Existence and local stability of stationary solutions for nonlinear Gilpin Ayala competition model with Dirichlet boundary value

In this paper, the existence of two nontrivial stationary solutions for the nonlinear Gilpin Ayala two species competition model is given by using the mountain pass lemma, and the local stability criterion of the trivial solution is given by using Lyapunov function method. Based on the local stability criterion, we give some suggestions on how to avoid the population extinction. This is, when the population is on the verge of extinction, we should try our best to avoid the diffusion behavior and reduce the diffusion coefficient, otherwise the species are easy to go extinct. Numerical example shows the effectiveness of the proposed method.


Introduction
In 1920, Lotka and Volterra proposed the famous population competition model ( [1,2]): (1.1) where x i (t) represents the population density of the ith population at time t(i = 1, 2), b i > 0 represents the birth rate of the population of the ith population, a ij > 0 represents the competition parameter of two populations, which is recognized and cited by many scholars. Diffusion is usually considered reasonably, for example in the reference [3] and related references: (1.2) In 2017, Yuanyuan Liu and Youshan Tao studied the linear competition model of cross diffusion of two populations under Neumann boundary conditions ( [4]): (1.4) In 1973, Gilpin and Ayala found that the linear competition model was not consistent with the experimental results ( [5]). Through accurate data analysis, they proposed a nonlinear competition model of two populations: where θ 1 , θ 2 represents the nonlinear density constraint parameter. As pointed out in [6][7][8][9], when the parameter θ i is much less than 1, the nonlinear density constrained model can well simulate the population ecology of Drosophila melanogaster, and the diffusion type Gilpin Ayala competition model under Neumann boundary value condition has also been studied by scholars: where µ 1 , µ 2 , a, b, c and d all are positive numbers. It is noted that the diffusion ecosystem with Neumann boundary value has been widely studied ( [3,4,8] and related references), but the diffusive ecosystem under Dirichlet boundary value is rarely studied. In fact, the Drichlet boundary value diffusion ecosystem can better reflect the actual population ecology. Therefore, this paper will study the dynamic behavior of nonlinear Gilpin Ayala competition model with Dirichlet zero boundary value, I will give the existence of two nonzero steady-state solutions for this model. Recently, the author has studied the double positive solutions of the following delay feedback Gilpin-Ayala competition model in [20], where the global stability of the positive solution was presented in [20].
So, in this paper, I only study the stability of zero solution, avoiding duplication with another article of mine This paper, I denote by λ 1 the first positive eigenvalue of the Laplace operator −∆ in H 1 0 (Ω). Denote by u = Ω |∇u(x)| 2 dx the norm of Sobolev space H 1 0 (Ω).

Preparation
Consider the nonlinear Gilpin-Ayala competition model under Dirichlet boundary value: where Ω is a domain in R 3 with the smooth boundary ∂Ω.

Remark 1.
Here, we assume Ω ⊂ R 3 . And if two species live in two dimensional plane, we can assume Besides, I need Mountain Pass Lemma as follows ( [12]).

Lemma 2.1 (Mountain Pass Lemma without the (PS) condition). Let
Let Γ be the set of all paths connecting 0 and e. That is, Then c * α, and Ψ possesses a critical sequence on c * .

Theorem 3.2.
Under the assumptions of Theorem 3.1, the zero solution (0, 0) is locally asymptotically stable .
Proof. Firstly, the condition b i < λ 1 d i yields,

14)
where Next, consider the following linear system:

Conlusions
In this paper, the existence of two nontrivial stationary solutions of the nonlinear Gilpin-Ayala model of two species competition is derived by using the mountain pass lemma. The local stability criteria of the trivial solutions are given by using the Lyapunov function method. The local stability conclusion of the double zero solution fully indicates that when the population is on the verge of extinction, the diffusion behavior should be avoided and the diffusion coefficient should be reduced, otherwise the species will be prone to extinction