scholarly journals Dynamical Analysis of a Stochastic Predator-Prey Model with an Allee Effect

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Feng Rao
Keyword(s):  
Asymptotic Stability ◽  
Lyapunov Functions ◽  
Allee Effect ◽  
Dynamical Analysis ◽  
Type Ii ◽  
Qualitative Properties ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Stochastic Boundedness ◽  
Prey Model

We present and analyze a modified Holling type-II predator-prey model that includes some important factors such as Allee effect, density-dependence, and environmental noise. By constructing suitable Lyapunov functions and applying Itô formula, some qualitative properties are given, such as the existence of global positive solutions, stochastic boundedness, and the global asymptotic stability. A series of numerical simulations to illustrate these mathematical findings are presented.

scholarly journals The Complex Dynamics of a Stochastic Predator-Prey Model

2012 ◽  
Vol 2012 ◽  
pp. 1-24 ◽  
Author(s):  
Xixi Wang ◽  
Huilin Huang ◽  
Yongli Cai ◽  
Weiming Wang
Keyword(s):  
Complex Dynamics ◽  
Deterministic Model ◽  
Qualitative Properties ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Stochastic Environments ◽  
Stochastic Boundedness ◽  
Prey Model ◽  
The Stability ◽  
Existence Of Equilibria

A modified stochastic ratio-dependent Leslie-Gower predator-prey model is formulated and analyzed. For the deterministic model, we focus on the existence of equilibria, local, and global stability; for the stochastic model, by applying Itô formula and constructing Lyapunov functions, some qualitative properties are given, such as the existence of global positive solutions, stochastic boundedness, and the global asymptotic stability. Based on these results, we perform a series of numerical simulations and make a comparative analysis of the stability of the model system within deterministic and stochastic environments.

scholarly journals Analysis of Stochastic Predator-Prey Model with Disease in the Prey and Holling Type II Functional Response

2020 ◽  
Vol 2020 ◽  
pp. 1-17
Author(s):  
C. Gokila ◽  
M. Sambath ◽  
K. Balachandran ◽  
Yong-Ki Ma
Keyword(s):  
Functional Response ◽  
Lyapunov Functions ◽  
Stochastic Systems ◽  
Uniform Continuity ◽  
Type Ii ◽  
Equilibrium Solutions ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Type Ii Functional Response

A stochastic predator-prey model with disease in the prey and Holling type II functional response is proposed and its dynamics is analyzed. We discuss the boundedness of the dynamical system and find all feasible equilibrium solutions. For the stochastic systems, we obtain the conditions for the existence of the global unique solution, boundedness, and uniform continuity. We derive the conditions for extinction and permanence of species. Moreover, we construct appropriate Lyapunov functions and discuss the asymptotic stability of equilibria. To illustrate our theoretical findings, we have performed numerical simulations and presented the results.

scholarly journals Dynamics Analysis of a Stochastic Leslie–Gower Predator-Prey Model with Feedback Controls

2019 ◽  
Vol 2019 ◽  
pp. 1-13 ◽  
Author(s):  
Huailan Ren ◽  
Wencai Zhao
Keyword(s):  
Lyapunov Functions ◽  
Type Ii ◽  
Functional Responses ◽  
Feedback Controls ◽  
Predator Prey ◽  
Noise Interference ◽  
Predator Prey Model ◽  
Prey Model ◽  
Global Positive Solution ◽  
Theoretical Results

This study focuses on the investigation of a stochastic Leslie–Gower predator-prey model with feedback controls and Holling type II functional responses. First, the existence and uniqueness of a global positive solution to the system under white noise interference are proved. Second, the conditions for the existence of the system’s positive recurrence are established by constructing suitable Lyapunov functions. Additionally, the persistence and extinction of prey and predator in the system are discussed, and the impacts of noise interference and feedback controls on the system are revealed. Finally, we validate the theoretical results by numerical simulations.

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