Dynamics of a stochastic Holling II predator-prey model with Lévy jumps and habitat complexity

2021 ◽  
pp. 2150077
Author(s):  
Guangjie Li ◽  
Qigui Yang
Keyword(s):  
Positive Solution ◽  
Habitat Complexity ◽  
Sufficient Conditions ◽  
Ultimate Boundedness ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Global Positive Solution ◽  
Lévy Jumps ◽  
Lyapunov Technique

This paper investigates a stochastic Holling II predator-prey model with Lévy jumps and habit complexity. It is first proved that the established model admits a unique global positive solution by employing the Lyapunov technique, and the stochastic ultimate boundedness of this positive solution is also obtained. Sufficient conditions are established for the extinction and persistence of this solution. Moreover, some numerical simulations are carried out to support the obtained results.

Analysis of a stochastic predator–prey model with prey subject to disease and Lévy noise

2019 ◽  
Vol 19 (05) ◽  
pp. 1950038
Author(s):  
Meihong Qiao ◽  
Shenglan Yuan
Keyword(s):  
Positive Solution ◽  
Asymptotic Properties ◽  
Sufficient Conditions ◽  
Lévy Noise ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Stochastic Boundedness ◽  
Prey Model ◽  
Global Positive Solution ◽  
Levy Noise

We consider a non-autonomous predator–prey model, with prey subject to the disease and Lévy noise. We show the existence of global positive solution and stochastic boundedness. Then, we examine the asymptotic properties of the solution. Finally, we offer sufficient conditions for persistence and extinction.

scholarly journals Permanence and global stability of positive solutions of a nonautonomous discrete ratio-dependent predator-prey model

2005 ◽  
Vol 2005 (2) ◽  
pp. 135-144 ◽  
Author(s):  
Hai-Feng Huo ◽  
Wan-Tong Li
Keyword(s):  
Lyapunov Function ◽  
Global Stability ◽  
Positive Solution ◽  
Positive Solutions ◽  
Sufficient Conditions ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Ratio Dependent

We first give sufficient conditions for the permanence of nonautonomous discrete ratio-dependent predator-prey model. By linearization of the model at positive solutions and construction of Lyapunov function, we also obtain some conditions which ensure that a positive solution of the model is stable and attracts all positive solutions.

scholarly journals A Stochastic Holling-Type II Predator-Prey Model with Stage Structure and Refuge for Prey

2021 ◽  
Vol 2021 ◽  
pp. 1-14
Author(s):  
Wanying Shi ◽  
Youlin Huang ◽  
Chunjin Wei ◽  
Shuwen Zhang
Keyword(s):  
Positive Solution ◽  
Existence And Uniqueness ◽  
Sufficient Conditions ◽  
Stage Structure ◽  
Predator Population ◽  
Type Ii ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Stochastically Ultimate Boundedness

In this paper, we study a stochastic Holling-type II predator-prey model with stage structure and refuge for prey. Firstly, the existence and uniqueness of the global positive solution of the system are proved. Secondly, the stochastically ultimate boundedness of the solution is discussed. Next, sufficient conditions for the existence and uniqueness of ergodic stationary distribution of the positive solution are established by constructing a suitable stochastic Lyapunov function. Then, sufficient conditions for the extinction of predator population in two cases and that of prey population in one case are obtained. Finally, some numerical simulations are presented to verify our results.

scholarly journals Persistence and Extinction of a Stochastic Modified Bazykin Predator-Prey System with Lévy Jumps

2018 ◽  
Vol 2018 ◽  
pp. 1-7 ◽  
Author(s):  
Zhangzhi Wei ◽  
Zheng Wu ◽  
Ling Hu ◽  
Lianglong Wang
Keyword(s):  
Positive Solution ◽  
Stochastic Analysis ◽  
Sufficient Conditions ◽  
Itô Formula ◽  
Ito Formula ◽  
Predator Prey ◽  
Prey System ◽  
Global Positive Solution ◽  
Lévy Jumps

This paper is devoted to the dynamics of a stochastic modified Bazykin predator-prey system with Lévy jumps. First, we show that the system has a unique global positive solution and give some properties of solutions. Then, some sufficient conditions for persistence and extinction are derived by Itô formula and some inequalities on stochastic analysis. At last, some simulations are provided to check the main results.

Effects of Toxicants on a Non-Autonomous Predator-Prey Model with Random Perturbation

2015 ◽  
Vol 737 ◽  
pp. 487-490
Author(s):  
Yan Zhang ◽  
Kuan Gang Fan ◽  
Qing Yun Wang
Keyword(s):  
Random Perturbation ◽  
Stochastic Perturbation ◽  
Ultimate Boundedness ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Persistence In The Mean ◽  
Global Positive Solution ◽  
The Mean ◽  
Stochastically Ultimate Boundedness

A new non-autonomous predator-prey model in a polluted environment with stochastic perturbation is considered in this paper. The existence of a global positive solution and stochastically ultimate boundedness are derived. Furthermore, some sufficient and necessary criteria for extinction, non-persistence in the mean, weak persistence in the mean, strong persistence in the mean are obtained. At last, a series of numerical simulations to illustrate our mathematical findings are presented.

Global stability of a stochastic predator–prey model with Allee effect

2015 ◽  
Vol 08 (04) ◽  
pp. 1550044 ◽  
Author(s):  
Baodan Tian ◽  
Liu Yang ◽  
Shouming Zhong
Keyword(s):  
Asymptotic Stability ◽  
Global Stability ◽  
Global Asymptotic Stability ◽  
Allee Effect ◽  
Sufficient Conditions ◽  
Initial Value ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Global Positive Solution

In this paper, we study a stochastic predator–prey model with Beddington–DeAngelis functional response and Allee effect, and show that there is a unique global positive solution to the system with the positive initial value. Sufficient conditions for global asymptotic stability are established. Some simulation figures are introduced to support the analytical findings.

scholarly journals Stochastic Predator-Prey System Subject to Lévy Jumps

2016 ◽  
Vol 2016 ◽  
pp. 1-13 ◽  
Author(s):  
Xinzhu Meng ◽  
Xiaohong Wang
Keyword(s):  
Theoretical Analysis ◽  
Differential Equations ◽  
Positive Solution ◽  
Sufficient Conditions ◽  
Stochastic Permanence ◽  
Predator Prey ◽  
Prey System ◽  
Global Positive Solution ◽  
Lévy Jumps ◽  
Permanence In Mean

This paper investigates a new nonautonomous impulsive stochastic predator-prey system with the omnivorous predator. First, we show that the system has a unique global positive solution for any given initial positive value. Second, the extinction of the system under some appropriate conditions is explored. In addition, we obtain the sufficient conditions for almost sure permanence in mean and stochastic permanence of the system by using the theory of impulsive stochastic differential equations. Finally, we discuss the biological implications of the main results and show that the large noise can make the system go extinct. Simulations are also carried out to illustrate our theoretical analysis conclusions.

BIFURCATION AND CHAOS IN A DISCRETE PREDATOR–PREY MODEL WITH HOLLING TYPE-III FUNCTIONAL RESPONSE AND HARVESTING EFFECT

2021 ◽  
pp. 1-28
Author(s):  
ANURAJ SINGH ◽  
PREETI DEOLIA
Keyword(s):  
Functional Response ◽  
Sufficient Conditions ◽  
Flip Bifurcation ◽  
Bifurcation Structure ◽  
Bifurcation And Chaos ◽  
Predator Prey ◽  
Type Iii ◽  
Predator Prey Model ◽  
Prey Model ◽  
Wide Range

In this paper, we study a discrete-time predator–prey model with Holling type-III functional response and harvesting in both species. A detailed bifurcation analysis, depending on some parameter, reveals a rich bifurcation structure, including transcritical bifurcation, flip bifurcation and Neimark–Sacker bifurcation. However, some sufficient conditions to guarantee the global asymptotic stability of the trivial fixed point and unique positive fixed points are also given. The existence of chaos in the sense of Li–Yorke has been established for the discrete system. The extensive numerical simulations are given to support the analytical findings. The system exhibits flip bifurcation and Neimark–Sacker bifurcation followed by wide range of dense chaos. Further, the chaos occurred in the system can be controlled by choosing suitable value of prey harvesting.

Dynamical behaviors of a diffusive predator–prey model with Beddington–DeAngelis functional response and disease in the prey

2017 ◽  
Vol 10 (08) ◽  
pp. 1750119 ◽  
Author(s):  
Wensheng Yang
Keyword(s):  
Lyapunov Function ◽  
Iterative Method ◽  
Functional Response ◽  
Local Stability ◽  
Sufficient Conditions ◽  
Positive Equilibrium ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Dynamical Behaviors ◽  
Prey Model

The dynamical behaviors of a diffusive predator–prey model with Beddington–DeAngelis functional response and disease in the prey is considered in this work. By applying the comparison principle, linearized method, Lyapunov function and iterative method, we are able to achieve sufficient conditions of the permanence, the local stability and global stability of the boundary equilibria and the positive equilibrium, respectively. Our result complements and supplements some known ones.

Qualitative analysis of a diffusive predator–prey model with Allee and fear effects

2021 ◽  
pp. 2150037
Author(s):  
Jia Liu
Keyword(s):  
Allee Effect ◽  
Sufficient Conditions ◽  
Allee Effects ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Fear Effect ◽  
Prey Model ◽  
Strong Allee Effect ◽  
Spatially Homogeneous ◽  
Fear Effects

In this study, we consider a diffusive predator–prey model with multiple Allee effects induced by fear factors. We investigate the existence, boundedness and permanence of the solution of the system. We also discuss the existence and non-existence of non-constant solutions. We derive sufficient conditions for spatially homogeneous (non-homogenous) Hopf bifurcation and steady state bifurcation. Theoretical and numerical simulations show that strong Allee effect and fear effect have great effect on the dynamics of system.

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