On a Predator-Prey Model Involving Age and Spatial Structure

In this paper, we study the mathematical analysis of a nonlinear age-dependent predator–prey system with diusion in a bounded domain with a non-standard functional response. Using the xed point theorem, we rst 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 nite dierence method to validate the results.


Introduction and Assumptions
e study of predator-prey systems has attracted the attention of many mathematicians in the last century. Pioneers like Voltera and Lotka were the rst to mathematically model the interaction between predators and preys. e standard model of Lotka and Voltera is: x ′ (t) rx(t) − p(x(t))y(t), y ′ (t) ηp(x(t))y(t) − σy(t), t> 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 coe cient for the conversion that predator intake to per capital prey and the predator mortality rate. And, p(x) 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 conguration, migratory movements (di usion coe cients) 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 di use predator-prey system with a mutually interfering predator and a nonlinear harvest in the predator with a Crowley-Martin functional response. ey analyze the existence and uniqueness of the solution of the system using the C 0 semigroup. ey 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. ey 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 di usive 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 di erential equations and a reaction-di usion model. e 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 di usive system analytically. e classical stability properties of the non-spatial counterpart of the system are also studied. e 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]. e 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]. e 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.
e global stability of the unique positive equilibrium solution of the autonomous model is determined by defining an appropriate Lyapunov function. e autonomous system exhibits complex dynamics via bifurcation scenarios, such as the period doubling bifurcation. ey 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][8][9][10][11][12][13][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 response F subject to the Klipschitz 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 nonstandard 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] [17] where the system models a nonlinear age and two-sex population dynamics.
Let us denote by p(a, t, x) and q(a, 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 agedependent population dynamics diffusive system: where T is a positive number, and Ω a bounded open subset of R n which boundary is assumed to be of class C 2 .
We denote by A 1 (resp. A 2 ) the maximum life expectancy of prey (resp. predators).
In (2), β 1 (a) and β 2 (a) are respectively the natural birthrate at age a of preys and predators, μ 1 and μ 2 are the functions describing the mortality rate respectively of preys and predators that depends on a, η is the external normal derivative on zΩ.
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 by F(p(a, t, x), q(α, 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. us, x))dα is the amount of prey of age a consumed by predator at time t > 0 and location x ∈ Ω. e function b(x, 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.
us, the biomass is transformed and influences the birth process through the quantity which is the supply. e function f 1 (a, t, x) (resp. f 2 (a, 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? e 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 biomass b 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 of b, 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 Section 2, 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. e 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 Which is the probability for a newborn to survive to age a and And assume that the following hypotheses hold: F 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 And F(0, 0) � 0.
For the biological meanings of assumptions (A 1 ), (A 2 ) and (A 3 ), the functions μ i , β i , π i and the constants R i , we refer the reader to books such [18,19].

Spatially Heterogeneous Solutions
Let us make the following assumptions: We have the following result: Moreover, there exist a constant C depending on T such that Proof. Let us fix T > 0. e proof of the theorem is based on the method used in [20]. Set u � e − λ 0 t p in Q 1 and v � e − λ 0 t q in Q 2 where λ 0 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.
International Journal of Mathematics and Mathematical Sciences For any nonnegative h ∈ L 2 (Q 1 ), we introduce the following cascade system: Using the Fubini's theorem, the function v solves the following system: loc (0, A 2 ) and β 2 is bounded.
Multiplying (10) by v h and integrating over Q 2 , we get Using the Young and Cauchy Schwarz's inequalities, we obtain 4 International Journal of Mathematics and Mathematical Sciences So, we have where For Multiplying (16) by w and integrating over (0, A) × (0, t) × Ω, and following the same calculations as before, one has erefore, we have where Now, as h and v h are known, the remainder of (9) can be rewritten as following where μ 1 (a) � λ 0 + μ 1 (a) and g(a, t, According to the results in [20], (20) has a unique nonnegative solution u h in L 2 (Q 1 ).

International Journal of Mathematics and Mathematical Sciences
Multiplying (20) by u h , integrating over Q 1 and using Young's inequality, we get Using now Cauchy Schwarz's inequality, en, one gets where Denote by Λ: , the application given by where u h is the unique solution of (20).
Multiplying (26) by V and integrating over And we also have By recalling the inequality (18) in (29), we deduce that, there exists a constant C 4 , such that where Let us define on L 2 + (Q) the metric d by: for any Using the Fubini's theorem, we conclude that: From here we have: en, Λ is a contraction on the complete metric space L 2 + (Q 1 ) and using Banach's fixed point theorem, we conclude the existence of a unique fixed point u h nonnegative International Journal of Mathematics and Mathematical Sciences 7 for Λ, so the unique couple (u h , v h ) is the unique solution of (9). Hence, we deduce that the couple (p, q) � (e λ 0 t u h , e λ 0 t v h )is the unique solution to the problem (2). From the explicit expression of the constant C 3 , we see that it is always possible to choose λ 0 so that C 3 < 1. Replace h by u h in (14) and in (23), summing the inequalities (14) and (17), we conclude that there exists a constant C independent on T such that And the inequalities (7) follow clearly.

Spatially Homogeneous Solutions
Let consider the following spatial homogeneous system deduced from (2): Proof. To simplify the calculations and without losing sight of the generality, we set where λ is a positive constant that will be choosed later. We fix p in Y. Consider now the following system Integrating the system (38) along the characteristic curves a − t � c, we obtain implicit formulas for the solutions of (2) stated below: 8 International Journal of Mathematics and Mathematical Sciences Let us fixed q∈ Y and define the mapping G: Y ⟶ Y by: for every q∈ Y and for all (t, a) ∈ Q, And for a < t, Summing of the previous inequalities, we get So, we have For λ large enough, thus G(q) ∈ Y. Now, for all q 1 , q 2 ∈ Y, for all (a, t) ∈ Q such that a < t; one has Finally, it follows that International Journal of Mathematics and Mathematical Sciences Multiplying the inequality (47) by e − λt , we get It is obvious that for all (a, t) ∈ Q such that a ≥ t, we have Combining the inequalities (46) and (47) it follows that For λ 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 mapping Λ: Y ⟶ Y by: for every p∈ Y and for all (t, a) ∈ Q, And for a < t, we have By adding the inequalities (52) and (53) and using (45), one gets

It is clear that
Multiplying the inequality (56) by e − λt , we get And for a < t, we have In other hand, one has Multiplying the inequality (59) by e − λt , we get Combining the inequalities (57) and (59), we deduce that where q 1 and q 2 are solutions of (39) and (40) associated respectively to p 1 and p 2 . en, we have for all (a, t) ∈ Q such that a < t, So, it follows that We deduce from (63) that From (64), we obtain By combining the inequalities (61) and (65), we get International Journal of Mathematics and Mathematical Sciences For λ large enough such that where K is a Lipschitz constant, one gets clearly that Λ is a contraction in Y. erefore, the (37) and (38) have a unique solution p in Y. e 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: and f ∈ L 2 ((0, A 2 )× (0, T)), the system (37) admits a unique solution in Proof. Let u � e − λt p and v � e − λt q. e system becomes Fix h in L 2 ((0, A 1 ) × (0, T)) and consider the following system: Multiplying the second equation of (70) by v, integrating over (0, A 2 ) × (0, T) and using Young and Cauchy-Schwarz's inequalities, we get 12 International Journal of Mathematics and Mathematical Sciences erefore, we deduce that where C is a constant that does not depend on λ. For For every v in L 2 ((0, A 2 ) × (0, T)), we define the mapping Ψ: where v satisfies the following equations: then w solves the system: Multiplying (74) by w, integrating over (0, A 2 ) × (0, T) and using Young and Cauchy-Schwarz's inequalities, we get at is, International Journal of Mathematics and Mathematical Sciences 13 Hence, for λ large enough, Ψ is a contraction in L 2 ((0, A 2 ) × (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 L 2 ((0, A 1 ) × (0, T)).
Multiplying the first equation of (70), integrating over (0, A 1 ) × (0, T) and using Cauchy-Schwarz and Young's inequalities, one gets us, where C is also a constant that does not depend on λ. Now, let us define the mapping Φ: For every h 1 and h 2 in L 2 ((0, A 1 ) × (0, T)) such that Φ(h 1 ) � u 1 and Φ(h 2 ) � u 2 , we set again w � u 1 − u 2 where u 1 and u 2 are the solutions of (79) corresponding respectively to h 1 and h 2 .
So, w solves the following system: where v 1 and v 2 are solutions of (73) corresponding respectively to h 1 and h 2 .
Multiplying the (80) by w, integrating over (0, A 1 ) × (0, T) and using Young and Cauchy-Scharz's inequalities, we deduce that 14 International Journal of Mathematics and Mathematical Sciences Finally, we obtain We also have So, one has Combining the inequalities (83)) and (84), we get where And, it is clear that for λ large enough, that is λ > λ 0 with International Journal of Mathematics and Mathematical Sciences is a unique solution of the system (82).

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.

Theorem 4. Under the hypothesis (A 1 )-(A 4 ), the system (88), admits at least one non-trivial positive solution in
Proof. Let (p, q) ∈ L 1 and denote by Γ: L 1 , the application given by: where (p; q) is a solution of the following uncoupled system e comparison theorem (see [20]) imply that, for any (p, q) ∈ L 1 where p and q satisfy respectively p a + μ 1 p(a) � 0 in Q A 1 , And Consider now the set L � (p, q) ∈ L 1 (0, A 1 )× L 1 (0, A 2 ) such that 0 ≤ p ≤ p and 0 ≤ q ≤ q}.
It clear that L is a closed convex set and Γ(L) ⊂ L because Γ(p, q) ∈ L for every (p, q) ∈ L.
We have shown that Γ is a continuous application in L 1 + (0, A 1 ) × L 1 + (0, A 2 ). Integrating the first equation of (94) over (0, A 1 ), we obtain (105) e right hand term of this equality is bounded then the left hand one is bounded too. is also implies that (μ 1 p n ) is bounded. erefore, by the first equation of (94), we deduce that (p n ′ ) is bounded. Since (p n ) and (p n ′ ) are both bounded. en, (p n ) is bounded in ∈ W 1,1 (0, A 1 ). Similarly, we show also that (q n ) is bounded ∈ W 1,1 (0, A 2 ).

Numericals Simulations
We use the finite difference method to approximate the solution of problem (35) ( [23,24]). Let us denote by u i (t) and v i (t) the approximations of u(a i , t) and v(a i , t) respectively, where a i � i△a, 0 ≤ i ≤ N, △a � (A/N).
All the numerical tests run on Matlab.
For numerical simulations, we take the functions With the initial conditions u 0 (a) � 10e − 2(a− (A/3)) 2 and v 0 (a) � 20e − 0.3(2a− A) 2 . At t � 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). is population generates significant births especially between the instants t � 0.03 and t � 0.8 (yellow zone).
No birth is observed in the population of predators between the instants t � 0 and t � 0.3 e 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. ese births are important at times t > 0.7. e 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 With the initial conditions u 0 (a) � 10e − 2(a− (A/3)) 2 and v 0 (a) � 10e − 0.2(2a− A) 2 . Preys are rare so the predators population does not develop. As soon as the preys population began to be abundant from t ≥ 3, that of predators also becomes important from t ≥ 4 (see Figure 3).
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 ( b 0 � 1.25 or b 0 � 0.01): e transformation of biomass is important, preys consumed benefits predators by considerably increasing their fertility, which increases their births (see Figure 4). So the population of predators increases but on the other hand that of prey decreases. e 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).
In Figure 6, 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.
An example of code under Matlab to get Figure 6.      International Journal of Mathematics and Mathematical Sciences e biomass is almost zero, so the preys are eaten without contribution on the fertility of the predators so the births are very low. e predator population is decreasing and the prey population is increasing see (Figure 7).

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]). us, we have proposed the model that we consider much more complete with a more general functional response subject to the condition of K− lipschitz. is 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. e numerical simulation section confirms the theoretical results and shows that the quantity of prey and predators present also depends strongly on the biomass conversion rate b: Indeed, a high biomass conversion rate increases the fertility of predators by the amount A 1 0 (b(a, x, t)F(p; q)(a; x, t))da 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 f 1 and f 2 . In other words, by taking the functions f 1 and f 2 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].

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.