Existence, Uniqueness, and Input-to-State Stability of Ground State Stationary Strong Solution of a Single-Species Model via Mountain Pass Lemma

Ruofeng Rao,Jialin Huang,Quanxin Zhu,Eulalia Mart Nez

doi:10.1155/2021/8855351

Abstract

In this study, the authors utilize mountain pass lemma, variational methods, regularization technique, and the Lyapunov function method to derive the unique existence of the positive classical stationary solution of a single-species ecosystem. Particularly, the geometric characteristic of saddle point in the mountain pass lemma guarantees that the equilibrium point is the ground state stationary solution of the ecosystem. Based on the obtained uniqueness result, the authors use the Lyapunov function method to derive the globally exponential stability criterion, which illuminates that under some suitable conditions, a certain internal competition is conducive to the global stability of the population, and a certain amount of family planning is conducive to the overall stability of the population. Most notably, the regularity technique of weak stationary solution employed in this study can also be applied to some existing literature related with time-delays reaction-diffusion systems for the purpose of regularization of weak solutions. Finally, an illuminative numerical example shows the effectiveness of the proposed methods.

Highlights

System DescriptionsWhere cij ≥ 0 is the j(j ≠ i) and transition probability cii − 􏽐nj 0 1,j≠i cij, rate from δ>0 i to and lim\limits δ ⟶ 0o(δ)/δ 0
In this study, the authors utilize mountain pass lemma, variational methods, regularization technique, and the Lyapunov function method to derive the unique existence of the positive classical stationary solution of a single-species ecosystem
The geometric characteristic of saddle point in the mountain pass lemma guarantees that the equilibrium point is the ground state stationary solution of the ecosystem

Summary

System Descriptions

Where cij ≥ 0 is the j(j ≠ i) and transition probability cii − 􏽐nj 0 1,j≠i cij, rate from δ>0 i to and lim\limits δ ⟶ 0o(δ)/δ 0. Because only the adult is competitive for survival and there is a mature period from the larva to the adult, we consider the time-delayed system in this study, which is better to simulate this maturity problem. Let u∗(x) be a stationary solution of system (4) implies that u∗(x) is a solution of the following equation:. U|zΩ 0, where Ω is a Ck+2,α domain of Rn, and f satisfies the following conditions:. En, the solution of equation (6) in H10(Ω) is the strong solution. In addition, Ψ satisfies the PS condition, c∗ is a critical value of Ψ. Let Ψ be the functional corresponding to equation (6), u∗(x) must be a ground state solution of equation (6) if u∗(x) is a critical point of the functional Ψ with Ψ(u∗(x)) c∗ defined in (10) of Lemma 3

Main Result

Conclusions