# Struggle for Existence: the models for Darwinian and non-Darwinian selection

## TL;DR

This paper explores various mathematical models of biological competition, revealing conditions under which Darwinian and non-Darwinian survival outcomes occur, and challenges classical principles like competitive exclusion.

## Contribution

It introduces and analyzes non-homogeneous logistic and birth-death models, showing diverse survival outcomes and providing exact distributions, thus expanding understanding of evolutionary dynamics.

## Key findings

- Non-homogeneous logistic models can lead to non-Darwinian survival.
- Distributed carrying capacity results in Darwinian survival of the fittest.
- Exact limit distribution of parameters in birth-death models is derived.

## Abstract

Classical understanding of the outcome of the struggle for existence results in the Darwinian survival of the fittest. Here we show that the situation may be different, more complex and arguably more interesting. Specifically, we show that different versions of non-homogeneous logistic-like models with a distributed Malthusian parameter imply non-Darwinian survival of everybody. In contrast, the non-homogeneous logistic equation with distributed carrying capacity shows Darwinian survival of the fittest. We also consider an non-homogeneous birth-and-death equation and give a simple proof that this equation results in the survival of the fittest. In addition to this known result, we find an exact limit distribution of the parameters of this equation. We also consider frequency-dependent non-homogeneous models and show that although some of these models show Darwinian survival of the fittest, there is not enough time for selection of the fittest species. We discuss the well-known Gauze Competitive exclusion principle that states that Complete competitors cannot coexist. While this principle is often considered as a direct consequence of the Darwinian survival of the fittest, we show that from the point of view of developed mathematical theory complete competitors can in fact coexist indefinitely.

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Source: https://tomesphere.com/paper/1812.03048