Existence of local and global solution for a spatio-temporal predator-prey model

In this paper we prove the existence and uniqueness of weak solutions for a kind of Lotka–Volterra system, by using successive linearization techniques. This approach has the advantage to treat two equations separately in each iteration step. Under suitable initial conditions, we construct an invariant region to show the global existence in time of solutions for the system. By means of Sobolev embeddings and regularity results, we ﬁnd estimates for predator and prey populations in adequate norms. In order to demonstrate the convergence properties of the introduced method, several numerical examples are given.


Introduction
This work is concerned with a special evolution variant of the predator-prey system with homogeneous Neumann boundary conditions where Ω is an open, bounded subset of n , and Ω T := Ω × [0, T ).
The functions f (u, v) and g (u, v) describe the interaction between the prey and predator densities u and v respectively. These are given by and The parameters d 1 and d 2 measure the tendency of each population to spread, a 1 is the birth rate of prey, a 2 is the death rate of the predators, b 1 is the decay rate of preys due to the competition among themselves for limited resources, and c 1 and c 2 are measures of the effect of the interaction between the two species. The maximal concentration of preys in the absence of predators (i.e. the carrying capacity of the environment) is γ := a 1 /b 1 . We assume that all these parameters are positive.
The interaction of predator and prey populations with logistic growth of prey, without considering any spatial variations in populations density, has been well studied in [1] and [2]. Concerning the solvability of the spatially extended predator-prey system (1) several techniques have been proposed. In [3] and [4] the existence of traveling wave solutions is studied and some possible biological implications are given. In [5], the authors employ the implicit function theorem and spectral theory to show existence and uniqueness of the positive steady-state solutions of a reaction-diffusion two-competition species model with advection term under the Dirichlet boundary condition. In [6], it was established the existence of a solution for a fractional differential Lotka-Volterra reaction-diffusion equation. The technique of upper and lower solutions is used in [7], to show the existence and uniqueness of a classical global time-dependent solution and its asymptotic relation with the steady-state solutions. Another line of research is presented in [8], where the global existence and boundedness of classical solutions is studied for a predator-prey model via semigroup theory. The existence and uniqueness of weak solutions of the system (1) in Ω ⊂ n with homogeneous Dirichlet boundary conditions have been studied in [9] using the semi-implicit Rothe method.
In this work, we use a different strategy in order to obtain the existence and uniqueness of a weak solution for system (1). We propose an iterative process, which solves at each iteration a linear problem. For each linear problem, we prove the existence and uniqueness of a weak solution. We obtain estimates for the iterative sequence. In the next step, we show that this sequence of solutions of linear problems is a Cauchy sequence in an appropriate Banach The main virtue of this iterative approach is that due to the structure of the resulting linear problems, it is easy to prove the existence and uniqueness of a weak solution as well as to obtain a priori uniform estimates for the generated sequence. Another virtue is that the technique developed here could be applied to more complex nonlinear problems and adapted to a numerical scheme that solves these kinds of problems.

Methods
We show for a fixed time T > 0, the system (1) has exactly one weak solution.
We will formulate the nonlinear system (1) as a decoupled linear system in a weaker space setting, and show that a weak solution exists as the limit of solutions to corresponding approximation systems. In addition, we use embedding theorems and classical regularity results to obtain estimates for the predator and prey densities.
The most important result of this paper is the following theorem.
In order to prove Theorem 2.2 we will use frequently the following wellknown result of linear partial differential equations. Assume a(x, t ), c(x, t ) ∈ L ∞ (Ω T ). Then for any u 0 (x) ∈ H 1 (Ω) and f (x, t ) ∈ L 2 (0, T ; L 2 (Ω)), there exists a unique weak solution u ∈ L 2 (0, T ; H 2 (Ω)) ∩ ∞ L (0, T ; H 1 (Ω)) with u t ∈ L 2 (0, T ; H −1 (Ω)) to the problem The strategy of the proof is: to construct a sequence of linear approximations to the problem (1); in this way, we reduce the nonlinear coupled system (1) into a sequence of linear systems (9)- (10). After that, we establish that the resulting solutions (u k , v k ) define a Cauchy sequence in a suitable Banach space, which converges to a solution of our problem. ; H 2 (Ω) × L 2 0, T ; H 2 (Ω) be a weak solution of (7) and (8), with u t ,v t ∈ L 0, T ; L 2 (Ω) solution of (9) and (10), with u t k+1 , v t k+1 ∈ L 2 0, T ; L 2 (Ω) for k ∈ be a weak , f (u k , v k ) = a 1 u k and g (u k , v k ) = c 2 u k v k : on Ω; on Ω. (10) The following lemmas guarantee the existence and uniqueness of the functions (u k , v k ) for k ∈ 0 in the sequence.
Lemma 2.5. Under the assumptions of Theorem 2.2, the weak solution (u 0 , v 0 ) of system (7)-(8) satisfies Proof. To prove that u 0 (x, t ) 0. Let us take φ = (u 0 ) − := min{0, u 0 } in the variational formulation of (7), we obtain Integrating this over time, we obtain As that is to say that (u 0 ) − = 0 almost everywhere in Ω T . Now we establish a upper bound for u 0 . If here we take φ = (u 0 − γ ) + := max{u 0 − γ , 0} in the variational formulation of (7), we obtain Integrating this over time, we get As The proof of 0 v 0 (x, t ) ρ does not differ from the one for u 0 (x, t ) and it is therefore omitted here.

Proof by induction
Lemma 2.6. (Properties of the Iterative Sequence). Under the assumptions of Theorem 2.2, for every k ∈ 0 , there is a unique weak solution to the system (9)-(10) such that Furthermore, for all k ∈ 0 , the functions u k , v k satisfy the following inequalities: and the standard regularity estimates ess sup with C an adequate embedding constant and Therefore, by Theorem 2.3, there is a unique weak solution (u 1 , v 1 ) of the system (9)-(10) that satisfies (15), (17) and (18).
Now, we only have to establish (16) for u 1 and v 1 . To show that u 1 0, we take as a test function (u 1 ) − := min{0, u 1 } in the weak formulation of (9), In view the right-hand side is negative, then Integrating this over time, we obtain To show that u 1 (x, t ) γ almost everywhere in Ω T , we take as a test function (u 1 − γ ) + := max{0, u 1 − γ } in the weak formulation of (9), this yields Now, we prove that v 1 (x, t ) 0 almost everywhere in Ω T . Multiplying (10) by (v 1 ) − = min{0, v 1 } and integrating over Ω, we get and thus This implies that v 1 (x, t ) 0 almost everywhere in Ω T .
Let us show that ρ is an upper bound of v 1 . Testing the weak formulation of the system (10) using (v 1 − ρ) + , one gets since u 0 γ a 2 /c 2 and 0 v 0 ρ. Then Integrating on (0, t ) gives Thus, we conclude that v 1 (x, t ) ρ almost everywhere in Ω T .
(ii) Induction hypothesis: the lemma holds for an arbitrary k ∈ 0 .
(iii) Induction step: by the induction hypothesis we have, u k , v k ∈ L ∞ (Ω T ) and f , g ∈ L 2 (0, T ; Ω). Then Theorem 2.3 guarantees the existence and uniqueness of u k+1 and v k+1 . The proof of 0 u k+1 γ and 0 v k+1 ρ follows the same steps used for the case k = 0 and it is therefore omitted here. Hence, Lemma 2.6 is satisfied for (k + 1) and this concludes the induction proof.

Results
Now we prove the main results of this paper. The existence and uniqueness of weak solution for system (1) (Theorem 2.2) as well as its global existence in time (Theorem 5.5).

Existence and uniqueness of weak solutions
Before the proof of Theorem 2.2, we subtract the (k + 1)-th and k-th terms of the iterative sequence (9) and (10) to obtain and After subtracting and adding b 1 u k u k + c 1 v k u k in (19) and doing the same with c 2 u k v k−1 in (20) we obtain and Now, let us denote by for k = 0, 1, . . . . Then, from equations (21) and (22) we get the following boundary value problems on Ω; on Ω.

(24)
Proof of Theorem 2.2. To prove existence, we show that the iterative sequences {u k }, {v k } obtained are Cauchy sequences. Since, H 1 (Ω) is a Banach space, there are u and v limiting functions that fulfill the system (1).
In order to ensure the uniqueness, we suppose that (u 1 , v 1 ) and (u 2 , v 2 ) are the weak solutions of system (1). Owing to (30), it follows that:

Global existence of solutions
In order to prove global existence in time of solutions for system (1), we apply the technique based on the bounded invariant regions [11]. With this purpose, we write system (1) with predator-prey interaction functions (2) and (3) in matrix form As in [11], we define an invariant region for (32).
Definition 5.1. Let Σ ⊂ m be a closed set. Σ is a (positively) invariant region for the local solution of (32), if any solution (u(x, t ), v(x, t )) having all of its boundary and initial values in Σ, satisfies (u(x, t ), v(x, t )) ∈ Σ for all x ∈ Ω and for all t ∈ [0, δ).
To find invariant regions, we use the following combined version of the well known results (see [11], Chapter 14, §B).
Theorem 5.2. Suppose that D is the diagonal matrix with entries d 1 and d 2 , then any region of the form is invariant for (32), whenever the following conditions are satisfied Then Σ is an invariant region for the system (32).

Numerical simulations
In this section, in order to illustrate the convergence of the iterative method introduced in this paper, we present two numerical examples. To generate the sequence of solutions (u k , v k ) of (9) and (10), we use the backward Euler method and the finite element method, for time direction and spatial direction, respectively.
Where the matrices M , A, D(u k , v k ) ∈ N × N are given by and and the right-hand sides F k (t ), G k (t ) ∈ N are given by For the time discretization we split the time interval into sub-intervals with constant step ∆t > 0, the backward Euler method for (37) reads   1 shows the convergence history of the generated sequence. The tolerance of the L 2 (0, T ; H 1 (Ω))-norm changes between two successive iterates, which monitors the convergence of the iterative method, has been set equal to 10 −8 . The method converges in 17 iterations. In Fig. 2 and 3, we observe the convergence of the iterative method towards the solution of the prey-predator system (1). The iterate (u k , v k ) for k = 1, 2, . . . , are denoted by the k in the plots.
From the qualitative point of view of the solutions, Fig. 2 and 3 show clearly the effect of the diffusive spatial dispersion of the two populations. At first, both preys and predators move towards the regions of low density until they are homogeneously distributed in the interval [0, 1]. After that, the population of preys grows logistically until they reach the steady-state, i.e the carrying capacity γ = 4/2. In contrast, even though the interaction with preys and its abundance, the predator population decay continuously until extinction (steady-state for predators).    We begin by providing the convergence history of the method in Fig. 4. The error has a fluctuating behavior and it takes 141 iteration to fulfill the stopping criteria of 10 −8 . Note that, in this case the requirement that a 1 /b 1 a 2 /c 2 in Theorem 2.2 is not fulfilled, however this is a sufficient and not necessary condition for convergence.
Looking at the results u 141 (x, t ) and v 141 (x, t ) in the Fig. 5 and 6, we see that after both population fall rapidly as they move away from their original locations, the prey and predator populations oscillate over time until they reach a coexistence steady-state. The inequality a 1 /b 1 a 2 /c 2 states that for there to be a steady state with prey and predator both present, the carrying capacity of the prey, a 1 /b 1 , must be high enough to support the predator.

Conclusions
Using an alternative method based on successive iterations of a linear approximation of the original problem and under mild regularity conditions, we proved the existence, uniqueness and positivity of the solution for the predator-prey model with logistic growth of preys. The proof is based on mathematical induction and takes full advantage of the standard theory of linear partial differential equations. In order to guarantee convergence of the iterative method (i.e. the existence of local solution) and the global existence of the solution; it is sufficient that, the carrying capacity of the environment being bounded by the ratio between the death rate of the predators and growth rate of predators due to the interaction with preys. Numerical tests verify that the proposed iteration is convergent to the solution of the model and that the solution exists globally in time.