Size structured populations: Dispersion effects due to stochastic variability of the individual growth rate

Let z = z(a), a = chronological age, be a biometric descriptor of the individuals of a population, such as weight, a characteristic length,..., of the individuals. The variable z may be considered as a physiological age and r = dz/da is defined as the growth rate (growth velocity) of the individuals. The z-size structure of the population is obtained by distributing the individuals into z-classes: (z(i), z(i+1)), i = 0, 1,..., n, where z(i) = z(0) + i Delta z and Delta z is the size class. The class (z(0), z(1)) is the recruitment class. The discrete model for the dynamics of a H-size structured population presented here is based on the following main assumptions. The growth plasticity of the individuals is taken into account by assuming that the growth rate r is a random variable, with values r(j) = j Delta z/Delta t, j is an element of J = {m, m + 1,..., M - 1, M}. If r(m) < 0, then processes of shrinking or fragmentation may occur, for example, in the case of organisms with highly variable development (as clonal invertebrates and plants). The basic feedback, due to the population size, only occurs in survival of the recruitment in the first z-class. We obtain that the evolution equations are based on a generalized nonlinear Leslie matrix operator. Necessary and sufficient conditions for the existence of positive steady state solutions are given. An algorithm for computing these solutions is described. A local stability analysis around the equilibrium has also been performed. The (t, z)-continuous analogue of the discrete model has been derived: it consists of a first-order hyperbolic system. (C) 2000 Elsevier Science Ltd. All rights reserved.