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

积分差分方程是一种简单的离散时间模型，出现在种群生物学中，作为具有离散非重叠世代和明确定义的生长和扩散阶段的生物体的模型。 这些方程具有许多连续时间反应扩散模型的属性。 同时，标量积分差分方程容易表现出倍周期和混沌，以及行波和行波循环。 积分差分方程系统，反过来，表现出一种新的扩散不稳定性和图案的形成。 研究人员建议研究行波和周期以及这些模型中出现的各种空间模式，采用各种分析和计算技术。 一些方程的解在空间或时间上表现出重复的行为，形成模式。 感兴趣的是确定在什么条件下可以出现这种图案形成。 这项研究将揭示一般模式的形成。 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，以及这些格局如何反过来约束或决定过程的速率。 研究人员研究了可能对空间模式负责的基本机制。 该项目具有实际影响，因为它可以更好地预测入侵河岸植物和入侵害虫的传播。

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.