SIAM Student Chapter Seminar

Forest ecosystems often experience sudden insect outbreaks in which populations remain low for long periods before rapidly exploding to damaging levels. These abrupt transitions cannot be explained by simple population growth models alone. In this talk, we introduce a dynamical systems model that incorporates both logistic growth and saturating predation to describe insect population dynamics. Using graphical and qualitative analysis, we show how the model can admit multiple equilibria corresponding to low and high population states. As environmental parameters change, these equilibria can appear or disappear through saddle-node bifurcations, leading to sudden shifts in population levels. We also discuss the phenomenon of hysteresis, which explains why outbreaks may persist even after environmental conditions improve. This example provides an accessible introduction to bifurcation theory and illustrates how dynamical systems can help explain sudden transitions in ecological systems.