Saddle-Point Approximations, Integrodifference Equations, and Invasions
| Autor: | Michael G. Neubert, Mark Kot |
|---|---|
| Rok vydání: | 2008 |
| Předmět: |
Mathematical optimization
General Mathematics Population Dynamics Immunology Population Models Biological General Biochemistry Genetics and Molecular Biology Saddle point Animals Quantitative Biology::Populations and Evolution Applied mathematics Population Growth education Ecosystem Saddle General Environmental Science Mathematics Population Density Pharmacology education.field_of_study Models Statistical Laplace transform Mathematical model General Neuroscience Nonlinear system Computational Theory and Mathematics Linear Models Method of steepest descent Biological dispersal General Agricultural and Biological Sciences Algorithms |
| Zdroj: | Bulletin of Mathematical Biology. 70:1790-1826 |
| ISSN: | 1522-9602 0092-8240 |
| Popis: | Invasion, the growth in numbers and spatial spread of a population over time, is a fundamental process in ecology. Governments and businesses expend vast sums to prevent and control invasions of pests and pestilences and to promote invasions of endangered species and biological control agents. Many mathematical models of biological invasions use nonlinear integrodifference equations to describe the growth and dispersal processes and to predict the speed of invasion fronts. Linear models have received less attention, perhaps because they are difficult to simulate for large times. In this paper, we use the saddle-point method, alias the method of steepest descent, to derive asymptotic approximations for the solutions of linear integrodifference equations. We work through five examples, for Gaussian, Laplace, and uniform dispersal kernels in one dimension and for asymmetric Gaussian and radially symmetric Laplace kernels in two dimensions. Our approximations are extremely close to the exact solutions, even for intermediate times. We also employ an empirical saddle-point approximation to predict densities using dispersal data. We use our approximations to examine the effects of censored dispersal data on estimates of invasion speed and population density. |
| Databáze: | OpenAIRE |
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