数学生态学中两个随机模型的平衡态
Equilibrium States of Two Stochastic Models in Mathematical Ecology
摘要：
这项工作涉及数学生态学中出现的两个问题。第一个问题是关于二倍体分支粒子模型和它的行为时，快速搅拌添加到相互作用。粒子模型涉及两种类型的粒子，男性和女性，并且只有当两种类型的粒子都存在时才能发生分支。我们表明，如果分支率是足够大的，这个粒子模型有一个非平凡的平稳分布，即一个不集中所有的重量的全0状态，使用比较参数由于R。杜雷特我们还表明，消光小分支率，从而建立一个相变的存在。然后，我们添加两种不同的快速搅拌机制的相互作用，并表明，快速搅拌的粒子模型，也存在非平凡的平稳分布（S）;为此，我们分析了限制PDE，并建立一个条件的PDE，保证存在的非平凡平稳分布足够快的搅拌。第二个问题涉及同域物种形成的模型，即没有地理分隔的物种形成。Dieckmann和M. 1999年的Doebeli我们修改了他们的原始模型，以获得几个常数粒子模型。我们集中在一个连续时间模型，收敛到一个确定性的动力系统的粒子数变得很大。我们建立了各种结果是否发生形态通过研究存在的双峰平稳分布的限制动力系统。
英文摘要：
This work deals with two problems arising in mathematical ecology. The first problem is concerned with diploid branching particle models and its behavior when rapid stirring is added to the interaction. The particle models involve two types of particles, male and female, and branching can only occur when both types of particles are present. We show that if the branching rate is sufficiently large, this particle model has a nontrivial stationary distribution, i.e. one that does not concentrate all weight on the all-0 state, using a comparison argument due to R. Durrett. We also show extinction for small branching rates, thereby establishing the existence of a phase transition. We then add two different rapid stirring mechanisms to the interactions and show that for the particle models with rapid stirring, there also exist nontrivial stationary distribution(s); for this, we analyze the limiting PDE and establish a condition on the PDE that guarantees existence of nontrivial stationary distributions for sufficient fast stirring. The second problem deals with a model of sympatric speciation, i.e. speciation in the absence of geographical separation, originally proposed by U. Dieckmann and M. Doebeli in 1999. We modify their original model to obtain several constant-population particle models. We concentrate on a continuous-time model that converges to a deterministic dynamical system as the number of particles becomes large. We establish various results regarding whether speciation occurs by studying the existence of bimodal stationary distributions for the limiting dynamical system.
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