Starvation-driven diffusion in predator-prey dynamics

Starvation driven diffusion (SDD) describes a cognitive strategy that starvation of a species leads to its stronger movement. In this paper, to better understand the effects of SDD, we propose and analyze a type of predator-prey systems with predator and prey both obeying SDD. By analyzing the linearized eigenvalue problem, we investigate the stability and instability of a semi-trivial steady state, which depends on the conversion efficiency of prey to predator as well as on the predator’s minimum motility rate when conversion efficiency is properly large. Predator and prey coexist if the unique semi-trivial steady state is unstable. Utilizing Crandall-Rabinowitz bifurcation theorem, we investigate the local existence, stability, and structure of a bifurcating nontrivial steady state. There exists only one critical conversion efficiency guaranteeing the occurrence of steady-state bifurcation at the unique semi-trivial steady state. The global existence and structure of a bifurcating nontrivial steady state are proven by the global bifurcation theorem. One nontrivial steady state always exists for sufficiently large conversion efficiency. As examples, we apply theoretical results to predator-prey models with Holling type II/IV functional response involving SDD, and verify them via numerical simulations. We numerically observe spatially inhomogeneous periodic solutions, which should arise from nontrivial steady states via Hopf bifurcation, or even via homoclinic bifurcation in the case of Holling type IV functional response. Notably, these solutions consistently mirror resource distribution patterns, aligning conceptually with the ideal free distribution.