Multiscale Dynamics in a Prey–Predator Model with Allee Effect on Predators

Multiple time scales have a profound impact on prey–predator dynamics and the formation of spatial patterns within spatiotemporal systems. Prey species, characterized by their high reproductive rates, typically exhibit a rapid population growth rate. In contrast, predators exhibit slower population dynamics because their growth depends on the availability of prey and other ecological factors. To underscore the necessity of considering separate time scales, we focus on analyzing a slow–fast prey–predator model, where the Allee effect influences the predator’s growth rate. The entry–exit function identifies the point at which the predator’s decline saturates, marking the transition from the slow time scale to the fast time scale where prey regrowth rapidly occurs. Through relaxation oscillations, we capture the cyclic behavior of the slow–fast dynamics, where the system rapidly switches between two distinct states, i.e. from high prey and low predator densities to low prey and high predator densities. To further explore the implications of time scales on pattern formation, we extend our temporal model to a spatiotemporal model by incorporating diffusion processes. This extension allows us to examine the emergence and evolution of various Turing and non-Turing patterns, offering deeper insights into their formation and dynamics, which are pivotal in understanding the spatial distribution and temporal fluctuations of populations. By extending our model to include diffusion, we uncover a diverse range of patterns like spot pattern, spiral and chaotic pattern and demonstrate how these patterns vary with variations in the time scale parameter. In the one-dimensional extension of the temporal model, we show the existence of traveling wave solutions that capture the spatial spread and invasion of predators into the habitat. Moreover, we find that the minimal wave speed of traveling-wave solutions increases with the time scale parameter.