scholarly journals Periodic solution and stationary distribution for stochastic predator-prey model with modified Leslie-Gower and Holling type II schemes

Filomat ◽  
2020 ◽  
Vol 34 (4) ◽  
pp. 1383-1402
Author(s):  
Qixing Han ◽  
Liang Chen ◽  
Daqing Jiang
Keyword(s):  
Periodic Solution ◽  
Computer Simulations ◽  
Stationary Distribution ◽  
Autonomous System ◽  
Sufficient Conditions ◽  
Type Ii ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System ◽  
Autonomous Case

In this paper, a stochastic predator-prey system with modified Leslie-Gower and Holling type II schemes is studied. For the autonomous case, we prove that the system has a stationary distribution under some parametric restrictions. We also obtain conditions for the non-persistence of the system, and the results are illustrated by computer simulations. For the non-autonomous system with continuous periodic coefficients, sufficient conditions which guarantee the existence of periodic solution of the system are established.

scholarly journals Dynamic behaviors of a nonautonomous predator–prey system with Holling type II schemes and a prey refuge

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Yumin Wu ◽  
Fengde Chen ◽  
Caifeng Du
Keyword(s):  
Lyapunov Function ◽  
Differential Equations ◽  
Comparison Theorem ◽  
Sufficient Conditions ◽  
Oscillation Theory ◽  
Type Ii ◽  
Prey Refuge ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System

AbstractIn this paper, we consider a nonautonomous predator–prey model with Holling type II schemes and a prey refuge. By applying the comparison theorem of differential equations and constructing a suitable Lyapunov function, sufficient conditions that guarantee the permanence and global stability of the system are obtained. By applying the oscillation theory and the comparison theorem of differential equations, a set of sufficient conditions that guarantee the extinction of the predator of the system is obtained.

scholarly journals Existence, Stationary Distribution, and Extinction of Predator-Prey System of Prey Dispersal with Stochastic Perturbation

2012 ◽  
Vol 2012 ◽  
pp. 1-24 ◽  
Author(s):  
Li Zu ◽  
Daqing Jiang ◽  
Fuquan Jiang
Keyword(s):  
Numerical Simulation ◽  
Stationary Distribution ◽  
Sufficient Conditions ◽  
Stochastic Perturbation ◽  
Ergodic Property ◽  
Unique Positive Solution ◽  
Initial Value ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System

We consider a predator-prey model in which the preys disperse amongnpatches (n≥2) with stochastic perturbation. We show that there is a unique positive solution and find out the sufficient conditions for the extinction to the system with any given positive initial value. In addition, we investigate that there exists a stationary distribution for the system and it has ergodic property. Finally, we illustrate the dynamic behavior of the system withn=2via numerical simulation.

Almost periodic solution of a modified Leslie–Gower predator–prey model with Holling-type II schemes and mutual interference

2014 ◽  
Vol 07 (03) ◽  
pp. 1450028 ◽  
Author(s):  
Shengbin Yu ◽  
Fengde Chen
Keyword(s):  
Periodic Solution ◽  
Sufficient Conditions ◽  
Mutual Interference ◽  
Almost Periodic Solution ◽  
Type Ii ◽  
Almost Periodic ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Globally Attractive

In this paper, we consider a modified Leslie–Gower predator–prey model with Holling-type II schemes and mutual interference. By applying the comparison theorem of the differential equation and constructing a suitable Lyapunov function, sufficient conditions which guarantee the permanence and existence of a unique globally attractive positive almost periodic solution of the system are obtained. Our results not only supplement but also improve some existing ones.

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 Hopf bifurcation and stability in a Beddington-DeAngelis predator-prey model with stage structure for predator and time delay incorporating prey refuge

2019 ◽  
Vol 17 (1) ◽  
pp. 141-159 ◽  
Author(s):  
Zaowang Xiao ◽  
Zhong Li ◽  
Zhenliang Zhu ◽  
Fengde Chen
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Sufficient Conditions ◽  
Stage Structure ◽  
Prey Refuge ◽  
Predator Species ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System ◽  
Bifurcation And Stability

Abstract In this paper, we consider a Beddington-DeAngelis predator-prey system with stage structure for predator and time delay incorporating prey refuge. By analyzing the characteristic equations, we study the local stability of the equilibrium of the system. Using the delay as a bifurcation parameter, the model undergoes a Hopf bifurcation at the coexistence equilibrium when the delay crosses some critical values. After that, by constructing a suitable Lyapunov functional, sufficient conditions are derived for the global stability of the system. Finally, the influence of prey refuge on densities of prey species and predator species is discussed.

scholarly journals Extinction and Persistence in Mean of a Novel Delay Impulsive Stochastic Infected Predator-Prey System with Jumps

Complexity ◽  
2017 ◽  
Vol 2017 ◽  
pp. 1-15 ◽  
Author(s):  
Guodong Liu ◽  
Xiaohong Wang ◽  
Xinzhu Meng ◽  
Shujing Gao
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Time Delays ◽  
Sufficient Conditions ◽  
Predator Prey ◽  
Persistence In Mean ◽  
Predator Prey Model ◽  
Prey System ◽  
Delay Differential ◽  
Lévy Jumps

In this paper, we explore an impulsive stochastic infected predator-prey system with Lévy jumps and delays. The main aim of this paper is to investigate the effects of time delays and impulse stochastic interference on dynamics of the predator-prey model. First, we prove some properties of the subsystem of the system. Second, in view of comparison theorem and limit superior theory, we obtain the sufficient conditions for the extinction of this system. Furthermore, persistence in mean of the system is also investigated by using the theory of impulsive stochastic differential equations (ISDE) and delay differential equations (DDE). Finally, we carry out some simulations to verify our main results and explain the biological implications.

Dynamics of a non-autonomous ratio-dependent predator—prey system

2003 ◽  
Vol 133 (1) ◽  
pp. 97-118 ◽  
Author(s):  
Meng Fan ◽  
Qian Wang ◽  
Xingfu Zou
Keyword(s):  
Periodic Solution ◽  
Asymptotic Stability ◽  
Positive Periodic Solution ◽  
Almost Periodic Solution ◽  
Almost Periodic ◽  
Predator Prey ◽  
Prey System ◽  
Autonomous Case ◽  
Positive Almost Periodic Solution ◽  
Ratio Dependent

We investigate a non-autonomous ratio-dependent predator–prey system, whose autonomous versions have been analysed by several authors. For the general non-autonomous case, we address such properties as positive invariance, permanence, non-persistence and the globally asymptotic stability for the system. For the periodic and almost-periodic cases, we obtain conditions for existence, uniqueness and stability of a positive periodic solution, and a positive almost-periodic solution, respectively.

DYNAMICS ON A HOLLING II PREDATOR–PREY MODEL WITH STATE-DEPENDENT IMPULSIVE CONTROL

2012 ◽  
Vol 05 (03) ◽  
pp. 1260006 ◽  
Author(s):  
BING LIU ◽  
YE TIAN ◽  
BAOLIN KANG
Keyword(s):  
Periodic Solution ◽  
Chemical Control ◽  
Impulsive Control ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System ◽  
State Dependent ◽  
Continuous Dynamic System ◽  
Continuous Dynamic ◽  
Holling Ii Functional Response

According to biological and chemical control strategy for pest control, a Holling II functional response predator–prey system concerning state-dependent impulsive control is investigated. We define the successor functions of semi-continuous dynamic system and give an existence theorem of order 1 periodic solution of such a system. By means of sequence convergence rules and qualitative analysis, we successfully get the conditions of existence and attractiveness of order 1 periodic solution. Our results show that our method used in this paper is more efficient and easier than the existing methods to prove the existence and attractiveness of order 1 periodic solution.

scholarly journals Dynamic Behaviors of a Nonautonomous Discrete Predator-Prey System Incorporating a Prey Refuge and Holling Type II Functional Response

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Yumin Wu ◽  
Fengde Chen ◽  
Wanlin Chen ◽  
Yuhua Lin
Keyword(s):  
Global Stability ◽  
Numerical Simulations ◽  
Functional Response ◽  
Sufficient Conditions ◽  
Type Ii ◽  
Prey Refuge ◽  
Predator Species ◽  
Predator Prey ◽  
Prey System ◽  
Type Ii Functional Response

A nonautonomous discrete predator-prey system incorporating a prey refuge and Holling type II functional response is studied in this paper. A set of sufficient conditions which guarantee the persistence and global stability of the system are obtained, respectively. Our results show that if refuge is large enough then predator species will be driven to extinction due to the lack of enough food. Two examples together with their numerical simulations show the feasibility of the main results.

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.

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