Dynamic patterns in herding predator-prey system: Analyzing the impact of inertial delays and harvesting.

This study expands traditional reaction-diffusion models by incorporating hyperbolic dynamics to explore the effects of inertial delays on pattern formation. The kinetic system considers a harvested predator-prey model where predator and prey populations gather in herds. Diffusion and inertial effects are subsequently introduced. Theoretical frameworks establish conditions for stability, revealing that inertial delay notably alters diffusion-induced instabilities and Hopf bifurcations. The inclusion of inertial effects narrows the stability region of the kinetic system by wave instability, which cannot arise in a two-variable spatiotemporal system without inertia. Computational simulations demonstrate that Turing and wave instabilities lead to diverse spatial and spatiotemporal patterns. This study highlights that initial conditions influence wave instability, generating distinct patterns based on different initial values, while other instabilities remain unaffected. Additionally, patterns, such as hot spots, cold spots, and stripes, are observed within the Turing region. The impact of harvesting on spatiotemporal system stability is also examined, showing that increased harvesting efforts can shift systems between unstable and uniform states. The findings provide practical implications for ecological modeling, offering insights into how inertial delays and harvesting practices affect pattern formation in natural populations.