Dynamic complexity in fractional multispecies ecological systems: A Caputo derivative approach

In this study, a novel implicit numerical approach is introduced by combining finite-difference techniques with innovative L1 schemes. This method is designed to solve time-fractional reaction–diffusion systems occurring in one and two dimensions. Specifically, the focus is on ecological systems with mixed boundary conditions, which are commonly found in biological and chemical processes. This research focuses on the spatiotemporal behavior of a predator–prey model with a Holling III functional response, taking into account the presence of prey refuges. This study revealed that this model does not exhibit a Turing pattern, which is typically associated with diffusion-driven instability. Consequently, this investigation explored alternative non-Turing patterns using extensive numerical simulations. In scenarios involving two-dimensional subdiffusion, the study observed a variety of spatiotemporal dynamics within the diffusive prey–predator model. When prey refuge availability was low, the system displayed a circular pattern that gradually expanded over time to encompass the entire spatial domain. As the availability of refugees decreased, the system transitioned from a spiral to a chaotic pattern. Furthermore, the research revealed that, as the ratio of predator-to-prey diffusion rates increased, the system exhibited a subdiffusive spiral pattern, which then transformed into a spot-like pattern. Eventually, these spots merged to form stripe-like patterns as the ratio increased. This investigation highlights the rich and intricate dynamics that can emerge in fractional predator–prey interactions when considering both spatial and temporal factors. To further confirm the complexity of the dynamical behaviors, Lyapunov exponents were estimated numerically.