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。 为了更好地了解群落的结构，特别是哪些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。我们的目标是调和 相信一般原则，例如，“竞争排斥原则”，基本上是说，如果竞争n个小生境，最多n个物种会持续存在（例如，资源），与实验观察相矛盾，例如，“浮游生物的Paramentary”，因此提出了新的或修改的原则。虽然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，我发现了预测“竞争者介导的共存”是可能的模型的各种例子，即更多的物种在某种竞争对手存在的情况下持续存在，而不是在竞争对手被消除的情况下持续存在。只要有可能，我都会尝试进行全局分析。分叉理论有助于确定所有适当的参数范围和初始状态的行为的全谱。它还确定了哪些关键参数需要测量，以及测量的准确程度，因为只有这些参数的幅度变化才会导致动态变化。计算机模拟被用来阐明复杂的动力学，测试结构，并揭示在开发分析证明有用的模型的属性。符号计算用于复杂的计算。当吸引不变集复杂且存在多个吸引子时，用于获得分叉图的专用软件特别有用。分析往往导致有趣的抽象数学问题的动力系统，普通，脉冲，积分和功能微分方程，微分方程的定性理论，包括分歧和稳定性理论。

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.