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The KPP equation with noise - a model exhibiting local, density dependent competition

带有噪声的 KPP 方程 - 一个展示局部、密度相关竞争的模型

基本信息

批准号：
393092071
负责人：
Dr. Sandra Kliem
金额：
--
依托单位：
Mathematics Institute
依托单位国家：
德国
项目类别：
Research Grants
财政年份：
2017
资助国家：
德国
起止时间：
2016-12-31 至 2020-12-31
项目状态：
已结题

来源：
https://gepris.dfg.de/gepris/projekt/393092071?language=en
关键词：
KPP equation noise model exhibiting

项目摘要

In this project, the main goal is to obtain a better understanding of the dynamics of solutions to the Kolmogorov-Petrovskii-Piskunov-(KPP) equation (also known as the Kolmogorov- or Fisher-equation) with noise. Solutions to this stochastic partial differential equation (SPDE) model the (random) density of a population in time and space that undergoes local, linear mass creation and local, density dependent competition. The competition term in the SPDE introduces non-trivial dependence in space and thereby makes the model a very challenging one.The rate of mass creation is parameter-dependent. There exists a critical value for the mass-creation parameter. If started in compactly supported initial conditions with the parameter fixed above the critical value, solutions have a positive probability of not going extinct in finite time. Is local survival possible? In particular, does a so-called complete convergence theorem hold as for the nearest neighbor contact process? As of yet, a positive answer only exists for translation invariant initial conditions or high parameter values. We study these questions in the regime of parameters close to criticality. Starting in Heavyside initial data, with the parameter fixed above the critical value, the speed of the right front marker of the solution (the supremum of the support) was recently shown to be deterministic and positive. What happens if we start compactly supported instead and condition on survival? It is our goal in this project to obtain new insights on the dynamics of solutions to the KPP equation with noise. In particular, we aim at a better understanding of the interplay of invasion of empty space at the front and progression of mass under competition at the back.Another objective of this work is to obtain particle representations for solutions of this SPDE to clarify the notion of the competition in terms of one-to-one interactions. Such representations should lead to a better understanding of the driving forces behind the paths of solutions and result in a new set of tools to investigate their behavior over time.

在这个项目中，主要目标是更好地理解Kolmogorov-Petrovskii-Piskunov-（KPP）方程（也称为Kolmogorov-或Fisher方程）的动态解。这个随机偏微分方程（SPDE）的解模拟了在时间和空间中经历局部线性质量创造和局部密度依赖竞争的种群的（随机）密度。SPDE中的竞争项引入了非平凡的空间依赖性，从而使模型成为一个非常具有挑战性的模型。体量创建参数存在一个临界值。如果在紧支撑的初始条件下开始，参数固定在临界值以上，解在有限时间内不灭绝的概率为正。当地的生存是可能的吗？特别是，所谓的完全收敛定理是否适用于最近邻接触过程？到目前为止，一个肯定的答案只存在于平移不变的初始条件或高参数值。我们研究这些问题的参数接近临界状态的政权。从Heavyside初始数据开始，参数固定在临界值以上，解决方案的右前标记（支持的上确界）的速度最近被证明是确定性的和积极的。如果我们以生存为条件，开始全面支持，会发生什么？这是我们在这个项目中的目标，以获得新的见解的动态KPP方程的解决方案与噪声。特别是，我们的目标是在一个更好地理解的相互作用的入侵的空的空间在前面和进展的质量下的竞争在back.Another这项工作的目的是获得粒子表示的解决方案，这个SPDE澄清的概念的竞争方面的一对一的相互作用。这种表示应该会让我们更好地了解解决方案路径背后的驱动力，并产生一套新的工具来研究它们随着时间的推移的行为。