Global dynamics of a stochastic ratio-dependent predator–prey system

2017 ◽  
Vol 10 (01) ◽  
pp. 1750002
Author(s):  
Xiaolin Fan ◽  
Zhidong Teng ◽  
Ahmadjan Muhammadhaji
Keyword(s):  
Lyapunov Functions ◽  
Global Attractivity ◽  
Sufficient Conditions ◽  
Upper Bounds ◽  
Global Dynamics ◽  
Predator Prey ◽  
Prey System ◽  
Dynamical Properties ◽  
Ratio Dependent ◽  
Theoretical Results

The dynamical properties of a stochastic non-autonomous ratio-dependent predator–prey system are studied by applying the theory of stochastic differential equations, Itô’s formula and the method of Lyapunov functions. First, the existence, the uniqueness and the positivity of the solution are discussed. Second the boundedness of the moments and the upper bounds for growth rates of prey and predator are studied. Moreover, the global attractivity of the system under some a weaker sufficient conditions are investigated. Finally, the theoretical results are confirmed by the special examples and the numerical simulations.

A delayed predator–prey system with impulsive diffusion between two patches

2016 ◽  
Vol 10 (01) ◽  
pp. 1750010 ◽  
Author(s):  
Hong-Li Li ◽  
Long Zhang ◽  
Zhi-Dong Teng ◽  
Yao-Lin Jiang
Keyword(s):  
Time Delays ◽  
Impulsive Differential Equation ◽  
Global Attractivity ◽  
Sufficient Conditions ◽  
Predator Prey ◽  
Analysis Techniques ◽  
Prey System ◽  
Impulsive Diffusion ◽  
Population Interactions ◽  
Theoretical Results

In most models of population dynamics, diffusion between two patches is assumed to be either continuous or discrete. However, in the real world, it is often the case that diffusion occurs at certain moment every year, impulsive diffusion can provide a more suitable manner to model the actual dispersal (or migration) behaviors for many ecological species. In addition, it is generally recognized that some kinds of time delays are inevitable in population interactions. In view of these facts, a delayed predator–prey system with impulsive diffusion between two patches is proposed. By using comparison theorem of impulsive differential equation and some analysis techniques, criteria on the global attractivity of predator-extinction periodic solution are established, sufficient conditions for the permanence of system are obtained. Finally, numerical simulations are presented to illustrate our theoretical results.

scholarly journals Dynamical behavior of a stochastic ratio-dependent predator-prey system with Holling type IV functional response

Filomat ◽  
2018 ◽  
Vol 32 (19) ◽  
pp. 6549-6562
Author(s):  
Jing Fu ◽  
Daqing Jiang ◽  
Ningzhong Shi ◽  
Tasawar Hayat ◽  
Baslur Abmad
Keyword(s):  
Functional Response ◽  
Stochastic Equation ◽  
Dynamical Behavior ◽  
Sufficient Conditions ◽  
Initial Value ◽  
Predator Prey ◽  
Type Iv ◽  
Prey System ◽  
Dynamical Properties ◽  
Ratio Dependent

In this paper, we investigate the dynamical properties of a stochastic ratio-dependent predatorprey system with Holling type IV functional response. The existence of the globally positive solutions to the system with positive initial value is shown employing comparison theorem of stochastic equation and It??s formula. We derived some sufficient conditions for the persistence in mean and extinction. This system has a stable stationary distribution which is ergodic. Numerical simulations are carried out for further support of present research.

scholarly journals Dynamics Behaviors of a Discrete Ratio-Dependent Predator-Prey System with Holling Type III Functional Response and Feedback Controls

2008 ◽  
Vol 2008 ◽  
pp. 1-19 ◽  
Author(s):  
Jinghui Yang
Keyword(s):  
Lyapunov Function ◽  
Functional Response ◽  
Global Attractivity ◽  
Sufficient Conditions ◽  
Feedback Controls ◽  
Predator Species ◽  
Predator Prey ◽  
Type Iii ◽  
Prey System ◽  
Ratio Dependent

A ratio-dependent predator-prey system with Holling type III functional response and feedback controls is proposed. By constructing a suitable Lyapunov function and using the comparison theorem of difference equation, sufficient conditions which ensure the permanence and global attractivity of the system are obtained. After that, under some suitable conditions, we show that the predator speciesywill be driven to extinction. Examples together with their numerical simulations show that the main results are verifiable.

scholarly journals Dynamic Behaviors of Holling Type II Predator-Prey System with Mutual Interference and Impulses

2014 ◽  
Vol 2014 ◽  
pp. 1-13 ◽  
Author(s):  
Hongli Li ◽  
Long Zhang ◽  
Zhidong Teng ◽  
Yaolin Jiang
Keyword(s):  
Global Attractivity ◽  
Sufficient Conditions ◽  
Positive Periodic Solution ◽  
Mutual Interference ◽  
Individual Growth ◽  
Type Ii ◽  
Variable Environment ◽  
Predator Prey ◽  
Prey System ◽  
Theoretical Results

A class of Holling type II predator-prey systems with mutual interference and impulses is presented. Sufficient conditions for the permanence, extinction, and global attractivity of system are obtained. The existence and uniqueness of positive periodic solution are also established. Numerical simulations are carried out to illustrate the theoretical results. Meanwhile, they indicate that dynamics of species are very sensitive with the period matching between species’ intrinsic disciplinarians and the perturbations from the variable environment. If the periods between individual growth and impulse perturbations match well, then the dynamics of species periodically change. If they mismatch each other, the dynamics differ from period to period until there is chaos.

Global dynamics of a ratio-dependent predator-prey system

2001 ◽  
Vol 43 (3) ◽  
pp. 268-290 ◽  
Author(s):  
Dongmei Xiao ◽  
Shigui Ruan
Keyword(s):  
Global Dynamics ◽  
Predator Prey ◽  
Prey System ◽  
Ratio Dependent

GLOBAL ASYMPTOTIC STABILITY FOR A TWO-SPECIES DISCRETE RATIO-DEPENDENT PREDATOR–PREY SYSTEM

2013 ◽  
Vol 06 (01) ◽  
pp. 1250064 ◽  
Author(s):  
XIANGLAI ZHUO
Keyword(s):  
Asymptotic Stability ◽  
Global Stability ◽  
Global Asymptotic Stability ◽  
Sufficient Conditions ◽  
Positive Equilibrium ◽  
Predator Prey ◽  
Prey System ◽  
Ratio Dependent ◽  
Analysis Of Stability ◽  
Appl Math

The dynamical behaviors of a two-species discrete ratio-dependent predator–prey system are considered. Some sufficient conditions for the local stability of the equilibria is obtained by using the linearization method. Further, we also obtain a new sufficient condition to ensure that the positive equilibrium is globally asymptotically stable by using an iteration scheme and the comparison principle of difference equations, which generalizes what paper [G. Chen, Z. Teng and Z. Hu, Analysis of stability for a discrete ratio-dependent predator–prey system, Indian J. Pure Appl. Math.42(1) (2011) 1–26] has done. The method given in this paper is new and very resultful comparing with papers [H. F. Huo and W. T. Li, Existence and global stability of periodic solutions of a discrete predator–prey system with delays, Appl. Math. Comput.153 (2004) 337–351; X. Liao, S. Zhou and Y. Chen, On permanence and global stability in a general Gilpin–Ayala competition predator–prey discrete system, Appl. Math. Comput.190 (2007) 500–509] and it can also be applied to study the global asymptotic stability for general multiple species discrete population systems. At the end of this paper, we present an open question.

Hopf Bifurcation of a Generalized Delay-Induced Predator–Prey System with Habitat Complexity

2020 ◽  
Vol 30 (06) ◽  
pp. 2050082
Author(s):  
Zhihui Ma
Keyword(s):  
Hopf Bifurcation ◽  
Habitat Complexity ◽  
Sufficient Conditions ◽  
Explicit Formulas ◽  
Predator Prey ◽  
Dynamical Behaviors ◽  
Prey System ◽  
The Stability ◽  
Theoretical Results ◽  
Positivity And Boundedness

A delay-induced nonautonomous predator–prey system with variable habitat complexity is proposed based on mathematical and ecological issues, and this system is more realistic than the published models. Firstly, the permanence of the nonautonomous predation system is studied and some sufficient conditions are obtained. Secondly, the dynamical behaviors of the corresponding autonomous predation system are investigated, including the positivity and boundedness, and local and global stabilities. Thirdly, the properties of Hopf bifurcation of the autonomous predation system without/with delay are investigated and the explicit formulas which determine the stability and the direction of periodic solutions are obtained. Finally, a numerical example is given to test our theoretical results.

scholarly journals Permanence for a Discrete Ratio-Dependent Predator-Prey System with Holling Type III Functional Response and Feedback Controls

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Jiangbin Chen ◽  
Shengbin Yu
Keyword(s):  
Feedback Control ◽  
Functional Response ◽  
Sufficient Conditions ◽  
Feedback Controls ◽  
Control Variables ◽  
Predator Prey ◽  
Type Iii ◽  
Prey System ◽  
Ratio Dependent

A new set of sufficient conditions for the permanence of a ratio-dependent predator-prey system with Holling type III functional response and feedback controls are obtained. The result shows that feedback control variables have no influence on the persistent property of the system, thus improving and supplementing the main result of Yang (2008).

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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