Spatial pattern of a class of SI models driven by cross diffusion

Abstract

Currently, most research on the spatial patterns of the SI model, both domestically and internationally, focuses on the impacts of self-diffusion and system parameters on pattern formation. However, there is a little study on how cross-diffusion influences the evolution of spatial patterns. In this paper, a spatial epidemic model with both self-diffusion and cross-diffusion is established. The study investigates the effects of cross-diffusion on the stability, rate of stability, and pattern structure of the SI model, while considering scenarios where self-diffusion either induces system instability or it does not. The stability of the non-diffusive system is analyzed, and the conditions for Turing instability in the presence of diffusion terms are elucidated. It is found that when the system is stable under self-diffusion-driven conditions, the introduction of cross-diffusion can change the system's local stability, and can produce Turing patterns. Furthermore, different cross-diffusion coefficients can generate patterns with different structures. When the system is unstable under self-diffusion-driven conditions, the introduction of cross-diffusion can change the pattern structure. Specifically, when the cross-diffusion coefficient $D_1$ for the susceptible individuals is negative, the pattern structure transitions from spot-stripe patterns to spot patterns, and when positive, it transitions from spot-stripe patterns to labyrinthine patterns, and eventually to a uniform solid color distribution. When the cross-diffusion coefficient $D_2$ for the infected individuals is positive, the pattern transformation is similar to when the cross-diffusion coefficient $D_1$ for susceptible individuals is negative, gradually changing to spot patterns, while when $D_2$ is negative, the pattern structure exhibits a porous structure, eventually transitioning to a uniform solid color distribution. Regarding the rate of stability of the SI model, in the case of a stable self-diffusion system, the introduction of cross-diffusion may alter the rate of system stability, and the larger the cross-diffusion coefficient $D_1$ for the susceptible individuals, the faster the system stabilizes. When the self-diffusion-driven system is unstable, cross-diffusion causes the system to transition from an unstable state to a locally stable state, and the smaller the susceptible individuals' cross-diffusion coefficient, the slower the rate of system stabilization. Therefore, cross-diffusion has a significant impact on the stability, rate of stability, and pattern structure of the SI model.