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

Firstly, the author do dynamic analysis for reaction-di ﬀ usion Gilpin-Ayala competition model with Dirichlet boundary value, involved in harmful species. Existence of multiple stationary solutions is veriﬁed by way of Mountain Pass lemma, and the local stability result of the null solution is obtained by employing linear approximation principle. Secondly, the author utilize variational methods and LMI technique to deduce the LMI-based global exponential stability criterion on the null solution which becomes the unique stationary solution of the ecosystem with delayed feedback under a reasonable boundedness assumption on population densities. Particularly, LMI criterion is involved in free weight coe ﬃ cient matrix, which reduces the conservatism of the algorithm. In addition, a new impulse control stabilization criterion is also derived. Finally, two numerical examples show the e ﬀ ectiveness of the proposed methods. It is worth mentioning that the obtained stability criteria of null solution presented some useful hints on how to eliminate pests and bacteria.


Introduction
In 1920, Lotka and Volterra originally proposed the famous population competition model ( [1,2]), which is recognized and cited by many scholars ( [3][4][5][6][7][8][9]). Because of the diffusion of population in space, some reaction-diffusion models were investigated in order to better simulate the real ecological situation( [3,4] and their references therein). Besides, practice has verified that nonlinear models are often able to better simulate the actual situation. 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, called as Gilpin-Ayala competition model (GACM) by adding the nonlinear density constraint parameters. As pointed out in [6][7][8][9], when each of nonlinear density constraint parameters is much less than 1, GACM can well simulate the population ecology of Drosophila melanogaster. Naturally, reaction-diffusion Gilpin-Ayala models had been studied (see, e.g. [20,24]). But most of the related literatures mainly focus on the study of the competition model under Neumann boundary Email address: ruofengrao@163.com; ruofengrao@cdnu.edu.cn (Ruofeng Rao ) value condition ( [3,4,8] and related references), and the diffusive ecosystem under Dirichlet zero-boundary value is rarely studied. In fact, the Drichlet boundary value diffusion ecosystem can better reflect the actual situation for a class of species that can not live on the edge of the biosphere, for example, rabbits don't live on the edge of the grass where the degree of desertification is very serious. Recently, Ruofeng Rao, Quanxin Zhu and Kaibo Shi investigated the stability of positive stationary solution of reaction-diffusion Gilpin-Ayala competition model (RDGACM), but the null solution of RDGACM is not studied still. In fact, if we wish that pest or bacterial populations are destroyed, the stability of the null solution should be investigated. This inspires us to write this paper.
In this paper, our paper has the following innovative points: (I1) This paper, the author consider the stability of the null solution of RDGACM. Seldom papers involved in the stability of the null solution of ecosystem( [1][2][3][4][5][6][7][8][9]20,24]), but it has to be studied due to some practical situation. For example, rabbits are rampant in Australia, and carp are rampant in the United States. The stability analysis of the zero solution of this paper is closely related to the extinction of harmful species.
(I2) RDGACM with Dirichlet zero-boundary value is considered in this paper, which is suitable to research on a class of species that can not live on the edge of the biosphere. But there are many literature only involved in Neumann zero-boundary value( [3,4,8]). Seldom papers involved in Dirichlet zero-boundary value.
(I3) Dirichlet zero-boundary value of this paper brings about mathematical difficulties in the dynamical analysis. According [11,Remark 9], non-zero constant function may be a stationary solution of reaction-diffusion system with Neumann zero-boundary value, which can always be obtained by solving a simple algebraic equations. But [11, statement 1] shows that non-zero constant function must be not a stationary solution in the case of Dirichlet zero-boundary value. So in this paper we have to use variational methods to prove the existence or unique existence of stationary solution in the case of Dirichlet zero-boundary value. This is a main difficulty different from previous literature related to dynamical system with Neumann zero-boundary value ( [3,4,8,25,26] and their references therein). In fact, except [25,26] involving more complex mathematical tools, such as Laplacian semigroup or variational methods, most of the literature related with Neumann zero-boundary value only involved in simple mathematical methods dealing with the unique existence of the equilibrium point. In this paper, on one hand, the author utilize Mountain Pass Lemma to derive the existence of multiple stationary solutions. On the other hand, under boundedness assumption on the population densities, variational technique is applied to deal with the uniqueness of the stationary solution.
(I4) LMI-based criterion and free weight coefficient matrix technique reduce the conservatism of the algorithm in this paper. In fact, we seldom find out the LMI-based criterion in previous literature involved in nonlinear ecosystem (see. e.g., [1][2][3][4][5][6][7][8][9]20] and their references therein). Although LMI-based criterion or free weight coefficient matrix technique is common in many literature related to reaction-diffusion nonlinear system, such as [23], but the nonlinear active function of [23] is globally Lipschitz continuous, and the "active function" of this paper is locally Lipschitz continuous.
Particularly, the growth period is always affected by weather, temperature, humidity and other random factors, which can be reduced to a limited number of modes by specific statistical data.
(I6) In this paper, we shall give the sufficient condition on the existence of multiple stationary solutions of GACM, which directly warrants the local stability of the zero solution. This shows vividly the perfection of the sufficient condition.
In next sections, we shall give some models description in the chapter 2, and dynamic analysis for the Gilpin-Ayala competition model in Section 3, including the existence of multiple equilibrium points, and particularly caring about stability of the zero solution. In Chapter 4, the author will care about the global stability of the zero solution of RDGACM, which should be the unique equilibrium point under some suitable assumptions. Next, two numerical examples will be presented to show the effectiveness of the obtained results. Finally, some interesting conclusions will be proposed in Section 6.
For convenience, we introduce the following notations in this paper: is a positive definite matrix for two symmetric matrices A and B; • Denote by λ 1 the first positive eigenvalue of the Laplace operator −∆ in H 1 0 (Ω), and by ∥u∥ = • Denote simplyτ r (t) = d dt τ(r(t), t) for r(t) = r ∈ S , where S = {1, 2, · · · , n 0 }; • The mark * in a symmetric matrix represents the symmetric terms in the symmetric matrix; • Denote Z + = {1, 2, · · · }, and t 0 = 0; • Denote A −1 for any invertible matrix A; • Denote by λ max A and λ min A the maximum eigenvalue and minimum eigenvalue of a symmetric matrix A, respectively.

Preliminaries
Consider the nonlinear Reaction-diffusion Gilpin-Ayala competition model (RDGACM) under Dirichlet boundary value: where Ω is a domain in R n (n ∈ {2, 3}) with the smooth boundary ∂Ω. For i = 1, 2, u i (t, x) represents the population density of the ith population at time t and the spatial location x , b i > 0 represents the birth rate of the population of the ith species, and a i j > 0 represents the competition parameter between the species i and the species j. d i > 0 represents the diffusion coefficient for the species i. Initial value function ξ i (x) is bounded and continuous. . Let X is a Banach space, Ψ ∈ C 1 (X, R), satisfying Ψ(0) = 0 , and there exists ρ > 0 such that Ψ| ∂Bρ(0) α > 0. Besides, there is e ∈ X \ B ρ (0) such that Ψ(e) 0. Let Γ be the set of all paths connecting 0 and e. That is, 1] Ψ(ψ(s)).
Then c * α, and Ψ possesses a critical sequence on c * .
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 we assume θ i ∈ (0, 1), and consider the following condition for the upcoming section 3: where p i and q i are a pair of Coprime odd numbers.
Below, we may firstly give a dynamical analysis for GACM (2.1) in the section 3. After giving it, we shall further study the stability of delayed feedback Markovian jumping RDGACM in view of the viewpoint (I5) in the introduction section.

Dynamical analysis for RDGACM
In this section, we firstly analyze the number of equilibrium points on the phase plane of the RDGACM (2.1) with Dirichlet zero-boundary value.
Proof. Due to Ω ⊂ R 2 or Ω ⊂ R 3 , we may consider the case of Ω ⊂ R 3 , and another case can be similarly proved.
Firstly, (0, 0) is a trivial solution of the system (2.1). Obviously, is the functional corresponding to the equation (3.1), and J ∈ C 1 (H 1 0 (Ω), R 1 ). Besides, J(0) = 0. And Sobolev embedding theorem yields that there is c > 0 such that Let ρ > 0 small enough such that 11,17]). Then Let Γ be the set of all paths connecting 0 and −s 0 φ 1 , i.e., Below, similarly as those of [18], we will prove the sequence {u 1n } ∞ n=1 ⊂ H 1 0 (Ω) satisfying (3.10) must be bounded. In fact, (3.10) yields and and for ε > 0 small enough suc that there exits a n big enough such that So we have which means the boundedness of {u 1n } ∞ n=1 . Now we shall prove that the bounded sequence {u 1n } ∞ n=1 must be compact sequentially. This is only a conventional proof. However, in view of the completeness of the proof, we are willing to give the proof: In fact, (H1) means 1 where c 1 , c 2 are positive numbers big enough. Due to Ω ⊂ R 3 , then the critical Sobolev exponent is 6, and hence the operator Moreover, and then the bounded sequence {u 1n } ∞ n=1 possesses a subsequence, say, (3.10) and the arbitrariness of φ implies This shows that {u 1n } ∞ n=1 is compact sequentially. And then there exists a subsequence of {u 1n } ∞ n=1 convergent to a point in H 1 0 (Ω), say, u 1 * ∈ H 1 0 (Ω), Due to J(u 1 * ) = c 0 Similarly, we can similarly prove there is at least another stationary solution (0, u 2 * ) (0, 0) for the system (2.1).
Remark 3. Due to the mathematical difficulty brought about by Dirichlet zero-boundary value (see (I3) for details), we have to employ variational methods to overcome it, in which the variational technique developed in [18] ploys an important role. In sum, the difficulty is much bigger than that of Neumann zero-boundary value in many previous literature.  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

Global Stability with boundedness assumption on population densities
Theorem 3.2 illuminates that under the conditions of Theorem 3.2 or Theorem 3.1, a suitable initial value will make two population densities tend to zero eventually. In this section, we want to find out such suitable conditions that two population densities eventually tend to zero, no matter what the initial value is. So in this section, we do not need the condition (H1). Another 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}.
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. and there is a positive number β, a sequences of positive definite diagonal matrices P r (r ∈ S ), Q i > 0(i ∈ {1, 2}) and W > 0 then the null solution of the delayed RDGACM (4.7) is globally exponentially stable with convergence rate β 2 , where τ r (t) ∈ [0, τ] withτ r (t) τ * , and Indeed, let u ≡ u(x) be a stationary solution, satisfying (4.1), then it is obvious that (4.15) 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 ∫ Ω e βs u T (s, x)Wu(s, x)dxds.
Remark 7. The LMI-based stability criterion and free weight coefficient matrix make Theorem 4.1 reduce the conservatism of the algorithm.

Remark 8.
No matter what the initial value is, both of two population densities must eventually tend to zero under the conditions of Theorem 4.1, which gives us effective tips on how to eliminate pests.

Remark 9.
In the neural networks system [27, (1)], the self-feedback term A is a positive definite matrix such that −A is a negative definite matrix. But in our ecosystem (4.7) or (2.1), there is not such a similar negative definite matrix. Note the LMI condition (4.10) means a big negative definite matrix is necessary. And so the LMI condition (4.10) is harsh to some extent. But in Theorem 3.2, the sufficient conditions of local stability is not harsh at all. This implies that if we want to exterminate destructive insects inevitably no matter what the initial value is, we need human intervention to some extent. Impulse control technique should be considered. Besides, it is also an unreasonable assumption that time delay must be differentiable. So we shall abandon the abovementioned two unreasonable assumptions about negative definite matrix and differentiable delay time functions in a new theorem as follows.
Consider the following impulsive RDGACM: where each M k is a positive definite diagonal matrix with λ max M k < 1 for k ∈ Z + .
Theorem 4.2. Suppose the boundedness condition (4.1) holds. If there exits a positive definite diagonal matrix Λ 1 > 0 such that Indeed, assume u ≡ u(x) is a stationary solution, satisfying (4.1), then On the other hand, since A is a diagonal matrix, Below, we only need to prove the global exponential stability of (0, 0) T under the impulse control. Consider the Lyapunov function as follows, Let L be the weak infinitesimal operator such that Moreover, Let ε → 0, then (4.26) leads to Next, we claim that where Indeed, employing mathematical induction will lead to (2).
It follows from (H3) that there exists a positive constant M > 0 satisfying And then At first, we need to prove which implies that we only need to show In fact, it is obvious from (4.25) that Thus, if (6) does not hold, there must exist some t ∈ (t 0 , t 1 ) such that which implies that there is t * ∈ (t 0 , t 1 ), satisfying which together with (4.25) and ( * ) means that EV(t 0 ) c 2 ∥ξ∥ 2 τ < M∥ξ∥ 2 τ e −λ(t 1 −t 0 ) , and hence, there is t * * ∈ [t 0 , t * ) such that EV(t * * ) = c 2 ∥ξ∥ 2 τ and On the other hand, (4.25) yield This contradiction implies that (6) holds, and then (5) holds.
Next, we assume that (2) holds for k = 1, 2, · · · , m, or Below, we shall conclude It is obvious that Indeed, since M m is a diagonal matrix, Then the continuity of EV(t) on [t m , t m+1 ) derives And ( * * * ) yields t m t b , and hence t m < t b < t m+1 .
On the other hand, Now, employing the methods in [28, (3.48)-(3.50)] results in which together with D + EV(t) (σ − λ)V(t) and the condition (H3) means . This contradiction verifies (11), and hence mathematical induction demonstrates the claim (2), which together with (4.25) means that the null solution of the impulsive system (4.20) is globally exponential with convergence rate λ 2 . Then Theorem 4.1 yields that the null solution of the ecosystem (4.7) is globally exponentially stable.

Conclusions
Gilpin-Ayala competition model with Dirichlet zero-boundary condition simulates well a class of actual ecological situation, but it brings out many difficulties on dynamical analysis of RDGACM. In this paper, the author employs mountain pass lemma,