Leslie Population Models in Predator-prey and Competitive populations: theory and applications by machine learning

TL;DR

This paper introduces a Leslie matrix-based predator-prey and competitive population model, analyzes its dynamics, proves a dominance theorem, and applies machine learning and quantum methods for modeling and computation.

Contribution

It extends classical Leslie models with matrix-based dynamics, introduces a new stability theorem, and demonstrates machine learning and quantum approaches for population prediction.

Findings

Population exhibits exponential growth or decay.

Machine learning effectively fits population data.

Quantum operators can model Leslie system dynamics.

Abstract

We introduce a new predator-prey model by replacing the growth and predation constant by a square matrix, and the population density as a population vector. The classical Lotka-Volterra model describes a population that either modulates or converges. Stability analysis of such models have been extensively studied by the works of Merdan (https://doi.org/10.1016/j.chaos.2007.06.062). The new model adds complexity by introducing an age group structure where the population of each age group evolves as prescribed by the Leslie matrix. The added complexity changes the behavior of the model such that the population either displays roughly an exponential growth or decay. We first provide an exact equation that describes a time evolution and use analytic techniques to obtain an approximate growth factor. We also discuss the variants of the Leslie model, i.e., the complex value predator-prey model and the competitive model. We then prove the Last Species Standing theorem that determines the dominant population in the large time limit. The recursive structure of the model denies the application of simple regression. We discuss a machine learning scheme that allows an admissible fit for the population evolution of Paramecium Aurelia and Paramecium Caudatum. Another potential avenue to simplify the computation is to use the machinery of quantum operators. We demonstrate the potential of this approach by computing the Hamiltonian of a simple Leslie system.