Unification theory of optimal life histories and linear demographic models in internal stochasticity.

Ryo Oizumi

doi:10.1371/journal.pone.0098746

Abstract

Life history of organisms is exposed to uncertainty generated by internal and external stochasticities. Internal stochasticity is generated by the randomness in each individual life history, such as randomness in food intake, genetic character and size growth rate, whereas external stochasticity is due to the environment. For instance, it is known that the external stochasticity tends to affect population growth rate negatively. It has been shown in a recent theoretical study using path-integral formulation in structured linear demographic models that internal stochasticity can affect population growth rate positively or negatively. However, internal stochasticity has not been the main subject of researches. Taking account of effect of internal stochasticity on the population growth rate, the fittest organism has the optimal control of life history affected by the stochasticity in the habitat. The study of this control is known as the optimal life schedule problems. In order to analyze the optimal control under internal stochasticity, we need to make use of “Stochastic Control Theory” in the optimal life schedule problem. There is, however, no such kind of theory unifying optimal life history and internal stochasticity. This study focuses on an extension of optimal life schedule problems to unify control theory of internal stochasticity into linear demographic models. First, we show the relationship between the general age-states linear demographic models and the stochastic control theory via several mathematical formulations, such as path–integral, integral equation, and transition matrix. Secondly, we apply our theory to a two-resource utilization model for two different breeding systems: semelparity and iteroparity. Finally, we show that the diversity of resources is important for species in a case. Our study shows that this unification theory can address risk hedges of life history in general age-states linear demographic models.

Highlights

Environmental stochasticity is one of the problems which organisms face in their life schedule because it brings uncertainty to their maturity and reproduction timing
Iteroparous species evolve to have optimal utilization and to survive as long as possible, as found for trees. This theory generalizes from the previous study [10] and extends to general optimal life schedule problem’’ (OLSP) in linear demographic models (LDM)
The path-integral formulation is more suited to address controlled life history than the others (PDE, transition matrix models (TMM), and integral projection models (IPM)) because it conserves continuity of states and does not require differentiablity of parameters: gð:Þ and sð:Þ. These parameters represent statistics of stage transition rate in TMM; they appear as components of Hamiltonian in the pathintegral formulation. This Hamiltonian has a significant meaning in OLSP because the optimal control of life history minimizes or maximizes it depending on fertility function

Summary

Introduction

Environmental stochasticity is one of the problems which organisms face in their life schedule because it brings uncertainty to their maturity and reproduction timing. Life history with internal stochasticity generates a population density function (population vector) that can be expressed by an age-size structured LDM. Oizumi and Takada (2013) asserted that the objective function should be the Laplace transform of the expectation of the reproductive success (ERS), because it is generated by the Euler–Lotka equation in the age-size structured model.

Results

Conclusion