Spatial patterns through diffusion-driven instability in modified predator–prey models with chaotic behaviors

Abstract

Understanding the connection between spatial patterns in population densities and ecological heterogeneity is significant to the understanding of population dynamics and the governance of species in a given domain. In this paper, the spatiotemporal complexity of prey–predator dynamics with fractional Laplacian derivative and migration is investigated. To provide good guidelines on the choice of parameters, the linear stability analysis of the models was investigated. The fractional Laplacian operator was defined in terms of the left- and right-handed Riemann–Liouville derivative which in turn were approximated by using the fourth-order compact difference scheme, and the resulting system of ODEs was advanced in time using the fourth-order exponential time-differencing Runge–Kutta method. In the simulation experiments, different Turing dynamics such as spots, stripes, and other chaotic patterns are observed. Overall, pattern formation in predator–prey models is useful for understanding the dynamics of ecological systems, predicting the long-term behavior of the system, and studying the impact of environmental factors on the dynamics of the system.