

^fj

(

^y

)

^{dy for allx}

∈ [

0

,

¹

]

and F_i^c

(λ) = − λ

^{for all}

λ ∈

_Λ

.

Whence, sinceB

(

^fi

) = −

Q_i⁻

(

^f

)

^, lim

h→0+

1

hd

(

^fi

−

hQ_i⁻

(

^f

) ;

D_i

) =

0

.

^(4.2)

SinceQ_i⁺

(

^f

) ∈

D_ifor allf

∈

D, it is easy to show,

d

(

^fi

+

hQ_i

(

^f

) ;

D_i

) ≤

d

(

^fi

−

hQ_i⁻

(

^f

) ;

D_i

).

^(4.3)

By using(3.1), we have d

(

^f

+

hQ

(

^f

) ;

D

) ≤

n

X

i=1

d

(

^fi

+

hQ_i

(

^f

) ;

D_i

)

which combined with(4.2)and(4.3)proves the desired result(4.1).

The following lemma is useful to establish the dissipativity property Lemma 4.3.

h

Q

(

^f

) −

Q

(

^g

),

^f

−

g

i

₋

≤

0, for all f

,

^g

∈

D

.

Proof. Now, for eachi

,

^j

=

1

, . . . ,

n, multiplyingQ_ijby a regular function

ϕ

, integrating on

[

0

,

¹

]

and using the symmetry of

η

ij, we get

Z

1

0

ϕ(

^x

)

^Qij

(

^fi

,

^fj

)(

^x

)

^dx

= Z

[0,1]³

(ϕ(

^x

) − ϕ(

^y

)) ψ

ij

(

^y

,

^z

,

^x

)

^fi

(

^y

)

^fj

(

^z

)η

ij

(

^y

,

^z

)

^dxdydz

.

In particular, choosing

ϕ(

^x

) =

1 we deduce the relation

Z

1

0

Q_ij

(

^fi

,

^fj

)(

^x

)

^dx

=

0

.

Then

Z

1

0

Q

(

^f

)(

^x

)

^dx

=

0

.

^(4.4)

It is shown in [17], that, for allf

,

^g

∈

X

, h

f

,

^g

i

₋

≤ k

g

k

min

Z

1

0

f_i

(

^x

)

^sgn

(

^gi

(

^x

))

^dx

:

iwith

Z

1

0

|

g_i

(

^x

) |

dx

= k

g

k

.

Now, forf

,

^g

∈

D

,

h

Q

(

^f

) −

Q

(

^g

),

^f

−

g

i

₋

≤ k

f

−

g

k Z

1

0

[ (

^Qi

(

^f

))(

^x

) − (

^Qi

(

^g

))(

^x

) ]

sgn

(

^fi

(

^x

) −

g_i

(

^x

))

^dx

.

By using(4.4), we have

h

Q

(

^f

) −

Q

(

^g

),

^f

−

g

i

₋

≤

0

,

^{for allf}

,

^g

∈

D

.

Now, we are ready to state the main result of this section

Theorem 4.1. For every f₀

∈

D, System(1.1)–(1.3), has a unique mild solution f

(

^t

,

^f0

)

on the interval

[

0

, ∞ ).

^{If we set} S

(

^t

)

^f0

=

f

(

^t

,

^f0

)

^{, then}

(

^S

(

^t

))

t≥0is a strongly continuous non-linear semigroup on D, generated by the operator A

+

Q . Moreover, the mild solutions f

(

^t

,

^f0

)

satisfying f₀

∈

D

(

^A

) ∩

D are strict solutions to problem(1.1)–(1.3).

Proof. First, the invariance ofDis proved inLemma 3.1. Secondly, the condition (ii) ofTheorem 2.1holds byLemma 4.2.

Concerning the dissipativity,Lemma 4.3 proves that the operator Q is dissipative on D. Moreover, the opertor Q is continuously differentiable onD. Finally, the application ofTheorem 2.1, combined withRemark 2.1, ends the proof.

5. Equilibrium profiles

First, assume the one-dimensional linear problem

∂

^f

(

^t

,

^x

)

∂

^t

+ ∂(v

0

(

^x

)

^f

(

^t

,

^x

))

∂

^x

+ µ

0

(

^x

)

^f

(

^t

,

^x

) =

0

, v

0

(

⁰

)

^f

(

^t

,

⁰

) =

Z

1

0

β

0

(

^x

)

^f

(

^t

,

^x

)

^dx

,

f

(

⁰

,

^x

) =

_f0

(

^x

).

If we put,g

(

^t

,

^x

) = v

0

(

^x

)

^f

(

^t

,

^x

)

^thengsatisfies the equation

∂

^g

(

^t

,

^x

)

∂

^t

+ v

0

(

^x

) ∂

^g

(

^t

,

^x

)

∂

^x

+ µ

0

(

^x

)

^g

(

^t

,

^x

) =

0

,

g

(

^t

,

⁰

) =

Z

1

0

β

0

(

^x

)

v

0

(

^x

)

^g

(

^t

,

^x

)

^dx

,

g

(

⁰

,

^x

) = v

0

(

^x

)

^f0

(

^x

) =

g₀

(

^x

).

Assumption

(

^Hv

)

^implies

Z

1

0

d

σ

v

0

(σ ) = ∞ .

^(5.1)

Let, now, us consider the following space transformation y

=

V₀

(

^x

) =

Z

x

0

d

σ

v

0

(σ) .

^(5.2)

By(5.1)and

(

^Hv

),

^V0

(

^x

)

is strictly increasing with lim_x↑1V₀

(

^x

) = ∞

. ThereforeV₀⁻¹exists, and is a strictly increasing map

[

0

, ∞ ) −→ [

0

,

¹

).

Then we obtain the new equivalent system for ally

∈ [

0

, ∞ ).

∂ψ(

^t

,

^y

)

∂

^t

+ ∂ψ(

^t

,

^y

)

∂

^y

+ µ

0

(

^V0⁻¹

(

^y

))ψ(

^t

,

^y

) =

0

,

^(5.3)

ψ(

^t

,

⁰

) = Z

∞

0

β

0

(

^V0⁻¹

(

^y

))ψ(

^t

,

^y

)

^dy

,

^(5.4)

ψ(

⁰

,

^y

) =

g₀

(

^V0⁻¹

(

^y

)) = ψ

0

(

^y

).

^(5.5)

LetY₀

=

L¹

( [

0

, ∞ ))

and define a linear operatorA₀inY₀by means of

(

^A0

ψ)(

^y

) = −

^d

ψ(

^y

)

dy

− µ

0

(

^V0⁻¹

(

^y

))ψ(

^y

),

^(5.6)

for

ψ ∈

D

(

^A0

)

, where the domainD

(

^A0

)

^ofA0is D

(

^A0

) =

ψ ∈

L¹

( [

0

, ∞ )) : ψ

absolutely continuous on

[

0

, ∞ ),

^A0

ψ ∈

L¹

( [

0

, ∞ )),

lim

y↑∞

ψ(

^y

) =

0

, ψ(

⁰

) =

Z

∞

0

β

0

(

^V0⁻¹

(

^y

))ψ(

^y

)

^dy

.

^(5.7)

By similar considerations as in [17] whered₀

(

^y

) = µ

0

(

^V0⁻¹

(

^y

))

^andb0

(

^y

) = β

0

(

^V0⁻¹

(

^y

))

, we also get that the essential spectrum

σ

e

(

^A0

)

^ofA0is contained in the half-space

σ

e

(

^A0

) ⊂ { λ ∈

C

:

Re

λ ≤ −

ess lim

y→∞infd₀

(

^y

) } ,

σ(

^A0

) ⊂ { λ ∈

C

:

Re

λ ≤ w

⁰∞

=

ess_ysup

(

^b0

(

^y

) −

d₀

(

^y

)) }

and, for allu

∈

_Y0and

λ > w

∞⁰, we have

ψ(

^y

) = (λ

^I

−

A₀

)

^−1u

(

^y

) =

exp

( − ϕ

_λ⁰

(

^y

))

ψ(

⁰

) + Z

y

0

u

(

^z

)

^exp

(ϕ

⁰_λ

(

^y

))

^dz

(5.8) where

ϕ

_λ⁰

(

^y

) = λ

^y

+ Z

y

0

d₀

(τ)

^d

τ.

Inserting into(5.4)we find

ψ(

⁰

)

1

−

Z

∞

0

b₀

(

^y

)

^exp

( − ϕ

_λ⁰

(

^y

))

^dy

= Z

∞

0

b₀

(

^y

)

^exp

( − ϕ

_λ⁰

(

^y

)) Z

y

0

u

(

^y

)

^exp

(ϕ

_λ⁰

(

^y

))

^dydx

.

^(5.9) Moreover,(5.8)and(5.9)imply

k (λ

^I

−

A₀

)

⁻¹

k ≤

¹

λ − w

⁰∞

,

^for

λ > w

∞⁰

,

henceA₀generates aC₀-semigroup

(

^T0

(

^t

))

t≥0positive of type

(

¹

, w

⁰∞

)

^inY0. Consequently,

Theorem 5.1. A, given by the components(5.6)and(5.7), generates a C₀-semigroup

(

^T

(

^t

))

t≥0, positive of type

(

¹

, w

∞

)

^such that

w

^∞

=

max

i

w

^∞i

, w

^∞i

=

ess_ysup

β

i V_i⁻¹

(

^y

)

− µ

i V_i⁻¹

(

^y

)

,

^y

=

V_i

(

^x

) = Z

x

0

d

σ v

i

(σ ) .

Now, we turn our attention to the question of existence of equilibrium solutions. The corresponding steady-states system to the model(1.1)–(1.3)is given by the following equations

d

(v

i

(

^x

)

^fi

(

^x

))

dx

+ µ

i

(

^x

)

^fi

(

^x

) =

n

X

j=1

Z

1

0

Z

1

0

η

ij

(

^y

,

^z

)ψ

ij

(

^y

,

^z

,

^x

)

^fi

(

^y

)

^fj

(

^z

)

^dydz

−

f_i

(

^x

)

n

X

j=1

Z

1

0

η

ij

(

^x

,

^y

)

^fj

(

^y

)

^dy

,

^(5.10)

v

i

(

⁰

)

^fi

(

⁰

) = Z

1

0

β

i

(

^x

)

^fi

(

^x

)

^dx

.

^(5.11)

More precisely, our purpose is to prove the existence of solutions to the following functional equation:

Af

+

Q

(

^f

) =

0

,

^f

= {

f_i

}

ⁿ_i

∈

D

(

^A

) ∩

D

.

^(5.12)

It is clair that the trivial solutionf

=

0 always satisfies(5.10)and(5.11).

Assuming the following hypothesis related to the boundedness of the birth and death rates,

β

i and

µ

i, for eachi

=

1

, . . . ,

ⁿ

,

w

^∞

=

max

i

w

^∞i

<

⁰

.

^(5.13)

Theorem 5.2. Under hypothesis(5.13), the trivial solution is the only stable equilibrium solution of the initial value problem (1.1)–(1.3).

Proof. Since

(

^T

(

^t

))

t≥0is theC₀-semigroup of type

(

¹

, w

^∞

)

, which leavesDinvariant. InLemma 4.3, it is proved thatQis dissipative onD

.

According to [12, Proposition 5.1, p. 357], into accountLemmas 4.2and4.3, if(5.13)hold, the problem (1.1)–(1.3)has a unique stable steady state inD

∩

D

(

^A

)

, which is a trivial steady state.

An important problem concerning(1.1)–(1.3), is that of the existence of nontrivial equilibrium solutions. Each nontrivial solution of(1.1)–(1.3)admits the representation

f_i

(

^x

) =

^exp

( −

_Φi

(

^x

)) v

i

(

^x

)

v

i

(

⁰

)

^fi

(

⁰

) + Z

x

0

Q_i⁺

(

^y

)

^exp

(

Φi

(

^y

))

^dy

,

ⁱ

=

₁

, . . . ,

^{n (5.14)}

where Φi

(

^x

) =

Z

x

0

µ

i

(

^y

) +

F_i

(

^y

) v

i

(

^y

)

^dy F_i

(

^x

) =

n

X

j=1

Z

1

0

η

ij

(

^x

,

^y

)

^fj

(

^y

)

^dy

,

Q_i⁺

(

^x

) =

n

X

j=1

Z

1

0

Z

1

0

η

ij

(

^y

,

^z

)ψ

ij

(

^y

,

^z

,

^x

)

^fi

(

^y

)

^fj

(

^z

)

^dydz

.

One of the most important considerations in the existence of nontrivial equilibrium solutions for the model, is to find a physically more realistic condition on

η

^and

ψ.

This question of the existence and the stability of nontrivial equilibrium solutions is under investigation.

6. Simulation problems for models

In this section deals with simulation of the dynamics of a two population model of the type classified as Model(1.1)–(1.3), that is characterized by the following assumptions.

Assumption 1.The system is constituted by two interacting cell populations: environmental and immune cells, labeled, respectively, by the indexesi

=

1 andi

=

2

.

Assumption 2.The functional state of each cell is described by the real variablex

∈ [

0

,

¹

]

. For the environmental cells, the valuex

=

0 is the natural state corresponding to normal cells and to the abnormal cells for the abnormal state which is the positive values ofxi.e.x

∈ (

⁰

,

¹

]

. On the other hand, for the immune cells, the valuex

=

0 corresponds to nonactivity and the positive values ofxcorrespond to activation.

Assumption 3.The encounter rate is assumed to be constant, and equal to unity for all interacting pairs i.e.

η

ij

=

1 for i

,

^j

=

1

,

²

.

Assumption 4.The transition probability density related to conservative interactions is assumed to be a Gaussian distribution function with the output defined by the mean valuem_ij, which may depend on the activation of the interacting pairs, and with a finite variances_ij:

ψ

ij

(

^x

,

^y

,

^z

) =

¹

p

2

π

^sij

exp

− (

^z

−

m_ij

(

^x

,

^y

))

² 2s_ij

,

wheres_ij

=

0

.

²

.

•

The environmental cells do not change their state when encountering other cells of the same population, and if an abnormal environmental cell encounters an active immune cell, its state decreases: i.e.

m₁₁

(

^x

,

^y

) =

x

,

^(6.1)

m₁₂

(

^x

,

^y

) =

x

− σ

12

,

^(6.2)

where

σ

12, is a parameter which indicates the ability of the immune system to reduce the state of cells of the first population.

•

It is assumed that interactions between cells of the second population have a trivial output and the only encounters with nontrivial output are those between active immune cells and abnormal environmental cells: i.e.

m₂₂

(

^x

,

^y

) =

x

,

^(6.3)

m₂₁

(

^x

,

^y

) =

x

− σ

21

,

^(6.4)

where

σ

21is a parameter which indicates the ability of abnormal cells to inhibit immune cells. The behavior is observed inFig. 1, when

σ

12

=

0 and

σ

21

=

0

.

⁹

.

For further discussion of these functions, we refer to [9].

(a) Density at different times. (b) Evolution of density.

Fig. 1. σ12=0, σ21=0.^9.

Assumption 5. The cells start progressing toward values ofx

∈ [ α, β ]

with velocity

v

^{, where}

α < β

are positive values. On the other hand, the progression velocity is decreased to zero asxgoes to 1. We take for simplicity the following natural vital growth rate

v

1

(

^x

) =

0

.

^{2 ifx}

∈ [

0

,

⁰

.

⁹

]

2

(

¹

−

x

)

^ifx

∈ [

0

.

⁹

,

¹

]

^and

v

2

(

^x

) =

(

₀

.

^{5 ifx}

∈ [

0

,

⁰

.

⁴

]

0

.

⁵

0

.

⁶

(

¹

−

x

)

^ifx

∈ [

0

.

⁴

,

¹

] .

^(6.5) Assumption 6. For our simulation study, we consider, for simplicity, the following natural vital birth and death rates

β

1

(

^x

) =

0

.

⁵

χ

^{0≤x≤0.⁴}

, β

2

(

^x

) =

0

.

⁹

χ

^{0.²≤x≤0.⁶} and

µ

i

(

^x

) =

^x

x

+

1

,

ⁱ

=

1

,

²

.

^(6.6)

7. Conclusion

This paper deals with the global existence, and the uniqueness of the state trajectories for a mathematical model of a population consisting of several species with coupled interactions and nonlocal boundary conditions. It has also been proved that the trajectories are positive. In addition, under some consideration, it has been proved that the trivial solution f

=

0 is the only stable equilibrium solution for such systems.

An important question is the existence and the stability analysis of nontrivial equilibrium profiles for system(1.1)–(1.3).

A further open question is the study of the asymptotic behavior of the state trajectories for system(1.1)–(1.3)with spatial structure and when the natural vital growth rate,

v = { v

i

}

ⁿ_i=1, is time varying and the natural vital (birth,

β = { β

i

}

ⁿ_i=1and death,

µ = { µ

i

}

ⁿ_i=1) rates are, also, time varying and depends on the total populationP

(

^t

) = P

n

i=1

R

1

0 f_i

(

^t

,

^x

)

dx. Moreover, the analysis of control problems is related to the expression of the natural vital rates.

Acknowledgements

This work is supported by INRIA Bordaux — Sud-Ouest ‘‘Institut National de Recherche en Informatique et en Automatique’’. The authors are very grateful to Prof. N. BELLOMO for invaluably interesting discussions.

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