Steady-state bifurcation analysis of a herbivore-vegetation reaction–diffusion model with memory effect

Spatial memory is a crucial cognitive ability that affects animals’ perception and judgment, and it has garnered widespread attention. However, its underlying mechanisms have not been fully understood. This paper constructs a herbivore-vegetation model with spatial memory effect, in which the memory effects are described by a distributed delayed diffusion term. Subsequently, steady-state bifurcation analysis is employed to investigate the influence of spatial memory effects on the structure of vegetation patterns. The conditions for spatial inhomogeneous steady-state solution are gained by linear stability analysis. Moreover, the structure of the non-constant positive steady-state solutions is analyzed. Furthermore, the numerical results demonstrate the theoretical findings and reveal that the memory-based diffusion coefficient has a significant impact on the vegetation pattern and spatial heterogeneity.