Spatiotemporal patterns in predator-prey system with free space and limited resources.

Habitat availability is a fundamental ecological factor that influences population growth, species persistence, and the spatial organization of ecosystems. Motivated by this principle, we employ a predator-prey model in which prey recruitment is governed by limited resources and is described by the Beverton-Holt function. The model explicitly incorporates space limitation into the population dynamics, providing a biologically meaningful frame that explores how habitat occupancy shapes ecological interactions and spatial self-organization. Extending the model to the reaction-diffusion framework, we derive the conditions for diffusion-driven instability and investigate how predator competition and resource limitation regulate the emergence of self-organized spatial structures. Numerical simulations reveal a rich spectrum of Turing patterns associated with a remarkable sequence of spatial transitions as predator intra-specific mortality increases. To provide a theoretical explanation for the numerically observed patterns, we further employ weakly nonlinear analysis to derive the amplitude equations near the Turing threshold and validate the numerical results through analytical predictions. The proposed framework offers valuable insights into the mechanisms through which space limitation and density-dependent population regulation shape biodiversity and pattern formation in heterogeneous ecosystems.