Electronic Journal of Differential Equations, Vol. 2002(2002), No. 28, pp. 1–9. ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp) OPTIMAL CONTROL FOR A NONLINEAR AGE-STRUCTURED POPULATION DYNAMICS MODEL BEDR’EDDINE AINSEBA, SEBASTIAN ANIT ¸ A, & MICHEL LANGLAIS Abstract. We investigate the optimal harvesting problem for a nonlinear age-dependent and spatially structured population dynamics model where the birth process is described by a nonlocal and nonlinear boundary condition. We establish an existence and uniqueness result and prove the existence of an optimal control. We also establish necessary optimality conditions. 1. Introduction and setting of the problem We consider a general mathematical model describing the dynamics of a single species population with age dependence and spatial structure. Let u(x, t, a) be the distribution of individuals of age a ≥ 0 at time t ≥ 0 and location x in Ω. Here Ω is a bounded open subset of R N , N ∈{1, 2, 3}, with a suitably smooth boundary ∂ Ω. Thus P (x, t)=  A † 0 u(x, t, a) da (1.1) is the total population at time t and location x, where A † is the maximal age of an individual. Let β(x, t, a, P (x, t)) ≥ 0 be the natural fertility-rate, and let μ(x, t, a, P (x, t)) ≥ 0 be the natural death-rate of individuals of age a at time t and location x. We also assume that the flux of population takes the form k∇u(x, t, a) with k> 0, where ∇ is the gradient vector with respect to the spatial variable x. In this paper we are concerned with the optimal harvesting problem on the time interval (0,T ), T> 0, subject to an external supply of individuals f (x, t, a) ≥ 0 and to a specific harvesting effort v(x, t, a), where (x, t, a) ∈ Q =Ω × (0,T ) × (0,A † ). So, we deal with the problem of finding the harvesting effort v in order to obtain the best harvest; i.e., Maximize, over all v ∈V , the value of  Q v(x, t, a)g(x, t, a)u v (x, t, a) dx dt da , (1.2) 2000 Mathematics Subject Classification. 35D10, 49J20, 49K20, 92D25. Key words and phrases. Optimal control, optimality conditions, age-structured population dynamics. c 2002 Southwest Texas State University. Submitted January 4, 2003. Published March 16, 2003. 1