Spatiotemporal Patterns in a Diffusive Predator-Prey System of Leslie-Gower Type with Smith Growth for Prey

Why It Matters

Identifying exact bifurcation thresholds clarifies how spatial patterns and oscillations arise in ecological systems, guiding both theoretical research and practical management of predator‑prey interactions.

Key Takeaways

• Hopf bifurcation requires predator capture > 1 + competition
• Spatially homogeneous Hopf emerges under diffusion
• Turing and Turing‑Turing bifurcations coexist with stable equilibrium
• Normal form theory determines bifurcation direction and stability
• Simulations confirm analytical predictions on one‑dimensional domain

Pulse Analysis

Predator‑prey dynamics have long served as a cornerstone of mathematical ecology, yet incorporating realistic growth laws and spatial diffusion remains challenging. By pairing Smith‑type growth for the prey with a modified Leslie‑Gower formulation for the predator, the authors capture density‑dependent effects that traditional Lotka‑Volterra models overlook. The inclusion of a Beddington‑DeAngelis functional response further accounts for predator interference, making the model highly relevant for species where hunting efficiency declines with predator density. This richer framework sets the stage for rigorous bifurcation analysis that bridges local population oscillations and emergent spatial structures.

The study’s analytical core pinpoints the exact parameter regime where a Hopf bifurcation can destabilize the coexistence equilibrium: the predator’s capture rate must surpass the sum of unity and its intraspecific competition coefficient. When diffusion is introduced, the same condition triggers a spatially homogeneous Hopf bifurcation, while the diffusion coefficients themselves give rise to classic Turing instabilities. Remarkably, the authors also uncover Turing‑Turing bifurcations—simultaneous activation of two distinct spatial modes—within the region where the ODE equilibrium remains stable. Employing normal‑form theory and center‑manifold reduction, they determine the direction and stability of the resulting periodic solutions, providing a complete classification of pattern‑forming mechanisms.

Beyond theoretical elegance, these findings have practical implications for ecosystem management and conservation. Understanding the thresholds that lead to oscillatory outbreaks or spatial patchiness can inform strategies to mitigate pest surges or preserve endangered predator populations. The accompanying numerical simulations on a bounded one‑dimensional domain validate the analytical predictions, demonstrating realistic pattern evolution. Future work may extend this framework to heterogeneous environments or stochastic perturbations, further enhancing its applicability to real‑world ecological challenges.