zbMATH — the first resource for mathematics

Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems. (English) Zbl 0821.34003
Mathematical Surveys and Monographs. 41. Providence, RI: American Mathematical Society (AMS). x, 174 p. (1995).
Ideas using monotonicity (or order preservence) have been appearing in study of evolution systems since a long time ago. However, not until the early 80’s were these ideas systematically developed and integrated with the theory of dynamical systems. At that time, the fundamental work of Hirsch opened an active area of research, now often referred to as the theory of monotone dynamical systems. After significant contributions of Matano, Smith and many other authors, the body of results in the theory and their applications was quickly expanded and an overview of the topic, such as the present monograph, became very desirable. The author presents existing basic results on continuous-time monotone systems in an up-to- date form with complete streamlined proofs. The account starts with abstract monotone dynamical systems giving the fundamentals (convergence criterion, limit set dichotomy) and addressing the central problems (stability, generic asymptotic behavior of solutions). A typical result proved here asserts that a generic initial condition has trajectories approaching to a set of equilibria, respectively, to a single equilibrium if the dynamical system is differentiable.
A major part of the text is devoted to applications of the abstract theory to differential equations. Ordinary, functional, as well as parabolic partial differential equations are discussed at various levels of generality. Using standard ideas of comparison (but also less standard methods such as exponential ordering), monotonicity of associated semiflows is shown for many concrete models of population dynamics and other fields of biology. An extensive bibliography quotes most of existing important contribution to the theory of continuous-time systems. Discrete-time monotone dynamical systems are (intentionally) not treated in the monograph. The book of P. Hess [Periodic-parabolic boundary value problems and positivity. Longman Scientific and Technical, New York (1991; Zbl 0731.35050)] is a good complementary reference for results then known. More recent results and references on discrete-time monotone dynamical systems can be found in the paper of the reviewer and L. Tereščák [J. Dyn. Differ. Equations 5, No. 2, 279-303 (1993; Zbl 0786.58002)].

34-02 Research exposition (monographs, survey articles) pertaining to ordinary differential equations
35K55 Nonlinear parabolic equations
34C11 Growth and boundedness of solutions to ordinary differential equations
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
37-XX Dynamical systems and ergodic theory
34K99 Functional-differential equations (including equations with delayed, advanced or state-dependent argument)
34K20 Stability theory of functional-differential equations
35B50 Maximum principles in context of PDEs