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<p>Many territorial animals shape their movement decisions using memory of past conflicts, yet the population-level consequences of such nonlocal conflict memory remain poorly understood. We propose and analyze a PDE-ODE hybrid model for a single population moving in response to spatial memory of territorial conflicts. The population density satisfies a diffusion equation with delayed, nonlocal advection generated by a convolution of a conflict variable, while the conflict intensity evolves according to a local ODE encoding a warning mechanism with decay and resetting. Linearization about the positive homogeneous steady state yields a non-self-adjoint operator whose spectrum reduces to a countable family of characteristic equations indexed by Fourier modes. For Gaussian and Laplacian perception kernels, the steady state is linearly asymptotically stable for all perceptual radii in the memoryless case, whereas a top-hat kernel admits diffusion-driven (Turing) instability below a critical radius, producing stationary spatial patterns. For positive memory delay, we identify Hopf bifurcation thresholds at the level of the linearized spectral problem for all three kernels. In the top-hat case, the stationary and oscillatory thresholds may meet in the <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\((R,\tau )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>R</mi> <mo>,</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-plane, giving a codimension-two Turing-Hopf spectral point. Numerical simulations illustrate these thresholds and the associated spatial and spatiotemporal patterns.</p>

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Pattern Formation from Nonlocal Conflict Memory in a PDE-ODE Model

  • 期刊论文

  • 发表日期：

    2026年7月16日

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  • Shu Li,
  • Binxiang Dai,
  • Hao Wang

摘要

Many territorial animals shape their movement decisions using memory of past conflicts, yet the population-level consequences of such nonlocal conflict memory remain poorly understood. We propose and analyze a PDE-ODE hybrid model for a single population moving in response to spatial memory of territorial conflicts. The population density satisfies a diffusion equation with delayed, nonlocal advection generated by a convolution of a conflict variable, while the conflict intensity evolves according to a local ODE encoding a warning mechanism with decay and resetting. Linearization about the positive homogeneous steady state yields a non-self-adjoint operator whose spectrum reduces to a countable family of characteristic equations indexed by Fourier modes. For Gaussian and Laplacian perception kernels, the steady state is linearly asymptotically stable for all perceptual radii in the memoryless case, whereas a top-hat kernel admits diffusion-driven (Turing) instability below a critical radius, producing stationary spatial patterns. For positive memory delay, we identify Hopf bifurcation thresholds at the level of the linearized spectral problem for all three kernels. In the top-hat case, the stationary and oscillatory thresholds may meet in the \((R,\tau )\) ( R , τ ) -plane, giving a codimension-two Turing-Hopf spectral point. Numerical simulations illustrate these thresholds and the associated spatial and spatiotemporal patterns.