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 = dzda 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: (zi, Zi + 1), i = 0,1,…, n, where zi = z0 + iΔz and Δz is the size class. The class (z0, z1) is the recruitment class. The discrete model for the dynamics of a z-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 rj = jΔzΔt, j ϵ J = {m, m + 1,…, M − 1, M}. If rm < 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. ain 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.