# On a Predator-Prey Model Involving Age and Spatial Structure

Authors: Okana S. Sougué; Oumar Traoré
Journal: International Journal of Mathematics and Mathematical Sciences
 (2022)

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## Abstract

In this paper, we study the mathematical analysis of a nonlinear age-dependent predator–prey system with diffusion in a bounded domain with a non-standard functional response. Using the fixed point theorem, we first show a global existence result for the problem with spatial variable. Other results of existence concerning the spatial homogeneous problem and the stationary system are discussed. At last, numerical simulations are performed by using finite difference method to validate the results.


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## Body

## 1. Introduction and Assumptions

The study of predator-prey systems has attracted the attention of many mathematicians in the last century. Pioneers like Voltera and Lotka were the first to mathematically model the interaction between predators and preys. The standard model of Lotka and Voltera is:(1)x′t=rxt−pxtyt,y′t=ηpxtyt−σyt,t>0,x0>0,y0>0,where x and y denote preys and predators density respectively; r, η and σ are positive constants stand respectively for the prey intrinsic growth rate, the coefficient for the conversion that predator intake to per capital prey and the predator mortality rate. And, px is the functional response which can take many forms [1].In order to be closer to biological reality, the Lotka-Volterra model (1) has been improved. Models taking into account both mortality and fertility of species, the carrying capacity of prey of the environment, environmental configuration, migratory movements (diffusion coefficients) have emerged. In [2], B. Ainseba, F. Heiser, and M. Langlais establish the existence of a solution of a predator-prey system in a highly heterogeneous environment but without the age variable and with a holling II type functional response. Note that in their model, prey dynamics is governed by logistic growth. In [3], the authors discuss a diffuse predator-prey system with a mutually interfering predator and a nonlinear harvest in the predator with a Crowley-Martin functional response. They analyze the existence and uniqueness of the solution of the system using the C0 semi-group. They show that the upper bound on the predator harvest rate for species coexistence can be guaranteed. Furthermore, they establish the existence and non-existence of a positive non-constant steady state. They give explicit conditions on predator harvesting for local and global stability of the interior equilibrium, as well as for the existence and non-existence of a non-constant steady state solution. In [4] the authors propose a diffusive prey-predator system with mutual interference between the predator (Crowley-Martin functional response) and the prey pool. In particular, they develop and analyze both a spatially homogeneous model based on ordinary differential equations and a reaction-diffusion model. The authors mainly study the global existence and limit of the positive solution, the stability properties of the homogeneous steady state, the non-existence of the non-constant positive steady state, the Turing instability and the Hopf bifurcation conditions of the diffusive system analytically. The classical stability properties of the non-spatial counterpart of the system are also studied. The analysis ensures that the prey pool leaves a stabilizing effect on the stability of the time system. A model of predator-prey interaction with Beddington-DeAngelis functional response and incorporating the cost of fear in prey reproduction is proposed and analyzed in [5]. The authors study the stability and existence of transcritical bifurcations. For the spatial system, the Hopf bifurcation around the inner equilibrium, the stability of the homogeneous steady state, the direction and stability of spatially homogeneous periodic orbits have been established. Using the normal form of the steady state bifurcation, they establish the possibility of a pitchfork bifurcation. A Leslie-Gower type prey-predator system with feedback is constructed in [6]. The authors systematically analyze the effects of feedback controls on ecosystem dynamics. In this study, they examine the global dynamics of non-autonomous and autonomous systems based on the Leslie-Gower type model using the Beddington-DeAngelis functional response with time independent and time dependent model parameters. The global stability of the unique positive equilibrium solution of the autonomous model is determined by defining an appropriate Lyapunov function. The autonomous system exhibits complex dynamics via bifurcation scenarios, such as the period doubling bifurcation. They then prove the existence of a globally stable quasiperiodic solution of the associated non-autonomous model. For the mathematical and qualitative study of some prey-predator models, the reader can also consult the following articles [7–14].In this article, we will mainly study the existence of solutions of a predator-prey system structured in age, time and space with a functional responseF subject to the K-lipschitz condition.Our motivation arose from the fact that, there is no existence result concerning predator-prey systems simultaneous structuring in age, time and space with a non-standard functional response. But there is works on population dynamics systems that take these three variables into account according to our best knowledge. For example in [15], an existence result is proved by Ainseba where the system models the transmission of an epidemic to holy individuals by carrier individuals. A result of existence and uniqueness and positivity of solution is also proved in [16] by Traoré et al. where their system models the population dynamics of Callosobruchus Maculatus. Traoré et al. have proved an existence result in [17] where the system models a nonlinear age and two-sex population dynamics.Let us denote bypa,t,x and qa,t,x respectively the distribution of preys and predators of age a being at time t>0 and location x over a bounded domain Ω.We consider in this paper the following nonlinear age-dependent population dynamics diffusive system:(2)∂tp+∂ap−Δp+μ1ap=f1−∫0A2Fpa,t,x,qα,t,xdαinQ1,∂tq+∂aq−Δq+μ2aq=f2inQ2,p0,t,x=∫0A1β1apa,t,xdainQT,q0,t,x=∫0A2β2aqa,t,xda+∫0A1∫0A2bx,a,αFpa,t,x,qα,t,xdαdainQT,pa,0,x=p0a,xinQA1,qa,0,x=q0a,xinQA2,∂ηp=0onΣ1,∂ηq=0onΣ2,where T is a positive number, and Ω a bounded open subset of ℝn which boundary is assumed to be of class C2.Actually,Qi=0,Ai×0,T×Ω, QAi=0,Ai×Ω, Σi=0,Ai×0,T×∂Ω for i=1,2 and QT=0,T×Ω.We denote byA1 (resp. A2) the maximum life expectancy of prey (resp. predators).In (2), β1a and β2a are respectively the natural birth-rate at age a of preys and predators, μ1 and μ2 are the functions describing the mortality rate respectively of preys and predators that depends on a, η is the external normal derivative on ∂Ω.In this model, predators and prey live on the same domain and any movement accross the boundaries is impossible.Our model is much more general because it simultaneously involves the notion of time, age and space. Moreover, the dynamics of prey and predators are governed by partial differential equations and not by the usual exponential or logistic growth laws. It is also a realistic model because in this model the prey is not the only source of food for the predators. Since in nature, it is almost impossible to find predators that feed exclusively on a single prey. Here, prey is not the only food source for predators. External food sources are also available. Not all of the prey that is consumed by predators is converted into predator energy (biomass), only a fraction is used.Consumption of prey directly affects prey density (direct decrease in prey numbers) but indirectly affects predator density through an increase in predator fertility (predator numbers do not increase immediately after consumption but over time predator density increases).We have denoted byFpa,t,x,qα,t,x the functional response to predation, that is the capture rate of prey having age a per predator of age α or the average number of prey having age a captured by predators of age α at times t>0 , and location x∈Ω.Thus,∫0A2Fpa,t,x,qα,t,xdα is the amount of prey of age a consumed by predator at time t>0 and location x∈Ω.The functionbx,a,α is the conversion rate of the biomass of captured prey having age a by predators of age α into predator offspring at location x∈Ω.Thus, the biomass is transformed and influences the birth process through the quantity(3)∫0A1∫0A2bx,a,αFpa,t,x,qα,t,xdαda,which is the supply.The functionf1a,t,x (resp. f2a,t,x) is the external supply for prey persistence (resp. for predator persistence) having age a at time t>0 and location x∈Ω.Our main goal in this paper is to answer some ecological questions:Is the cohabitation of predators and prey modeled by the model (2) possible?The answer to this question will bring us back to the notion of a well posedness problem or to the notion of the existence of a solution in suitably chosen spaces.Does the biomassb influence the size of the two populations?Will the predators succeed in consuming all the prey? Can predators or prey disappear into the environment?To answer these last questions, we will use numerical simulations by varying the values ofb, that is to say we will take small and large values of b to observe the behavior of the two populations.Our work will be structured as follows:In Section2, we give a global existence result of solution of system (2) with the space variable in appropriate spaces. We will also study the existence of solutions of the spatially homogeneous problem in Section 3. The Section 4 is devoted to the analysis of the stationary problem. Results of numerical simulations are given in Section 5 and we will end in Section 6 with a conclusion and some perspectives.Before starting, let(4)πia=exp−∫0aμiσdσ,a∈0;Ai,i∈1,2.Which is the probability for a newborn to survive to agea and(5)Ri=∫0Aiβiaπiada,i∈1,2.And assume that the following hypotheses hold:A1∀i∈1,2,βi∈L∞0,Ai,βia≥0a.ea∈0,AiA2∀i∈1,2,μi∈Lloc10,Ai,μia≥0a.ea∈0,AiA3∫0Aiμiada=+∞,i=1,2.A4F is a positive and mesurable function on 0;∞2 and satisfies the usual locally boundedness and Lipschitz continuity conditions with respect to the pair variable, that is(6)∃K>0∀p1,q1,p2,q2∈0;∞2,Fp1,q1−Fp2,q2≤Kp1−p2+q1−q2.AndF0,0=0.For the biological meanings of assumptionsA1, A2 and A3, the functions μi, βi, πi and the constants Ri, we refer the reader to books such [18, 19].


## 2. Spatially Heterogeneous Solutions

Let us make the following assumptions:H1p0,q0∈L∞QA1×L∞QA2, p0≥0a.einQA1 and. q0≥0a.einQA2H2f1,f2∈L∞Q1×L∞Q2, f1≥0a.einQ1 and. f2≥0a.einQ2We have the following result:Theorem 1.
Under the hypothesiesA1—A4, H1, H2 the system (2) admits a unique solution in L2Q1×L2Q2. Moreover, there exist a constant C depending on T such that(7)pL2Q12,pL2Q22≤Cp0L2QA12+q0L2QA22+f1L2Q12+f2L2Q22.Proof.
Let us fixT>0. The proof of the theorem is based on the method used in [20].
Setu=e−λ0tpinQ1 and v=e−λ0tqinQ2 where λ0 is a positive parameter that we will be fixed later.
Hence the system (2) admits a unique solution if and only the following system admits a unique solution.(8)∂tu+∂au−Δu+λ0+μ1au=e−λ0tf1−e−λ0t∫0A2Feλ0tua,t,x,eλ0tvα,t,xdαinQ1,∂tv+∂av−Δv+λ0+μ2av=e−λ0tf2inQ2,u0,t,x=∫0A1β1aut,a,xdainQT,v0,t,x=∫0A2β2ava,t,xda+e−λ0t∫0A1∫0A2bx,a,αFeλ0tua,t,x,eλ0tvα,t,xdαdainQT,ua,0,x=p0a,xinQA1,va,0,x=q0a,xinQA2,∂ηu=0onΣ1,∂ηv=0onΣ2.
For any nonnegativeh∈L2Q1, we introduce the following cascade system:(9)∂tu+∂au−Δu+λ0+μ1au=e−λ0tf1−e−λ0t∫0A2Feλ0tua,t,x,eλ0tvα,t,xdαinQ1,∂tv+∂av−Δv+λ0+μ2av=e−λ0tf2inQ2,u0,t,x=∫0A1β1aut,a,xdainQT,v0,t,x=∫0A2β2ava,t,xda+e−λ0t∫0A1∫0A2bx,a,αFeλ0tua,t,x,eλ0tvα,t,xdαdainQT,ua,0,x=p0a,xinQA1,va,0,x=q0a,xinQA2,∂ηu=0onΣ1,∂ηv=0onΣ2.
Using the Fubini’s theorem, the functionv solves the following system:(10)∂tv+∂av−Δv+μ2^av=e−λ0tf2inQ2,v0,t,x=∫0A2β2^a,t,x,va,t,xva,t,xdainQT,va,0,x=q0a,xinQA2,∂ηv=0onΣ2.
Withμ2^a=λ0+μ2a and β2^a,t,x,va,t,x=β2a+e−λ0t∫0A1bx,α,aFeλ0thα,t,x,eλ0tva,t,xdα.
Asλ0 is fixed, e−λ0tf2∈L∞Q2, b∈L∞A1×A2×Ω, F is bounded then μ2^∈Lloc10,A2 and β2^ is bounded.
Hence, (10) admits a unique nonnegative solution vh in L2Q2 (see [20]).
Multiplying (10) by vh and integrating over Q2, we get(11)λ0∫Q2vha,t,x2dadtdx≤12q0L2QA22+12f2L2Q22+12∫Q2vh2a,t,xdadtdx+12∫Q2vh20,t,xdtdx.
Using the Young and Cauchy Schwarz’s inequalities, we obtain(12)λ0∫Q2vha,t,x2dadtdx≤12q0L2QA22+12f2L2Q22+12∫Q2vh2a,t,xdadtdx+β2∞2A2∫Q2vha,t,x2dadtdx+A1A22K2b∞2∫Q1h2a,t,xdadtdx+A12A2K2b∞2∫Q1vh2α,t,xdαdtdx.
So, we have(13)λ0−β2∞2A2−A12A2K2b∞2−12∫Q2vh2a,t,xdadtdx≤12q0L2QA22+12f2L2Q22+A1A22K2b∞2∫Q1h2a,t,xdadtdx.
That is to say(14)vhL2Q22≤C1q0L2QA22+f2L2Q22+hL2Q12,where(15)C1=1λ0−β2∞2A2−A12A2K2b∞2−1/2×max12,A1A22K2b∞2.
For anyh1,h2∈L+2Q1, set w=e−λ0tvh1−vh2a.einQ2. So, w solves the following system(16)∂tw+∂aw−Δw+λ0+μ2aw=0inQ2,w0,t,x=∫0A2β2awa,t,xda+∫0A1∫0A2e−λ0tbx,a,α×Feλ0th1a,t,x,eλ0tvh1α,t,xdαda−∫0A1∫0A2e−λ0tbx,a,αFeλ0th2a,t,x,eλ0tvh2α,t,xdαdainQT,wa,0,x=0inQA2,∂ηw=0onΣ2.
Multiplying (16) by w and integrating over 0,A×0,t×Ω, and following the same calculations as before, one has(17)λ0−β2∞2A2−A2A12K2b∞2∫0twa,s,xL2Q22ds≤A1A22K2b∞2∫0th1−h2a,s,xL2Q1ds.
Therefore, we have(18)∫0tvh1−vh1a,s,xL20,A2×Ω2ds≤C2∫0th1−h2a,s,xL20,A2×Ω2ds,where(19)C2=1λ0−β2∞2A2−A2A12K2b∞2×A1A22K2b∞2.
Now, ash and vh are known, the remainder of (9) can be rewritten as following(20)∂tu+∂au−Δu+μ1^au=ga,t,xinQ1,u0,t,x=∫0A1β1aut,a,xdainQT,ua,0,x=p0a,xinQA1,∂ηu=0onΣ1,where μ1^a=λ0+μ1a and ga,t,x=e−λ0tf1−e−λ0t∫0A2Feλ0tua,t,x,eλ0tvα,t,xdα.
Asλ0 is fixed, μ1^∈Lloc10,A1 and g∈L∞Q1 because f1∈L∞Q1 and F is bounded.
According to the results in [20], (20) has a unique nonnegative solution u˜h in L2Q1.
Multiplying (20) by u˜h, integrating over Q1 and using Young’s inequality, we get(21)λ0∫Q1u˜h2a,t,xdadtdx≤12p0L2QA12+12f1L2Q12+∫Q1u˜h2a,t,xdadtdx+12∫0T∫Q1u˜h20,t,xdadtdx+12∫Q2∫0A2Feλ0tha,t,x,eλ0tvha,t,xdα2dαdtdx.
Using now Cauchy Schwarz’s inequality,λ0∫Q1u˜h2a,t,xdadtdx≤1/2p0L2QA12+1/2f1L2Q12+∫Q1u˜h2a,t,xdadtdx+1/2∫0T∫Ωu˜h20,t,xdadtdx+1/2e−2λ0t∫Q1∫0A2Feλ0tha,t,x,eλ0tvhα,t,xdα2dadtdx. we obtain(22)λ0−12A1β1∞2−1∫Q1u˜h2a,t,xdadtdx≤12p0L2QA12+12f1L2Q12+12A22K2b∞2∫Q1h2a,t,xdadtdx+12A1A2K2b∞2∫Q2vh2α,t,xdαdtdx.
Then, one gets(23)∫Q1u˜h2a,t,xdadtdx≤C3p0L2QA12+q0L2QA22+f1L2Q12+f2L2Q22+hL2Q12,where(24)C3=1λ0−1/2A1β1∞2−1×max12,12A22K2b∞2+12A1A2K2b∞2C1.
Denote byΛ:L+2Q1⟶L+2Q1, the application given by(25)Λh=u˜h,where u˜h is the unique solution of (20).
For anyh1,h2∈L+2Q1, we denote by V=e−γ0tu˜h1−u˜h2inQ1, V satisfies the following equations(26)∂tV+∂aV−ΔV+γ0+μ1aV=−e−γ0t∫0A2Feλ0th1a,t,x,eλ0tvh1α,t,xdα+e−γ0t∫0A2Feλ0th2a,t,x,eλ0tvh2α,t,xdαinQ1,V0,t,x=∫0A1β1aVt,a,xdainQT,Va,0,x=0inQA1,∂ηV=0onΣ1,where vhi, i=1,2 is solution of(27)∂tvhi+∂avhi−Δvhi+λ0+μ2avhi=e−λ0tf2inQ2,vhi0,t,x=∫0A2β2avhia,t,xda+e−λ0t∫0A1∫0A2bx,a,αFeλ0thia,t,x,eλ0tvhiα,t,xdαdainQT,vhia,0,x=q0a,xinQA2,∂ηvhi=0onΣ2.
Multiplying (26) by V and integrating over 0,A1×0,t×Ω, on gets(28)λ0∫0tVa,s,xL2QA12dadtdx≤12β1∞2A1∫0t∫0A1∫ΩVa,s,x2dtdx+12∫0t∫0A1∫ΩV2a,s,xdadsdx+∫0t∫0A1∫ΩA22b∞2K2h1−h2a,s,x2+A2b∞2K2∫0A2vh1−vh2α,s,x2dαdadsdx.
And we also have(29)λ0−12A1β1∞2+12∫0tVa,s,xL2QA12≤A22b∞2K2∫0th1−h2a,s,xL2QA12+A2A1b∞2K2∫0tvh1−vh2α,s,xL2Q22ds.
By recalling the inequality (18) in (29), we deduce that, there exists a constant C4, such that(30)∫0tu˜h1−u˜h2a,s,xL2QA12≤C4∫0th1−h2a,s,xL2QA12,where(31)C4=1λ0−1/2A1β1∞2−1/2×A22b∞2K2+A2A1b∞2K2C2.
Let us define onL+2Q the metric d by: for any h1,h2∈L+2Q1,(32)dh1,h2=∫0Th1−h2tL2QA12exp−2C4tdt1/2.
So,(33)dΛh1,Λh22=∫0Tu˜h1−u˜h2tL2QA12exp−2C4tdt≤C4∫0Texp−2C4t∫0th1−h2sL2QA12dsdt.
Using the Fubini’s theorem, we conclude that:(34)dΛh1,Λh22≤C4∫0Texp−2C4t∫0th1−h2sL2QA12dsdt=∫0T∫sTC4exp−2C4th1−h2sL2QA12dtds=∫0Th1−h2sL2QA12∫sTC4exp−2C4tds.
From here we have:(35)dΛh1,Λh2≤12dh1,h2.
Then,Λ is a contraction on the complete metric space L+2Q1 and using Banach’s fixed point theorem, we conclude the existence of a unique fixed point uh nonnegative for Λ, so the unique couple uh,vh is the unique solution of (9). Hence, we deduce that the couple p,q=eλ0tuh,eλ0tvhis the unique solution to the problem (2). From the explicit expression of the constant C3, we see that it is always possible to choose λ0 so that C3<1. Replace h by uh in (14) and in (23), summing the inequalities (14) and (17), we conclude that there exists a constant C independent on T such that(36)uhL2Q12,vhL2Q22≤Cp0L2QA12+q0L2QA22+f1L2Q12+f2L2Q22.
And the inequalities (7) follow clearly.


## 3. Spatially Homogeneous Solutions

Let consider the following spatial homogeneous system deduced from (2):(37)∂tp+∂ap+μ1ap=−∫0A2Fpa,t,qα,tdαinQ,∂tq+∂aq+μ2aq=finQ,p0,t=∫0A1β1apt,adain0,T,q0,t=∫0A2β2aqa,tda+∫0A1∫0A2ba,αFpa,t,qα,tdαdain0,T,pa,0=p0ain0,A,qa,0=q0ain0,A.Theorem 2.
Under the assumptionsA1-A4, for all p0,q0∈L10,A2 and f∈C0,T,L10,A1, the system (37) admits a unique solution in C0,T,L10,A1×C0,T,L10,A2.Proof.
To simplify the calculations and without losing sight of the generality, we setA=maxA1,A2. Let set Y=C0,T,L10,A endowed with the norm hY=supt∈0,Te−λthL10,Awhere λ is a positive constant that will be choosed later.
We fixp¯ in Y. Consider now the following system(38)∂tp+∂ap+μ1ap=−∫0A2Fp¯a,t,qα,tdαinQ,∂tq+∂aq+μ2aq=finQ,p0,t=∫0A1β1apt,adain0,T,q0,t=∫0A2β2aqa,tda+∫0A1∫0A2ba,αFp¯a,t,qα,tdαdain0,T,pa,0=p0ain0,A,qa,0=q0ain0,A.
Integrating the system (38) along the characteristic curves a−t=c, we obtain implicit formulas for the solutions of (2) stated below:(39)pa,t=∏1a∏1a−tp0a−t−∫0t∏1a∏1s+a−t∫0AFp¯s+a−t,s,qα,sdαds,a≥t,pa,t=∏1a∫0Aβ1sps,t−ads−∫0a∏1a∏1s∫0AFp¯s,s+t−a,qα,s+t−adαds,a<t,(40)qa,t=∏2a∏2a−tq0a−t+∫0t∏1a∏1s+a−tfs+a−t,sds,a≥t,qa,t=∏2a∫0Aβ2sqs,t−ads+∫0A∫0Abs,αFp¯s,t−a,qα,t−adαds+∫0a∏1a∏1sfs,t−a+sds,a<t.
Let us fixedq¯∈Y and define the mapping G:Y⟶Y by: for every q¯∈Y and for all t,a∈Q,(41)Gq¯a,t=qa,t=∏2a∏2a−tq0a−t+∫0t∏1a∏1s+a−tfs+a−t,sds,a≥t∏2a∫0Aβ2sqs,t−ads+∫0A∫0Abs,αFp¯s,t−a,q¯α,t−adαds+∫0a∏1a∏1sfs,t−a+sds,a<t.
For alla,t∈Q such that a≥t,(42)e−λt∫tAGq¯a,tda≤q0L10,A+1λfY,
And fora<t,(43)e−λt∫0tGq¯.,tda≤β2∞e−λt∫0teλl.e−λlGq¯.,lL10,Adl+AKb∞p¯Y+AKb∞q¯Y1λ.
Summing of the previous inequalities, we get(44)e−λtGq¯.,tL10,A≤β2∞e−λt∫0teλl.e−λlGq¯.,lL10,Adl+AKb∞p¯Y+AKb∞q¯Y1λ+q0L10,A+1λfY.
So, we have(45)Gq¯Y≤λq0L10,A+AKb∞p¯Y+AKb∞q¯Y+fYλ−β2∞<∞,
Forλ large enough, thus Gq¯∈Y.
Now, for allq1¯,q2¯∈Y, for all a,t∈Q such that a<t; one has(46)∫0tGq1¯−Gq2¯a,tda≤β2∞∫0t∫0Aq1s,t−a−q2s,t−adsda+b∞∫0t∫0A∫0AFp¯s,t−a,q1¯α,t−a−Fp¯s,t−a,q2¯α,t−adαdsda≤β2∞∫0teλle−λl∫0Aq1s,l−q2s,ldsdl+AKb∞∫0teλle−λl∫0Aq1¯α,l−q2¯α,ldαdl.
Finally, it follows that(47)∫0tGq1¯−Gq2¯a,tda≤β2∞q1−q2Y+Kb∞Aq1¯−q2¯Y×1−e−λtλ.
Multiplying the inequality (47) by e−λt, we get(48)e−λt∫0tGq1¯−Gq2¯a,tda≤1λβ2∞q1−q2Y+Kb∞Aq1¯−q2¯Y.
It is obvious that for alla,t∈Q such that a≥t, we have(49)e−λt∫tAGq1¯−Gq2¯a,tda≤1λβ2∞q1−q2Y+Kb∞Aq1¯−q2¯Y.
Combining the inequalities (46) and (47) it follows that(50)Gq1¯−Gq2¯Y≤2AKb∞λ−2β2∞q1¯−q2¯Y.
Forλ large enough, it is clear that G is a contraction in Y and the (40) have a unique solution q in Y.
Now, define the mappingΛ:Y⟶Y by: for every p¯∈Y and for all t,a∈Q,(51)Λp¯a,t=pa,t=∏1a∏1a−tp0a−t−∫0t∏1a∏1s+a−t∫0AFp¯s+a−t,s,qa′,sda′ds,a≥t,∏1a∫0Aβ1sps,t−ads−∫0a∏1a∏1s∫0AFp¯s,s+t−a,qa′,s+t−ada′ds,a<t.
For everya,t∈Q such that a≥t, one has(52)e−λt∫tAΛp¯a,tda≤p0L10,A+KAλp¯Y+qY,
And fora<t, we have(53)e−λt∫0tΛp¯a,tda≤β1∞e−λt∫0teλle−λlΛp¯.,lL10,Adl+AKλp¯Y+qY.
By adding the inequalities (52) and (53) and using (45), one gets(54)Λp¯Y≤λp0L10,A+2AKp¯Y+qYλ−β1∞≤λp0L10,A+2AKp¯Yλ−β1∞+2KAλq0L10,A+AKb∞p¯Y+fYλ−β1∞λ−β2∞+AKb∞<∞.
Therefore,Λp¯∈Y for p¯ fixed in Y.
For allp1¯,p2¯∈Y, for all a,t∈Q such that a≥t; we have(55)∫tAΛp1¯−Λp2¯a,tda≤∫tA∫0t∫0Ap1¯s+a−t,s,q1α,s−Fp2¯s+a−t,s,q2α,sdαdsda≤KA∫0t∫sA+s−tp1¯l,s−p2¯l,sdlds+K∫0t∫tA∫0Aq1α,s−q2α,sdαdads≤KA∫0teλse−λs∫0Ap1¯l,s−p2¯l,sdlds+KA∫0teλse−λs∫0Aq1α,s−q2α,sdαds.
It is clear that(56)∫tAΛp1¯−Λp2¯a,tda≤KAp1¯−p2¯Y+q1−q2Y1−e−λtλ.
Multiplying the inequality (56) by e−λt, we get(57)e−λt∫tAΛp1¯−Λp2¯a,tda≤KAλp1¯−p2¯Y+q1−q2Y.
And fora<t, we have(58)∫0tΛp1¯−Λp2¯a,tda≤β1∞∫0t∫0Ap1s,l−p2s,ldsdl+AK∫0t∫0ap1¯s,s+t−a−p2¯s,s+t−adsda+K∫0t∫0a∫0Aq1α,s+t−a−q2α,s+t−adαdsda.
In other hand, one has(59)∫0tΛp1¯−Λp2¯a,tda≤β1∞p1−p2Y+AKp1¯−p2¯Y+AKq1−q2Y1−e−λtλ.
Multiplying the inequality (59) by e−λt, we get(60)e−λt∫0tΛp1¯−Λp2¯a,tda≤1λβ1∞p1−p2Y+AKp1¯−p2¯Y+AKq1−q2Y.
Combining the inequalities (57) and (59), we deduce that(61)Λp2¯−Λp2¯Y≤2KAλ−β1∞p1¯−p2¯Y+q1−q2Y,where q1 and q2 are solutions of (39) and (40) associated respectively to p1¯ and p2¯.
Then, we have for alla,t∈Q such that a<t,(62)∫0tq1−q2a,tda≤β2∞∫0t∫0Aq1−q2s,ldsdl+KAb∞∫0t∫0Ap1¯−p2¯s,ldsdl+KAb∞∫0t∫0Aq1−q2α,ldαdl≤β2∞q1−q2Y+Kb∞Ap1¯−p2¯Y+Kb∞Aq1−q2Y1−e−λtλ.
So, it follows that(63)e−λt∫0tq1−q2a,tda≤1λβ2∞q1−q2Y+Kb∞Ap1¯−p2¯Y+Kb∞Aq1−q2Y.
We deduce from (63) that(64)q1−q2Y≤2λβ2∞q1−q2Y+Kb∞Ap1¯−p2¯Y+Kb∞Aq1−q2Y.
From (64), we obtain(65)q1−q2Y≤2AKb∞λ−2β2∞+AKb∞p1¯−p2¯Y.
By combining the inequalities (61) and (65), we get(66)Λp2¯−Λp2¯Y≤2KAλ−2β2∞λ−β1∞λ−2β2∞+AKb∞p1¯−p2¯Y.
Forλ large enough such that(67)λ>12β1∞+2β2∞+2AKb∞+2AK+12β1∞−2β2∞−2AKb∞+2AK2+16A2K2b∞1/2,where K is a Lipschitz constant, one gets clearly that Λ is a contraction in Y. Therefore, the (37) and (38) have a unique solution p in Y. The couple p,q is the unique solution to the system (35) because the problem (35) is equivalent to solve the equations (39)–(42). (see [21])
We have also the following result:Theorem 3.
Under the assumptionsA1—A4, for all p0,q0∈L20,A1×L20,A2 and f∈L20,A2×0,T, the system (37) admits a unique solution in L20,A1×0,T×L20,A2×0,T. Moreover, there exists a positive constant C=Cβ1∞,β2∞,b∞,A1,A2,K,T, such that(68)pL20,A1×0,T2,qL20,A2×0,T2≤Cp0L20,A12+q0L20,A22+fL20,A2×0,T2.Proof.
Letu=e−λtp and v=e−λtq. The system becomes(69)∂tu+∂au+λ+μ1au=−e−λt∫0A2Feλtua,t,eλtvα,tdαinQ1,∂tv+∂av+λ+μ2av=e−λtfinQ2,u0,t=∫0A1β1aut,adain0,T,v0,t=∫0A2β2ava,tda+e−λt∫0A1∫0A2ba,αFeλtua,t,eλtvα,tdαdain0,T,ua,0=p0ain0,A1,va,0=q0ain0,A2.
Fixh in L20,A1×0,T and consider the following system:(70)∂tu+∂au+λ+μ1au=−e−λt∫0A2Feλtha,t,eλtvα,tdαinQ1,∂tv+∂av+λ+μ2av=e−λtfinQ2,u0,t=∫0A1β1aut,adain0,T,v0,t=∫0A2β2ava,tda+e−λt∫0A1∫0A2ba,αFeλtha,t,eλtvα,tdαdain0,T,ua,0=p0ain0,A1,va,0=q0ain0,A2.
Multiplying the second equation of (70) by v, integrating over 0,A2×0,T and using Young and Cauchy-Schwarz’s inequalities, we get(71)λ∫0T∫0A2va,t2dadt≤12∫0A2v2a,0da+12∫0Tv20,tdt+12∫0T∫0A2f2a,tdadt+12∫0T∫0A2v2a,tdadt+∫0T∫0A2β2ava,tda2+∫0Te−2λtb∞2A1A2∫0A1∫0A2Feλtha,t,eλtva,t2da′dadt≤12q0L20,A22+12fL20,A2×0,T2+12∫0T∫0A2v2a,tdadt+β2∞2A2∫0T∫0A2v2a,tdadt+b∞2A12A2K2∫0T∫0A2v2a,tdadt+b∞2A1A22K2∫0T∫0A1h2a,tdadt.
Therefore, we deduce that(72)∫0T∫0A2v2a,tdadt≤Cq0L20,A22+fL20,A2×0,T2+hL20,A1×0,Tλ−β2∞2A2+b∞2A12A2k2+1/2<∞,where C is a constant that does not depend on λ.
Forλ large enough, we showed that v∈L20,A2×0,T.
For everyv¯ in L20,A2×0,T, we define the mapping Ψ:L20,A2×0,T⟶L20,A2×0,T by Ψv¯=v, where v satisfies the following equations:(73)∂tv+∂av+λ+μ2av=−e−λtfinQ2,v0,t=∫0A2β2ava,tda+e−λt∫0A1∫0A2ba,a′Feλtha,t,eλtv¯α,tdαdain0,T,va,0=q0a,0in0,A2.
Setw=v1−v2, then w solves the system:(74)∂tw+∂aw+λ+μ1aw=−e−λtfinQ2,w0,t=∫0A1β2awa,tda+e−λt∫0A1∫0A2ba,αFeλtha,t,eλtv1α,t−Feλtha,t,eλtv2α,tdαdain0,T,wa,0=0in0,A2.
Multiplying (74) by w, integrating over 0,A2×0,T and using Young and Cauchy-Schwarz’s inequalities, we get(75)λ∫0T∫0A2w2a,tdadt≤∫0T∫0A2β2awa,tdadt2+∫0Te−λt∫0A1∫0A2ba,αFeλtha,t,eλtv1¯α,t−Feλtha,t,eλtv2¯α,tdαda2dt≤A2β2∞2∫0T∫0A2w2a,tdadt+K2b∞2A12A2∫0T∫0A2v1¯α,t−v2¯α,t2dαdt.
That is,(76)Ψv1¯−Ψv2¯L20,A2×0,T2≤k2A12A2b∞2λ−A2β2∞2v1¯−v2¯L20,A2×0,T2.
Hence, forλ large enough, Ψ is a contraction in L20,A2×0,T and using Banach’s fixed point theorem, Ψ has a unique fixed point v which is a unique solution to the system (73).
Now,v and h are being known. So, the first equation of (70) has a unique solution in L20,A1×0,T.
Multiplying the first equation of (70), integrating over 0,A1×0,T and using Cauchy—Schwarz and Young’s inequalities, one gets(77)λ∫0T∫0A1u2a,tdadt≤12p0L20,A12+12β1∞2A1∫0T∫0A1u2a,tdadt+12∫0T∫0A1u2a,tdadt+12A22K2∫0T∫0A1h2a,tdadt+12A2A1K2∫0T∫0A1v2a,tdadt.
Thus,(78)∫0T∫0A1u2a,tdadt≤Cλ−1/2β1∞2A1+1/2p0L20,A12+∫0T∫0A1h2a,tdadt+∫0T∫0A1v2a,tdadt<∞,where C is also a constant that does not depend on λ.
Now, let us define the mappingΦ:L20,A1×0,T⟶L20,A1×0,T by:
For everyh∈L20,A1×0,T, Φh=u where u is a unique solution of the system(79)∂tu+∂au+λ+μ1au=−e−λt∫0A2Feλtha,t,eλtva′,tda′inQ1,u0,t=∫0A1β1aut,adain0,T,ua,0=p0ain0,A1.
For everyh1 and h2 in L20,A1×0,T such that Φh1=u1 and Φh2=u2, we set again w=u1−u2 where u1 and u2 are the solutions of (79) corresponding respectively to h1 and h2.
So,w solves the following system:(80)∂tw+∂aw+λ+μ1aw=e−λt∫0A2Feλth2a,t,eλtv2α,t−Feλth1a,t,eλtv1α,tdαinQ1,w0,t=∫0A1β1awt,adain0,T,wa,0=0in0,A1,where v1 and v2 are solutions of (73) corresponding respectively to h1 and h2.
Multiplying the (80) by w, integrating over 0,A1×0,T and using Young and Cauchy-Scharz’s inequalities, we deduce that(81)λ∫0T∫0A1w2a,tdadt≤12∫0Tw20,tdadt+12∫0T∫0A1w2a,tdadt+12∫0T∫0A1e−2λt∫0A2Feλth2a,t,eλtv2α,t−Feλth1a,t,eλtv1α,tdα2dadt≤12A1β1∞2∫0T∫0A1w2a,tdadt+12∫0T∫0A1w2a,tdadt+K2A22∫0T∫0A1h1a,t−h2a,t2a,tdadt+K2A2A1∫0T∫0A2v1α,t−v2α,t2dαdt.
Finally, we obtain(82)λ−12β1∞2A1+12∫0T∫0A1wa,t2a,tdadt≤K2A22∫0T∫0A1h1a,t−h2a,t2a,tdadt+K2A2A1∫0T∫0A2v1α,t−v2α,t2dαdt.
We also have(83)λ∫0T∫0A2v1−v2a,t2a,tdadt≤∫0T∫0A2β2awa,tda2dt+∫0Te−λt∫0A1∫0A2ba,αFeλtha,t,eλtv1α,t−Feλtha,t,eλtv2α,tdαda2dt≤A2β2∞2∫0T∫0A2w2a,tdadt+K2b∞2A12A2∫0T∫0A2v1α,t−v2α,t2dαdt+K2b∞2A22∫0T∫0A1h1a,t−h2a,t2dadt.
So, one has(84)λ−β2∞2A2+A12A2K2b∞2∫0T∫0A2v1a,t−v2a,t2a,tdadt≤K2b∞2A22∫0T∫0A1h1a,t−h2a,t2dadt.
Combining the inequalities (83)) and (84), we get(85)Φh1−Φh2L20,A1×0,T2≤Cλh1−h2L20,A1×0,T2,where(86)Cλ=K2A22λ−β2∞2A2+A12A2K2b∞2+K4A23A1b∞2λ−1/2β1∞2A1+1/2λ−β2∞2A2+A12A2K2b∞2.
And, it is clear that forλ large enough, that is λ>λ0 with(87)λ0=1212β1∞2A1+β2∞2A2+A12A2K2b∞2+K2A22+12+1212β1∞2A1−β2∞2A2−A12A2K2b∞2+K2A22+122+4K2A23A1b∞21/2,
Cλ<1. So, Φ is a contraction in L20,A1×0,T, hence Φ has a unique fixed point u. So, the couple u,v∈L20,A1×0,T×L20,A1×0,T is a unique solution of the system (82).
Replacingh by u in (85) and in (78) and summing the inequalities (85) and (90), we deduce (81).


## 4. Spatially Homogeneous Stationary Solutions

We now consider problem (2) and we look for spatial homogeneous stationary solutions i.e for solutions that are constant in time and space. The System (2) becomes:(88)pa+μ1pa=−∫0A2Fpa;qαdαinQA1,qa+μ2qa=0inQA2,p0=∫0A1β1apada,q0=∫0A2β2αqαdα+∫0A1∫0A2ba,αFpa;qαdαda.Theorem 4.
Under the hypothesisA1—A4, the system (88), admits at least one non-trivial positive solution in L+10,A1×L+10,A2.Proof.
Letp¯,q¯∈L+10,A1×L+10,A2⟶L+10,A1×L+10,A2 and denote by Γ:L+10,A1×L+10,A2⟶L+10,A1×L+10,A2, the application given by:(89)Γp¯,q¯=p,q,where p;q is a solution of the following uncoupled system(90)pa+μ1pa=−∫0A2Fp¯a;q¯αdαinQA1,qa+μ2qa=0inQA2,p0=∫0A1β1apada,q0=∫0A2β2αqαdα+∫0A1∫0A2ba,αFp¯a;q¯αdαda.
The comparison theorem (see [20]) imply that, for any p¯,q¯∈L+10,A1×L+10,A2,(91)0≤p≤p˜and0≤q≤q˜,where p˜ and q˜ satisfy respectively(92)p˜a+μ1p˜a=0inQA1,p˜0=β1∞∫0A1p˜ada,
And(93)q˜a+μ2q˜a=0inQA2,q˜0=β2∞+A1b∞F∞∫0A2q˜ada.
Consider now the setL=p¯,q¯∈L10,A1×L10,A2such that0≤p¯≤p˜and0≤q¯≤q˜.
It clear thatL is a closed convex set and ΓL⊂L because Γp¯,q¯∈L for every p¯,q¯∈L.
We fixη¯,ζ¯∈L10,A1×L10,A2 and set η,ζ the corresponding solution of (90). Let p¯n and q¯n be two convergent sequences respectively in L10,A1s and in L10,A2 such that p¯n⟶η¯ in L10,A1 and q¯n⟶ζ¯ in L10,A2. Set pn,qn the corresponding solution of (90). Then pn,qn solves(94)pn′a+μ1pna=−fnainQA1,qn′a+μ2pna=0inQA2,p0=∫0A1β1apnada,q0=∫0A2β2αqnαdα+gn,where fna=∫0A2Fp¯na;q¯nαdαa.e in0,A1 and gn=∫0A1∫0A2ba,αFp¯na;q¯nαdαda.
Denote byfa=∫0A2Fη¯a;ζ¯αdαa.e in0,A1 and g=∫0A1∫0A2ba,αFη¯a;ζ¯αdαda.
One has(95)fn−fL10,A1≤A2Kp¯n−η¯L10,A1+A1Kq¯n−ζ¯L10,A2⟶0,
And(96)gn−g≤KA2b∞p¯n−η¯L10,A1+KA1b∞q¯n−ζ¯L10,A2⟶0.
So,fn⟶finL10,A1 and gn⟶ginℝ+.
By elementary calculations, the first equation of (94) yields:(97)pna=π1a∫0A1β1apnada−∫0afnτexp−∫τaμ1θdθdτ,a∈a;A1.
Multiplying (93) by β1 and integrating over 0,A1, we get(98)1−∫0A1π1sβ1sds∫0A1β1apnada=−∫0A1β1s∫0sfnτexp−∫τsμ1θdθdτ.
The sequencejn=∫0sfnτexp−∫τsμ1θdθdτ converges to j=∫0sfτexp−∫τsμ1θdθdτ in ℝ+ because jn−j≤∫0sfn−fτexp−∫τsμ1θdθdτ≤fn−fL10,A1⟶0.
So,∫0A1β1apnada⟶l. Thus, pna⟶γaa.e0,A1 and(99)γa=π1al−∫0afτexp−∫τaμ1θdθdτ,a∈a;A1.
So, we see thatγ solves(100)γ′a+μ1aγ=−fainQA1,γ0=l=∫0A1β1aγada.
The uniqueness properties of the solutions of (100) give that γ=η.
The solution of the second equation in (92) can be rewritten as(101)qna=π2a∫0A2β2aqnada+gn,a∈0,A2.
As before, we multiply (97) by β2 and integrate over 0,A2. We get(102)1−∫0A2β2sπ2sds∫0A2β2aqnada=gn∫0A2β2aπ2ada.
Then,∫0A2β2aqnada converges to h. So, (97) implies that qna converges to φa. And, one has(103)φa=π2ah+gπ2a.
By derivation of (105), we see that φ satisfies(104)φ′a+μ2aφ=0inQA2,φ0=h=∫0A2β2aφada+g.
Using again uniqueness properties, we conclude thatφ=ζ.
We have shown thatΓ is a continuous application in L+10,A1×L+10,A2.
Integrating the first equation of (94) over 0,A1, we obtain(105)∫0A1μ1apnada=pnA1−∫0A1β1apnada,−∫0A1fnada.
The right hand term of this equality is bounded then the left hand one is bounded too. This also implies thatμ1pn is bounded. Therefore, by the first equation of (94), we deduce that pn′ is bounded. Since pn and pn′ are both bounded. Then, pn is bounded in ∈W1,10,A1. Similarly, we show also that qn is bounded ∈W1,10,A2.
From the compact injections ofW1,10,A1 into L10,A1 and W1,10,A2 into L10,A2, we deduce that ΓL is relatively compact. By invoking Schauder’s theorem [22], Γ has at least one fixed point p,q which is the solution of (90).


## 5. Numericals Simulations

We use the finite difference method to approximate the solution of problem (35) ([23, 24]). Let us denote by uit and vit the approximations of uai,t and vai,t respectively, where ai=i△a, 0≤i≤N, △a=A/N.We consider thatFp,q≡p+q and ba,α≡b0. And we use the discrete approximation(106)∂uit∂a≅uit−ui−1t△a,and∂vit∂a≅vit−vi−1t△a;i=1,2,…,N.We approximate the boundary condition by replacing the integrals with the series [25]. This yields(107)∫0Aβ1aua,tda≅AN∑k=1Nβ1iuit+β10u0,t+β1AuA,t2and∫0Aβ2ava,tda+∫0A∫0Aba,αua,t+vα,tdαda≅AN∑k=1Nβ2i+Ab0Nvit+A2b0N∑k=1Nukt+β20v0,t+β2AvA,t2.Finally, the initial conditions are replaced by(108)ui0=u0ai,vi0=v0aifori=1,2,…,N.Substituting (106)–(109) into (37) and replacing the approximate equalities with equalities, yields,(109)duitdtt=1△aui−1t−uit−μ1iuit−Auit−AN∑k=1Nvkt,dvitdtt=1△avi−1t−vit−μ2ivit,u0t=AN∑k=1Nβ1iuit,i=1,…,N,v0t=AN∑k=1Nβ2i+Ab0Nvit+A2b0N∑k=1Nukt,ui0=u0ai,vi0=v0ai,where we set μ1i=μ1ai, β1i=β1ai, μ2i=μ2ai, β2i=β2ai for i=1,2,…,N.SetUt=u1t,…,uNt,v1t,…,vNt⊥ and U0=U0=u10,…,uN0,v10,…,vN0⊥; the linear system (5) becomes(110)dUtdt=GUt,i=1,…,N,U0=U0,where G=CDEF, see (Figure 1).Figure 1
MatrixG.The matrixC, D, E and F are square matrix of order N with(111)C=1/△aA/Nβ11+k1A/Nβ12…A/Nβ1N−1A/Nβ1N1k20⋯00⋱⋱⋱⋮⋮⋱1kN−100⋯01KN,D=−A/N1…1⋮⋱⋮1…1,E=A2b0/N……A2b0/N00…0⋮⋱⋱⋮0…00,F=1/△ad1+c1c2…cN−1cN1d20⋯00⋱⋱⋱⋮⋮⋱1dN−100⋯01dN.Andki=−△aμ1i+A−1; ci=A/Nβ2i+b0A2/N, di=−△aμ2i−1 for i=1,2,…,N.We visualize in Matlab the general form of the matrixG (see Figure 1).All the numerical tests run on Matlab.For numerical simulations, we take the functions(112)μ1a=2A−a−1,μ2a=A−a−1,β1a=105a2A−a2e−3a−A/22,β2a=100a−5A4e−0.52100a−5A2,0.4≤a≤1.7,0,otherwise.With the initial conditionsu0a=10e−2a−A/32 and v0a=20e−0.32a−A2.Att=0, we have a large prey population, in particular the prey which has an age between a=0.2 and a=1.1 (yellow zone, see (Figure 2). This population generates significant births especially between the instants t=0.03 and t=0.8 (yellow zone).Figure 2
Approximate solution of the problem (37) with T=1, A=2.No birth is observed in the population of predators between the instantst=0 and t=0.3 The consumption of prey under the action of biomass increases the fertility of predators so there are births between the instants t=0.3 and t=0.6. These births are important at times t>0.7. The population of predators then increases from t>0.8 and at the same time leads to a decrease in that of preys.We also take the following functions(113)μ1a=0.5A−a−1;μ2a=1.05A−a−2;β1a=105a2A−a2e−3a−A/32;β2a=100a−5A4e−0.5210a−5A2/Γ5,0.4≤a≤1.7,0,otherwise.With the initial conditionsu0a=10e−2a−A/32 and v0a=10e−0.22a−A2.Preys are rare so the predators population does not develop. As soon as the preys population began to be abundant fromt≥3, that of predators also becomes important from t≥4 (see Figure 3).Figure 3
Approximate solution of the problem (37) with T=4 and A=2.To account for the effect of predation on the evolution of the two populations, we present cases where the transformation of the biomass is more or less important (b0=1.25 or b0=0.01):The transformation of biomass is important, preys consumed benefits predators by considerably increasing their fertility, which increases their births (see Figure4). So the population of predators increases but on the other hand that of prey decreases.The biomass is important, but the predators do not live long so they do not have the time to procreate which leads to their decrease therefore the prey population increases with many births (see Figure 5).Figure 4
Approximate solution of the problem (37) with T=1 and A=2 and b0=1.25.Figure 5
Approximate solution of the problem (37) with T=1 and A=2 and b0=1.25.In Figure6, the biomass is low so the prey consumed does not influence the fertility of predators so births are very low. So the predator population is decreasing and that of the prey too because the prey does not live long enough to procreate.Figure 6
Approximate solution of the problem (37) with b0=0.01.An example of code under Matlab to get Figure6.The biomass is almost zero, so the preys are eaten without contribution on the fertility of the predators so the births are very low. The predator population is decreasing and the prey population is increasing see (Figure7).Figure 7
Approximate solution of the problem (37) with b0=0.01.


## 6. Conclusion and Perspectives

We have analyzed in this work existence results of a predator-prey model. Existence results already exist on predator-prey models but these models do not simultaneously take into account the variables of space, time and age and use classical functional response functions (see [1, 2]). Thus, we have proposed the model that we consider much more complete with a more general functional response subject to the condition of K−lipschitz. This model has been analyzed in the different previous sections under these different variants proving that the cohabitation of predators and prey in our model is possible.The numerical simulation section confirms the theoretical results and shows that the quantity of prey and predators present also depends strongly on the biomass conversion rateb: Indeed, a high biomass conversion rate increases the fertility of predators by the amount ∫0A1ba,x,tFp;qa;x,tda therefore leads to a large population of predators that consume almost all prey from at a given time see (Figure 4). And if the biomass is very low then the consumption of prey does not benefit the birth rate of predators so their number does not increase and end up disappearing under the effect of mortality see (Figure 7).In practice, it will be difficult to control the behavior of these two populations by acting on the biomass conversion rate since it is an intrinsic and biological factor of predators. So the investigation of (2) is not yet complete. We believe that it is possible to control the model (2) through the external functions f1 and f2. In other words, by taking the functions f1 and f2 as controls in a bounded domain of Ω, it is possible to have the extinction either the population of prey or that of predators or both simultaneously from of a time T as we did in [17].



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