MATHEMATICAL MODELLING OF POPULATION DYNAMICS BANACH CENTER PUBLICATIONS, VOLUME 63 INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES WARSZAWA 2004 STRUCTURED POPULATION DYNAMICS GLENN F. WEBB Department of Mathematics, Vanderbilt University Nashville, TN 37235, U.S.A. E-mail: glenn.f.webb@vanderbilt.edu Abstract. The objective of these lectures is to apply the theory of linear and nonlinear semi- groups of operators to models of structured populations dynamics. The mathematical models of structured populations are typically partial differential equations with variables correspond- ing to such properties of individual as age, size, maturity, proliferative state, quiescent state, phenotype expression, or other physical properties. The main goal is to connect behavior at the individual level to behavior at the population level. Theoretical results from semigroup theory are applied to analyze such population behaviors as extinction, growth, stabilization, oscillation, and chaos. 1. General theory of operator semigroups in Banach spaces. In this section we provide basic definitions and theorems in the theory of semigroups of operators in Banach spaces and illustrate the concepts with some examples relevant to structured populations. Definition 1.1. Let X be a Banach space and let Y ⊂ X.A strongly continuous semi- group of operators in Y is a set of operators (linear or nonlinear) T (t),t ≥ 0 satisfying (i) T (t) is continuous from Y to Y ∀t ≥ 0, (ii) T (0)φ = φ ∀φ ∈ Y , (iii) T (t + s)φ = T (t)T (s)φ ∀t,s ≥ 0 and φ ∈ Y , (iv) t → T (t)φ is continuous ∀φ ∈ Y . We remark that if Y = X and T (t) ∈ B(X) ∀t ≥ 0, where B(X) is the Banach algebra of bounded linear operators in X, then T (t),t ≥ 0 is called a strongly continuous semigroup of bounded linear operators in X. Otherwise, T (t),t ≥ 0 is called a nonlinear semigroup in Y . Example 1.1. Let X = C [0, 1] or C 0 [0, 1], where C [0, 1] is the Banach space of continuous real-valued functions on [0, 1] with supremum norm and C 0 [0, 1] is the subspace of C [0, 1] 2000 Mathematics Subject Classification : Primary 92D25; Secondary 34G10, 34G20, 47D03. The paper is in final form and no version of it will be published elsewhere. [123]