交叉扩散羊群行为中熵与分岔的一个交汇点
A meeting point of entropy and bifurcations in cross-diffusion herding
摘要：
在有界区域中，研究了一类描述个体信息羊群行为的交叉扩散系统。变量是物种的密度和一个影响函数，它改变了个体的信息状态。交叉扩散项可以使系统稳定或不稳定。此外，它允许一个正式的梯度流或熵结构。利用这一结构，证明了在一定的参数范围内弱解的整体存在性和指数衰减到常数平衡态。这种方法并不适用于所有参数。我们研究局部分歧从均匀的稳定状态分析，以确定这是否定义的有效性边界。这一分析表明，一般有一个差距之间的熵方法的有效性和第一个局部分歧的参数制度。接下来，我们使用数值延拓方法来跟踪全球的分叉非齐次稳态，并确定非平凡的固定解决方案有关的羊群行为。总之，我们发现参数区域的主边界是由第一局部歧点、扩散矩阵的退化和一定的熵衰减有效性条件给出的。我们研究了几个参数的限制解析以及数值，重点是改变线性阻尼参数以及控制交叉扩散的参数的作用。我们建议，我们的范式比较分叉产生的障碍，全局功能方法的参数有效性也可能是相关的许多其他模型超出了这里研究的。
英文摘要：
A cross-diffusion system modeling the information herding of individuals is analyzed in a bounded domain with no-flux boundary conditions. The variables are the species' density and an influence function which modifies the information state of the individuals. The cross-diffusion term may stabilize or destabilize the system. Furthermore, it allows for a formal gradient-flow or entropy structure. Exploiting this structure, the global-in-time existence of weak solutions and the exponential decay to the constant steady state is proved in certain parameter regimes. This approach does not extend to all parameters. We investigate local bifurcations from homogeneous steady states analytically to determine whether this defines the validity boundary. This analysis shows that generically there is a gap in the parameter regime between the entropy approach validity and the first local bifurcation. Next, we use numerical continuation methods to track the bifurcating non-homogeneous steady states globally and to determine non-trivial stationary solutions related to herding behaviour. In summary, we find that the main boundaries in the parameter regime are given by the first local bifurcation point, the degeneracy of the diffusion matrix and a certain entropy decay validity condition. We study several parameter limits analytically as well as numerically, with a focus on the role of changing a linear damping parameter as well as a parameter controlling the cross-diffusion. We suggest that our paradigm of comparing bifurcation-generated obstructions to the parameter validity of global-functional methods could also be of relevance for many other models beyond the one studied here.
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