Optimal life schedule with stochastic growth in age-size structured models: theory and an application.

Reproduction timing is one of the most important factors for the life history because it is closely related to subsistence of species. On the other hand, ecological demographers recently noted the effects of environmental stochasticity on the population dynamics by using linear demographic models because stochasticity reduces the population growth rate. Linear demographic models are generally composed of reproduction timing, several life history traits and stochasticity. The stochasticity is generated by twofold stochasticity, that is, internal and external stochasticities. In transition matrix models, the internal stochasticity gives a species a set of transition probabilities to other states, whereas the external stochasticity annually variegates the value of these transition probabilities. If the population vector has only the internal stochasticity, it satisfies a partial differential equation, in which it is described by a stochasticity in body-size growth rate. In this paper, we focus on the stochasticity which affects the body-size growth rate under r-selection. We construct a mathematical model of stochastic life history of each individual by using a stochastic differential equation, and analyze the relationship between optimal life schedule and the population dynamics by finding Euler-Lotka equation. Then, we use the formalism of path-integral expression to the population dynamics and show that the expression is consistent with other expressions in linear demographic models. Finally, we apply our method to a simple example, and obtain a characteristic of the stochasticity which has not only negative effect on the fitness but also positive effect from our model.