Pattern self-organization and pattern transition on the route to chaos in a spatiotemporal discrete predator–prey system

Abstract

Abstract A spatiotemporal discrete predator–prey system is investigated for understanding the pattern self-organization on the route to chaos. The discrete system is modelled by a coupled map lattice and shows advection of populations in space. Based on the conditions of stable stationary states and Hopf bifurcation, Turing pattern formation conditions are determined. As the parameter value is changed, self-organization of diverse patterns and complex phase transition among the patterns on the route to chaos are observed in simulations. Ordered patterns of stripes, bands, circles, and various disordered states are revealed. When we zoom in to observe the pattern transition in smaller and smaller parameter ranges, subtle structures for transition process are found: (1) alternation between self-organized structured patterns and disordered states emerges as the main nonlinear characteristic; (2) when the parameter value varies in the level from 10−3 to 10−4, a cyclic pattern transition process occurs repeatedly; (3) when the parameter value shifts in the level of 10−5 or below, stochastic pattern fluctuation dominates as essential regularity for pattern variations. The results obtained in this research promote comprehending pattern self-organization and pattern transition on the route to chaos in spatiotemporal predator–prey systems.

Keywords

• Self-organization
• Chaos
• Coupled map lattice
• Bifurcation
• Turing instability
• Predator–prey system