Asymptotic Stability Of Positive Equilibrium Research Articles

Bifurcation analysis and chaos control in a host‐parasitoid model

We investigate the qualitative behavior of a host‐parasitoid model with a strong Allee effect on the host. More precisely, we discuss the boundedness, existence and uniqueness of positive equilibrium, local asymptotic stability of positive equilibrium and existence of Neimark–Sacker bifurcation for the given system by using bifurcation theory. In order to control Neimark–Sacker bifurcation, we apply pole‐placement technique that is a modification of OGY method. Moreover, the hybrid control methodology is implemented in order to control Neimark–Sacker bifurcation. Numerical simulations are provided to illustrate theoretical discussion. Copyright © 2017 John Wiley & Sons, Ltd.

一类具有双时滞效应的溶解氧-浮游生物模型的动力学分析

基于海洋生态系统内在动态变化特性和海洋生物种群之间的动态作用机制，本论文构建了一类具有双时滞效应的溶解氧-浮游生物动态模型，并对其相关动力学性质进行理论分析，解析出模型内平衡点具有局部渐近稳定性和发生Hopf分支的阈值条件，并详细探析了Hopf分支的相关动力学性质。本研究成果有利于从动力学的角度揭示溶解氧与海洋浮游生物之间的相互作用机制，有助于深入理解时滞效应如何影响海洋生态系统的动态运行趋势，为进一步研究海洋浮游生物之间相互制约、相互协调的生长动态机制提供一定的理论支撑。 Based on the internal dynamic characteristics of marine ecosystem and the dynamic mechanism of marine biological population, this paper constructed a dissolved oxygen-plankton dynamic model involving two time delays, and made theoretical analysis of corresponding dynamics. We obtained the locally asymptotic stability of positive equilibrium and the threshold conditions of occurring Hopf bifurcation were gained. Therefore, we analyzed the dynamics of Hopf bifurcation in detail. This result provides a great help for the interaction between dissolved oxygen and marine plankton in dynamics, and is helpful to deeply understand how the delay affects the dynamic trend of the marine ecosystem, and furthermore provide certain theoretical support for the study of dynamic growth mechanism of mutual restriction and mutual coordination in marine plankton.

A Novel Method to Determine the Local Stability of the n-Species Lotka-Volterra System with Multiple Delays

The n-species Lotka-Volterra system with discrete delays is considered. The local asymptotic stability of positive equilibrium is investigated based on a contour integral method. The main purpose of this paper is to propose a new and general algorithm to study the local asymptotic stability of the positive equilibrium for then-dimensional Lotka-Volterra system. Some numerical experiments are carried out to show the effectiveness of the proposed method.

Global analysis of a delayed density dependent predator–prey model with Crowley–Martin functional response

In this paper, we present and study a two-dimensional continuous time dynamical system modelling a predator–prey with discrete delay incorporating Crowley–Martin functional response. Crowley–Martin functional response is similar to the Beddington–DeAngelis functional response but contains an extra term describing mutual interference by predator at high prey density. We consider the permanence, non-permanence, local asymptotic stability behaviour of various equilibrium points and global asymptotic stability of positive equilibrium to understand the dynamics of both delayed and non-delayed model systems. Global asymptotic stability is discussed constructing a suitable Lyapunov functional. We also show that increasing delays may cause bifurcations into periodic solutions. It is found that fluctuations in population levels arising due to gestation delay of predator can be prevented under certain parametric conditions. The direction and stability of Hopf bifurcation is also discussed by using normal form method and center manifold theory. In the end, some numerical simulations have been performed to substantiate our analytical findings.

The Stability of Solutions for a Fractional Predator-Prey System

We study a class of fractional predator-prey systems with Holling II functional response. A unique positive solution of this system is obtained. In order to prove the asymptotical stability of positive equilibrium for this system, we study the Lyapunov stability theory of a fractional system.

On the stability in a discrete two-species competition system

The paper discusses a two-species discrete competition system. The local stability of positive equilibrium is obtained. Further, a sufficient condition for the global asymptotic stability of positive equilibrium is established by using an iteration scheme and the comparison principle of difference equations.

Global asymptotic stability of positive equilibrium of three-species Lotka–Volterra mutualism models with diffusion and delay effects

In the mutualism system with three species if the effects of dispersion and time delays are both taken into consideration, then the densities of the cooperating species are governed by a coupled system of reaction–diffusion equations with time delays. The aim of this paper is to investigate the asymptotic behavior of the time-dependent solution in relation to a positive uniform solution of the corresponding steady-state problem in a bounded domain with Neumann boundary condition, including the existence and uniqueness of a positive steady-state solution. A simple and easily verifiable condition is given to ensure the global asymptotic stability of the positive steady-state solution. This result leads to the permanence of the mutualism system, the instability of the trivial and all forms of semitrivial solutions, and the nonexistence of nonuniform steady-state solution. The condition for the global asymptotic stability is independent of diffusion and time-delays as well as the net birth rate of species, and the conclusions for the reaction–diffusion system are directly applicable to the corresponding ordinary differential system and 2-species cooperating reaction–diffusion systems. Our approach to the problem is based on inequality skill and the method of upper and lower solutions for a more general reaction–diffusion system. Finally, the numerical simulation is given to illustrate our results.

STABILITY AND HOPF BIFURCATIONS IN A DELAYED PREDATOR–PREY SYSTEM WITH A DISTRIBUTED DELAY

This paper is concerned with a delayed Lotka–Volterra two-species predator–prey system with a distributed delay. By linearizing the system at the positive equilibrium and analyzing the associated characteristic equation, the asymptotic stability of positive equilibrium is investigated and Hopf bifurcations are demonstrated. It is found that the positive equilibrium of the system is always locally asymptotically stable when the delay kernel is the weak kernel while there is a stability switch of positive equilibrium when the delay kernel is the strong kernel and the system can undergo a Hopf bifurcation at the positive equilibrium when the average time delay in the delay kernel crosses certain critical values. In particular, by applying the normal form theory and center manifold reduction to functional differential equations (FDEs), the explicit formula determining the direction of Hopf bifurcations and the stability of bifurcated periodic solutions is given. Finally, some numerical simulations are also included to support the analytical results obtained.

STABILITY AND HOPF BIFURCATION FOR A DELAYED COOPERATIVE SYSTEM WITH DIFFUSION EFFECTS

The main purpose of this paper is to investigate the stability and Hopf bifurcation for a delayed two-species cooperative diffusion system with Neumann boundary conditions. By linearizing the system at the positive equilibrium and analyzing the corresponding characteristic equation, the asymptotic stability of positive equilibrium and the existence of Hopf oscillations are demonstrated. It is shown that, under certain conditions, the system undergoes only a spatially homogeneous Hopf bifurcation at the positive equilibrium when the delay crosses through a sequence of critical values; under the other conditions, except for the previous spatially homogeneous Hopf bifurcations, the system also undergoes a spatially inhomogeneous Hopf bifurcation at the positive equilibrium when the delay crosses through another sequence of critical values. In particular, in order to determine the direction and stability of periodic solutions bifurcating from spatially homogeneous Hopf bifurcations, the explicit formulas are given by using the normal form theory and the center manifold reduction for partial functional differential equations (PFDEs). Finally, to verify our theoretical predictions, some numerical simulations are also included.

Stability and Hopf bifurcation for a delayed cooperation diffusion system with Dirichlet boundary conditions

This paper is concerned with a delayed cooperation diffusion system with Dirichlet boundary conditions. By applying the implicit function theorem, the normal form theory and the center manifold reduction, the asymptotic stability of positive equilibrium and Hopf bifurcation are investigated. It is shown that an increase in delay will destabilize the positive equilibrium and lead to the occurrence of a supercritical Hopf bifurcation when the delay crosses through a sequence of critical values. Based on the normal form theory and the center manifold reduction for partial functional differential equations (PFDEs), we find that the bifurcating periodic solution occurring from the first Hopf bifurcation point is stable on the center manifold and those occurring from the other bifurcation points are unstable. Finally, some numerical simulations are given to illustrate our results.