Emergence of Rich Patterns in a Discrete System with Migration and Diffusion

Abstract

This study employs the pattern formation of a discrete system with migration and diffusion. By analyzing the stability of fixed points and flip and Neimark-Sacker bifurcations, we demonstrate the non-spatial form of the system can generate complex dynamical behaviors, including periodic, quasiperiodic and chaotic orbits. Numerical simulations reveal the emergence of rich patterns which show different ways of transitions, especially on the routes to chaos with the variation of bifurcation parameter. We find the migration in the discrete system enhances spatial coupling and reshapes pattern stability. This work connects bifurcation with spatiotemporal dynamics, revealing how migration-diffusion interaction drives pattern self-organization.