Global weak solution for a multi-stage physiologically structured population model with resource interaction

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Nonlinear Analysis: Real World Applications

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

Global weak solution for a multistage physiologically structured population model with resource interaction

B. Aylaj

^a,∗

, A. Noussair

^b

aINRIA Bordeaux - Sud-Ouest, Université Bordeaux 1, 351 cours de la Libération Talence 33400, France

bIMB Université Bordeaux 1, 351 cours de la Libération Talence 33400, France

a r t i c l e i n f o

Article history:

Received 15 October 2008 Accepted 20 March 2009 Keywords:

Population dynamics Kinetic models

Age structured population Fully implicit scheme Stability

Helly-compactness

a b s t r a c t

We construct a multistage kinetic model of a physiologically structured insect population whose life history consists of fourth stages of development termed eggs, larval, pupal and adult moth (male and female). The model is a system of weakly coupled hyperbolic partial differential equations with nonlocal boundary conditions. The vital rates depend on the resource which satisfy an ordinary differential equation. We discretize the physiological space and formulate an implicit scheme and we prove the existence and uniqueness of the solution. The numerical simulation provides an analytical tool to improve the understanding of the moth’s biology.

©2010 Published by Elsevier Ltd

1. Introduction

In recent years, various mathematical models for one-sex age-dependent population growth have appeared in the literature, see the books by Webb [1], Metz and Diekmann [2] and Calsina [3] (see also [4]) for a comprehensive treatment of the mathematical aspects. However, few research papers describe partial differential equation models for age-dependent populations whose life history consists of several stages. The first models describing the time evolution of age-dependent populations structured in several stages appear in the literature [5]. In [5], it is assumed that a life history consists of three stages: the eggs, the juveniles and the adults in the first, second and third stages respectively.

The model proposed by McNair and Goulden [5] has been carefully studied by Matucci [6] by using semigroup theory of bounded linear operators in Banach space. Further investigations on a two-stage population model under two different types of boundary conditions can be found in Busoni and Matucci [7]. In these last two works, the existence and uniqueness of a global in time nonnegative strong solution is proved, and the asymptotic behavior is described in terms of the biological parameters by using spectral analysis.

In this work we consider a boundary value problem that describes the growth of a grapevine moth whose life history consists of several stages termed egg, larvae, pupae, adult (male or female). The model is a time- and age-dependent system of coupled Lokta–MacKendrick equations which describes the population throughout these different stages.

Models and simulations of multistage population dynamics (for example multistage of grapevine moth growth) can reduce the amount of necessary experimentation. Moreover, the mathematical theory developed might not only provide a detailed description of nonlinear multistage evolution problem, but also may help us understand and manipulate aspects of the process that are difficult to access experimentally.

The development of numerical schemes which give good calculation of our model are desirable (see [8–10]) and hopefully the work presented here is a step towards obtaining such algorithms. Analytical understanding of the continuous problem

∗Corresponding author.

E-mail addresses:[email protected](B. Aylaj),[email protected](A. Noussair).

1468-1218/$ – see front matter©2010 Published by Elsevier Ltd doi:10.1016/j.nonrwa.2009.03.021

is used to design a transport projection numerical scheme. The question how to discretize the given model, in particular, the treatment of the discrete encounter operator together with the special treatment of the nonlocal boundary condition represents an interesting part. Investigations of convergence properties 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.

For a variety of reasons we therefore often want to distinguish individual from each other on the basis of a number of physiological characteristics that influence its life history in terms of its chance to grow, migrate, reproduce or die. The environmental conditions that the population are exposed to are important for its dynamics, while it usually sets the limits for it development. Environmental conditions can pertain to biotic and abiotic factors, for example temperature (or degree days), humidity, food abundance and the number of predators or competitors around. In the case of abiotic modulation, the state of environment changes over time in a predetermined manner. In this case one has to specify a priori the pattern of the daily temperature and relative humidity variation as functions of time and days. In this work, each equation is related to a specific stage with vital rates depending on the given resourceR(habitat, forage, or other situations), temperature

θ(

^t

)

^and relative humidityH

(

^t

)

. To formulate the model we first introduce a physiological age variable denoteda

(

^t

)

which define the oldness of an insect at each stage (egg, larvae and pupae). This age is calculated by the following formulae

a

(

^t

) = Z

t

t0

v( ¯

^R

(τ), θ(τ),

^H

(τ),

^a

(τ))

^d

τ

⁽¹⁾

where

v ¯

is the velocity growth (rate of development). Time and age do not increase at the same rate. For constant environment there will be a fixed relation between physiological age and chronological age. Therefore we can also approach the problem from the angle of classical age-dependent theory.

We denote with the indexethe egg stage, with the indexlthe larval stage, with the indexpthe pupal stage, with the indexmthe adult male stage and by the indexf the adult female stage. For every stage

κ ∈ {

e

,

^l

,

^p

,

^m

,

^f

}

, the individual distribution depends on the age as well as on the timet. The rate of development is given by(1).

We define at timetand located ageathe densitiesu^e

(

^t

,

^a

)

^{of egg,ul}

(

^t

,

^a

)

^{of larval,up}

(

^t

,

^a

)

^{of pupae,uf}

(

^t

,

^a

)

^{of female} insect andu^m

(

^t

,

^a

)

of male insect.



 

 

 

 

 

 



 

 

 

 

 

 



∂

∂

^tue

+ ∂

∂

^a

( v ¯

^e

(

^R

, θ,

^H

,

^a

)

^ue

) = − ( µ ¯

^e

+ ¯ β

^e

)(

^R

, θ,

^H

,

^a

)

^ue

∂

∂

^tul

+ ∂

∂

^a

( v ¯

^l

(

^R

, θ,

^H

,

^a

)

^ul

) = − ( µ ¯

^l

+ ¯ β

^l

)(

^R

, θ,

^H

,

^a

)

^ul

∂

∂

^tup

+ ∂

∂

^a

( v ¯

^p

(

^R

, θ,

^H

,

^a

)

^up

) = − ( µ ¯

^p

+ ¯ β

^p

)(

^R

, θ,

^H

,

^a

)

^up

∂

∂

^tum

+ ∂

∂

^a

( v ¯

^m

(

^R

, θ,

^H

,

^a

)

^um

) = − ¯ µ

^m

(

^R

, θ,

^H

,

^a

)

^um

∂

∂

^tuf

+ ∂

∂

^a

( v ¯

^f

(

^R

, θ,

^H

,

^a

)

^uf

) = − ¯ µ

^f

(

^R

, θ,

^H

,

^a

)

^uf

(2)

associated to the given abiotic environment

θ(

^t

)

and the dynamic of the biotic resource with d

dtR

(

^t

) =

k

(

^R

) − X

κ f^κ

(

^R

)

Z

ω

^κ

(

^a

)

^uκ

(

^t

,

^a

)

^{da (3)}

with the nonlocal boundary conditions



 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

v ¯

^e

(

^R

, θ,

^H

,

⁰

)

^ue

(

^t

,

⁰

) = Z

a^f+

a^f−

β ¯

^f

(

^R

, θ,

^H

,

^s

)

^uf

(

^t

,

^s

)

^ds

v ¯

^l

(

^R

, θ,

^H

,

⁰

)

^ul

(

^t

,

⁰

) = Z

a^e+

a^e−

β ¯

^e

(

^R

, θ,

^H

,

^s

)

^ue

(

^t

,

^s

)

^ds

,

v ¯

^p

(

^R

, θ,

^H

,

⁰

)

^up

(

^t

,

⁰

) = Z

a^l+

a^l−

β ¯

^l

(

^R

, θ,

^H

,

^s

)

^ul

(

^t

,

^s

)

^ds

,

v ¯

^m

(

^R

, θ,

^H

,

⁰

)

^um

(

^t

,

⁰

) = Z

a^p+

a^p−

(

¹

− τ) β ¯

^p

(

^R

, θ,

^H

,

^s

)

^up

(

^t

,

^s

)

^ds

,

v ¯

^f

(

^R

, θ,

^H

,

⁰

)

^uf

(

^t

,

⁰

) = Z

a^p+

a^p−

τ β ¯

^p

(

^R

, θ,

^H

,

^s

)

^up

(

^t

,

^s

)

^ds

(4)

and given initial data

u^κ

(

⁰

,

^a

) =

u^κ₀

(

^a

), κ ∈ {

e

,

^l

,

^p

,

^m

,

^f

} .

For the sake of simplicity, it will be assumed that for all

κ ∈ {

e

,

^l

,

^p

,

^m

,

^f

}

,

v

^κ

(

^R

(

^t

),

^t

,

^a

) = ¯ v

^κ

(

^R

(

^t

), θ(

^t

),

^H

(

^t

),

^a

),

β

^κ

(

^R

(

^t

),

^t

,

^a

) = ¯ β

^κ

(

^R

(

^t

), θ(

^t

),

^H

(

^t

),

^a

), µ

^κ

(

^R

(

^t

),

^t

,

^a

) = ¯ µ

^κ

(

^R

(

^t

), θ(

^t

),

^H

(

^t

),

^a

).

The functions

β

^f

(

^R

(

^t

),

^t

, .) :

a

∈ [

a^f₋

,

^af+

] → β

^f

(

^R

(

^t

),

^t

,

^a

) ∈

R+is the age fertility function which is the age moth specific fertility which can be defined as the number of new egg, in one unit time from a single moth whose age is in the infinitesimal age interval

[

a

,

^a

+

da

] .

β

^e

(

^R

(

^t

),

^t

, .) :

a

∈ [

a^e₋

,

^ae+

] → β

^e

(

^R

(

^t

),

^t

,

^a

) ∈

R+is the age maturation function which is the age maturation at which an egg becomes a new larvae.

β

^l

(

^R

(

^t

),

^t

, .) :

a

∈ [

a^l₋

,

^al+

] → β

^l

(

^R

(

^t

),

^t

,

^a

) ∈

R+is the age maturation function which is the age maturation at which an larvae becomes a pupae.

β

^p

(

^R

(

^t

),

^t

, .) :

a

∈ [

a^p₋

,

^ap+

] → β

^p

(

^R

(

^t

),

^t

,

^a

) ∈

R+is the age maturation function which is the age maturation at which a pupae becomes an adult.

In each stage,

µ

^κ

(

^R

(

^t

),

^t

, .) :

a

∈ [

0

,

^a+

] → µ

^κ

(

^R

(

^t

),

^t

,

^a

) ∈

R+define the mortality function,

µ

^κ

(., .,

^a

)

is the per capita rate at which individual in stage

κ

^{of ageadie.}

The model belongs to the class of physiologically structured population and its aim is to know how individual variability influences the dynamics of the whole population. Ecologists have for a long time pointed the importance of physiological structure in populations, see [2,11–15,20]. This has led to a general understanding of how size-based interactions can shape the population and community dynamics.

The Eq.(3)is the simplest assumption we can make about the resource dynamic. The functionk

(

^R

)

describes the dynamics of the resource. A typical resource growth law is

k

(

^R

) = α(

^RM

−

R

),

^RM

>

⁰

, α >

^{0 (5)}

which corresponds to a constant replenishing of food particles and a constant food loss. Another is the classical logistic law.

k

(

^R

) = γ

1

−

^R K

R

, γ >

⁰

,

^K

>

^{0 (6)}

for a self-renewing resource.

We assume that the ingestion of resource (food) by an individual of physiological ageaat resourceR, equals

[Resource intake]

=

f^κ

(

^R

)ω

^κ

(

^a

).

⁽⁷⁾

The resource uptake rate functional f^κ

(

^R

)

is dependent on resource densityR. It is assumed that this rate is positive whenever the resource density is nonzero and Lipschitz with respect toR:

f^κ

(

⁰

) =

0

,

^fκ

(

^R

) >

⁰

, for

R

>

⁰

, |

f^κ

(

^R

) −

f^κ

(

_R

˜ ) | ≤

L

|

R

− ˜

R

| .

⁽⁸⁾ A typical such relationship is the Michaelis–Menten or Holling type II expression

f^κ

(

^R

) =

c^κ R

R

+

a^κ

,

^aκ

>

⁰

,

^cκ

>

⁰

.

⁽⁹⁾

It is further assumed that ingested food is allocated between metabolic maintenance, individual growth, and reproduction.

For further discussion of the biotic resourceR, we refer to [13].

1.1. Weak formulation

To summarize, we write the model



 

 

 



 

 

 



∂

∂

^tuκ

+ ∂

∂

^a

(v

^κ

(

^R

,

^t

,

^a

)

^uκ

) = −

m^κ

(

^R

,

^t

,

^a

)

^uκ d

dtR

(

^t

) =

k

(

^R

) − X

κ f^κ

(

^R

)

Z

ω

^κ

(

^a

)

^uκ

(

^t

,

^a

)

^da

v

^κ

(

^R

,

^t

,

⁰

)

^uκ

(

^t

,

⁰

) =

Z

a^q+ a^q−

α

^κq

β

^q

(

^R

,

^t

,

^s

)

^uq

(

^t

,

^s

)

^ds

(10)

where

α

^κq

=

(

₁

(κ,

^q

) ∈ { (

^e

,

^f

), (

^l

,

^e

), (

^p

,

^l

) } ,

1

− τ (κ,

^q

) = (

^m

,

^p

),

τ (κ,

^q

) = (

^f

,

^p

).

⁽¹¹⁾

By weak solution to the problem(10)we mean a bounded and measurable functionu

(

^t

,

^a

) = (

^ue

,

^ul

,

^up

,

^um

,

^uf

)

^satisfying R

(φ,

^uκ

) =

0 for all smooth function

φ ∈

C¹

( [

0

,

^T

[ ×[

0

,

^a+

] )

with compact support on

[

0

,

^T

[

, where

R

(φ,

^uκ

) = Z

T

0

Z

a+

0

u^κ

(φ

t

+ v

^κ

(

^R

(

^t

),

^t

,

^a

)φ

a

−

m^κ

(

^R

(

^t

),

^t

,

^a

)φ)

^dadt

+ Z

T

0

B^κ

(

^t

)φ(

^t

,

⁰

)

^dt

+ Z

a+

0

u^κ₀

(

^a

)φ(

⁰

,

^a

)

^{da (12)}

where we denote by B^κ

(

^t

) =

Z

a^q+ a^q−

α

^κq

β

^q

(

^R

(

^t

),

^t

,

^s

)

^uq

(

^t

,

^s

)

^{ds (13)}

the inflow of newborns in the

κ

^thsubpopulation at timet, with

(κ,

^q

) ∈ { (

^e

,

^f

), (

^l

,

^e

), (

^p

,

^l

), (

^m

,

^p

), (

^f

,

^p

) } .

The rate of loss process (due to mortality or mutation to next stage) by m^κ

=

µ

^κ

+ β

^κ

κ =

e

,

^l

,

^p

µ

^κ

κ =

m

,

^f

.

⁽¹⁴⁾

The following regularity conditions will be imposed on our model parameters throughout the paper

(

^H1

)

We assume that these functions,

β

^κ

(

^R

(.), ., .) ∈

_L^∞

( [

₀

,

^T

] × [

_a^κ₋

,

^aκ+

] ),

^mκ

(

^R

(.), ., .) ∈

_L^∞

( [

₀

,

^T

] × [

₀

,

^a+

] )

^are nonnegative Uniformly Lipschitz continuous functions with respect toR andt

.

^mκ

(

^R

(

^t

),

^t

, .)

have bounded total variation with respect toai.e.m^κ

(

^R

(

^t

),

^t

, .) ∈

BV

(

⁰

,

^a+

), κ ∈ {

e

,

^l

,

^p

,

^m

,

^f

} .

(

^H2

)

^uκ0

≥

0

,

^uκ0

∈

BV

(

⁰

,

^a+

)

^anduκ0

∈

L^∞

( [

0

,

^a+

] ) ∩

L¹

( [

0

,

^a+

] ).

(

^H3

) v

^κ

(

^R

(

^t

),

^t

, .) ≥

0 is ofC²

( [

0

,

^a

+] ), v

^κ

(

^R

(.), .,

^a

) ∈

L^∞

( [

0

,

^T

] ), v

^κ Uniformly Lipschitz continuous function with respect toRandt

. v

^κ

(

^R

(

^t

),

^t

,

⁰

) ≥ v

^where

v

being constant independent oft

∈ [

0

,

^T

] , κ ∈ {

e

,

^l

,

^p

,

^m

,

^f

} .

This paper is organized as follows. In Section2, we give an implicit numerical scheme for computing the solution of(10). In Section3, we prove stability, BV compactness and the convergence of the scheme. In Section5, we perform some numerical experiments to demonstrate the properties of the scheme.

2. Numerical approximation of the model

In this section, we construct approximate solutions to(10). To establish the scheme, we define a mesh of rectangles in the at-plane. Let choose any1^a

=

a+

/

^ILbe a mesh length in age, and1^t

=

r1^abe a mesh length in time. The mesh points are given by:a_i

=

i1^a

,

ⁱ

=

0

,

¹

, . . .

^IL. We approximate the solution across the bottom of each rectangle by adjacent constant states. Inside each rectangle, we define, the approximate solutionU¹

=

U¹

(

^tn

,

^ai

) =

U_i^1,n, approximateU

(

^tn

,

^ai

)

^{the exact} value of the exact solutionUof(10)at timet_nand located agea_i. We introduce the following operatorsS

(

1^t

)

^andB1such thatUⁿ⁺¹

=

U¹

(

^tn

+

₁t

) =

S

(

1^t

)

^Unwith the boundary data, at age zero,U¹

(

^tn

,

⁰

) =

B^1,n

(

^U1,n

)

. These operators are defined as follows:

To replace the continuous problems(10)by discrete approximation, we denote the values of the given functions at the mesh points by

Rⁿ

=

R

(

^tn

), ω

i^κ

= ω

^κ

(

^ai

),

^uκ,i ⁿ

=

u^κ_i

(

^tn

,

^ai

)

and

v

^κ,i ⁿ

= v

^κ

(

^R

(

^tn

),

^tn

,

^ai

), β

i^κ,n

= β

^κ

(

^R

(

^tn

),

^tn

,

^ai

),

^mκ,i ⁿ

=

m^κ

(

^R

(

^tn

),

^tn

,

^ai

).

To approximate and update the resources at each timet_n, we use the Runge–Kutta scheme



 

 

 

 



 

 

 

 



Rⁿ⁺¹

−

Rⁿ

1^t

=

k

(

^Rn

) − X

κ IL

X

i=1

f^κ

(

^Rn

)ω

^κiu^κ,_i ⁿ1^a u^κ,_i ⁿ⁺¹

−

u^κ,_i ⁿ

1^t

+ (v

^u

)

^κ,i ⁿ⁺¹

− (v

^u

)

^κ,i−ⁿ1⁺¹

1^a

=

m^κ,_i ⁿ⁺¹u^κ,_i ⁿ⁺¹

(v

^u

)

^κ,0ⁿ

=

i=a^q+

X

i=a^q−

α

^κq

β

i^q,nu^q_i^,n1^a

which on simplification gives



 

 

 

 

 

 

 

 

 

 

Rⁿ⁺¹

=

Rⁿ

+

₁tk

(

^Rn

) −

₁t

X

κ IL

X

i=1

f^κ

(

^Rn

)ω

^κiu^κ,_i ⁿ1^a

u^κ,_i ⁿ⁺¹

=

^u

κ,n

i

+

r

(v

^u

)

^κ,i−ⁿ1⁺¹

1

+

r

v

^κ,i ⁿ⁺¹

+

m^κ,_i ⁿ⁺¹1^t

(v

^u

)

^κ,0ⁿ

=

i=a^q+

X

i=a^q−

α

^κq

β

i^q,nu^q_i^,n1^a

(15)

and with the initial condition u^κ,_i ⁰

=

¹

1^a

Z

ai

ai−1

u^κ

(

⁰

,

^a

)

^da

.

If we defineU^1,n

=

U^e,1,n

,

^Ul,1,n

,

^Up,1,n

,

^Um,1,n

,

^Uf,1,n

T

, withU^κ,1,n

= (

^uκ,1ⁿ

, . . .

^uκ,ILⁿ

), κ ∈ {

e

,

^l

,

^p

,

^m

,

^f

} ,

^{then the} scheme can be equivalently written as the following system of linear equations

A^1,n+1U^1,n+1

=

U^1,n (16)

whereA^1,nis the following block diagonal matrix:

A^1,n

=

























D^e,n 0 B^f,n

B^e,n D^l,n 0 0

0 B^l,n D^p,n 0 0

0 0

(

¹

− τ)

^{Bp,n Dm,n 0} 0 0

τ

^{Bp,n 0 Df,n}

























(17)

with the lower triangular matrix

D^κ,n

=























d^κ,₁ⁿ 0 0 0 0

−

^1t

1^a

v

^κ,1ⁿ d^κ,₂ⁿ 0 0 0

0

−

^1t

1^a

v

^κ,2ⁿ d^κ,₃ⁿ 0 0

.. .. .. .. ..

0 0

.. −

^1t

1^a

v

I^κ,L−ⁿ1 d^κ,_I ⁿ

L























IL×IL

(18)

whered^κ,_i ⁿ

=

1

+

₁^1t

a

v

i^κ,n

+

₁tm^κ,_i ⁿ

,

ⁱ

=

1

,

²

. . .

^ILand

B^κ,n

=















0

.. −

^1t

1^a

β

_a^κ,kⁿ

−

.. −

^1t

1^a

β

_a^κ,kⁿ +

0

−− −− −− −− −− −−

0 0

..

^{0 0}

: .. .. .. ..

0

..

⁰















IL×IL

.

⁽¹⁹⁾

This linear system is solved using Gauss–Seidel iterations.

3. Convergence of approximations

3.1. Bounds for the approximate solution

In this section, we give some estimates satisfied by the scheme. We introduce the followingl^∞andl¹discrete norms of u¹by

|

u¹

|

_∞,1

=

sup

i

|

u_i

|

and

|

u¹

|

_1,1

= X

i

|

u_i

| .

⁽²⁰⁾

First one can easily show, from the linear system(16)the following lemma:

Lemma 1. The scheme is positive in the sense that for all i

=

1

, . . . ,

^IL

(

^uei^,n

,

^uli^,n

,

^upi^,n

,

^umi^,n

,

^ufi^,n

) ≥

0

H⇒ (

^uei^,n+1

,

^uli^,n+1

,

^upi^,n+1

,

^umi ^,n+1

,

^ufi^,n+1

) ≥

₀

.

⁽²¹⁾ Next, we will show that the difference approximation is bounded inl¹-norm.

Lemma 2. Let u^κ₀

∈

L¹

( [

0

,

^a+

] )

. Under the positiveness of the scheme, the discrete L¹-norm satisfies maxκ

|

u^κ,n+1

|

_1,1

≤

e^{Cmaxκ |βκ|∞T}max

κ

|

u^κ₀

|

_1,1 (22)

with C being constant independent of n

.

We then establish anl^∞bound on the difference approximation:

Lemma 3. The scheme defined by(15)L^∞stable i.e. the following estimate holds maxκ

|

u^κ,n+1

|

_∞,1

≤

e^C

0max κ |v^κ|_Lip,1T

maxκ

|

u^κ

(

⁰

) |

_∞,1 (23)

with C⁰being a constant independent of n

.

Now we will show that our approximationu^κ,1has bounded total variation. The total variation bound plays a crucial role in establishing the subsequential convergence of the approximate solution to weak solution of(10).

TV

(

^un

) =

IL−1

X

i=0

|

uⁿ_i+1

−

uⁿ_i

| .

⁽²⁴⁾

So we can show

Lemma 4. Let u^κ₀

∈ (

^L1

∩

L^∞

)( [

0

,

^a+

] )

with the total variation be bounded locally in

[

0

,

^a+

]

then we have

maxκ TV

(

^uκ,n+1

) ≤

C

(

^T

)

^max_κ

(

^TV

(

^uκ0

)) +

C₁

(

^T

)

⁽²⁵⁾

with C

(

^T

)

^{and C}1

(

^T

)

being constants independent of n.

The next result shows that the difference approximations satisfy the Lipschitz-type continuity in time Lemma 5. There exists a constant C^te

>

^{0such that}

I(L)

X

i=1

|

u^κ,_i ⁿ⁺¹

−

u^κ,_i ⁿ

|

₁a

≤

C^te1^{t (26)}

which implies the L¹Lipschitz continuity in time.

3.2. Global existence of weak solution in L¹_loc

In this section, we deal with the existence of global weak solution using Helly-compactness argument. We first prove the existence of a limitU¹toUas the meshsize1^agoes to zero.

Theorem 6.Let u^κ₀

∈ (

^L1

∩

L^∞

)( [

0

,

^a+

] )

with total variation bounded locally in

[

0

,

^a+

]

, u^κ₀

≥

0, for

κ ∈ {

e

,

^l

,

^p

,

^m

,

^f

}

, then as the meshsize1a tends to zero, there is a subsequence of

(

^uκ,1

)

1^a>0, the family of approximate solutions, converging in L¹_loc

( [

0

,

^T

] × [

0

,

^a+

] )

to a function u^κ

∈

L¹_loc

( [

0

,

^T

] × [

0

,

^a+

] )

^.

It remains to show that the scheme is consistent.

Theorem 7.Under the same assumptions as inTheorem6, the limiting function

(

^uκ

(

^t

,

^x

))

^for

κ ∈ {

e

,

^l

,

^p

,

^m

,

^f

}

, just obtained is a weak solution for the problem(10).

The following theorem guarantees the continuous dependence of the solution

(

^uκ

)

^of(10)with the respect of the initial data

(

^uκ0

(

^x

))

^.

Theorem 8.Let

(

^uκ

(

^t

,

^x

))

^and

(

_u

˜

^κ

(

^t

,

^x

))

^for

κ ∈ {

e

,

^l

,

^p

,

^m

,

^f

}

, be two bounded variation weak solutions for the problem(10) corresponding to initial condition

(

^uκ0

(

^x

))

^,

(

u

˜

^κ₀

(

^x

))

respectively, then there exist a constant C^t

= ( {

C_i

(

^T

) }

³_i=1

, { ˜

C_i

(

^T

) }

³_i=1

)

^such that



 

  Z

|

u^κ

− ˜

u^κ

|

da

≤

C₁

(

^T

) Z

|

u^κ₀

− ˜

u^κ₀

|

da

+

C₂

(

^T

) |

R₀

− ˜

R₀

| +

C₃

(

^T

)

|

R

− ˜

R

|

_∞

≤ ˜

C₁

(

^T

) Z

|

u^κ₀

− ˜

u^κ₀

|

da

+ ˜

C₂

(

^T

) |

R₀

− ˜

R₀

| + ˜

C₃

(

^T

).

⁽²⁷⁾

4. Proofs

4.1. Proof ofLemma 2

Proof. Let us rewrite the scheme(15)under the following form

u^κ,_i ⁿ⁺¹

=

u^κ,_i ⁿ

−

r

(v

i^κ,n+1u^κ,_i ⁿ⁺¹

− v

i^κ,−ⁿ1⁺¹u^κ,_i−ⁿ₁⁺¹

) −

₁tm^κ,_i ⁿ⁺¹u^κ,_i ⁿ⁺¹

.

⁽²⁸⁾ Summing over the indices,i

=

1

,

²

, . . .

^IL, we obtain easily the following estimate

IL

X

i=1

u^κ,_i ⁿ⁺¹

=

IL

X

i=1

u^κ,_i ⁿ

+

r

v

0^κ,n+1u^κ,₀ⁿ⁺¹

−

₁t

IL

X

i=1

m^κ,_i ⁿ⁺¹u^κ,_i ⁿ⁺¹

≤

IL

X

i=1

u^κ,_i ⁿ

+

₁t

| β

^q

|

_∞,1

X

i

u^q_i^,n+1

.

⁽²⁹⁾

LetFⁿ

=

max_κ

|

u^κ,n

|

_1,1and multiplying the above inequality by1^aand using the discrete Gronwall lemma we obtain Fⁿ⁺¹

≤

Fⁿ

+

₁tmax

κ

| β

^κ

|

_∞,1Fⁿ⁺¹

≤

^F

n

1

−

₁tmax κ

| β

^κ

|

_∞,1

≤

e^{Cmaxκ |βκ|∞,1T}F⁰ (30)

which ends the proof.

4.2. Proof ofLemma 3

Proof. Let us defineMⁿ

=

max_κ

|

u^κ,n

|

_∞and let

κ

0

∈ {

e

,

^l

,

^p

,

^m

,

^f

}

such thatu^k_i^0,n+1

0

=

max_κ,i

(

^uκ,i ⁿ⁺¹

)

, then from the scheme (28)fori

=

i₀and

κ = κ

0,

(

¹

+

r

v

i^k0^0,n+1

)

^uki0^0,n+1

=

u^k_i^0,n

0

+

r

v

i^k0^0,n+1u^k_i^0,n+1

0−1

+

r

(v

i^k0⁰−^,n1⁺¹

− v

^ki0^0,n+1

)

^uki0⁰−^,n1⁺¹

−

₁tm^k_i^0,n+1

0 u^k_i^0,n+1

0 (31)

sincem^k_i^0,n+1

0 1^tis positive then we have

n (

¹

+

r

v

^ki0^0,n+1

)

^uki0^0,n+1

≤

u^k_i^0,n

0

+

r

v

i^k0^0,n+1u^k_i^0,n+1

0−1

+

r

| v

^κ0,n+1

|

_Lip1au^k_i^0,n+1

0−1 (32)

from using the discrete Gronwall lemma, we deduce

Mⁿ⁺¹

≤

^M

n

1

−

max

κ

| v

^κ,n+1

|

_Lip,11t

≤

e^C

0(n+1)1tmaxκ |v^κ,n+1|_Lip M⁰

≤

e^C

0maxκ |v^κ,n+1|_LipT

M⁰ which implies the estimate.

4.3. Proof ofLemma 4

Proof. Using the expression of the scheme(28)we have u^κ,_i+ⁿ₁⁺¹

−

u^κ,_i ⁿ⁺¹

=

u^κ,_i+ⁿ₁

−

u^κ,_i ⁿ

−

r

h v

^κ,i+ⁿ1⁺¹u^κ,_i+ⁿ₁⁺¹

− v

^κ,i ⁿ⁺¹u^κ,_i ⁿ⁺¹

i

+

r

h v

i^κ,n+1u^κ,_i ⁿ⁺¹

− v

^κ,i−ⁿ1⁺¹u^κ,_i−ⁿ₁⁺¹

i

−

₁t

m^κ,_i+ⁿ₁⁺¹u^κ,_i+ⁿ₁⁺¹

−

m^κ,_i ⁿ⁺¹u^κ,_i ⁿ⁺¹

.

⁽³³⁾