scholarly journals Asymptotic Behaviors of a Delayed Nonautonomous Predator-Prey System Governed by Difference Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-15 ◽  
Author(s):  
Lili Liu ◽  
Zhijun Liu
Keyword(s):  
Numerical Simulations ◽  
Positive Solutions ◽  
Differential System ◽  
Difference Equations ◽  
Global Attractivity ◽  
Asymptotic Behaviors ◽  
Predator Prey ◽  
Prey System ◽  
Discretization Technique ◽  
Theoretical Results

Based on a predator-prey differential system with continuously distributed delays, we derive a corresponding difference version by using the method of a discretization technique. A good understanding of permanence of system and global attractivity of positive solutions of system is gained. An example and its numerical simulations are presented to substantiate our theoretical results.

scholarly journals Dynamical behaviors in a discrete fractional-order predator-prey system

Filomat ◽  
2018 ◽  
Vol 32 (17) ◽  
pp. 5857-5874 ◽  
Author(s):  
Yao Shi ◽  
Qiang Ma ◽  
Xiaohua Ding
Keyword(s):  
Numerical Simulations ◽  
Fixed Points ◽  
Fractional Order ◽  
Control Strategies ◽  
Flip Bifurcation ◽  
Predator Prey ◽  
Local Asymptotic Stability ◽  
Dynamical Behaviors ◽  
Prey System ◽  
Theoretical Results

This paper is related to the dynamical behaviors of a discrete-time fractional-order predatorprey model. We have investigated existence of positive fixed points and parametric conditions for local asymptotic stability of positive fixed points of this model. Moreover, it is also proved that the system undergoes Flip bifurcation and Neimark-Sacker bifurcation for positive fixed point. Various chaos control strategies are implemented for controlling the chaos due to Flip and Neimark-Sacker bifurcations. Finally, numerical simulations are provided to verify theoretical results. These results of numerical simulations demonstrate chaotic behaviors over a broad range of parameters. The computation of the maximum Lyapunov exponents confirms the presence of chaotic behaviors in the model.

A delayed predator–prey system with impulsive diffusion between two patches

2016 ◽  
Vol 10 (01) ◽  
pp. 1750010 ◽  
Author(s):  
Hong-Li Li ◽  
Long Zhang ◽  
Zhi-Dong Teng ◽  
Yao-Lin Jiang
Keyword(s):  
Time Delays ◽  
Impulsive Differential Equation ◽  
Global Attractivity ◽  
Sufficient Conditions ◽  
Predator Prey ◽  
Analysis Techniques ◽  
Prey System ◽  
Impulsive Diffusion ◽  
Population Interactions ◽  
Theoretical Results

In most models of population dynamics, diffusion between two patches is assumed to be either continuous or discrete. However, in the real world, it is often the case that diffusion occurs at certain moment every year, impulsive diffusion can provide a more suitable manner to model the actual dispersal (or migration) behaviors for many ecological species. In addition, it is generally recognized that some kinds of time delays are inevitable in population interactions. In view of these facts, a delayed predator–prey system with impulsive diffusion between two patches is proposed. By using comparison theorem of impulsive differential equation and some analysis techniques, criteria on the global attractivity of predator-extinction periodic solution are established, sufficient conditions for the permanence of system are obtained. Finally, numerical simulations are presented to illustrate our theoretical results.

Global dynamics of a stochastic ratio-dependent predator–prey system

2017 ◽  
Vol 10 (01) ◽  
pp. 1750002
Author(s):  
Xiaolin Fan ◽  
Zhidong Teng ◽  
Ahmadjan Muhammadhaji
Keyword(s):  
Lyapunov Functions ◽  
Global Attractivity ◽  
Sufficient Conditions ◽  
Upper Bounds ◽  
Global Dynamics ◽  
Predator Prey ◽  
Prey System ◽  
Dynamical Properties ◽  
Ratio Dependent ◽  
Theoretical Results

The dynamical properties of a stochastic non-autonomous ratio-dependent predator–prey system are studied by applying the theory of stochastic differential equations, Itô’s formula and the method of Lyapunov functions. First, the existence, the uniqueness and the positivity of the solution are discussed. Second the boundedness of the moments and the upper bounds for growth rates of prey and predator are studied. Moreover, the global attractivity of the system under some a weaker sufficient conditions are investigated. Finally, the theoretical results are confirmed by the special examples and the numerical simulations.

scholarly journals A Stochastic Predator-Prey System with Stage Structure for Predator

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Shufen Zhao ◽  
Minghui Song
Keyword(s):  
Global Existence ◽  
Global Stability ◽  
Numerical Simulations ◽  
Functional Response ◽  
Sufficient Conditions ◽  
Stage Structure ◽  
Predator Prey ◽  
Stability In Probability ◽  
Prey System ◽  
Theoretical Results

The authors introduce stochasticity into a predator-prey system with Beddington-DeAngelis functional response and stage structure for predator. We present the global existence and positivity of the solution and give sufficient conditions for the global stability in probability of the system. Numerical simulations are introduced to support the main theoretical results.

A Markov-switching predator–prey model with Allee effect for preys

2020 ◽  
Vol 13 (03) ◽  
pp. 2050018
Author(s):  
Xiaoxia Guo ◽  
Zhiming Guo
Keyword(s):  
Numerical Simulations ◽  
Allee Effect ◽  
Global Attractivity ◽  
Markov Switching ◽  
Sufficient Conditions ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Theoretical Results ◽  
Prey Populations

This paper concerns with a Markov-switching predator–prey model with Allee effect for preys. The conditions under which extinction of predator and prey populations occur have been established. Sufficient conditions are also given for persistence and global attractivity in mean. In addition, stability in the distribution of the system under consideration is derived under some assumptions. Finally, numerical simulations are carried out to illustrate theoretical results.

Global attractivity of the predator–prey system for cotton aphid and seven-spot ladybird beetle with stage structure

2018 ◽  
Vol 11 (01) ◽  
pp. 1850012
Author(s):  
Lifei Zheng ◽  
Guixin Hu ◽  
Huiyan Zhao ◽  
M. K. D. K. Piyaratne ◽  
Aying Wan
Keyword(s):  
Population Dynamics ◽  
Numerical Simulations ◽  
Natural Enemies ◽  
Global Attractivity ◽  
Equilibrium Points ◽  
Stage Structure ◽  
Cotton Aphid ◽  
Ladybird Beetle ◽  
Predator Prey ◽  
Prey System

It is well known that the cotton aphid is the major pest in cotton fields of Northwest China, and seven-spot ladybird is an important natural enemy among the various possible natural enemies of cotton aphid. In order to increase the applications of population dynamics in integrated pest management and control the cotton aphids biologically, we need to understand the population dynamics of cotton aphid and their natural enemies. A delay predator–prey system on cotton aphid and seven-spot ladybird beetle are proposed in this paper. Based on the comparison theorem and an iterative method, we investigate the global attractivity of the equilibrium points which have important biological meanings. Furthermore, some numerical simulations were carried out to illustrate and expand our theoretical results, in which a conjecture to generalize the well-known Theorem 16.4 in H. R. Thiemes book was put forward, which was taken as the open problem. The numerical simulations show coexistence of periodic solution, confirming the theoretical prediction.

scholarly journals Dynamic Behaviors of Holling Type II Predator-Prey System with Mutual Interference and Impulses

2014 ◽  
Vol 2014 ◽  
pp. 1-13 ◽  
Author(s):  
Hongli Li ◽  
Long Zhang ◽  
Zhidong Teng ◽  
Yaolin Jiang
Keyword(s):  
Global Attractivity ◽  
Sufficient Conditions ◽  
Positive Periodic Solution ◽  
Mutual Interference ◽  
Individual Growth ◽  
Type Ii ◽  
Variable Environment ◽  
Predator Prey ◽  
Prey System ◽  
Theoretical Results

A class of Holling type II predator-prey systems with mutual interference and impulses is presented. Sufficient conditions for the permanence, extinction, and global attractivity of system are obtained. The existence and uniqueness of positive periodic solution are also established. Numerical simulations are carried out to illustrate the theoretical results. Meanwhile, they indicate that dynamics of species are very sensitive with the period matching between species’ intrinsic disciplinarians and the perturbations from the variable environment. If the periods between individual growth and impulse perturbations match well, then the dynamics of species periodically change. If they mismatch each other, the dynamics differ from period to period until there is chaos.

Stability and Bifurcation in a State-Dependent Delayed Predator–Prey System

2016 ◽  
Vol 26 (04) ◽  
pp. 1650060 ◽  
Author(s):  
Aiyu Hou ◽  
Shangjiang Guo
Keyword(s):  
Hopf Bifurcation ◽  
Numerical Simulations ◽  
Local Stability ◽  
Perturbation Methods ◽  
Positive Equilibrium ◽  
Predator Prey ◽  
Prey System ◽  
State Dependent ◽  
The Stability ◽  
Theoretical Results

In this paper, we consider a class of predator–prey equations with state-dependent delayed feedback. Firstly, we investigate the local stability of the positive equilibrium and the existence of the Hopf bifurcation. Then we use perturbation methods to determine the sub/supercriticality of Hopf bifurcation and hence the stability of Hopf bifurcating periodic solutions. Finally, numerical simulations supporting our theoretical results are also provided.

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