Hopf- Bifurcation Analysis of Delayed Prey-Predator System

Author: Nishant Juneja* and Kulbhushan Agnihotri**

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Abstract

The present paper deals with a SIS (Susceptible Infected Susceptible) predator-prey model with disease in prey species only. It is assumed that the predator species only predates the infected prey with Holling type - II functional response. It is assumed that predator growth is not instantaneous after consuming the prey and a discrete time lag for gestation of predator is required. The conditions for Hopf bifurcation around the interior equilibrium point are also derived. Finally, numerical simulations supporting the theoretical results are given.

Keywords

Local stability, Carrying capacity, Hopf bifurcation, Predation rate, Delay

Conclusion

An eco-epidemiological mathematical model incorporating time delay with infection in prey species is formed. It is found that the time delay can remove the limit cycle oscillations from the system. It is found that coexistence of all the three species is possible through periodic solutions due to Hopf bifurcation. Numerical simulations have been carried out to defend the theoretical results obtained. The numerical simulations have revealed that the dynamics of the system is largely affected by considering delay in the gestation period of the predator species. Juneja and Agnihotri 270 The introduction of small delay in gestation period can make the system oscillation free and thus showing the stabilizing nature of delay.

References

The existence of periodic solution / limit cycle oscillations due to predator functional response is well studied behavior in prey-predator models (Freedman, 1980 and Kot, 2001). A lot of work has already been done on different types of functional responses depending upon the densities of prey, predator and other significant factors (May, 2001, Murray, 2002, Agnihotri and Juneja, 2015, Agnihotri and Juneja, 2015). These functional responses are mostly classified as classical prey dependent and ratio dependent responses. Presently, the researchers are showing immense and continuing curiosity in studying the dynamics of predator–prey systems with time delay, stage structure, functional response, etc. In recent years, a number of researchers (Wang, 1998 and Huang, et al., 2006) studied delay induced prey–predator model to discuss the stability of the system. Several investigations reported that under the influence of these factors, the system exhibits more complex and richer dynamics

How to cite this article

Nishant Juneja and Kulbhushan Agnihotri (2017). Hopf- Bifurcation Analysis of Delayed Prey-Predator System , Biological Forum – An International Journal 9(2): 265-270(2017)