Mathematical Sciences: Spatial and Temporal Patterns in Discrete-Time Growth-Dispersal Models

积分差分方程是在种群生物学中出现的简单的离散时间模型，作为具有离散的非重叠世代和明确的生长和扩散阶段的生物体的模型。这些方程具有连续时间反应扩散模型的许多特性。同时，标量积分差分方程很容易表现出周期加倍和混沌、行循环和行波。而积分差分方程组则表现出一种新的扩散不稳定性和模式形成。研究者建议使用各种分析和计算技术来研究这些模型中出现的行波和周期以及不同的空间模式。有些方程的解在空间或时间上显示出重复的行为，形成模式。确定在什么条件下会出现这种模式形成是很有趣的。这一研究将阐明一般模式的形成。这里研究的方程与生态过程模型有关。景观生态学正开始研究生态过程如何产生景观上观察到的复杂空间格局，以及这些格局反过来又如何制约或决定这些过程的速率。研究者研究了一种可能导致空间格局的基本机制。该项目具有实际的影响，因为它可以更好地预测入侵的河岸植物和入侵的害虫的传播。

Integrodifference equations are simple discrete-time models that arise in population biology as models for organisms with discrete nonoverlapping generations and well-defined growth and dispersal stages. These equations possess many of the attributes of continuous-time reaction-diffusion models. At the same time, scalar integrodifference equations readily exhibit period-doubling and chaos, and traveling cycles as well as traveling waves. Systems of integrodifference equations, in turn, exhibit a new variety of diffusive instability and pattern formation. The investigator proposes to study the traveling waves and cycles and the varied spatial patterns that arise in these models, employing a variety of analytical and computational techniques. Some equations have solutions that show repetitive behavior either in space or in time, developing patterns. It is of interest to determine under what conditions such pattern formations can arise. This research will shed light on pattern formation in general. The equations studied here are related to models of ecological processes. Landscape ecology is beginning to investigate how ecological processes generate the complex spatial patterns observed on the landscape and how these patterns, in turn, constrain or determine the rates of the processes. The investigator examines a fundamental mechanism that may be responsible for spatial patterns. The project has practical ramifications in that it may allow for better prediction of the spread of invading riparian plants and invading insect pests.