A RATIO-DEPENDENT PREDATOR-PREY MODEL WITH DELAY AND HARVESTING

2010 ◽  
Vol 18 (02) ◽  
pp. 437-453 ◽  
Author(s):  
A. K. MISRA ◽  
B. DUBEY
Keyword(s):  
Hopf Bifurcation ◽  
Numerical Simulations ◽  
Functional Response ◽  
Bifurcation Parameter ◽  
Discrete Delay ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Ratio Dependent

In this paper a predator-prey model with discrete delay and harvesting of predator is proposed and analyzed by considering ratio-dependent functional response. Conditions of existence of various equilibria and their stability have been discussed. By taking delay as a bifurcation parameter, the system is found to undergo a Hopf bifurcation. Numerical simulations are also performed to illustrate the results.

Spatial pattern in a diffusive predator–prey model with sigmoid ratio-dependent functional response

2014 ◽  
Vol 07 (05) ◽  
pp. 1450047 ◽  
Author(s):  
Lakshmi Narayan Guin ◽  
Prashanta Kumar Mandal
Keyword(s):  
Spatial Pattern ◽  
Numerical Simulations ◽  
Functional Response ◽  
Turing Instability ◽  
Spatial Pattern Analysis ◽  
Predator Prey ◽  
Interior Equilibrium ◽  
Predator Prey Model ◽  
Prey Model ◽  
Ratio Dependent

In this paper, spatial patterns of a diffusive predator–prey model with sigmoid (Holling type III) ratio-dependent functional response which concerns the influence of logistic population growth in prey and intra-species competition among predators are investigated. The (local and global) asymptotic stability behavior of the corresponding non-spatial model around the unique positive interior equilibrium point in homogeneous steady state is obtained. In addition, we derive the conditions for Turing instability and the consequent parametric Turing space in spatial domain. The results of spatial pattern analysis through numerical simulations are depicted and analyzed. Furthermore, we perform a series of numerical simulations and find that the proposed model dynamics exhibits complex pattern replication. The feasible results obtained in this paper indicate that the effect of diffusion in Turing instability plays an important role to understand better the pattern formation in ecosystem.

Bifurcation analysis and control of a delayed stage-structured predator–prey model with ratio-dependent Holling type III functional response

2019 ◽  
Vol 26 (13-14) ◽  
pp. 1232-1245
Author(s):  
Miao Peng ◽  
Zhengdi Zhang
Keyword(s):  
Hopf Bifurcation ◽  
Functional Response ◽  
Control Method ◽  
Equilibrium Points ◽  
Predator Prey ◽  
Center Manifold Theorem ◽  
Type Iii ◽  
Predator Prey Model ◽  
Prey Model ◽  
Ratio Dependent

A delayed stage-structured predator–prey model with ratio-dependent Holling type III functional response is proposed and explored in this study. We discuss the positivity and the existence of equilibrium points. By choosing time delay as the bifurcation parameter and analyzing the relevant characteristic equations, the local stability of the trivial equilibrium, the predator-extinction equilibrium, and the coexistence equilibrium of the system is investigated. In accordance with the normal form method and center manifold theorem, the property analysis of Hopf bifurcation of the system is obtained. Furthermore, for the purpose of protecting the stability of such a biological system, a hybrid control method is presented to control the Hopf bifurcation. Finally, numerical examples are given to verify the theoretical findings.

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