Effects of seasonal growth on delayed prey–predator model Sunita Gakkhar * , Saroj Kumar Sahani, Kuldeep Negi Department of Mathematics, IIT Roorkee, Roorkee 247667, India Accepted 2 January 2007 Communicated by Prof. Ji-Huan He Abstract The dynamic behavior of a delayed predator–prey system with Holling II functional response is investigated. The stability analysis has been carried out and existence of Hopf bifurcation has been established. The complex dynamic behavior due to time delay has been explored. The effects of seasonal growth on the complex dynamics have been sim- ulated. The model shows a rich variety of behavior, including period doubling, quasi-periodicity, chaos, transient chaos, and windows of periodicity. Ó 2007 Elsevier Ltd. All rights reserved. 1. Introduction Predator–prey dynamics will continue to be of interest to both applied mathematicians and ecologists due to its uni- versal existence and importance. Many complex models for two or more interacting species have been proposed on the basis of Lotka–Volterra models by taking into account the effects of crowding, age structure, time delay, functional response, switching, etc. [4,10,8,12,14]. Factors that introduce time delay may include age structure of the population (influencing the birth and death rates), maturation periods (thresholds), feeding times, reaction times, food storage, resource regeneration times and hunger coefficients in predator–prey interactions. Models incorporating time delays in diverse biological models are extensively reviewed by MacDonald [9], Kuang [8], Gopalsamy [6], Cushing [3], Beretta and Kuang [2]. The population communities are imbedded in periodically varying environments. Therefore, it is quite natural to identify the functional role that play on the behaviour of population communities. The basic problem is to understand the relationship between the magnitude of the seasonal variations and the complexity of the system. Numbers of studies have been performed on the interactions between seasons and internal biological rhythms of the simple prey–predator ecosystems [1,5,7,13]. Naturally, more realistic and interesting models of population interactions should take into account both the seasonality of the changing environment and the effects of time delay. In this paper, the effects of seasonality in delayed prey–predator system with Michaelis–Menten or Holling type II functional response is discussed. The effects of time delays due to gestation [11] and negative feedbacks of prey species to the growth of species itself [8] are included in the model. 0960-0779/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.01.141 * Corresponding author. Tel.: +91 01332 285171. E-mail addresses: sungkfma@iitr.ernet.in (S. Gakkhar), sarojdma@iitr.ernet.in (S.K. Sahani), negikdma@iitr.ernet.in (K. Negi). Chaos, Solitons and Fractals 39 (2009) 230–239 www.elsevier.com/locate/chaos