# Inequality in resource allocation and population dynamics models

## TL;DR

This paper derives a Hassell population model from first principles, linking the model's exponent to resource allocation inequality, and shows how different assumptions lead to variations including Beverton-Holt and Ricker models.

## Contribution

It provides a first-principles derivation of the Hassell model that connects the exponent to resource inequality, introducing new variants under modified assumptions.

## Key findings

- Hassell model exponent relates to resource inequality.
- Derived models include Beverton-Holt and Ricker as special cases.
- Changing resource unit size alters inequality and the model's exponent.

## 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. The Beverton-Holt and Ricker models can be regarded as special cases of the derived Hassell model. Two additional Hassell models are derived under some modified assumptions.