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Local and global stability analysis for Gilpin-Ayala competition model involved in harmful species via LMI approach and variational methods

In this paper, stability of reaction-diffusion Gilpin-Ayala competition model with Dirichlet boundary value, involved in harmful species, was investigated. Employing Mountain Pass Lemma and linear approximation principle results in the local stability criterion of the null solution of the ecosystem which owns at least three stationary solutions. On the other hand, globally asymptotical stability criterion for the null solution of the ecosystem was derived by variational methods and LMI approach. It is worth mentioning that the stability criteria of null solution presented some useful hints on how to eliminate pests and bacteria. Finally, two numerical examples show the effectiveness of the proposed method.


Introduction
In 1920, Lotka and Volterra proposed the famous population competition model ( [1,2]): 11 x 1 (t) − a 12 x 2 (t)], 22 x 2 (t)], (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 i j > 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) and (1. 3) 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: 5) 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 ( [20,21]). Recently, the double positive solutions of the following delay feedback Gilpin-Ayala competition model has been studied in [20]: Therefore, in this paper, the author will study the dynamic behavior of nonlinear Gilpin Ayala competition model with Dirichlet zero boundary value. This paper includes two main purposes: Under some assumptions, the author will give the existence of two nonzero steady-state solutions for this model, and the global asymptotical stability of null solution can not hold so that the local stability criterion was derived ( [23]). On the other hand, under another reasonable assumptions, the author will consider the globally asymptotical stability, which may present some good suggestions on how to eliminate pests and bacteria. This is the second main objective of this paper.

Preparation
Consider the nonlinear Gilpin-Ayala competition model under Dirichlet boundary value: where Ω is a domain in R n (n ∈ {2, 3}) with the smooth boundary ∂Ω.
Then c * α, and Ψ possesses a critical sequence on c * . Remark 1. Lemma 2.1 is the Mountain Pass Lemma without the (PS) condition (see,e.g. [11,12]). If, in addition, Ψ satisfies the (PS) condition, then c * is a critical value of Ψ.

Main results
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. So I assume θ i ∈ (0, 1), and where p i and q i are a pair of Coprime odd numbers.
Theorem 3.1 tells us that under the assumptions of Theorem 3.1, (0, 0) can not be global stable. And so the local stability is considered.
Theorem 3.2. Under the assumptions of Theorem 3.1, the zero solution (0, 0) of the system (2.1) is locally asymptotically stable .
Proof. Firstly, the condition b i < λ 1 d i yields, where Next, consider the following linear system: Consider the Lyapunov function: The condition (3.14) yields dV dt where u = (u 1 , u 2 ) T . Then (3.17) yields that the zero solution (0, 0) of the linear system (3.16) is asymptotically stable ( [19]). And hence, the zero solution (0, 0) of the nonlinear system (2.1) is locally asymptotically stable.
The main results of this section originate in the author's another work ( [23]).

Global Stability with boundedness assumption on population densities
In this section, a suitable assumption may proposed on population densities u 1 , u 2 : where M i is a given positive number for a given i ∈ {1, 2}. Remark, this assumption is reasonable due to the limit resources ( [20]). Next, the system (2.1) can be rewrite as follows, where (4.4) And the system (4.2) can be rewritten as follows, where the matrices D and B is defined in (3.15), u = (u 1 , where γ i j 0 is transition probability rate from i to j ( j i) and γ ii = − ∑ n 0 j=1, j i γ i j , δ > 0 and lim Consider the following delayed feedback system : where k 1 (r(t)) and k 2 (r(t)) are feedback benefit coefficients at mode r(t) = r ∈ S . Denote k 1 (r(t)) = k 1r , k 2 (r(t)) = k 2r for simple. there are a sequences of positive definite diagonal matrices P r (r ∈ S ), Q i > 0(i ∈ {1, 2}) and W > 0 such that then the null solution of the ecosystem (4.7) is globally exponentially stable, where and (4.14) Next, one can see that under the restrictive condition (4.1) on the state variable u, the null solution (0, 0) is the unique stationary solution of the system (4.7) Indeed, let u ≡ u(x) be a stationary solution, satisfying (4.1), then it is obvious that which together with the definition of f , Poincare inequality and boundary value condition implies ∫ Let P r (r ∈ S ) and W be positive definite matrices such that |u|, one can deduce On the other hand, it follows from (4.13) and (4.14) that Let L be the weak infinitesimal operator (see, e.g. [15]) such that Moreover, Let ε → 0, then combining (4.17) and (4.9) leads to Obviously there exist positive constants σ 1 , σ 2 > 0 such that where Therefore, one can see it from (4.18) and (4.19) that the null solution of the ecosystem (4.7) is globally exponentially stable.  Then Theorem 4.1 yields that the null solution of the ecosystem (4.7) is globally exponentially stable.

Conclusions
In this paper, the author investigated the local and global stability for Gilpin-Ayala competition model involved in harmful species via LMI approach and variational methods. Using the mountain pass lemma, the existence of multiple stationary solutions of ecosystem is given, which shows that the global stability of the zero solution of the pest model is difficult to achieve. And applying the principle of linear approximation results in the local stability criterion. Moreover, due to the limited resources of nature, the population densities of species are reasonably assumed to be limited. Based on this boundedness assumption, the author employ variational methods to prove that the null solution is the unique stationary solution of the ecosystem, and so the global stability can be considered. Furthermore, utilizing LMI technique gives the globally asymptotical stability criterion of the unique stationary solution. The obtained Theorems and numerical examples illuminate that improving the diffusion of bacteria or pests is conducive to the elimination of pests or bacteria. For example, more ventilation in the area where bacteria are located is conducive to preventing the multiplication of bacteria and ultimately eliminating them.