A Study on 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 <inline-formula><tex-math id="M1">\begin{document}$D_1$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20231877_M1.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20231877_M1.png"/></alternatives></inline-formula> 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 <inline-formula><tex-math id="M2">\begin{document}$D_2$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20231877_M2.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20231877_M2.png"/></alternatives></inline-formula> for the infected individuals is positive, the pattern transformation is similar to when the cross-diffusion coefficient <inline-formula><tex-math id="M3">\begin{document}$D_1$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20231877_M3.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20231877_M3.png"/></alternatives></inline-formula> for susceptible individuals is negative, gradually changing to spot patterns, while when <inline-formula><tex-math id="M4">\begin{document}$D_2$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20231877_M4.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20231877_M4.png"/></alternatives></inline-formula> 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 <inline-formula><tex-math id="M5">\begin{document}$D_1$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20231877_M5.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20231877_M5.png"/></alternatives></inline-formula> 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.