Transition and coexistence of Turing pattern, Turing-like pattern and spiral waves in a discrete-time predator-prey model

Turing patterns and spiral waves, which are spatiotemporal ordered structures, are a common occurrence in complex systems, manifesting in a variety of forms. Investigations on these two types of patterns primarily concentrate on different systems or different parameter ranges, respectively. Turing's theory, which postulates the presence of both a long-range inhibitor and a short-range activator, is used to explain the variety of Turing patterns in nature. Generally, Turing patterns are the result of Turing instability (including subcritical Turing instability), and research in this field is usually conducted within the parameter regions of Turing instability. Here, we observed the transition and coexistence phenomena of Turing pattern, Turing-like pattern and spiral wave, and discovered a mechanism for generating Turing-like patterns in discrete-time systems. Specifically, as the control parameter changes, the spiral wave gradually loses its dominant position and is eventually replaced by the Turing-like pattern, experiencing a state of coexistence of Turing/Turing-like pattern and spiral wave. The decrease in the move-state-effects results in the system's incapacity to generate spiral waves, which are ultimately replaced by Turing/Turing-like patterns. Outside the parameter intervals of Turing instability, we obtained a type of Turing-like patterns in a discrete-time model. The patterns can be excited through the application of a strong impulse noise (exceeding a threshold) to a homogeneous stable state. Analysis reveals that the Turing-like patterns are the consequence of the competition between two stable states, and the excitation threshold is determined by the relative position of the states. Our findings shed light on the pattern formation for Turing/Turing-like patterns and spiral waves in discrete-time systems, and reflect the diversity of mechanisms behind emergence and self-organization.