Intraguild Predation Communities with Anti-predator Behavior

Spatial heterogeneity plays a pivotal role in shaping the dynamics of ecosystems, with ecologists recognizing its importance in promoting the stability of population dynamics. In this study, we introduce a cross-diffusion model to describe the dynamics of an intraguild predation community. The model considers intraguild prey exhibiting anti-predator behavior, dispersing along local gradients in predator density, while the Lotka--Volterra functional form describes the local dynamics. Our primary focus is on understanding the mechanisms driving species coexistence. Beginning with an overview of existing results concerning the existence and stability of the steady state of homogeneous coexistence, we rigorously construct the Hopf bifurcation curve that separates oscillations from stationary coexistence in the local system. In addition, we demonstrate that the local dynamics support the bistability of the spatially homogeneous equilibrium with oscillations due to a subcritical Hopf bifurcation. We prove that the predator avoidance strategy described by cross-diffusion is crucial for pattern formation in the reaction-diffusion system and characterizes the cross-diffusion-driven Turing bifurcation. Using the formalism of amplitude equations, we derive the asymptotic profiles of the stationary solutions, revealing that anti-predator behavior can account for segregation patterns between IntraGuild Prey and IntraGuild Predator observed in field studies. Through a combination of analytical and numerical tools, we demonstrate that the predator avoidance strategy serves as a mechanism that stabilizes coexistence states in intraguild predation communities beyond the conditions imposed by the corresponding spatially homogeneous model.