Exactly Solvable Models of Biological Invasion

by Petrovskii; Sergei V.

• ISBN13: 9781584885214
• ISBN10: 1584885211
• Format: Hardcover
• Copyright: 2005-07-28
• Publisher: Chapman & Hall/

Summary

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.

Author Biography

Sergei V. Petrovskii is a Senior Scientist in the Theoretical Department of the Shirshov Institute for Oceanology, Russian Academy of Science, Moscow Bai-Lian Li is Professor of Mathematical and Theoretical Ecology at the University of California, Riverside, USA, and the Founding Editor-in-Chief of an international journal: Ecological Complexity

Table of Contents

1 Introduction	1	(16)
1.1 Why exactly solvable models are important	1	(4)
1.2 Intra- and inter-species interactions and local population dynamics	5	(6)
1.3 Basic mechanisms of species transport	11	(3)
1.4 Biological invasion: main facts and constituting examples	14	(3)
2 Models of biological invasion	17	(28)
2.1 Diffusion-reaction equations	17	(7)
2.2 Integral-difference models	24	(6)
2.3 Space-discrete models	30	(9)
2.4 Stochastic models	39	(3)
2.5 Concluding remarks	42	(3)
3 Basic methods and relevant examples	45	(36)
3.1 The Cole-Hopf transformation and the Burgers equation as a paradigm	46	(10)
3.1.1 Exact solutions for a forced Burgers equation	50	(6)
3.2 Further application of the Cole-Hopf transformation	56	(4)
3.3 Method of piecewise linear approximation	60	(8)
3.3.1 Exact solution for a population with logistic growth	60	(3)
3.3.2 Exact solution for a population with a strong Allee effect	63	(5)
3.4 Exact solutions of a generalized Fisher equation	68	(6)
3.4.1 Ansatz	69	(2)
3.4.2 The Ablowitz-Zeppetella method	71	(3)
3.5 More about ansatz	74	(7)
4 Single-species models	81	(36)
4.1 Impact of advection and migration	82	(7)
4.1.1 Advection	84	(1)
4.1.2 Density-dependent migration	85	(3)
4.1.3 General case	88	(1)
4.2 Accelerating population waves	89	(13)
4.2.1 Self-similar exact solution	93	(9)
4.3 The problem of critical aggregation	102	(15)
4.3.1 Practical stability concept	105	(6)
4.3.2 The Wilhelmsson "blow-up" solution	111	(6)
5 Density-dependent diffusion	117	(20)
5.1 The Aronson-Newman solution and its generalization	117	(9)
5.1.1 A general case	120	(6)
5.2 Stratified diffusion and the Allee effect	126	(11)
6 Models of interacting populations	137	(22)
6.1 Exact solution for a diffusive predator-prey system	137	(17)
6.1.1 Properties of the local system	140	(3)
6.1.2 Exact solution and its properties	143	(6)
6.1.3 Formal derivation of the exact solution	149	(5)
6.2 Migration waves in a resource-consumer system	154	(5)
7 Some alternative and complementary approaches	159	(12)
7.1 Wave speed and the eigenvalue problem	160	(3)
7.2 Convergence of the initial conditions	163	(2)
7.3 Convergence and the paradox of linearization	165	(3)
7.4 Application of the comparison principle	168	(3)
8 Ecological examples and applications	171	(24)
8.1 Invasion of Japanese beetle in the United States	172	(6)
8.2 Mount St. Helens recolonization and the impact of predation	178	(9)
8.3 Stratified diffusion and rapid plant invasion	187	(8)
9 Appendix: Basic background mathematics	195	(10)
9.1 Ordinary differential equations and their solutions	195	(3)
9.2 Phase plane and stability analysis	198	(2)
9.3 Diffusion equation	200	(5)
References	205	(10)
Index	215