SPATIAL DYNAMICS OF THE DIFFUSIVE LOGISTIC EQUATION WITH A SEDENTARY COMPARTMENT

K. P. HADELER AND M. A. LEWIS

Abstract. We study an extension of the diffusive logistic equation or Fisher's equation for a situation where one part of the population is sedentary and reproducing, and the other part migrating and subject to mortality. We show that this system is essentially equivalent to a semi-linear wave equation with viscous damping. With respect to persistence in bounded domains with absorbing boundary conditions and with respect to the rate of spread of a locally introduced population, there are two distinct scenarios, depending on the choice of parameters. In the first scenario the population can survive in sufficiently large domains and the linearization at the leading edge of the front yields a unique candidate for the spread rate. In the second scenario the population can survive in arbitrarily small domains and there are two possible candidates for the spread rate. Analysis shows it is the larger candidate which gives the correct spread rate. The phenomenon of spread is also investigated using travelling wave theory. Here the minimal speed of possible travelling front solutions equals the previously calculated spread rate. The results are explained in biological terms.