Dynamical Systems in Population Biology

by Zhao, Xiao-Qiang

• ISBN13: 9780387003085
• ISBN10: 0387003088
• Format: Hardcover
• Copyright: 2003-07-01
• Publisher: Springer Nature

Summary

The conjoining of nonlinear dynamics and biology has brought about significant advances in both areas, with nonlinear dynamics providing a tool for understanding biological phenomena and biology stimulating developments in the theory of dynamical systems. This research monograph provides an introduction to the theory of nonautonomous semiflows with applications to population dynamics. It develops dynamical system approaches to various evolutionary equations such as difference, ordinary, functional, and partial differential equations, and pays more attention to periodic and almost periodic phenomena. The presentation includes persistence theory, monotone dynamics, periodic and almost periodicsemiflows, traveling waves, and global analysis of typical models in population biology. Research mathematicians working with nonlinear dynamics, particularly those interested in applications to biology, will find this book useful. It may also be used as a textbook or as supplementary reading for a graduate special topics course on the theory and applications of dynamical systems.Dr. Xiao-Qiang Zhao is a professor in applied mathematics at Memorial University of Newfoundland, Canada. His main research interests involve applied dynamical systems, nonlinear differential equations, and mathematical biology. He is the author of more than 40 papers and his research has played an important role in the development of the theory of periodic and almost periodic semiflows and their applications.

Author Biography

 Dr. Xiao-Qiang Zhao is a professor in applied mathematics at Memorial University of Newfoundland, Canada. His main research interests involve applied dynamical systems, nonlinear differential equations, and mathematical biology. He is the author of more than 40 papers and his research has played an important role in the development of the theory of periodic and almost periodic semiflows and their applications.

Table of Contents

Preface	vii
Dissipative Dynamical Systems	1	(36)
Limit Sets and Global Attractors	2	(2)
Chain Transitivity and Attractivity	4	(11)
Chain Transitive Sets	5	(6)
Attractivity and Morse Decompositions	11	(4)
Strong Repellers and Uniform Persistence	15	(11)
Strong Repellers	15	(3)
Uniform Persistence	18	(2)
Coexistence States	20	(4)
Order Persistence	24	(2)
Persistence Under Perturbations	26	(7)
Perturbation of a Globally Stable Steady State	27	(1)
Persistence Uniform in Parameters	28	(1)
Robust Permanence	29	(4)
Notes	33	(4)
Monotone Dynamics	37	(26)
Attracting Order Intervals and Connecting Orbits	38	(4)
Global Attractivity and Convergence	42	(4)
Subhomogeneous Maps and Skew-Product Semiflows	46	(6)
Competitive Systems on Ordered Banach Spaces	52	(3)
Exponential Ordering Induced Monotonicity	55	(5)
Notes	60	(3)
Nonautonomous Semiflows	63	(38)
Periodic Semiflows	64	(10)
Reduction to Poincare Maps	64	(2)
Monotone Periodic Systems	66	(8)
Asymptotically Periodic Semiflows	74	(10)
Reduction to Asymptotically Autonomous Processes	74	(2)
Asymptotically Periodic Systems	76	(8)
Monotone and Subhomogeneous Almost Peodic Systems	84	(9)
Continuous Processes	93	(5)
Notes	98	(3)
A Discrete-Time Chemostat Model	101	(10)
The Model	102	(2)
The Limiting System	104	(3)
Global Dynamics	107	(3)
Notes	110	(1)
N-Species Competition in a Periodic Chemostat	111	(22)
Weak Periodic Repellers	112	(3)
Single Population Growth	115	(7)
N-Species Competition	122	(4)
3-Species Competition	126	(6)
Notes	132	(1)
Almost Periodic Competitive Systems	133	(26)
Almost Periodic Attractors in Scalar Equations	134	(9)
Competitive Coexistence	143	(4)
An Almost Periodic Chemostat Model	147	(5)
Nonautonomous 2-Species Competitive Systems	152	(6)
Notes	158	(1)
Competitor--Competitor--Mutualist Systems	159	(30)
Weak Periodic Repellers	161	(2)
Competitive Coexistence	163	(8)
Competitive Exclusion	171	(4)
Bifurcations of Periodic Solutions: A Case Study	175	(13)
Notes	188	(1)
A Periodically Pulsed Bioreactor Model	189	(28)
The Model	190	(3)
Unperturbed Model	193	(11)
Conservation Principle	194	(1)
Single Species Growth	195	(5)
Two Species Competition	200	(4)
Perturbed Model	204	(11)
Periodic Systems with Parameters	205	(3)
Single Species Growth	208	(4)
Two Species Competition	212	(3)
Notes	215	(2)
A Nonlocal and Delayed Predator--Prey Model	217	(24)
The Model	218	(5)
Global Coexistence	223	(3)
Global Extinction	226	(2)
Global Attractivity: A Fluctuation Method	228	(3)
Threshold Dynamics: A Single Species Model	231	(7)
Notes	238	(3)
Traveling Waves in Bistable Nonlinearities	241	(20)
Existence of Periodic Traveling Waves	242	(6)
Attractivity and Uniqueness of Traveling Waves	248	(5)
Exponential Stability of Traveling Waves	253	(3)
Autonomous Case: A Spruce Budworm Model	256	(3)
Notes	259	(2)
References	261	(14)
Index	275