Long-time behavior of a nonautonomous stochastic predator–prey model with jumps

Abstract This work aims to study some dynamical properties of a stochastic predator–prey model, which is considered under the combination of Crowley–Martin functional response, disease in predator, and saturation incidence. First, we discuss the existence and uniqueness of positive solution of the concerned stochastic model. Second, we prove that the solution is stochastically ultimate bounded. Then, we investigate the extinction and the long-time behavior of the solution. Furthermore, we establish some conditions for the global attractivity of the model. Finally, we propose some numerical simulations to illustrate our main results.

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Long-time behavior of a stochastic predator–prey model with modified Leslie–Gower and Holling-type II schemes

International Journal of Biomathematics ◽

10.1142/s179352451650039x ◽

2016 ◽

Vol 09 (03) ◽

pp. 1650039 ◽

Cited By ~ 4

Author(s):

Yuguo Lin ◽

Daqing Jiang

Keyword(s):

Time Behavior ◽

Type Ii ◽

Invariant Density ◽

Long Time Behavior ◽

Singular Measure ◽

Predator Prey ◽

Predator Prey Model ◽

Prey Model ◽

Long Time

In this paper, we consider a stochastic predator–prey model with modified Leslie–Gower and Holling-type II schemes. We analyze long-time behavior of densities of the distributions of the solution. We prove that the densities can converge in [Formula: see text] to an invariant density or can converge weakly to a singular measure under appropriate conditions.

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ANALYSIS OF A PREDATOR–PREY MODEL WITH DISEASE IN THE PREY

International Journal of Biomathematics ◽

10.1142/s1793524513500125 ◽

2013 ◽

Vol 06 (03) ◽

pp. 1350012 ◽

Cited By ~ 4

Author(s):

CHUNYAN JI ◽

DAQING JIANG

Keyword(s):

White Noise ◽

Stochastic System ◽

Stochastic Perturbation ◽

Deterministic System ◽

Time Behavior ◽

Large White ◽

Predator Prey ◽

Predator Prey Model ◽

Prey Model ◽

Long Time

In this paper, we discuss the behavior of a predator–prey model with disease in the prey with and without stochastic perturbation, respectively. First, we briefly give the dynamic of the deterministic system, by analyzing stabilities of its four equilibria. Then, we consider the asymptotic behavior of the stochastic system. By Lyapunov analysis methods, we show the stochastic stability and its long time behavior around the equilibrium of the deterministic system. We obtain there are similar properties between the stochastic system and its corresponding deterministic system, when white noise is small. But large white noise can make a unstable deterministic system to be stable.

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Long-time behavior of a class of diffusive predator–prey systems with fear cost and protection zone

Applied Mathematics Letters ◽

10.1016/j.aml.2021.107388 ◽

2021 ◽

pp. 107388

Author(s):

Yaying Dong ◽

Shanbing Li

Keyword(s):

Time Behavior ◽

Long Time Behavior ◽

Predator Prey ◽

Protection Zone ◽

Long Time

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Dynamics of a Discrete Prey–Predator Model with Mixed Functional Response

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127419501992 ◽

2019 ◽

Vol 29 (14) ◽

pp. 1950199

Author(s):

Mohammed Fathy Elettreby ◽

Aisha Khawagi ◽

Tamer Nabil

Keyword(s):

Type I ◽

Time Behavior ◽

Stable Fixed Point ◽

Functional Responses ◽

Predator Prey ◽

Predator Prey Model ◽

Dynamical Behaviors ◽

Long Time ◽

Predator Model ◽

The Stability

In this paper, we propose a discrete Lotka–Volterra predator–prey model with Holling type-I and -II functional responses. We investigate the stability of the fixed points of this model. Also, we study the effects of changing each control parameter on the long-time behavior of the model. This model contains a lot of complex dynamical behaviors ranging from a stable fixed point to chaotic attractors. Finally, we illustrate the analytical results by some numerical simulations.

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MODEL PREDATOR-PREY DENGAN DUA PREDATOR

Jurnal Ilmiah Matematika dan Pendidikan Matematika ◽

10.20884/1.jmp.2013.5.1.2915 ◽

2013 ◽

Vol 5 (1) ◽

pp. 43

Author(s):

Danar Agus Nugroho ◽

Rina Reorita

Keyword(s):

Equilibrium Point ◽

Brown Planthopper ◽

Local Extinction ◽

Predator Prey ◽

Predator Prey Model ◽

Prey Model ◽

Long Time ◽

The Stability ◽

Brown Plant Hopper ◽

Coexistence State

This paper discussed about the predator-prey model with two predators. This model is a development of the model given by Korobeinikov and Wake (1999). Dynamic behavior of the model can be determined based on the stability of the equilibrium point. The stability of the equilibrium point of predator-prey model with two predators on the general ecosystem shows that there is no coexistence state (grown in tandem) on both predators and for a long time one of the predators will lead to the local extinction even though there is no competition between the two predators. Furthermore, this model is applied to the brown plant hopper predator, mirid prey and tomcat prey. The result shows that the population of brown planthopper and both of the predators will oscillate towards a particular value with a shorter span of time. In the long term, the number of brown planthopper and mirid will be heading to the equilibrium point, while the tomcat will lead to local extinction.

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QUALITATIVE ANALYSIS OF A STOCHASTIC PREDATOR–PREY SYSTEM WITH DISEASE IN THE PREDATOR

International Journal of Biomathematics ◽

10.1142/s1793524512500684 ◽

2013 ◽

Vol 06 (01) ◽

pp. 1250068 ◽

Cited By ~ 3

Author(s):

SHUANG LI ◽

XINAN ZHANG

Keyword(s):

White Noise ◽

Predator Population ◽

Deterministic System ◽

Time Behavior ◽

Long Time Behavior ◽

Predator Prey ◽

Prey System ◽

Long Time ◽

Global Positive Solution ◽

Disease Free

A stochastic predator–prey system with disease in the predator population is proposed, the existence of global positive solution is derived. When the white noise is small, there is a stationary distribution. In addition, conditions of global stability for the deterministic system are also established from the above result. By Lyapunov function, the long time behavior of solution around the disease-free equilibrium of deterministic system is derived. These results mean that stochastic system has the similar property with the corresponding deterministic system. When the white noise is small, however, large environmental noise makes the result different. Finally, numerical simulations are carried out to support our findings.

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