Nonlocal interactions are ubiquitous in nature and play a central role in many biological systems. In this paper, we perform a bifurcation analysis of a widely-applicable advection–diffusion model with nonlocal advection terms describing the species movements generated by inter-species interactions. We use linear analysis to assess the stability of the constant steady state, then weakly nonlinear analysis to recover the shape and stability of non-homogeneous solutions. Since the system arises from a conservation law, the resulting amplitude equations consist of a Ginzburg–Landau equation coupled with an equation for the zero mode. In particular, this means that supercritical branches from the Ginzburg–Landau equation need not be stable. Indeed, we find that, depending on the parameters, bifurcations can be subcritical (always unstable), stable supercritical, or unstable supercritical. We show numerically that, when small amplitude patterns are unstable, the system exhibits large amplitude patterns and hysteresis, even in supercritical regimes. Finally, we construct bifurcation diagrams by combining our analysis with a previous study of the minimizers of the associated energy functional. Through this approach we reveal parameter regions in which stable small amplitude patterns coexist with strongly modulated solutions.

中文翻译：

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两种非局域平流扩散系统的弱非线性分析

非局域相互作用在自然界中普遍存在，并在许多生物系统中发挥着核心作用。在本文中，我们对广泛适用的平流扩散模型进行了分叉分析，其中非局部平流项描述了物种间相互作用产生的物种运动。我们使用线性分析来评估恒定稳态的稳定性，然后使用弱非线性分析来恢复非齐次解的形状和稳定性。由于该系统源于守恒定律，因此所得的振幅方程由金兹堡-朗道方程和零模式方程组成。特别是，这意味着金兹堡-朗道方程的超临界分支不必是稳定的。事实上，我们发现，根据参数的不同，分岔可以是亚临界（总是不稳定）、稳定超临界或不稳定超临界。我们通过数值表明，当小振幅模式不稳定时，即使在超临界状态下，系统也会表现出大振幅模式和滞后现象。最后，我们通过将我们的分析与先前对相关能量泛函的极小值的研究相结合来构建分岔图。通过这种方法，我们揭示了稳定的小幅度模式与强调制解共存的参数区域。