Predator-Prey
Models
Part 6: Summary
- 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?
- How do you find an equilibrium
solution to a system of differential equations? What does an equilibrium
solution mean for interacting populations?
- What is the biological
meaning of each of the four parameters in the Lotka-Volterra Model?
- 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.
- What special characteristic
is shared by all trajectories of the Lotka-Volterra Model? What is the
biological significance of this characteristic? Is this biologically reasonable?
- (Optional) What is likely to happen
to the trajectories if the model is altered, say, by hunting or stocking
one of the populations?
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 logistic growth models for a single population. 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.
- 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.
