Modeling Spiral Chaos in Predator–Prey Dynamics Using Implicit Integration Factor Schemes

Résumé fourni par la source

This paper presents efficient implicit integration factor (IIF) methods as robust alternatives to classical finite difference schemes for the numerical solution of nonlinear time‐dependent reaction–diffusion equations in one and two spatial dimensions. The proposed framework combines computational efficiency with methodological simplicity while maintaining high accuracy for stiff nonlinear systems. A rigorous convergence and stability analysis is carried out, complemented by stability‐region plots that provide insight into the qualitative behavior of the proposed schemes. As a biologically relevant application, the methods are employed to investigate a two‐dimensional predator–prey reaction–diffusion model exhibiting diffusion‐driven instability. Extensive numerical simulations accurately capture the emergence of Turing patterns, spot and stripe formations, rotating spiral waves, and spiral chaos, illustrating how species dispersal and nonlinear interactions generate complex spatiotemporal population distributions. These ecological patterns provide valuable insight into species coexistence, spatial self‐organization, and long‐term ecosystem dynamics. Detailed implementation guidelines, discretization strategies, and parameter selection procedures are provided to facilitate reproducibility and practical adoption. The numerical experiments confirm the theoretical convergence and stability properties of the proposed schemes while demonstrating their efficiency for large‐scale ecological simulations. To encourage further research and reproducibility, open‐access code implementing the second‐order IIF scheme for predator–prey reaction–diffusion systems is also provided.

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