Turing Instability of Hopf Bifurcation Periodic Solutions and Stability Analysis in a Diffusive Forest Kinematic Model

In this paper, we investigate the asymptotic behavior of solutions to a diffusive forest kinematic model, which describes the interactions among young trees, old trees, and airborne seeds. Our study focuses on the stability of the positive equilibrium, the occurrence of Hopf bifurcation yielding spatially homogeneous periodic solutions, and the subsequent Turing instability induced by diffusion in these periodic states. The analysis highlights that the juvenile tree mortality rate, represented by a quadratic function of mature tree density, plays a central dynamical role. Specifically, the parameter corresponding to the mature tree density at which juvenile mortality is minimized serves as a key Hopf bifurcation parameter. This indicates that the system’s transition to periodic solutions and later to diffusion-driven pattern formation can be effectively regulated through this parameter. From an ecological perspective, these results suggest that forest management strategies capable of indirectly influencing factors related to this critical parameter could help control the emergence of spatial patterns, such as forest patches. Furthermore, the functional form of the mortality rate offers a useful foundation for future studies examining how different assumptions regarding tree interaction morphology may influence ecosystem patterning.