scholarly journals Asymptotic Analysis of Impulsive Dispersal Predator-Prey Systems with Markov Switching on Finite-State Space

2019 ◽  
Vol 2019 ◽  
pp. 1-18 ◽  
Author(s):  
Guodong Liu ◽  
Zhengbo Chang ◽  
Xinzhu Meng
Keyword(s):  
White Noise ◽  
Regime Switching ◽  
Prey Species ◽  
Stochastic Dynamics ◽  
Sufficient Conditions ◽  
Positive Periodic Solution ◽  
Impulsive System ◽  
Impulsive Effect ◽  
Predator Prey ◽  
Finite State

In this paper, we investigate the stochastic dynamics of two dispersal predator-prey systems perturbed by white noise, impulsive effect, and regime switching. For the system just interrupted by white noise, we first prove that the stochastic impulsive system has a nontrivial positive periodic solution. Then the sufficient conditions for persistence in mean and extinction of the system are obtained. For the system with Markov regime switching, we verify it is ergodic and has a stationary distribution. And conditions for extinction of the prey species are established. Finally, we provide a series of numerical simulations to illustrate the theoretical analysis.

scholarly journals Positive Periodic Solutions of a Neutral Impulsive Predator-Prey Model

2015 ◽  
Vol 2015 ◽  
pp. 1-11
Author(s):  
Guirong Liu ◽  
Xiaojuan Song
Keyword(s):  
Periodic Solutions ◽  
Sufficient Conditions ◽  
Impulsive Differential Equations ◽  
Impulsive System ◽  
Impulsive Effect ◽  
Neutral System ◽  
Positive Periodic Solutions ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model

We investigate a ratio-dependent predator-prey model with Holling type III functional response based on system of neutral impulsive differential equations. Sufficient conditions for existence of positive periodic solutions are obtained by applying continuation theorem. Our main results demonstrate that under the suitable periodic impulse perturbations, the neutral impulsive system preserves the periodicity of the corresponding neutral system without impulse. In addition, our results can be applied to the corresponding system without impulsive effect, and thus, extend previous results.

The global dynamics of stochastic Holling type II predator-prey models with non constant mortality rate

Filomat ◽  
2017 ◽  
Vol 31 (18) ◽  
pp. 5811-5825
Author(s):  
Xinhong Zhang
Keyword(s):  
Mortality Rate ◽  
Stochastic Model ◽  
Sufficient Conditions ◽  
Positive Periodic Solution ◽  
Global Dynamics ◽  
Type Ii ◽  
Predator Prey ◽  
Persistence In The Mean ◽  
The Mean ◽  
Predator Prey Models

In this paper we study the global dynamics of stochastic predator-prey models with non constant mortality rate and Holling type II response. Concretely, we establish sufficient conditions for the extinction and persistence in the mean of autonomous stochastic model and obtain a critical value between them. Then by constructing appropriate Lyapunov functions, we prove that there is a nontrivial positive periodic solution to the non-autonomous stochastic model. Finally, numerical examples are introduced to illustrate the results developed.

Stochastic dynamics for the solutions of a modified Holling–Tanner model with random perturbation

2014 ◽  
Vol 25 (11) ◽  
pp. 1450105 ◽  
Author(s):  
Zhenjie Liu
Keyword(s):  
Stochastic System ◽  
Stochastic Dynamics ◽  
Deterministic Model ◽  
Random Perturbation ◽  
Sufficient Conditions ◽  
Unique Positive Solution ◽  
Initial Value ◽  
Environmental Forcing ◽  
Predator Prey ◽  
Predator Prey Model

In this paper, we consider a stochastic nonautonomous predator–prey model with modified Leslie–Gower and Holling II schemes in the presence of environmental forcing. The deterministic model is the modified Holling–Tanner model which is an extension of the classical Leslie–Gower model. We show that there is a unique positive solution to the stochastic system for any positive initial value. Sufficient conditions for strong persistence in mean and extinction to the stochastic system are established.

THE PERIODIC PREDATOR-PREY LOTKA–VOLTERRA MODEL WITH IMPULSIVE EFFECT

2002 ◽  
Vol 02 (03n04) ◽  
pp. 267-296 ◽  
Author(s):  
SANYI TANG ◽  
LANSUN CHEN
Keyword(s):  
Periodic Solution ◽  
Positive Solutions ◽  
Sufficient Conditions ◽  
Coincidence Degree ◽  
Volterra Model ◽  
Impulsive Effect ◽  
Biological Pest Control ◽  
Globally Asymptotically Stable ◽  
Predator Prey ◽  
Special Cases

In this paper, a classical periodic Lotka–Volterra predator-prey system with impulsive effect is investigated. We analyze the dynamics of positive solutions of such models. Among other results we show that if some trivial or semi-trivial positive solution is linearly stable, then it is globally asymptotically stable with respect to the positive solutions. By using the method of coincidence degree, a set of sufficient conditions are derived for the existence of at least one strictly positive (componentwise) periodic solution. We use bifurcation theorem to show the existence of coexistence states which arise near the sem-trivial periodic solution. As an application, we also examine some special cases of the system which can be used in the biological pest control.

scholarly journals Periodicity in a ratio-dependent predator-prey system with stage structure for predator

2005 ◽  
Vol 2005 (2) ◽  
pp. 153-169 ◽  
Author(s):  
Fengde Chen
Keyword(s):  
Periodic Solution ◽  
Sufficient Conditions ◽  
Coincidence Degree ◽  
Stage Structure ◽  
Positive Periodic Solution ◽  
Positive Periodic Solutions ◽  
Priori Estimate ◽  
Predator Prey ◽  
Prey System ◽  
Ratio Dependent

With the help of a continuation theorem based on Gaines and Mawhin's coincidence degree, easily verifiable criteria are established for the global existence of positive periodic solutions of a delayed ratio-dependent predator-prey system with stage structure for predator. The approach involves some new technique of priori estimate. For the system without delay, by constructing a suitable Lyapunov function, some sufficient conditions which guarantee the existence of a unique global attractive positive periodic solution are obtained. Those results have further applications in population dynamics.

Periodic Solution of a Class Predator-Prey System with Rate Stocking and Time Delay

2013 ◽  
Vol 291-294 ◽  
pp. 2412-2415
Author(s):  
Hui Li ◽  
Yi Fei Wang
Keyword(s):  
Periodic Solution ◽  
Time Delay ◽  
Periodic System ◽  
Sufficient Conditions ◽  
Coincidence Degree ◽  
Degree Theory ◽  
Positive Periodic Solution ◽  
Coincidence Degree Theory ◽  
Predator Prey ◽  
Prey System

In this paper, we investigate of a class of predator-prey system with rate stocking and time delay, the existence positive periodic solution by using coincidence degree theory. We obtain the sufficient conditions which guarantee existence of the positive periodic solution of the periodic system. Some new results obtained.

EXISTENCE AND GLOBAL ATTRACTIVITY OF A POSITIVE PERIODIC SOLUTION FOR A NON-AUTONOMOUS PREDATOR-PREY MODEL UNDER VIRAL INFECTION

2009 ◽  
Vol 02 (04) ◽  
pp. 419-442 ◽  
Author(s):  
FENGYAN ZHOU
Keyword(s):  
Numerical Simulation ◽  
Periodic Solution ◽  
Lyapunov Function ◽  
Global Attractivity ◽  
Sufficient Conditions ◽  
Coincidence Degree ◽  
Positive Periodic Solution ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System

A new non-autonomous predator-prey system with the effect of viruses on the prey is investigated. By using the method of coincidence degree, some sufficient conditions are obtained for the existence of a positive periodic solution. Moreover, with the help of an appropriately chosen Lyapunov function, the global attractivity of the positive periodic solution is discussed. In the end, a numerical simulation is used to illustrate the feasibility of our results.

scholarly journals Dynamic Behavior of Positive Solutions for a Leslie Predator-Prey System with Mutual Interference and Feedback Controls

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Cong Zhang ◽  
Nan-jing Huang ◽  
Chuan-xian Deng
Keyword(s):  
Global Attractivity ◽  
Sufficient Conditions ◽  
Positive Periodic Solution ◽  
Mutual Interference ◽  
Periodic Case ◽  
Feedback Controls ◽  
Predator Prey ◽  
Prey System ◽  
Inequality Theory ◽  
Nonautonomous Case

We consider a Leslie predator-prey system with mutual interference and feedback controls. For general nonautonomous case, by using differential inequality theory and constructing a suitable Lyapunov functional, we obtain some sufficient conditions which guarantee the permanence and the global attractivity of the system. For the periodic case, we obtain some sufficient conditions which guarantee the existence, uniqueness, and stability of a positive periodic solution.

scholarly journals Permanence and periodic solution for a modified Leslie-Gower type predator-prey model with diffusion and non constant coefficients

BIOMATH ◽  
2017 ◽  
Vol 6 (1) ◽  
pp. 1707107
Author(s):  
Moussaoui Ali ◽  
M. A. Aziz Alaoui ◽  
R. Yafia
Keyword(s):  
Periodic Solution ◽  
System Modeling ◽  
Sufficient Conditions ◽  
Environmental Parameters ◽  
Positive Periodic Solution ◽  
Auxiliary Function ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Constant Coefficients

In this paper we study a predator-prey system, modeling the interaction of two species with diffusion and T-periodic environmental parameters. It is a Leslie-Gower type predator-prey model with Holling-type-II functional response. We establish some sufficient conditions for the ultimate boundedness of solutions and permanence of this system. By constructing an appropriate auxiliary function, the conditions for the existence of a unique globally stable positive periodic solution are also obtained. Numerical simulations are presented to illustrate the results.

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