Exactly Solvable Models of Biological Invasion

Exactly Solvable Models of Biological Invasion

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Much of our current knowledge on biological invasion was derived from field studies, but many recent advances relied heavily on mathematics and computing, particularly mathematical modeling. While numerical simulations are clearly a useful approach, they have some serious drawbacks. Approximations errors and the number of parameter values can have a significant impact on the simulation results, the extent of which often remains obscure. Such difficulties do not arise, however, when the problem can be solved analytically. Exactly Solvable Models of Biological Invasion demonstrates the advantages and methods of obtaining exact solutions of partial differential equations that describe nonlinear problems encountered in the study of invasive species spread. With emphasis on PDEs of diffusion-reaction type, the authors present a comprehensive collection of exactly solvable models and a unified, self-contained description of the relevant mathematical methods. In doing so, they also provide new insight into important issues such as the impact of the Allee effect, the impact of predation, and the interplay between different modes of species dispersal. Full calculation details make this presentation accessible to biologists as well as applied mathematicians, and a range of ecological examples and applications demonstrate the utility of exact methods in practice. Exact solutions provide an immediate, complete description of system dynamics for a wide class of initial conditions and serve as a convenient tool for testing numerical algorithms and codes used in more specialized studies. This book lays the groundwork for bringing the power of exactly solvable models to bear on real-world ecological problems.show more

Product details

  • Hardback | 232 pages
  • 154 x 236 x 20mm | 498.96g
  • Taylor & Francis Inc
  • Chapman & Hall/CRC
  • Boca Raton, FL, United States
  • English
  • 62 black & white illustrations, 4 black & white halftones
  • 1584885211
  • 9781584885214
  • 1,888,991

Table of contents

INTRODUCTION Why exactly solvable models are important Intra- and inter-species interactions and local population dynamics Basic mechanisms of species transport Biological invasion: main facts and constituting examples MODELS OF BIOLOGICAL INVASION Diffusion-reaction equations Integral-difference models Space-discrete models Stochastic models Concluding remarks BASIC METHODS AND RELEVANT EXAMPLES The Cole-Hopf transformation and the Burgers equation as a paradigm Further application of the Cole-Hopf transformation Method of piecewise linear approximation Exact solutions of a generalized Fisher equation More about ansatz SINGLE-SPECIES MODELS Impact of advection and migration Accelerating population waves The problem of critical aggregation DENSITY-DEPENDENT DIFFUSION The Aronson-Newman solution and its generalization Stratified diffusion and the Allee effect MODELS OF INTERACTING POPULATIONS Exact solution for a diffusive predator-prey system Migration waves in a resource-consumer system SOME ALTERNATIVE AND COMPLEMENTARY APPROACHES Wave speed and the eigenvalue problem Convergence of the initial conditions Convergence and the paradox of linearization Application of the comparison principle ECOLOGICAL EXAMPLES AND APPLICATIONS Invasion of Japanese beetle in the United States Mount St. Helens recolonization and the impact of predation Stratified diffusion and rapid plant invasion APPENDIX: BASIC BACKGROUND MATHEMATICS Ordinary differential equations and their solutions Phase plane and stability analysis Diffusion equation References Indexshow more