Exploring ecological dynamics through mathematical models. This research delves into the complex dynamics of a two-component reaction-diffusion model, inspired by Leslie-Gower type predator-prey interactions. The study employs numerical continuation to compute solution branches, focusing on how spatially localized patterns emerge and evolve over time. Understanding these patterns sheds light on ecological stability and species distribution. The research examines two distinct regimes: one where the homogeneous state loses stability to uniform oscillations and another where it loses stability to Turing patterns. In the first regime, spatially localized states embedded in an oscillating background are observed. The second regime reveals the presence of spatially localized states even when the homogeneous state is stable to Turing patterns. Disconnected segments of oscillatory states zip up into continuous snaking branches of time-periodic localized states. The study uncovers novel mechanisms driving pattern formation in predator-prey systems, demonstrating behavior that deviates from what's expected from primary bifurcations. These findings have implications for understanding the stability and complexity of ecological systems and for predicting how populations will respond to environmental changes. This will benefit ecosystems.
This paper's publication in Chaos: An Interdisciplinary Journal of Nonlinear Science reflects its focus on mathematical modeling and nonlinear dynamics. The study fits within the journal's scope by applying mathematical tools to explore complex phenomena in a predator-prey system. The journal is a place for cutting-edge research that delves into the intricate dynamics of nonlinear systems.