In mathematical ecology models, the existence of trajectories that exhibit cyclic behavior is a very important result. These trajectories are also called limit cycles.
In general, finding limit cycles, or showing that there are no limit cycles to a system are difficult problems. In this modeling scenario, we propose a gentle introduction to limit cycles using nullcline analysis and a spreadsheet graphical approach via the fourth-order Runge-Kutta method.
We will be offering three predator-prey models, each more complex and realistic than the previous. We choose the nullcline method as a way to support students coming from Calculus and we choose the use of a spreadsheet, instead of commercial software, to make the activity accessible and in support of open access to educational resources.
In analyzing systems of predator-prey equations, these two methods along with the equilibrium analysis present a way to detect and analyze special solutions of differential equations.