相互作用种群聚集与扩散的对策论机制
A game-theoretic mechanism for aggregation and dispersal of interacting populations
摘要：
我们适应的适应度函数从演化博弈论作为一种机制，在偏微分方程（PDE）模型的两个相互作用的人口，描述的密度函数$u$和$v$的聚集和分散。我们考虑一个空间模型，其中个人迁移到当地的适应度梯度，寻找他们的给定特征更有利的位置。由此产生的系统的健身梯度方程是一个退化的系统，具有空间结构，光滑，稳定状态的解决方案，其特征在于在整个域的恒定健身。当种群被视为捕食者和猎物，我们的模型捕捉猎物聚集行为符合汉密尔顿的自私的羊群假说。我们还提出了弱稳定状态的解决方案，在1D是连续的，但在一般情况下并不光滑的地方，与相关的健身是不连续的，分段常数。我们给出了数值例子的解决方案，朝着这样弱的稳定状态。我们还给出了一个例子的空间洛特卡-沃尔泰拉模型，其中的健身梯度通量创建不稳定性，导致空间结构的稳定状态。我们的研究结果还表明，当健身有一定的依赖于当地的相互作用，健身为基础的扩散机制可能会采取行动，以创造整个栖息地的空间变化。
英文摘要：
We adapt a fitness function from evolutionary game theory as a mechanism for aggregation and dispersal in a partial differential equation (PDE) model of two interacting populations, described by density functions $u$ and $v$. We consider a spatial model where individuals migrate up local fitness gradients, seeking out locations where their given traits are more advantageous. The resulting system of fitness gradient equations is a degenerate system having spatially structured, smooth, steady state solutions characterized by constant fitness throughout the domain. When populations are viewed as predator and prey, our model captures prey aggregation behavior consistent with Hamilton's selfish herd hypothesis. We also present weak steady state solutions in 1d that are continuous but in general not smooth everywhere, with an associated fitness that is discontinuous, piecewise constant. We give numerical examples of solutions that evolve toward such weak steady states. We also give an example of a spatial Lotka--Volterra model, where a fitness gradient flux creates instabilities that lead to spatially structured steady states. Our results also suggest that when fitness has some dependence on local interactions, a fitness-based dispersal mechanism may act to create spatial variation across a habitat.
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