J. Math. Biol. (1991) 29:363-378 •Journal of Mathematical 61ol09y © Springer-Verlag 1991 Periodic solutions for a population dynamics problem with age-dependence and spatial structure M. Kubo* and M. Langlais U.F.R. Sciences Humaines Appliqu~es, UA CNRS 226, Universit~ de Bordeaux II, 146, rue Leo Saignat, F-33076 Bordeaux Cedex, France Received March 19, 1990; received in revised form September 29, 1990 Akstraet. Using a linear model with age-dependence and spatial structure we show how a periodical supply of individuals will transform an exponentially decaying distribution of population into a non-trivial asymptotically stable periodic distribution. Next we give an application to an epidemic model. Key words: Periodic solutions - Stability - Age-dependence - Spatial structure - Epidemic model 1. Introduction Let u(x, t, a) >1 0 be the distribution of individuals in a single species population having age a > 0 at time t > 0 and position x in f2 an open and bounded domain in R N. We assume that the flux of population is co-linear to the spatial gradient and given by kVxu(x, t, a), k being a positive constant. Suppose now that the net change in density is negative and takes the form s(x, t, a)=-#(a).u(x, t, a) where/~(a)/> 0 is the natural death-rate. Then, following Gurtin [5], a mathematical model describing the evolution of the distribution of population u starting at time t = 0 with initial distribution u(x, O, a) = uo(x, a) >>. O, x e I2, a>0 (1.1) is d,u+O~u-kAu+#(a)u=O, x~t2, t>0, a>0 (1.2) to which we add the birth process u(x, t, O) = fl(a)u(x, t, a) da, x ~ I2, t > 0, (1.3) * Permanent address: Department of Mathematics, Faculty of Science and Engineering, Saga Univeristy, I Honjo-machi, Saga 840, Japan