Convergence of Numerical Schemes to a Nonlinear Kinetic Model of Population Dynamics with Nonlocal Boundary Conditions

CONVERGENCE OF NUMERICAL SCHEMES TO A NONLINEAR KINETIC MODEL OF POPULATION DYNAMICS WITH

NONLOCAL BOUNDARY CONDITIONS

B. AYLAJ^† AND A. NOUSSAIR^†

Abstract. We examine existence and uniqueness of a global solution and some basic mathemat- ical issues associated with the development of a numerical scheme for a model of a tumor-immune system interaction. The system consists of nonlinear transport equations with a bilinear operator like a Boltzmann type and with a nonlocal boundary condition. We construct a numerical scheme as a combination of a S(t)-operator semigroup associated with the continuous problem and P^Δ projection operator on the discrete space. The three estimates, (1)L^∞bound, (2) a uniform total variation bound, and (3)L¹ continuity in time of the approximate solution, are established with a self-contained treatment of the stability and convergence properties. Numerical calculations are reported.

Key words. population dynamics, Boltzman interactions, transport-projection scheme, Helly compactness, kinetic models

AMS subject classifications.65M06, 15A15, 15A09, 65M12 DOI.10.1137/08073665X

1. Introduction and problem statement. In the past two decades, the the- ory of mathematical kinetics of large systems of active particles has been developed extensively, and the achieved results have found many applications in the evolution of several sociobiological systems. One of the first models of this type was proposed by Jagger and Segel [12] with the aim of studying the evolution of the physical state, called dominance, which characterizes certain populations of insects. This method has been subsequently developed by other authors ([3] and bibliography therein), to model population dynamics of interacting individuals whose microscopic state is related to their social and/or biological behavior.

Recently, increasing interest has been observed in applications of the analysis of competition between cell of an aggressive host cell of a corresponding immune system (see [5, 4]). These new classes of models of population dynamics with stochastic interaction are characterized by a mathematical structure similar to the one of the Boltzmann equations and are based on a framework related to a system of integro- differential equations, defining the evolution of the distribution function over the microscopic state of each element in a given system.

The interest of the above approach to cancer modeling is documented in [2, 4, 11], while additional mathematical structures are reported in [14]. Here, we aim to propose the development of a numerical scheme for a Boltzmann-like model of population dynamics of several species with special attention to the competition between tumor and immune cells.

The development of numerical schemes, which are accurate computational schemes of this model, is desirable (see [10, 6, 7]), and hopefully the work presented here is a step toward obtaining such algorithms. Analytical understanding of the continuous

∗Received by the editors September 29, 2008; accepted for publication (in revised form) June 21, 2010; published electronically September 29, 2010.

http://www.siam.org/journals/sinum/48-5/73665.html

†INRIA, Universit´e Bordeaux 1, 351 cours de la Lib´eration, 33400 Talence, France (bouchraylaj@

yahoo.fr, noussair@math.u-bordeaux1.fr).

1707

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problem is used to design a transport projection numerical scheme. The question of how to discretize the given model; in particular, the treatment of the discrete encounter operator together with the special treatment of the nonlocal boundary con- dition represents a major difficulty in the analysis. Investigations of convergence prop- erties of numerical schemes for this model seem to be a rather unexplored area, and the important point of this paper is to present a convergence result. More precisely, the system consists of nonlinear transport equations with a bilinear operator and with a nonlocal boundary condition. In detail, in our model, the following assumptions are proposed.

Assumption 1.1. The physical system is constituted by N

∈ N

interacting populations. 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 variable x that denotes the intensity with which the activities are performed. For the sake of simplicity, it will be assumed that x is a scalar variable with a range in [0, 1]. The value x = 1 corresponds to the highest value of the peculiar activities of the cell, while x = 0 corresponds to complete functional inhibition (no activities).

Assumption 1.2. The probability density functions are defined, for each popula- tion, by

f

= f

(t, x) : [0, ∞ ) × [0, 1] −→ R

.

Therefore, the probability of finding, at time t, an individual of the k population in the state interval [0, 1] is given by

P

(t) =

0

f

(t, x)dx.

The interactions can be divided into conservative ones, which preserve the total num- ber of cells, and proliferative destructive ones, which cause an increase or a decrease of the total number of cells (see the following Assumptions).

Assumption 1.3. Conservative encounters modify the state of the cells but not their number. The evolution due to conservative encounters modifies the progression of tumor cells and the activation of immune cells. Cell interactions in the case of mass conservative encounters will be defined by means of the two quantities: the encounter rate η

and the transition probability density ψ

, where η

(y, z) denotes the number of encounters per unit volume and unit time between cell pairs of the (k, l)th populations with states y and z, respectively, and ψ

(y, z, x) denotes the probability of transition of the kth cell to the state x, given its initial state y and the state z of the encountering cells belonging to the lth population.

Assumption 1.4. Proliferating cells will be described by the vital birth rate β

(., x), where β

denotes the number of cells produced per unit volume and unit time of the (k)th species with state x. We will assume that the inflow of the newborns is in the inhibition state. Destructive cells will be described by the vital death rate μ

(., x), which is the number of kth species with state x destroyed (see [15]).

In this case, these functions depend also on the set of all number density P(t) =

P

(t), . . . , P

(t)

.

The mathematical problem with initial and boundary conditions, obtained using the above assumptions, consists of the following system of N

coupled

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integro-differential equations:

⎧ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎨

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎩

∂f

(t, x)

∂t + ∂

∂x v

(P(t), x)f

(t, x)

+ μ

(P(t), x)f

(t, x)

=

l=1

Q

(f

, f

)(t, x), v

(P(t), 0)f

(t, 0) =

0

β

(P(t), x)f

(t, x)dx, f

(0, x) = f

(x),

(1.1)

where the operator Q

(f

, f

) = Q

(f

, f

) − Q

(f

, f

) with the gain part Q

(f

, f

)(t, x) =

0

0

ψ

(y, z, x)η

(y, z)f

(t, y)f

(t, z)dydz (1.2)

and the loss part

Q

(f

, f

)(t, x) = f

(t, x)

0

η

(x, y)f

(t, y)dy.

(1.3)

The function v

is the activation velocity (rate of growth).

We point out that the model we have built can be regarded as a superposition of a kinetic and a population dynamics model. It retains some features of both classes of models. In the case N

= 1, when v

= β

= μ

= 0, Jagger and Segel [12] analyzed the evolution of a physical state, called dominance, which characterizes a certain population of insects. This latter result has been generalized by [1], 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

= 0), the model is a particular case of [13, 17]. Pr¨ uß [13] established the global existence as well as the stability of equilibrium solutions of age specific population dynamics of several species. When no diffusion and no convection terms are considered, Bellomo and co-workers in a series of papers analyzed, analytically and/or numerically, the previous model, where various examples and applications are proposed and critically analyzed [3, 4]. In particular, Bellouquid and Delitala [5]

proposed a new biological model for the analysis of competition between cells of an aggressive host and cells of a corresponding immune system.

The paper is organized as follows. We first make some assumptions on the co- efficients of the system and define what we mean by the weak solution of (1.1). In section 3, we define the discrete problem, and in sections 4 we give some properties of the discrete operators of encounters. In sections 5 and 5.2, we construct explicit and implicit numerical schemes based on the finite volumes method, which are really a combination of the transport operator and the projection one. In section 6, we give some estimates satisfied by the approximate solutions. In section 7, we prove the convergence of the schemes. In section 8, we report some numerical results.

2. Assumptions and definition.

Assumption on the rate of encounters and the transition probability.

We make the following assumptions:

η

(x, y) = η

(y, x), η

∈ L

∩ BV ([0, 1] × [0, 1]); 0 ≤ η

≤ η

, (2.1)

ψ

∈ L

∩ BV ([0, 1] × [0, 1]); ψ

≥ 0 and

0

ψ

(y, z, x)dx = 1, (2.2)

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where η

is positive constant and BV ([0, 1] × [0, 1]) is the set of all locally bounded variation functions in [0, 1] × [0, 1] in the sense of Tonelli–Cesari. This means that for any bounded open set Ω in [0, 1] × [0, 1],

V ar(η

, Ω) =

0

V ar(η

(., y), Ω

)dy +

0

V ar(η

(x, .), Ω

)dx is bounded. In fact, we only need for all Δx > 0

0

| η

(x + Δx, y) − η

(x, y) | dx ≤ C

Δx for all y ∈ [0, 1), where C

is independent of Δx.

ψ

is also assumed to be a locally bounded variation on [0, 1] × [0, 1], and therefore verifies, for all Δx > 0

0

| ψ

(x + Δx, y, z) − ψ

(x, y, z) | dx ≤ C

Δx for all y, z ∈ [0, 1], where C

is independent of Δx.

Assumption on the activity growth factor. We assume that, for each k, (H

) v

(P(.), x) ∈ L

([0, T ]), v

Lipschitzian with respect to P

and strictly positive for all x ∈ [0, 1) with v

(P, 1) = 0, v

(P, 0) ≥ v, where v is independent of t, and for all Δx > 0

| v

(., x+Δx) − 2v

(., x)+v

(., x − Δx) | ≤ C

Δx

for all x ∈ [0, 1], where C

is independent of t and Δx.

(H

) μ

(P(.), .), β

(P(.), .) ∈ L

([0, T ] × [0, 1]) are non-negative Lipschitzian functions with respect to P and μ

(P(t), .) ∈ BV ([0, 1]).

Assumption on the initial data. The initial condition f

is supposed to be in L

([0, 1]) ∩ L

([0, 1]) with a locally bounded variation on [0, 1], and therefore verifies, for all Δx > 0,

0

| f

(x + Δx) − f

(x) | dx ≤ C

Δx, where C

is independent of Δx.

Weak solution. We need weak formulation to make sense of the solution of (1.1) even if the solution may be discontinuous. Since the operator Q

(f

, f

) satisfies the following property,

(2.3)

0

Q

(f

, f

)(x)φ(x)dx =

[0,1]³

(φ(x) − φ(y))ψ

(y, z, x)η

(y, z)f

(y)f

(z)dxdydz, we can give the following definition of a weak solution.

Definition 2.1. A measurable function f

(t, x)

is a weak solution for the problem (1.1) if R(φ, f

) = 0 for all smooth functions φ ∈ C

([0, T [ × [0, 1]) with

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compact support on [0,T[, where

R(φ, f

) =

0

0

(f

φ

)(t, x) + v

( P (t), x) (f

φ

)(t, x)dxdt +

0

0

⎛

⎝

l=1

Q

(f

, f

) − μ

( P (t), x)f

(t, x)

⎞

⎠ φ(t, x)dxdt

+

0

0

β

( P (t), x)f

(t, x)φ(t, 0)dxdt +

0

f

(x)φ(0, x)dx.

(2.4)

3. Discrete space and discrete operators of conservative encounters.

Discrete space. Given T > 0, we show how to obtain the solution in the interval [0, T ] × [0, 1]. We denote by Δx the space mesh size, and we set x

= 0 and x

= x

+ iΔx, for any i ∈ N

. We define the cells M

= ]x

, x

[ with the middle point x

= x

+ Δx/2 for any i ∈ N

. In this way [0, 1] is discretized with the cells M

. Let Δt be the time step size. We introduce the sequence t

= nΔt, and define the ratio r = Δ t

Δx , which will be used to control the stability. At each time t

, we build a discrete space,

V

= { f (t

, .) ∈ L

([0, 1]), f (t

, x) = f

on M

, i ∈ N } ,

with the “discrete” L

, L

norms, and the associated total variations are given by

| f

|

= Δx

i∈N

| f

| , | f

|

= sup

i∈N

| f

| and T V (f

) =

i∈N^∗

| f

− f

| .

Let P

be the orthogonal projection on V

associated with the usual L

inner product. This projection gives the usual mean value on each cell of any function f

as follows:

For all f

∈ L

([0, 1]) P

f

(x) = 1 Δx

xi−1/2

f

(y)dy if x ∈ M

. Discrete operators of conservative encounters. We start by replacing η

with the “discrete” η

η

(x, y) = η

= 1 (Δx)

xi−1/2

xj−1/2

η

(x, y)dxdy

for x, y in M

× M

and the total variation T V (η

) =

i,j∈N

η

− η

+

i,j∈N

η

− η

,

bounded, and let us approximate also the function ψ

with ψ

ψ

(x, y, z) = ψ

= 1

(Δx)

xi−1/2

xj−1/2

xp−1/2

ψ

(x, y, z)dxdydz

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for (x, y, z) in M

× M

× M

. With the discrete total variation of ψ

T V (ψ

) =

i,j,p∈N

ψ

− ψ

+

i,j,p∈N

ψ

− ψ

+

i,j,p∈N

ψ

− ψ

bounded. Now we define discrete conservative encounter operators for the piecewise constant function f

, defined in (1.2)–(1.3), by

Q

=

IL

j=1 IL

p=1

η

ψ

f

f

(Δx)

and Q

= f

L

,

with L

=

j=1

η

f

Δx.

4. Properties of the discrete conservative encounter operator. Now we give some properties satisfied by the following discrete conservative encounter operator:

(4.1) Q

(f

, f

)(x) = Q

− Q

.

Lemma 4.1. For any sequence (φ

)

the discrete conservative encounter oper- ator (4.1) satisfies

IL

i=1

φ

Q

Δx =

IL

i=1 IL

j=1 IL

p=1

(φ

− φ

) ψ

η

f

f

(Δx)

.

In particular, choosing φ

= 1 for all i, and taking into account

i=1

ψ

Δx = 1, we have

(4.2)

IL

i=1

Q

Δx = 0.

Now we give some estimates for Q

.

Lemma 4.2. Let f

∈ L

([0, 1]) ∩ L

([0, 1]) and η

, ψ

satisfy (2.1) and (2.2), respectively, and then Q

satisfy

Q

∞,Δ

≤ η

f

1,Δ

+ f

∞,Δ

f

1,Δ

, Q

1,Δ

≤ 2η

f

1,Δ

f

1,Δ

and L

∞,Δ

≤ η

f

1,Δ

. These estimates are straightforward. Now we shall give an estimate satisfied by the total variation of Q

.

Lemma 4.3. Under the assumption f

∈ L

([0, 1]), and noting T V

(ψ

) = sup

1≤j,p≤IL

I

ψ

− ψ

, T V

(η

) = sup

1≤j≤IL

I

η

− η

,

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we have

T V

(Q

) ≤ η

| f

|

T V

(ψ

) | f

|

+ T V (f

) + T V

(η

) | f

|

| f

|

,

T V

(Q

) ≤ η

T V

(ψ

) | f

|

| f

|

, T V

(L

) ≤ T V

(η

) | f

|

,

(4.3)

5. Formulation of the schemes.

5.1. Explicit scheme. The approximate solution f

, generated by the scheme, at time t

is denoted f

(x) = f

(t

, x) and f

(.) is built in such a way to lie in V

. The numerical scheme is really a transport projection method, that is, a combination of a S

(t) operator and a projection operator on the discrete space. It corresponds to the following formulation:

f

= P

S

(Δt)f

. More explicitly, the scheme take the following form:

f

= f

− Δt Δx

v

f

− v

f

+ Δt

⎛

⎝

l=1

Q

− μ

f

⎞

⎠ v

f

=

IL

i=1

β

f

Δx.

(5.1)

This scheme works under the following CFL condition:

(5.2) max

k

sup

v

(x) Δt

Δx ≤ 1.

5.2. An implicit scheme. In this section we just give an implicit scheme, of transport-projection type, for the first density equation

(5.3)

⎧ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎨

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎩

f

=

f

+ rv

f

+ Δt

l=1

Q

1 + r v

+ Δt(L

+ μ

) , with

v

f

=

IL

i=1

β

f

Δx and L

=

Nf

l=1

L

.

This scheme works without CFL condition and the positivity of the solution is ob- tained without restriction on Δt.

6. Bounds for the solution.

6.1. Bounds for the explicit scheme. In this section, we give some estimates satisfied by the scheme. We have the following lemma concerning the positiveness satisfied by the solutions of the scheme.

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Lemma 6.1. Starting with f

≥ 0, for all i, k and v

≥ 0. Choosing Δt so small that

(6.1) Δt

Δx ≤ 1

sup

L

v

+ Δx

μ

+ η

l=1

| f

|

,

for all p ≤ n, we obtain f

positive. Moreover, let f

∈ L

([0, 1]) ∩ L

([0, 1]); the discrete L

-norm satisfies

(6.2) | f

|

≤ e

| f

|

, and the discrete L

-norm satisfies

(6.3)

| f

|

≤ e

max

| v

|

+ η

N

e

max

| f

|

| f

|

+C(T ) with C(T ) being constant independent of n.

The total variation bound plays a crucial role in establishing the subsequential convergence of the difference approximation scheme. In the next lemma we give the boundness of the total variation of the approximate solution.

Lemma 6.2. Let f

∈ (L

∩ L

)([0, 1]) with the total variation be bounded locally in [0, 1] and under the CFL condition (5.2), and then we have

(6.4) T V (f

) ≤ e

T V (f

) + M

with M and K being constants independent of n. The total variation in time satisfies (6.5)

IL

i=1

| f

− f

| ≤ T V (f

) + rC(T ), which implies the L

Lipschitz continuity in time.

6.2. Bounds for the implicit scheme.

Lemma 6.3. Let f

∈ L

([0, 1]) and f

≥ 0, and the discrete L

-norm satisfies (6.6) | f

|

≤ e

| f

|

,

(6.7) | f

|

≤ e

| f

|

+ T C

(T ) with A, C, and C

(T ) being constants independent of n.

In the next lemma we give the boundness of the total variation of the approximate solution.

Lemma 6.4. Let f

∈ (L

∩ L

)([0, 1]) with the total variation be bounded locally in [0, 1], and then we have

(6.8) T V (f

) ≤ e

T V (f

) + T K

(T ) and the total variation in time satisfies

(6.9)

IL

i=1

| f

− f

| ≤ r max

k

| v

|

T V (f

) + r C(T )

with K, K

(T ), and C being constants independent of n, which implies the L

Lipschitz continuity in time.

For the proof of these lemmas see Appendix A.

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7. Global existence of weak solution in L

.

Passing to the limit. In this section, we deal with the existence of a global weak solution using the Helly compactness argument, based on the compact canonical imbedding from W

(Ω) into L

(Ω). We first prove the existence of a limit f

to f

as the mesh size Δx goes to zero. Then we prove that this limit is a weak solution of the problem (1.1).

Theorem 7.1. Let f

∈ (L

∩ L

)([0, 1]) with total variation bounded locally in [0, 1], f

≥ 0; then as the mesh size Δx tends to zero, there is a subsequence of (f

)

, the family of approximate solution, converging in L

([0, T ] × [0, 1]) to a function f

∈ L

([0, T ] × [0, 1]) ∩ L

([0, T ] × [0, 1]).

Consistency of the scheme. It remains to show that the scheme is consistent.

Theorem 7.2. Under the same assumptions as in Theorem (7.1), the limiting function f

(t, x) just obtained is a weak solution for the problem (1.1).

Uniqueness of the weak solution. The following theorem guarantees the continuous dependence of the solution (f

)

of (1.1) with the respect of the initial data (f

(x))

.

Theorem 7.3. Let (f

(t, x))

and ( ˜ f

(t, x))

be two bounded variation weak so- lutions for the problem (1.1) corresponding to initial conditions (f

(x))

and ( ˜ f

(x))

, respectively; then there exists a constant C(T ) such that

0

| f

− f ˜

| dx ≤ C(T )

0

| f

− f ˜

| dx.

(7.1)

For the proof of these theorems see Appendix B.

8. Numerical experiments. In this section, we report some numerical exper- iments. We use a uniform spatial grid over the interval [0, 1] with Δx = 0.01. The time mesh size is Δt = rΔx, with a rate r = 0.5 destined to control stability, and our stability criterion (5.2) is never exceeded.

We deal with the simulation of the dynamics of a two population model of the type classified as model (1.1) that is characterized by the following assumptions.

Assumption 8.1. The system is constituted by two interacting cell populations:

environmental and immune cells, labeled, respectively, by the indexes k = 1 and k = 2.

Assumption 8.2. The functional state of each cell is described by the real variable x ∈ [0, 1]. For the environmental cells, the value x = 0 is the natural state correspond- ing to normal cells and to the abnormal cells for the abnormal state which are the positive values of x, i.e., x ∈ (0, 1]. On the other hand, for the immune cells, the value x = 0 corresponds to nonactivity and the positive values of x correspond to activation.

Assumption 8.3. The encounter rate is assumed to be constant and equal to unity for all interacting pairs; i.e., η

= 1 for k, l = 1, 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:

m

(x, y) = x, m

(x, y) = (1 − σ

)x,

where σ

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

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• It is assumed that the 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:

m

(x, y) = x, m

(x, y) = (1 − σ

)x,

where σ

is a parameter which indicates the ability of abnormal cells to inhibit immune cells.

For further discussion of these functions, we refer to [5].

Assumption 8.4. The transition probability density related to conservative in- teractions is assumed to be a Gaussian distribution function with the output defined by the mean value m

, which may depend on the activation of the interacting pairs, and with a finite variance s

= 0.2

ψ

(x, y, z) = 1

√ 2πs

exp

− (z − m

(x, y))

2s

.

In Figures 8.1–8.8, the initial condition is the Gaussian distribution function with the mean value μ = 0.3 and a finite variance σ = 0.01, for both densities. The graphs in Figures 8.2, 8.4, 8.6, and 8.8 (resp., Figures 8.1, 8.3, 8.5, and 8.7) show the evolution in times of f

and f

(resp., the evolution in times of the densities of the system):

the continuous line is the evolution of f

(resp., the evolution of the density of the immune cells), while the dashed line is the evolution of f

(resp., the evolution of the abnormal density).

In experiments 2, 3, and 4, we assume that a fraction α

∈ (0, 1) of ingested food is channeled to growth and maintenance, and a fraction (1 − α

) to reproduction.

We also assume that for an individual of size x in the kth subpopulation, α

g

(P)x is the rate at which maintenance needs energy and α

g

(P)(1 − x) is what remains for growth (see [16, 8]). Thus we have the following submodels for the growth and reproduction rates for each subpopulation

(8.1) v

(P, x) = ¯ v

α

g

(P)(1 − x), and β

(P, x) = ¯ β

(1 − α

)g

(P)x.

g = (g

, g

) will be a smooth quasimonotone decreasing, in R

× R

, g will be a positive function satisfying g(0) = (1, 1) and tending to (0, 0) when P goes to ∞ , where ¯ v

and ¯ β

are positive constants. These functions were taken from Thieme [16]

(see also [8]). In addition, we assume that the mortality rate for an individual in the kth subpopulation is given by

μ

(P) = γ

− d

+ a

P

+ b

P

c

P

+ d

P

+ a

, γ

= − v ¯

α

+

(¯ v

α

)

+ 4 ¯ β

(1 − α

)¯ v

α

2 .

The function g

here is the following:

g

(P) = 1 − a

P

+ b

P

c

P

+ d

P

+ c

with (d

, a

, b

, c

, d

, a

) and (a

, b

, c

, d

, c

) are the constant positives.

The result of the simulation is reported in Figures 8.3, 8.4, 8.5, 8.5, 8.7, and 8.8, which correspond to different values of the parameters (see Table 8.1).

In Experiment 1, it is, also, interesting to notice that, for β

(P) = ¯ β

(1 − α

) P

c

+ P

x, β

(P) = ¯ β

(1 − α

)

1 − P

c

x

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Table 8.1

Different values of the model parameters.

Parameter/variable

Experiment 1 Experiment 2 Experiment 3 Experiment 4

v¯¹ 10 10 10 10

v¯² 0.3 0.3 0.3 0.3

β¯¹ 0.2 20 0.20 0.20

β¯² 0.3 30 30 0.3

α¹ 0.1 0.1 0.1 0.1

α² 0.2 0.2 0.2 0.2

c¹ 2 0.8 2 2

c² 2 9 9 2

d¹ 3.10⁻⁴ 3.10⁻⁵ 3.10⁻⁵ 0

d² 0 10⁻⁵ 10⁻⁵ γ2

a¹ 1 10 0.1 1

a² 1 20 0.2 1

q 0.2 √ √

0,2

e 0.9 √ √

0.9

σ12 0.9 0.1 0.9 0.9

σ21 0.1 0.9 0.1 0.1

(a1, b1, c1, d1) √

(0,1,0,1) (1,0,1,0) (0,0,0,0) (a2, b2, c2, d2) √

(1,0,1,0) (0,1,0,1) (e,0,0, q) (a₁, b₁, c₁, d₁) √ (0,1,0,1) (1,0,1,0) (0,1,0,1)) (a₂, b₂, c₂, d₂) √

(1,0,1,0) (0,1,0,1) (0,1,0,1)

and

μ

(P) = eP

1 + qP

, μ

(P) = γ

− d

,

that is, the predator-prey system (see Experiment 1 in Table 8.1), as shown in Figures 8.1 and 8.2.

Here, the growth (birth) rate of the kth predator and the parameters intrinsic rate of increase, respectively, depends on space x. If σ

< σ

, the ability of abnormal cells to inhibit immune cells is greater than the ability of immune cells to reduce the state of abnormal cells. The final output, if σ

> σ

, is a reduction of the state of abnormal cells until their complete depletion and a final survival of immune cells, as shown in Figures 8.2 and 8.6. In Figures 8.4 and 8.8, the evolution is not ruled by the parameters σ

and σ

.

In Figures 8.1 and 8.7, both abnormal and immune cells reduced during the competition. However, in Figures 8.3 and 8.5, only one cell population survives and the other is completely depleted: the probability of the 2-population is an increasing function of t and the probability of the 1-population is a decreasing function of t.

Appendix A.

A.1. Proof of Lemma 6.1: L

stability. The scheme can be rewritten as f

= (1 − r v

)f

+ r v

f

+ Δt

⎛

⎝

l=1

Q

− μ

f

⎞

⎠ .

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0 0.2 0.4 0.6 0.8 1 0.3

0.4 0.5 0.6 0.7 0.8 0.9 1

t

Density

P¹(t) P²(t)

Fig. 8.1. “Experiment 1”:Total depletion of cells.

Since

i=1

Q

Δx = 0 and under the positiveness condition (6.1) we have

IL

i=1

f

Δx ≤

i=1

f

Δx + r

IL

i=1

v

f

− v

f

Δx

≤ (1 + Δt | β

|

)

IL

i=1

f

Δx,

since rΔx = Δt and nΔt ≤ T and using the Gronwall lemma we assume easily (6.2).

A.2. Proof of the L

stability. We rewrite the scheme (5.1) as f

=

1 − Δt

Δx v

f

+ Δt

Δx v

f

− Δt

Δx (v

− v

)f

+ Δt

⎛

⎝

l=1

Q

− μ

f

⎞

⎠ .

From (6.1) and under the CFL condition (5.2), we have

| f

|

≤

⎛

⎝1 + Δt( | v

|

+ η

Nf

l=1

| f

|

)

⎞

⎠ | f

|

+ Δtη

| f

|

l=1

| f

|

.

Using (6.2) and the fact that nΔt ≤ T , the estimate (6.3) follows.

A.3. Proof of Lemma 6.2.

A.3.1. Proof of the BV estimate. Using the expression given in scheme (5.1), and setting δ

= f

− f

, we have

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0 0.2 0.4 0.6 0.8 1 0

0.5 1 1.5 2 2.5 3 3.5

t Space 25 −− f¹(t), − f²(t)

0 0.2 0.4 0.6 0.8 1

0.4 0.6 0.8 1 1.2 1.4

t Space 50 −− f¹(t), − f²(t)

0 0.2 0.4 0.6 0.8 1

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

t Space 75 −− f¹(t), − f²(t)

0 0.2 0.4 0.6 0.8 1

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

t Space 100 −− f¹(t), − f²(t)

σ₁₂=0.9, σ₂₁=0.1

0 0.2 0.4 0.6 0.8 1

0 0.5 1 1.5 2 2.5 3 3.5

t Space 25 −− f¹(t), − f²(t)

0 0.2 0.4 0.6 0.8 1

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

t Space 50 −− f¹(t), − f²(t)

0 0.2 0.4 0.6 0.8 1

0 0.1 0.2 0.3 0.4

t Space 75 −− f¹(t), − f²(t)

0 0.2 0.4 0.6 0.8 1

0 0.05 0.1 0.15 0.2 0.25

t Space 100 −− f¹(t), − f²(t)

σ₁₂=0.1, σ₂₁=0.9

Fig. 8.2. “Experiment 1”: Effect of the type of parameter σ for density evolution of both populations at different spaces: x= 25,50,75,100.

δ

=

1 − rv

δ

+ rv

δ

− r

v

− 2v

+ v

f

+ r

v

− v

f

− f

+ Δt

Nf

l=1

Q

− Q

− Δtμ

f

− f

− Δt

μ

− μ

f

.

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0 0.2 0.4 0.6 0.8 1 0

0.5 1 1.5 2 2.5 3

t

Density ^P1(t)

P²(t)

Fig. 8.3. “Experiment 2”: Depletion of abnormal cells and immune activation.

Under the CFL condition (5.2), which implies that (1 − rv

) ≥ 0, we obtain

I

| δ

| ≤

IL

i=2

| δ

| + rv

| δ

| + ΔtC

| f

|

+ Δt | v

|

i=1

| δ

|

+ Δt

Nf

l=1

T V

(Q

) + Δt | μ

|

i=1

| δ

| + ΔtT V (μ

) | f

|

. (A.1)

On the other hand, using the scheme for i = 1, we have f

− f

= (1 − rv

)δ

− (f

− f

)

− r

v

− v

f

+ Δt

⎛

⎝

l=1

Q

− μ

f

⎞

⎠ , (A.2)

we recall the expression of the total variation of f

(t

, .) T V (f

) =

I

| δ

| + | f

− f

| ,

and then, adding the terms (A.1) and (A.2), we obtain T V (f

) ≤ 1 + Δt( | μ

|

+ | v

|

)

T V (f

) + ΔtC

| f

|

+ Δt | v

|

+ T V (μ

) + | μ

|

| f

|

+ | f

− f

| + Δt

Nf

l=1

T V

(Q

) + | Q

|

.

From the boundary condition and assumption (H

), we have

| f

− f

| ≤ | β

|

i=1

| f

− f

| Δx,

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