Effect of small time delay in a predator-prey model within random environment

2008 ◽  
Vol 16 (3) ◽  
pp. 225-250 ◽  
Author(s):  
Tapan Saha ◽  
M. Banerjee
Keyword(s):  
Time Delay ◽  
Random Environment ◽  
Small Time ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Small Time Delay

scholarly journals Pattern Formation in Predator-Prey Model with Delay and Cross Diffusion

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Xinze Lian ◽  
Shuling Yan ◽  
Hailing Wang
Keyword(s):  
Time Delay ◽  
Pattern Formation ◽  
Turing Instability ◽  
Predator Prey ◽  
Cross Diffusion ◽  
Predator Prey Model ◽  
Prey Model ◽  
Diffusion Controlled ◽  
The Stability ◽  
Self Replication

We consider the effect of time delay and cross diffusion on the dynamics of a modified Leslie-Gower predator-prey model incorporating a prey refuge. Based on the stability analysis, we demonstrate that delayed feedback may generate Hopf and Turing instability under some conditions, resulting in spatial patterns. One of the most interesting findings is that the model exhibits complex pattern replication: the model dynamics exhibits a delay and diffusion controlled formation growth not only to spots, stripes, and holes, but also to spiral pattern self-replication. The results indicate that time delay and cross diffusion play important roles in pattern formation.

scholarly journals Uniformly Strong Persistence for a Delayed Predator-Prey Model

2012 ◽  
Vol 2012 ◽  
pp. 1-7 ◽  
Author(s):  
Changjin Xu ◽  
Yuanfu Shao ◽  
Peiluan Li
Keyword(s):  
Time Delay ◽  
Sufficient Conditions ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Asymptotically Periodic

An asymptotically periodic predator-prey model with time delay is investigated. Some sufficient conditions for the uniformly strong persistence of the system are obtained. Our result is an important complementarity to the earlier results.

A Predator-Prey Model Incorporating Prey Refuge and Allee Effect

2015 ◽  
Vol 713-715 ◽  
pp. 1534-1539 ◽  
Author(s):  
Rui Ning Fan
Keyword(s):  
Time Delay ◽  
Time Lag ◽  
Functional Responses ◽  
Prey Refuge ◽  
Predator Prey ◽  
Interior Equilibrium ◽  
Predator Prey Model ◽  
Prey Model ◽  
Discrete Time Delay ◽  
Prey Interaction

The effect of refuge used by prey has a stabilizing impact on population dynamics and the effect of time delay has its destabilizing influences. Little attention has been paid to the combined effects of prey refuge and time delay on the dynamic consequences of the predator-prey interaction. Here, a predator-prey model with a class of functional responses was studied by using the analytical approach. The refuge is considered as protecting a constant proportion of prey and the discrete time delay is the gestation period. We evaluated both effects with regard to the local stability of the interior equilibrium point of the considered model. The results showed that the effect of prey refuge has stronger influences than that of time delay on the considered model when the time lag is smaller than the threshold. However, if the time lag is larger than the threshold, the effect of time delay has stronger influences than that of refuge used by prey.

A fractional-order predator–prey model with Beddington–DeAngelis functional response and time-delay

2018 ◽  
Vol 27 (2) ◽  
pp. 525-538 ◽  
Author(s):  
Rajivganthi Chinnathambi ◽  
Fathalla A. Rihan ◽  
Hebatallah J. Alsakaji
Keyword(s):  
Time Delay ◽  
Functional Response ◽  
Fractional Order ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model

Hopf Bifurcation of a Delayed Predator–Prey Model with Nonconstant Death Rate and Constant-Rate Prey Harvesting

2018 ◽  
Vol 28 (14) ◽  
pp. 1850179 ◽  
Author(s):  
Fengrong Zhang ◽  
Xinhong Zhang ◽  
Yan Li ◽  
Changpin Li
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Differential Equations ◽  
Positive Constant ◽  
Death Rate ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Constant Rate ◽  
The Stability

This paper is concerned with a delayed predator–prey model with nonconstant death rate and constant-rate prey harvesting. We mainly study the impact of the time delay on the stability of positive constant solution of delayed differential equations and positive constant equilibrium of delayed diffusive differential equations, respectively. By choosing time delay [Formula: see text] as a bifurcation parameter, we show that Hopf bifurcation can occur as the time delay passes some critical values. In addition, the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are determined by using the normal form theory and center manifold theorem. Finally, some numerical simulations are carried out to depict our theoretical results.

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