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<p>A diffusive Lotka–Volterra competition system with memory-based diffusion is analyzed. The stability of the boundary steady states depends on memory-based self-diffusion but is independent of memory-based cross-diffusion. However, memory-based cross-diffusion induces instability in the constant coexistence state, increasing the complexity of the spatiotemporal dynamics of the system. We establish the critical conditions under which the constant coexistence state loses its stability through Turing bifurcation, Hopf bifurcation, and double-Hopf bifurcation. The normal form of the double-Hopf bifurcation is also derived. This allows us to prove the existence of an unstable spatially inhomogeneous quasi-periodic solution, and the coexistence of two types of stable spatially inhomogeneous periodic solutions with different wave numbers. In particular, a repulsive memory-based diffusion induces competing species to coexist in spatially segregated steady-state patterns, and an attractive memory-based diffusion can induce competing species to coexist in spatially segregated time-periodic patterns.</p>

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Spatially segregated coexistence and bistable spatiotemporal oscillatory patterns in the competition model with memory-based diffusion

  • 期刊论文

  • 发表日期：

    2025年1月16日

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  • Meng Liu,
  • Weihua Jiang,
  • Hongbin Wang

摘要

A diffusive Lotka–Volterra competition system with memory-based diffusion is analyzed. The stability of the boundary steady states depends on memory-based self-diffusion but is independent of memory-based cross-diffusion. However, memory-based cross-diffusion induces instability in the constant coexistence state, increasing the complexity of the spatiotemporal dynamics of the system. We establish the critical conditions under which the constant coexistence state loses its stability through Turing bifurcation, Hopf bifurcation, and double-Hopf bifurcation. The normal form of the double-Hopf bifurcation is also derived. This allows us to prove the existence of an unstable spatially inhomogeneous quasi-periodic solution, and the coexistence of two types of stable spatially inhomogeneous periodic solutions with different wave numbers. In particular, a repulsive memory-based diffusion induces competing species to coexist in spatially segregated steady-state patterns, and an attractive memory-based diffusion can induce competing species to coexist in spatially segregated time-periodic patterns.