Spatiotemporal Dynamics of a Predator‐Prey Model With Direct Prey‐Taxis and Indirect Predator‐Taxis

This paper introduces a novel predator‐prey model that synthesizes indirect and direct taxis mechanisms. The model incorporates the prey's chemorepulsion from a predator‐emitted signal (indirect predator‐taxis) together with the predator's tactic response to prey gradients (direct prey‐taxis). We establish the global existence and uniform boundedness of solutions. Stability analysis reveals that large self‐diffusion coefficients enhance stability, whereas small direct prey‐taxis promotes global stability and enlarges the parameter range of indirect predator‐taxis intensity that stabilizes the system. In contrast, large indirect predator‐taxis intensity drives the formation of spatial patterns, including both stationary spatially periodic and time‐periodic patterns. We show that the bifurcation branch around the positive constant steady state is of pitchfork type, and we further prove that the system undergoes at most a 2‐multiple Hopf bifurcation. By comparing the December 1988 distribution map of Thomson's gazelle on the Serengeti grassland (Duran et al., Afr. J. Ecol., 1988), we conclude that the chemotaxis‐enabled model provides a more accurate description of observed population dynamics than its non‐chemotaxis counterpart.