Global stability and bifurcation of time delayed prey–predator system incorporating prey refuge

2012 ◽  
Vol 85 ◽  
pp. 57-77 ◽  
Author(s):  
Soovoojeet Jana ◽  
Milon Chakraborty ◽  
Kunal Chakraborty ◽  
T.K. Kar
Keyword(s):  
Global Stability ◽  
Prey Refuge ◽  
Stability And Bifurcation ◽  
Predator System

Dynamics of an ecological system

2019 ◽  
Vol 10 (4) ◽  
pp. 355-376
Author(s):  
Shashi Kant
Keyword(s):  
Saddle Point ◽  
Numerical Simulations ◽  
Equilibrium Point ◽  
Local Stability ◽  
Deterministic Model ◽  
Equilibrium Points ◽  
Prey Refuge ◽  
Trivial Equilibrium ◽  
Stability Results ◽  
Predator System

AbstractIn this paper, we investigate the deterministic and stochastic prey-predator system with refuge. The basic local stability results for the deterministic model have been performed. It is found that all the equilibrium points except the positive coexisting equilibrium point of the deterministic model are independent of the prey refuge. The trivial equilibrium point, predator free equilibrium point and prey free equilibrium point are always unstable (saddle point). The existence and local stability of the coexisting equilibrium point is related to the prey refuge. The permanence and extinction conditions of the proposed biological model have been studied rigourously. It is observed that the stochastic effect may be seen in the form of decaying of the species. The numerical simulations for different values of the refuge values have also been included for understanding the behavior of the model graphically.

Modeling the Effect of Fear in a Prey–Predator System with Prey Refuge and Gestation Delay

2019 ◽  
Vol 29 (14) ◽  
pp. 1950195 ◽  
Author(s):  
Ankit Kumar ◽  
Balram Dubey
Keyword(s):  
Hopf Bifurcation ◽  
Prey Population ◽  
Reproduction Rate ◽  
Predator Population ◽  
Maximum Lyapunov Exponent ◽  
Stable Limit ◽  
Prey Refuge ◽  
Gestation Delay ◽  
Fear Effect ◽  
Predator System

Recently, some field experiments and studies show that predators affect prey not only by direct killing, they induce fear in prey which reduces the reproduction rate of prey species. Considering this fact, we propose a mathematical model to study the fear effect and prey refuge in prey–predator system with gestation time delay. It is assumed that prey population grows logistically in the absence of predators and the interaction between prey and predator is followed by Crowley–Martin type functional response. We obtained the equilibrium points and studied the local and global asymptotic behaviors of nondelayed system around them. It is observed from our analysis that the fear effect in the prey induces Hopf-bifurcation in the system. It is concluded that the refuge of prey population under a threshold level is lucrative for both the species. Further, we incorporate gestation delay of the predator population in the model. Local and global asymptotic stabilities for delayed model are carried out. The existence of stable limit cycle via Hopf-bifurcation with respect to delay parameter is established. Chaotic oscillations are also observed and confirmed by drawing the bifurcation diagram and evaluating maximum Lyapunov exponent for large values of delay parameter.

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