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.
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:where and denote preys and predators density respectively; , and 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, 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 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 response 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 by and respectively the distribution of preys and predators of age being at time and location over a bounded domain .
We consider in this paper the following nonlinear age-dependent population dynamics diffusive system:where is a positive number, and a bounded open subset of which boundary is assumed to be of class .
Actually, , , for and .
We denote by (resp. ) the maximum life expectancy of prey (resp. predators).
In (2), and are respectively the natural birth-rate at age of preys and predators, and are the functions describing the mortality rate respectively of preys and predators that depends on , is the external normal derivative on .
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 the functional response to predation, that is the capture rate of prey having age per predator of age or the average number of prey having age captured by predators of age at times , and location .
Thus, is the amount of prey of age consumed by predator at time and location .
The function is the conversion rate of the biomass of captured prey having age by predators of age into predator offspring at location .
Thus, the biomass is transformed and influences the birth process through the quantitywhich is the supply.
The function (resp. ) is the external supply for prey persistence (resp. for predator persistence) having age at time and location .
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 biomass 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 , that is to say we will take small and large values of 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. 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
Which is the probability for a newborn to survive to age and
And assume that the following hypotheses hold: . is a positive and mesurable function on and satisfies the usual locally boundedness and Lipschitz continuity conditions with respect to the pair variable, that is And .
For the biological meanings of assumptions , and , the functions , , and the constants , we refer the reader to books such [18, 19].
2. Spatially Heterogeneous Solutions
Let us make the following assumptions: , and. , and.
We have the following result:
Theorem 1. Under the hypothesies —, , the system (2) admits a unique solution in . Moreover, there exist a constant depending on such that
Proof. Let us fix . The proof of the theorem is based on the method used in [20].
Set and where 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.For any nonnegative , we introduce the following cascade system:Using the Fubini’s theorem, the function solves the following system:With and .
As is fixed, , , is bounded then and is bounded.
Hence, (10) admits a unique nonnegative solution in (see [20]).
Multiplying (10) by and integrating over , we getUsing the Young and Cauchy Schwarz’s inequalities, we obtainSo, we haveThat is to saywhereFor any , set . So, solves the following systemMultiplying (16) by and integrating over , and following the same calculations as before, one hasTherefore, we havewhereNow, as and are known, the remainder of (9) can be rewritten as followingwhere and .
As is fixed, and because and is bounded.
According to the results in [20], (20) has a unique nonnegative solution in .
Multiplying (20) by , integrating over and using Young’s inequality, we getUsing now Cauchy Schwarz’s inequality, we obtainThen, one getswhereDenote by , the application given bywhere is the unique solution of (20).
For any , we denote by , satisfies the following equationswhere , is solution ofMultiplying (26) by and integrating over , on getsAnd we also haveBy recalling the inequality (18) in (29), we deduce that, there exists a constant , such thatwhereLet us define on the metric by: for any ,So,Using the Fubini’s theorem, we conclude that:From here we have:Then, is a contraction on the complete metric space and using Banach’s fixed point theorem, we conclude the existence of a unique fixed point nonnegative for , so the unique couple is the unique solution of (9). Hence, we deduce that the couple is the unique solution to the problem (2). From the explicit expression of the constant , we see that it is always possible to choose so that . Replace by in (14) and in (23), summing the inequalities (14) and (17), we conclude that there exists a constant independent on such thatAnd the inequalities (7) follow clearly.
3. Spatially Homogeneous Solutions
Let consider the following spatial homogeneous system deduced from (2):
Theorem 2. Under the assumptions -, for all and , the system (37) admits a unique solution in .
Proof. To simplify the calculations and without losing sight of the generality, we set . Let set endowed with the norm where is a positive constant that will be choosed later.
We fix in . Consider now the following systemIntegrating the system (38) along the characteristic curves , we obtain implicit formulas for the solutions of (2) stated below:Let us fixed and define the mapping by: for every and for all ,For all such that ,And for ,Summing of the previous inequalities, we getSo, we haveFor large enough, thus .
Now, for all , for all such that ; one hasFinally, it follows thatMultiplying the inequality (47) by , we getIt is obvious that for all such that , we haveCombining the inequalities (46) and (47) it follows thatFor large enough, it is clear that is a contraction in and the (40) have a unique solution in .
Now, define the mapping by: for every and for all ,For every such that , one hasAnd for , we haveBy adding the inequalities (52) and (53) and using (45), one getsTherefore, for fixed in .
For all , for all such that ; we haveIt is clear thatMultiplying the inequality (56) by , we getAnd for , we haveIn other hand, one hasMultiplying the inequality (59) by , we getCombining the inequalities (57) and (59), we deduce thatwhere and are solutions of (39) and (40) associated respectively to and .
Then, we have for all such that ,So, it follows thatWe deduce from (63) thatFrom (64), we obtainBy combining the inequalities (61) and (65), we getFor large enough such thatwhere is a Lipschitz constant, one gets clearly that is a contraction in . Therefore, the (37) and (38) have a unique solution in . The couple 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 assumptions —, for all and , the system (37) admits a unique solution in . Moreover, there exists a positive constant , such that
Proof. Let and . The system becomesFix in and consider the following system:Multiplying the second equation of (70) by , integrating over and using Young and Cauchy-Schwarz’s inequalities, we getTherefore, we deduce thatwhere is a constant that does not depend on .
For large enough, we showed that .
For every in , we define the mapping by , where satisfies the following equations:Set , then solves the system:Multiplying (74) by , integrating over and using Young and Cauchy-Schwarz’s inequalities, we getThat is,Hence, for large enough, is a contraction in and using Banach’s fixed point theorem, has a unique fixed point which is a unique solution to the system (73).
Now, and are being known. So, the first equation of (70) has a unique solution in .
Multiplying the first equation of (70), integrating over and using Cauchy—Schwarz and Young’s inequalities, one getsThus,where is also a constant that does not depend on .
Now, let us define the mapping by:
For every , where is a unique solution of the systemFor every and in such that and , we set again where and are the solutions of (79) corresponding respectively to and .
So, solves the following system:where and are solutions of (73) corresponding respectively to and .
Multiplying the (80) by , integrating over and using Young and Cauchy-Scharz’s inequalities, we deduce thatFinally, we obtainWe also haveSo, one hasCombining the inequalities (83)) and (84), we getwhereAnd, it is clear that for large enough, that is with. So, is a contraction in , hence has a unique fixed point . So, the couple is a unique solution of the system (82).
Replacing by 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:
Theorem 4. Under the hypothesis —, the system (88), admits at least one non-trivial positive solution in .
Proof. Let and denote by , the application given by:where is a solution of the following uncoupled systemThe comparison theorem (see [20]) imply that, for any ,where and satisfy respectivelyAndConsider now the set .
It clear that is a closed convex set and because for every .
We fix and set the corresponding solution of (90). Let and be two convergent sequences respectively in and in such that in and in . Set the corresponding solution of (90). Then solveswhere and
Denote by and .
One hasAndSo, and .
By elementary calculations, the first equation of (94) yields:Multiplying (93) by and integrating over , we getThe sequence converges to in because .
So, . Thus, andSo, we see that solvesThe uniqueness properties of the solutions of (100) give that .
The solution of the second equation in (92) can be rewritten asAs before, we multiply (97) by and integrate over . We getThen, converges to . So, (97) implies that converges to . And, one hasBy derivation of (105), we see that satisfiesUsing again uniqueness properties, we conclude that .
We have shown that is a continuous application in .
Integrating the first equation of (94) over , we obtainThe right hand term of this equality is bounded then the left hand one is bounded too. This also implies that is bounded. Therefore, by the first equation of (94), we deduce that is bounded. Since and are both bounded. Then, is bounded in . Similarly, we show also that is bounded .
From the compact injections of into and into , we deduce that is relatively compact. By invoking Schauder’s theorem [22], has at least one fixed point 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 and the approximations of and respectively, where , , .
We consider that and . And we use the discrete approximation
We approximate the boundary condition by replacing the integrals with the series [25]. This yields
Finally, the initial conditions are replaced by
Substituting (106)–(109) into (37) and replacing the approximate equalities with equalities, yields,where we set , , , for .
Set and ; the linear system (5) becomeswhere , see (Figure 1).
The matrix , , and are square matrix of order with
And ; , for .
We visualize in Matlab the general form of the matrix (see Figure 1).
All the numerical tests run on Matlab.
For numerical simulations, we take the functions
With the initial conditions and .
At , we have a large prey population, in particular the prey which has an age between and (yellow zone, see (Figure 2). This population generates significant births especially between the instants and (yellow zone).
No birth is observed in the population of predators between the instants and The consumption of prey under the action of biomass increases the fertility of predators so there are births between the instants and . These births are important at times . The population of predators then increases from and at the same time leads to a decrease in that of preys.
We also take the following functions
With the initial conditions and .
Preys are rare so the predators population does not develop. As soon as the preys population began to be abundant from , that of predators also becomes important from (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 ( or ):
The 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.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).
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.
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 (Figure 7).
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 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 rate : Indeed, a high biomass conversion rate increases the fertility of predators by the amount 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 and . In other words, by taking the functions and as controls in a bounded domain of , it is possible to have the extinction either the population of prey or that of predators or both simultaneously from of a time as we did in [17].
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This article has not been funded by a structure or by a person.