Predator-Prey Models

Part 5: Summary

1. What features of the lynx and hare data suggest that the Lotka-Volterra model might be an appropriate mathematical description of the interaction? What features suggest this would not be an appropriate model?
2. How do you find an equilibrium solution to a system of differential equations? What does an equilibrium solution mean for interacting populations?
3. When you have dx/dt = a function of x and y and dy/dt = another function of x and y, how do you construct a slope field or direction field as a picture of the system? How do trajectories relate to the direction field? What do trajectories tell you about the solution functions x(t) and y(t)?
4. Describe in general terms how Euler's Method works for a system of differential equations. What keeps Euler's Method from being an effective solver for the Lotka-Volterra equations?
5. For solutions x(t) and y(t) of the Lotka-Volterra equations, explain why the peaks of the y(t) graph should lag about a quarter-period behind those of the x(t) graph.

There are many extensions of the Lotka-Volterra model to make the equations more realistic. For example, one might add a carrying capacity constraint for the prey population, as in the Limited Population Growth module. Here are some links to other sites at which you can study other models of interacting populations.

• Two Species Sharing a Habitat, at our CCP sister site at Montana State University. In addition to predator-prey interactions, this includes competitive interactions, aggressive interactions, cooperation or symbiosis, and weak-strong interactions.
• The Simulation Server, Lund, Sweden. The simulator lets you control the parameters and generate phase plane plots (similar to our direction field and trajectory plots) and time series plots (like our graphs of solutions) for predator-prey, competition, and plant-pollinator models.
• Predator-Prey with a finite-food constraint on the prey, Worcester Polytechnic Institute. Analysis of the outcomes with various combinations of parameters.
• Lotka-Volterra Model and an extended Predator-Prey Model, Virginia Tech. An entomological perspective that includes experiments for determining reasonable parameter values and applications to pest control.
• The Dynamics of Predation and Parasitism, University of California at Riverside. Another entomological perspective that links to other models.
• Predator-Prey Dynamics, University of Alberta, Canada. An outline for a biology lecture that links various biological and mathematical properties.
• Predation, SUNY at Geneseo. Interesting ecological background on parasites, primary and secondary consumers, and other matters related to predation. However, watch out for mathematical misinformation. There is a graph of populations with growing oscillations that is claimed to come from the Lotka-Volterra model. The author evidently used Euler's Method without knowing why it doesn't work for solving the Lotka-Volterra equations.

Last modified: November 11, 1997