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

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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.