Short and Long Range Population Dynamics of the Monarch

TL;DR

This paper develops a comprehensive mathematical model using differential equations to analyze the impact of habitat changes and herbicide use on the population dynamics of monarch butterflies throughout their migration cycle.

Contribution

It introduces a multi-stage differential equation model capturing the monarch's migration, breeding, and overwintering phases, incorporating age structure and predator-prey interactions.

Findings

Model predicts population decline under increased herbicide use.

Long-term behavior analysis shows potential population stabilization or decline.

Provides a framework for assessing conservation strategies.

Abstract

The monarch butterfly annually migrates from central Mexico to southern Canada. During recent decades, its population has been reduced due to human interaction with their habitat. We examine the effect of herbicide usage on the monarch butterfly's population by creating a system of linear and non-linear ordinary differential equations that describe the interaction between the monarch's population and its environment at various stages of migration: spring migration, summer loitering, and fall migration. The model has various stages that are used to describe the dynamics of the monarch butterfly population over multiple generations. In Stage 1, we propose a system of coupled ordinary differential equations that model the populations of the monarch butterflies and larvae during spring migration. In Stage 2, we propose a predator-prey model with age structure to model the population dynamics at the summer breeding site. In Stages 3 and 4, we propose exponential decay functions to model the monarch butterfly's fall migration to central Mexico and their time at the overwintering site. The model is used to analyze the long-term behavior of the monarch butterflies through numerical analysis, given data available in the research literature.