Inequality in resource allocation and population dynamics models.

Masahiro Anazawa

doi:10.1098/rsos.182178

Abstract

The Hassell model has been widely used as a general discrete-time population dynamics model that describes both contest and scramble intraspecific competition through a tunable exponent. Since the two types of competition generally lead to different degrees of inequality in the resource distribution among individuals, the exponent is expected to be related to this inequality. However, among various first-principles derivations of this model, none is consistent with this expectation. This paper explores whether a Hassell model with an exponent related to inequality in resource allocation can be derived from first principles. Indeed, such a Hassell model can be derived by assuming random competition for resources among the individuals wherein each individual can obtain only a fixed amount of resources at a time. Changing the size of the resource unit alters the degree of inequality, and the exponent changes accordingly. As expected, the Beverton–Holt and Ricker models can be regarded as the highest and lowest inequality cases of the derived Hassell model, respectively. Two additional Hassell models are derived under some modified assumptions.

Highlights

The population dynamics of seasonally reproducing species with non-overlapping generations, such as insects, are often described using discrete-time population models Nt+1 = f(Nt), which express the population size at one generation Nt+1 as a function of the population size at the previous generation Nt
Is well known that when a population is subject to mortality that linearly increases with density, the population size after a certain period is described by a Beverton–Holt model as a function of the original population size
The exponent is expected to relate to the degree of inequality in the resource allocation, but no first-principles derivation consistent with this expectation has ever been reported

Summary

Introduction

The second approach assumes a system of differential equations describing continuous-time dynamics within a year, from which a discrete-time model for the between-year dynamics is derived Is well known that when a population is subject to mortality that linearly increases with density, the population size after a certain period is described by a Beverton–Holt model as a function of the original population size Extending this idea, Nedorezov et al [19] derived a Hassell model by assuming discrete reproduction at the end of the year with fecundity exponentially decreasing with the mean population size over the year. This paper explores whether a Hassell model whose exponent is related to the inequality can be derived from first principles Such a Hassell model can be derived by assuming random competition for resources among the individuals wherein each individual can obtain only a fixed amount of resources at a time. Two additional Hassell models are derived when some assumptions are modified

Hassell model

Beverton–Holt and Ricker models

Interpretation of the derivation

Non-constant resource threshold

Non-constant fecundity

Discussion