Global Stability for a Semidiscrete Logistic System with Feedback Control

Li Xu,Shanshan Lou,Ruiwen Han

doi:10.1155/2020/3189515

Abstract

In this paper, a semidiscrete logistic model with the Dirichlet boundary conditions and feedback controls is proposed. By means of the sub- and supper-solution method and eigenvalue theory, the unique positive equilibrium is proved. By constructing a suitable Lyapunov function, the global asymptomatic stability of the unique positive equilibrium is investigated. Finally, numerical simulations are presented to verify the effectiveness of the main results.

Highlights

PreliminariesFrom the view point of biology, we only need to discuss the positive solution of systems (5) and (6)
It is well known that spatial heterogeneity and dispersal play an important role in the dynamics of populations, such as the role of dispersal in the maintenance of patchiness, or spatial population variation [6,7,8,9]; space can be taken into account in all fundamental aspects of ecological organization
If the diffusion-driven instability should be avoided in some situations, and system parameters are not easy to adjust, some other ways should be adopted to achieve the stabilization aim [13]. ere may exist a situation where the equilibrium of the dynamical model is not the desirable one and a smaller value of the equilibrium is required. en, altering the model structure so as to make the population stabilize at a lower value is necessary [14]

Summary

Preliminaries

From the view point of biology, we only need to discuss the positive solution of systems (5) and (6). E solution of system (6) and (7) with the initial conditions (8) remains positive and bounded for all t > 0. We will consider the uniqueness and existence of nontrivial positive equilibrium solutions system (6) and (7). En, there exist solutions u and v for problem (15) such that x ≤ u ≤ v ≤ x. En, problem (15) admits at most one positive solution. System (17) has a unique positive solution if r > 4 D sin2(π/2(n + 1)). En, we only will consider the uniqueness and existence of nontrivial positive equilibrium solutions system (17) with the discrete Dirichlet boundary conditions (7). By using Lemma 2, problem (17) has a positive solution when the condition r > 4 D sin2(π/2 On the basis of Lemma 3, problem (17) admits at most one positive solution. On the basis of Lemma 3, problem (17) admits at most one positive solution. e proof is finished

Global Stability of Positive Equilibrium

Numerical Example

Conclusions