Int. J. Nonlinear Sci. Numer. Simul., Vol. 13 (2012), pp. 299–309 Copyright © 2012 De Gruyter. DOI 10.1515/ijnsns-2012-0115 Complex Population Dynamics in Heterogeneous Environments: Effects of Random and Directed Animal Movements Vikas Rai, 1 Ranjit Kumar Upadhyay 2; and Nilesh Kumar Thakur 2 1 Department of Mathematics, Faculty of Science, Jazan University, Jazan, Kingdom of Saudi Arabia 2 Department of Applied Mathematics, Indian School of Mines, Dhanbad, India Abstract. In this paper, we have investigated the complex dynamics of a one-dimensional spatial nonlinear coupled reaction-diffusion system with a Holling type IV functional response, akin to standard Michaelis-Menten inhibitory ki- netics. Prey-taxis is included in a general reaction-diffusion equation to incorporate the active movement of predator species towards regions with high prey concentrations or if the predator is following some sort of cue (such as odor) to find the prey. We have carried out stability analysis of both the non-spatial model without diffusive spreading and of the spatial model. We performed extensive com- puter simulations to identify various parameter ranges for stable homogeneous solution. Our findings specifically elu- cidate the role of predator diffusion and prey-taxis in con- trolling emergent structures, and transitions towards spatio- temporal chaos. We observe that the increasing predator random movement and moderate value of prey-taxis stabi- lize the system. Keywords. Spatial plankton system, heterogeneous en- vironment, prey-taxis, spatiotemporal pattern, directional movement. PACS ® (2010). 92B05, 92C16. 1 Introduction Reaction-diffusion equations have been the subject of in- tense research due to their rich variety of patterns. Con- ceptual predator- prey models have successfully been used to elucidate mechanisms of spatiotemporal pattern forma- tion [1–3]. Wolpert [4] gave a clear and non-technical de- scription of mechanisms of pattern formation in animals. * Corresponding author: Ranjit Kumar Upadhyay, Department of Applied Mathematics, Indian School of Mines, Dhanbad, India; E-mail: ranjit_ism@yahoo.com. Received: January 18, 2012. Accepted: March 19, 2012. Lee et al. [5, 6] investigate the necessary conditions for pat- tern formation in prey-taxis systems. They have also de- tected continuous travelling wave for prey-taxis in a model with Allee effect. One of the most efficient approaches for modeling the spatio-temporal dynamics of the interacting populations is based on the reaction-diffusion- advection equation [7]. The appearance of advection-driven hetero- geneity in relation to multispecies interaction was studied by many authors [8–9]. Sapoukhina et al. [10] consider a reaction-diffusion-advection model for the dynamics of populations and investigated the role of prey-taxis in biolog- ical control. The advection term represents the movement of predator according to a basic prey-taxis assumption i.e., acceleration of predators is proportional to the prey den- sity gradient. The predation process is divided into random movement described by diffusion and directed movement represented by prey-taxis. Random movement of plankton populations with different velocities can give rise to spatial patterns [11] and the directional movement of zooplankton plays a role in generating patterns in a plankton community model [12] due to the foraging behavior of zooplankton that move towards high phytoplankton density. Lewis [13] stud- ied pattern formation in plant and herbivore dynamics and herbivore-taxis were seen to reduce the likelihood of pattern formation. The pattern formation in prey taxis models is still open to wide investigations. Recently, spatial heterogeneity of species has attracted much attention because it is closely re- lated to the stability and coexistence of species in ecological systems. Two factors concern spatial heterogeneity as well as spatial pattern in which population distributed spatially and individuals interact locally. The first is internal noise which induce spatio-temporal pattern of species in concern- ing range. The second is predation intensity of species. Spa- tial heterogeneity and diffusion introduce qualitatively new types of behavior in predator-prey interaction [14]. Random diffusion alone does not usually explain well the movement of animals. Rapid dispersal of predators, modeled as ran- dom diffusion, has a stabilizing effect on community dy- namics. Spatial heterogeneities [15–19] such as prey den- sity gradients, give rise to prey-taxis. This phenomenon has been found to have a stabilizing effect on the dynam- ics [6] although these authors employ a different modeling approach. Many biological factors ought to alter the form of preda- tor’s functional response and thereby alter the dynamics of the predator and prey populations. The functional response Authenticated | ranjit_ism@yahoo.com author's copy Download Date | 9/6/12 7:35 AM