<p>Spatiotemporal patterns are crucial for understanding population distribution dynamics, providing significant theoretical guidance for the conservation of both plants and animals. Delays and the Allee effect often lead to complex dynamical phenomena in temporal population models. Therefore, this paper incorporates the delay and Allee effects into the reaction-diffusion predator-prey model and explore the impact of such ecological effects on spatial-temporal dynamics. In theory, we obtain the critical value of Turing bifurcation and the condition of formation of the pattern, derive the amplitude equation, and calculate the delay-induced Hopf bifurcation and its properties. In numerical simulations, we found that the Allee effect and delays can alter the spatial distribution of populations. Specifically, as the Allee effect threshold increases, patterns evolve from spot patterns to mixed patterns, ultimately forming stripe patterns, while delays can induce the formation of spiral patterns, whose emergence depends on initial conditions. These findings contribute to our understanding of complex, non-equilibrium self-organization phenomena among populations.</p>

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Spatiotemporal patterns induced by delay in a predator-prey model with Allee effect

  • Danyang Li,
  • Weide Li,
  • Jiaying Zhou,
  • Hua Liu

摘要

Spatiotemporal patterns are crucial for understanding population distribution dynamics, providing significant theoretical guidance for the conservation of both plants and animals. Delays and the Allee effect often lead to complex dynamical phenomena in temporal population models. Therefore, this paper incorporates the delay and Allee effects into the reaction-diffusion predator-prey model and explore the impact of such ecological effects on spatial-temporal dynamics. In theory, we obtain the critical value of Turing bifurcation and the condition of formation of the pattern, derive the amplitude equation, and calculate the delay-induced Hopf bifurcation and its properties. In numerical simulations, we found that the Allee effect and delays can alter the spatial distribution of populations. Specifically, as the Allee effect threshold increases, patterns evolve from spot patterns to mixed patterns, ultimately forming stripe patterns, while delays can induce the formation of spiral patterns, whose emergence depends on initial conditions. These findings contribute to our understanding of complex, non-equilibrium self-organization phenomena among populations.