Mathematical modelling in ecology - A resource-based approach

在我的研究中，我使用基于资源的方法来研究人口动态。为了更好地理解群落的结构，特别是什么因素促进或限制了自然生态系统的多样性，我开发和分析了物种相互作用的数学模型，这些模型可以在实验室中进行测试。目标是调和普遍认为的一般原则，例如“竞争排斥原则”，基本上是说，如果竞争n个生态位（例如，资源），最多有n个物种持续存在，与矛盾的实验观察，例如“浮游生物悖论”，因此提出新的或修改的原则。尽管捕食通常被认为是自然生态系统多样性的因素之一，但竞争通常被认为是限制多样性的因素。然而，在我的研究中，我发现了各种模型的例子，这些模型预测“竞争介导共存”是可能的，即在某个竞争对手存在的情况下，更多的物种会持续存在，而不是在该竞争对手被消灭的情况下。只要有可能，我就会尝试进行全局分析。分岔理论帮助人们确定所有适当的参数范围和初始状态的全部行为谱。它还确定了需要测量哪些关键参数以及测量的准确度，因为只有它们的大小变化才会导致动力学的变化。计算机模拟被用来阐明复杂的动力学、检验猜想和揭示模型的特性，这些特性对发展分析证明是有用的。符号计算用于复杂的计算。当吸引不变集比较复杂且存在多个吸引子时，专门的分岔图获取软件尤为有用。这些分析常常导致动力系统、普通、脉冲、积分和泛函微分方程、微分方程的定性理论（包括分岔和稳定性理论）中有趣的抽象数学问题。

In my research I use a resource-based approach to study population dynamics.  In an attempt to better understand the structure of communities, and in particular, what factors promote or limit the diversity of natural ecosystems, I develop and analyze  mathematical models of  species interactions that make predictions that can be  tested in the laboratory. The goal is  to reconcile what were commonly   believed general principles, e.g., the  "Principle of Competitive Exclusion," that basically states that at most n species persist if competing for n niches (e.g., resources), with conflicting experimental observations e.g., the "Paradox of the Plankton," and hence  suggest new or modified principles. Whereas predation is generally considered to be one of the factors responsible for the diversity in natural ecosystems,  competition is usually thought of as limiting diversity.  However, in my research, I have found diverse examples of models that predict "Competitor-Mediated Coexistence" is possible, i.e. more species persist in the presence of a certain competitor than would persist if that competitor is eliminated. Whenever possible I try to do global analyses. Bifurcation theory helps one to determine the full spectrum of behaviour for all appropriate parameter ranges and initial states. It also identifies which key parameters need to be measured and how accurately, since only changes in their magnitude result in changes in the dynamics. Computer simulations are used to elucidate complicated dynamics, to test conjectures and to reveal properties of the models that are useful in developing analytic proofs. Symbolic computation is used for complicated calculations. Specialized software for obtaining bifurcation diagrams is especially useful when the attracting invariant sets are complicated and there are multiple attractors. The analyses often lead to interesting abstract mathematical problems in dynamical systems, ordinary, impulsive, integro- and functional differential equations, the qualitative theory of differential equations including bifurcation and stability theory.