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Book Review

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Book Information:

Authors: R. S. Cantrell and C. Cosner
Title: Spatial ecology via reaction-diffusion equations
Additional book information: John Wiley & Sons Ltd., Chichester, UK, 2003, 428 pp., ISBN 0-471-49301-5, $155.00$

Geoffrey Butler and Paul Waltman, Persistence in dynamical systems, J. Differential Equations 63 (1986), no. 2, 255–263. MR 848269, DOI 10.1016/0022-0396(86)90049-5

Geoffrey Butler, H. I. Freedman, and Paul Waltman, Uniformly persistent systems, Proc. Amer. Math. Soc. 96 (1986), no. 3, 425–430. MR 822433, DOI 10.1090/S0002-9939-1986-0822433-4

Robert Stephen Cantrell, Chris Cosner, and Vivian Hutson, Permanence in ecological systems with spatial heterogeneity, Proc. Roy. Soc. Edinburgh Sect. A 123 (1993), no. 3, 533–559. MR 1226616, DOI 10.1017/S0308210500025877

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Jack K. Hale, Asymptotic behavior of dissipative systems, Mathematical Surveys and Monographs, vol. 25, American Mathematical Society, Providence, RI, 1988. MR 941371, DOI 10.1090/surv/025

Jack K. Hale and Paul Waltman, Persistence in infinite-dimensional systems, SIAM J. Math. Anal. 20 (1989), no. 2, 388–395. MR 982666, DOI 10.1137/0520025

Daniel Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, vol. 840, Springer-Verlag, Berlin-New York, 1981. MR 610244

Peter Hess and Alan C. Lazer, On an abstract competition model and applications, Nonlinear Anal. 16 (1991), no. 11, 917–940. MR 1106994, DOI 10.1016/0362-546X(91)90097-K

Morris W. Hirsch, Differential equations and convergence almost everywhere in strongly monotone semiflows, Nonlinear partial differential equations (Durham, N.H., 1982) Contemp. Math., vol. 17, Amer. Math. Soc., Providence, R.I., 1983, pp. 267–285. MR 706104

10.

M. W. Hirsch and H. L. Smith, Monotone Dynamical Systems, in preparation.

Morris W. Hirsch, Hal L. Smith, and Xiao-Qiang Zhao, Chain transitivity, attractivity, and strong repellors for semidynamical systems, J. Dynam. Differential Equations 13 (2001), no. 1, 107–131. MR 1822214, DOI 10.1023/A:1009044515567

Josef Hofbauer and Karl Sigmund, Evolutionary games and population dynamics, Cambridge University Press, Cambridge, 1998. MR 1635735, DOI 10.1017/CBO9781139173179

S. B. Hsu, H. L. Smith, and Paul Waltman, Competitive exclusion and coexistence for competitive systems on ordered Banach spaces, Trans. Amer. Math. Soc. 348 (1996), no. 10, 4083–4094. MR 1373638, DOI 10.1090/S0002-9947-96-01724-2

V. Hutson, A theorem on average Liapunov functions, Monatsh. Math. 98 (1984), no. 4, 267–275. MR 776353, DOI 10.1007/BF01540776

Vivian Hutson and Klaus Schmitt, Permanence and the dynamics of biological systems, Math. Biosci. 111 (1992), no. 1, 1–71. MR 1175114, DOI 10.1016/0025-5564(92)90078-B

Olga Ladyzhenskaya, Attractors for semigroups and evolution equations, Lezioni Lincee. [Lincei Lectures], Cambridge University Press, Cambridge, 1991. MR 1133627, DOI 10.1017/CBO9780511569418

Xing Liang and Jifa Jiang, Discrete infinite-dimensional type-$K$ monotone dynamical systems and time-periodic reaction-diffusion systems, J. Differential Equations 189 (2003), no. 1, 318–354. MR 1968324, DOI 10.1016/S0022-0396(02)00062-1

Hiroshi Matano, Existence of nontrivial unstable sets for equilibriums of strongly order-preserving systems, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 30 (1984), no. 3, 645–673. MR 731522

Robert M. May and Warren J. Leonard, Nonlinear aspects of competition between three species, SIAM J. Appl. Math. 29 (1975), no. 2, 243–253. MR 392035, DOI 10.1137/0129022

Sebastian J. Schreiber, Criteria for $C^r$ robust permanence, J. Differential Equations 162 (2000), no. 2, 400–426. MR 1751711, DOI 10.1006/jdeq.1999.3719

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Hal L. Smith, Monotone dynamical systems, Mathematical Surveys and Monographs, vol. 41, American Mathematical Society, Providence, RI, 1995. An introduction to the theory of competitive and cooperative systems. MR 1319817

23.

H. L. Smith, Dynamics of competition, Mathematics Inspired by Biology, Springer Lecture Notes in Math. 1714 (1999), 191-240.

Hal L. Smith and Horst R. Thieme, Convergence for strongly order-preserving semiflows, SIAM J. Math. Anal. 22 (1991), no. 4, 1081–1101. MR 1112067, DOI 10.1137/0522070

H. L. Smith and H. R. Thieme, Stable coexistence and bi-stability for competitive systems on ordered Banach spaces, J. Differential Equations 176 (2001), no. 1, 195–222. MR 1861187, DOI 10.1006/jdeq.2001.3981

Horst R. Thieme, Persistence under relaxed point-dissipativity (with application to an endemic model), SIAM J. Math. Anal. 24 (1993), no. 2, 407–435. MR 1205534, DOI 10.1137/0524026

Yi Wang and Jifa Jiang, The general properties of discrete-time competitive dynamical systems, J. Differential Equations 176 (2001), no. 2, 470–493. MR 1866283, DOI 10.1006/jdeq.2001.3989

28.

X.-Q. Zhao, Dynamical Systems in Population Biology, CMS Books in Mathematics, Springer, New York, 2003.

Hal L. Smith and Xiao-Qiang Zhao, Robust persistence for semidynamical systems, Proceedings of the Third World Congress of Nonlinear Analysts, Part 9 (Catania, 2000), 2001, pp. 6169–6179. MR 1971507, DOI 10.1016/S0362-546X(01)00678-2

Review Information:

Reviewer: H. L. Smith
Affiliation: Arizona State University
Email: halsmith@asu.edu