Linear Processes in Stochastic Population Dynamics: Theory and Application to Insect Development

Abstract:
We consider stochastic population processes (Markov jump processes)

that develop as consequence of the occurrence of randon events at

random time-inervals. The population is divided into sub-populations or compartments. The events occur at rates that depend linearly with the number of individuals in the different described compartments. The dynamics is presented in terms of a Kolmogorov Forward Equation in the space of events and projected onto the space of populations when needed. The general properties of the problem are discussed. Solutions are obtained using a revised version of the Method of Characteristics. After a few examples of exact solutions we systematically develop short-time-approximations to the problem. While the lowest order approximation matches the Poisson and multinomial heuristics previously

proposed, higher-order approximations are completely new. Further, we

analyse a model for insect development as a sequence of E developmental

stages regulated by rates that are linear in the implied subpopulations. Transitions to the next stage compete with death at all times. The process ends at a predetermined stage, for example pupation or adult emergence. In its simpler version all the stages are distributed with the same characteristic time.