Unique existence of globally asymptotical input-to-state stability of positive stationary solution for impulsive Gilpin-Ayala competition model with diffusion and delayed feedback under Dirichlet zero boundary value

By partly generalizing the Lipschitz condition of existing results to the generalized Lipschitz one, the author utilizes a fixed point theorem, variational method and Lyapunov function method to derive the unique existence of globally asymptotical input-to-state stability of positive stationary solution for Gilpin-Ayala competition model with diffusion and delayed feedback under Dirichlet zero boundary value. Remarkably, it is the first paper to derive the unique existence of the stationary solution of reaction-diffusion Gilpin-Ayala competition model, which is globally asymptotical input-to-state stability. And numerical examples illuminate the effectiveness and feasibility of the proposed methods.


Introduction
Delayed ecosystem or reaction-diffusion ecosystem has been investigated for a long time (see, e.g. [1][2][3][4]10,[12][13][14]16] and the references therein). But most of the related literature only involved in the Neumann zero boundary value. In real world, Dirichlet zero boundary value can sometimes better simulate the population ecology, for example, the population density of deep-sea fish at the edge of their life circle is zero, and out of the circle may mean that they cannot adapt to the environment. Besides, the delayed feedback model is introduced in this paper, for the larval individuals in the population often have a certain growth period, and only adults can participate in the food competition among populations. Such delayed feedback models are not only suitable for biological population competition model, but also common to other dynamic models ( [14][15][16]). In addition, Markov models can always simulate the competition systems of biological population with random factors and other dynamical systems ( [9,17]). In addition, multiple-species competition models are always linear ones . For example, even in 2017, Yuanyuan Liu and Youshan Tao investigated the following the following two-species linear competition model with cross-diffusion for one species under Neumann boundary value ( [4]): u(x, 0) =u 0 (x), x ∈ Ω. (1.1) Until 1973, Gilpin and Ayala found that the model did not match a series of experimental data well( [5]). Via accurate data analysis, they proposed the following nonlinear competition model with two-species: , (1.2) in which θ 1 , θ 2 represent the nonlinear density restrictions. As pointed out in [6][7][8] that the nonlinear density restrictions model can match well the experimental data on drosophila melanogasters when θ i was far less than 1. From then, Gilpin-Ayala ecosystems have been investigated extensively (see, e.g. [3,12,13,17]). Even various reaction-diffusion Gilpin-Ayala competition models were investigated under Neumann boundary value (see, e.g. [2][3][4]). But seldom reaction-diffusion two-species competition models were studied under Dirichlet boundary value. In fact, there are many cases suitable to the Dirichlet boundary problem. For example, deep sea fish live in a certain range of three-dimensional waters, and in their area edge, the population density of deep sea fish is zero. Besides, the living range of some pollens is also affected by their regional environment. They only spread in a certain area, and the population density of the living pollens on the edge of the area is zero. Furthermore, input-to-state stability was studied in many literature involved in various dynamical systems (see [18][19][20][21][22] ), which is also suitable to ecosystem. In fact, putting a certain amount of food and small fry in the fish pond can be seen as the external input, which can make the dynamic of the ecosystem stabilized at a positive equilibrium point. By employing the methods used in my another paper [11], I shall utilize a fixed point theorem, variational method, and Lyapunov function method to derive the unique existence of the stationary solution of reaction-diffusion Gilpin-Ayala competition model, which is globally asymptotical input-to-state stability. This paper involves in the following innovations or novelties: It is the first paper to derive the unique existence of the stationary solution of reaction-diffusion Gilpin-Ayala competition model, which is globally asymptotical input-to-state stability.
Different from Neumann boundary problem, the non-zero constant equilibrium point is not the solution for the ecosystem with Dirichlet boundary value (see [11]), which brings about more mathematical difficulties.
Partly generalizing the Lipschitz condition of [11,Theorem 3.1] or [11,Theorem 3.2] to the Lipschitz one in the broad sense.

Remark 1.
Here, we assume Ω ⊂ R 3 . And if two species live in two dimensional plane, we can assume ) is a positive stationary solution of the system (2.1). Set

2)
and the stationary solution (u * 1 (x), u * 2 (x)) of the system (2.1) corresponds to the zero solution (0, 0) of the following system: where we denote U = (U 1 , U 2 ) T , and (2.4) The following system is the system (2.3) in form of vector-matrix: Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 3 September 2020 doi:10.20944/preprints202009.0052.v1 Under impulse control on (2.5), one can get the following system Besides, the bounded initial value of the system (2.7) is proposed as follows,
Next, the following assumption on the population density may be necessary: (H1) There are positive numbers M i , N i such that Everyone knows the fact that the population density of any species must have the bounded below, or the species will die out. For example, When the population density of whales is lower than a certain degree, it will be difficult for male and female whales to meet each other in the vast sea, leading to the extinction of the species. Besides, due to the limited resource, the population density of any species must have an supper boundedness. Next, the following existence of positive stationary solution comes mainly from [11, Theorem 3.1]. Of course, the ecosystem (2.5) is involved in non-Lipschitz functions, and so the author has to generalize the first conclusion of [11, Theorem 3.1] from the Lipschitz condition to the generalized Lipschitz condition.
Proof. Firstly definite the so-called generalized Lipschitz condition as follows, f (u 1 , u 2 ) is said to satisfy the generalized Lipschitz condition if there are constantsl 1 ,l 2 > 0 such that Indeed, let (u 1 (x), u 2 (x)) is the stationary solution, satisfying

5)
The condition (H1) yields that there are four positive constants l 1 , l 2 , l 3 and l 4 such that

7)
where If the stationary solution of the system (2.1) exists, I may denote it by u * (x) = (u * 1 (x), u * 2 (x)) T . Define the operator M : [C(Ω σ )] 2 → [C(Ω σ )] 2 as follows, The operator M has the inverse operator M −1 as follows, is a linear compact positive operator (see, e.g. [11]), and It is obvious that D −1 g(u * (x)) + D −1 χ is continuous for all the variables x, u * 1 , u * 2 . Define then K is a positive cone, which must be a closed convex subset of [C(Ω σ )] 2 . Define an operator T : K → K such that Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 3 September 2020 doi:10.20944/preprints202009.0052.v1 Because M −1 is the linear positive compact operator, and D −1 g(u * (x)) + D −1 χ is positive continuous, one can conclude that T : K → K is a positive compact operator. Next, completely similar as the proof of [11, Theorem 3.1], one can utilize the fixed point theorem ([11, Lemma 2.1]) to prove that T satisfies all the assumption conditions of [11, Lemma 2.1], which implies that T has at least a fixed point in K . And u * is a bounded positive solution of the system (2.1).
Next, [11,Theorem 3.2] proposes the methods which may be helpful to conclude the following uniqueness result: Theorem 3.2. Based on the assumptions of Theorem 3.1, and suppose, in addition, the following condition is satisfied, (H2) for any mode r(t) = r, there exists a scalar ε > 0 such that Proof. Assume both u(x) and v(x) are the stationary solutions of the system (2.1). Then we claim and v(x) are the stationary solutions of the system (2.1), one can see it from (3.10), variational method and the Poincare inequality that (3.11) Now the condition (H2) yields the claim via the proof by contradiction. And so the system (2.1) possesses a unique positive bounded stationaru solution u * (x) for x ∈ Ω σ with u * | ∂Ω σ = 0.
Below, I shall prove that the above-mentioned positive bounded vector function u * (x) is globally exponentially stable, which is the unique stationary solution of the system (2.1), corresponding to the null solution of the system (2.7). Theorem 3.3. Suppose the conditions (H1),(H2) and (3.2) hold. In addition, there is a sequence positive definite matrices P r (r ∈ S), positive numbers w r , π r (r ∈ S), ε, ε 1 , ε 2 ,γ, ς, λ such that (3.13) 0 < w r I P r π r I, ∀ r ∈ S, (3.14) where γ then the unique positive bounded stationary solution u * (x) is globally exponential input-to-state stability for 0 < |χ| < J. At the same time, the null solution of the impulsive system (2.7) with initial value (2.8) is globally exponential input-to-state stability with the convergence rate λ 2 .
Proof. Consider the following Lyapunov function: Below, the Poincare inequality is employed to deal with the diffusion item, just like the related literature (see, e.g. [23]). Let L be the weak infinitesimal operator (see, e.g. [23]) such that On the other hand, LV(t, r) Ω U T [−2λ 1 DP r + 2(B + K r )P r + ∑ j∈S γ rj P j ]U + [|U| T P r |Φ(U)| + |Φ(U)| T P r |U|] 20) which implies that for a small enough positive number ,

ELV(s)ds,
and letting → 0, it leads to Due to the conditions (3.12)-(3.14) and the proof of [15, Theorem 3.3], we can similarly prove and obtain the following inequality: where M > 1 is a constant. Moreover, it follows by (3.22) and (3.14) that (min in which the positive constant max r∈S π r min r∈S w r M is independent of any r ∈ S. Therefore, the unique positive bounded stationary solution u * (x) is globally exponential input-to-state stability for 0 < |χ| < J. At the same time, the null solution of the impulsive system (2.7) with initial value (2.8) is globally exponential input-to-state stability with the convergence rate λ 2 .

Numerical example
Example 4.1. Consider the following system: [11,Remark 14]). Let and w 1 = 0.98, π 1 = 1.005, w 2 = 0.99, π 2 = 1.07, then the condition (3.14) holds obviously. Assume the pulse interval (t k+1 − t k ) = 0.5, for all k ∈ Z + , and Now, we set γ = 26, then we get γ = 26 25 = 1 λ max A T k A k , k ∈ Z + . Set ς = 5 and λ = 1, then the direct calculation makes the condition (3.13) hold. Besides, which makes the condition (3.12) holden. Now, all the conditions of Theorem 3.3 are satisfied. According to Theorem 3.3, the system (4.1) possesses the unique positive bounded stationary solution u * (x), which is globally exponential input-to-state stability with the convergence rate λ 2 = 0.5. Example 4.2. In Example 4.1, replace the impulse quantity (4.2) with the following stronger pulse amplitude: which makes the condition (3.12) hold. Now, all the conditions of Theorem 3.3 are satisfied. According to Theorem 3.3, the system (4.1) possesses the unique positive stationary solution (u * 1 , u * 2 ), which is globally exponentially stabilized under impulse control with the convergence rate λ 2 = 1.  Table 1 illuminates that under the same pulse frequency, the higher the pulse intensity, the faster the convergence speed.
which makes the condition (3.12) hold. Now, all the conditions of Theorem 3.3 are satisfied. According to Theorem 3.2, the system (4.1) possesses the unique positive stationary solution (u * 1 , u * 2 ), which is globally exponentially stabilized under impulse control with the convergence rate λ 2 = 0.75.   Table 2 reveals that under the same pulse amplitude, the higher the pulse frequency, the faster the convergence speed.

Conclusions and further consideration
The ecosystem with Dirichelt zero boundary value represents that the nature has limited resources, and population density of the species is zero on the edge of the limited ecological resources, which is entirely in line with some actual situations. Gilpin and Ayala in [5] pointed out that the model did not match a series of experimental data well. Via accurate data analysis, they proposed the nonlinear competition model with two-species, in which θ 1 , θ 2 represent the nonlinear density restrictions. As pointed out in [6][7][8] that the nonlinear density restrictions model can match well the experimental data on drosophila melanogasters when θ i was far less than 1. So, in this paper, the author considers the nonlinear density restrictions model with θ i < 1. Utilizing the fixed point theorem, variational method and Lyapunov function method results in the unique existence of the stationary solution of reaction-diffusion Gilpin-Ayala competition model, which is globally asymptotical input-to-state stability. Numerical examples illustrate that improving pulse frequency and pulse amplitude is helpful to make the ecosystem stabilized quickly. Now, the further consideration is, how to study the bi-stabilization of reaction-diffusion two species competition model with Dirichlet boundary value under invasion of infectious diseases. Especially in the novel coronavirus pneumonia epidemic today, it is an interesting problem.