Darwinian evolution in Malthusian population growth model and Markov chains

TL;DR

This work analyzes Darwinian evolution within two discrete mathematical frameworks. It first augments a Verhulst carrying-capacity model with mutation via $f_{mut}$ and selection via $f_{mort}$ to produce trait-structured population dynamics, proving boundedness, an extinction condition, and showing numerically that selection favors higher-velocity traits with an increasing mean velocity $E_n$. It then adopts a Markov-chain approach with a velocity-based state space, establishing the existence of a unique stationary distribution $pi^*$ and introducing the expected state $E_{pi^*}$ as an evolutionary indicator; in a Hessenberg truncation, explicit formulas yield $ lim_{delta o 0^+} E_{pi^*}(delta) = (2n-1)/2$. Across both frameworks, the authors provide preliminary theoretical results and simulations that align with Darwinian intuitions, and they pose open questions about generalizations and quantitative measures of evolutionary direction.

Abstract

The paper is devoted to the study of Darwinian evolution in two mathematical models. The first one is a variation on the Malthusian population growth model with Verhulst's environmental capacity. The second model is grounded in the theory of Markov chains and their stationary distributions. We prove preliminary results regarding both models and pose conjectures, which are supported by computer simulations.

Figures (5)

• Figure 1: World population according to UN DESA booklet "The World at Six Billion"
• Figure 2: Population dynamics in model \ref{['model']}
• Figure 3: Size of population at different velocities after 5000 generations
• Figure 4: Expected average velocity
• Figure 5: Stationary distributions in Markov chains

Theorems & Definitions (7)

• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3
• Theorem 4
• proof