  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.
