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The Scientific World Journal
Volume 2014 (2014), Article ID 873624, 15 pages
http://dx.doi.org/10.1155/2014/873624
Research Article

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

1Departamento de Física, FCEN-UBA and IFIBA-CONICET, C1428EGA Buenos Aires, Argentina
2Centre for Mathematical Sciences, Lund University, P.O. Box 118, 221 00 Lund, Sweden

Received 15 August 2013; Accepted 23 October 2013; Published 17 February 2014

Academic Editors: S. Casado, W. Fei, and T. Nguyen

Copyright © 2014 Hernán G. Solari and Mario A. Natiello. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We consider stochastic population processes (Markov jump processes) that develop as a consequence of the occurrence of random events at random time intervals. The population is divided into subpopulations or compartments. The events occur at rates that depend linearly on the number of individuals in the different described compartments. The dynamics is presented in terms of 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 developmental stages regulated by rates that are linear in the implied subpopulations. Transition to the next stage competes 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.