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This research investigates pattern self-organization along the route to chaos in a space- and time-discrete predator–prey system, where the prey shows convection movement in space. Through analysis on Turing instability of the system, pattern self-organization conditions are determined. Based on the conditions, simulations are performed under two initial conditions, demonstrating two pattern transitions along the route to chaos. In the first pattern transition, the patterns start from regular stripes, experiencing twisted stripes, then return to regular stripes again. The second pattern transition is much more complex and shows three stages. Especially, an alternation between ordered patterns and disordered chaos is found, contributing greatly to the spatiotemporal complexity of the system. When the system stays at the homogeneous chaotic states, Turing instability driven by convection and diffusion can still force the self-organization of regular striped patterns. The finding in this research provides a new comprehending for pattern self-organization and transition in spatially extended predator–prey systems. [ABSTRACT FROM AUTHOR]

The spatial pattern formation of predator-prey systems is an important issue widely concerned. In this research, we address this issue by developing a new space- and time-discrete predator-prey model, with predation relationship described by Beddington-DeAngelis functional response. The discrete model is given by a coupled map lattice, taking a nonlinear relationship between predator-prey “reaction” stage and dispersal stage. Through analysis of Turing instability and Hopf instability for the discrete model, the parametric conditions for pattern formation are determined. Numerical simulations reveal a surprising variety of spatiotemporal patterns, including regular and irregular patterns of spots, stripes, labyrinth, gaps, mosaics, spirals, circles, and many intermediate patterns in-between. These patterns cover a majority of predator-prey pattern types recorded in literature. Besides, the discrete model predicts the occurrence of spatiotemporal chaos, which is responsible for the formation of irregular patterns. This research demonstrates that the nonlinear mechanisms of the discrete model better capture the complexity of pattern formation of predator-prey systems. [ABSTRACT FROM AUTHOR]