Monday, October 17, 2011

Exercise 5.1 - Modeling preditor-prey populations

1. Can you think of any examples of other systems that demonstrate the kind of behavior that we found in the predator-prey model?

Parasite-host dynamics are likely similar, where parasite population increases until the host’s capacity is reduced, at which point the parasite population declines until it finds a new host or the original host recovers to the point where the parasite can again expand.

2. In some predator-prey systems the prey can take refuge to hide from predators and avoid being consumed. Usually there is only a certain fixed number of individuals that the refuge can house. When the population of prey is large there is not enough refuge for all and the prey that could not find a place to hide gets consumed like in the standard predator-prey formalism. However when there are just a few preys their consumption slows down because they can find enough refuge places to hide. Build a predator-prey model with refuge and describe the dynamics that you observe. What equations did you modify and how? How can you explain the effect of refuge on the overall system dynamics?

To accomplish this, I made predation an exponential function:

V = y*a*eb*x

Where V is predation, y is # wolves, x is # rabbits, and a and b are parameters that represent, respectively, the magnitude of the refuge available to the rabbits and how easily rabbits beyond the refuge are taken by wolves.

In general, this has the effect of stabilizing the system’s equilibrium. Effectively, the rabbit population is limited to the size of the refuge and the wolf population consequently limited by the size of the rabbit population. This causality can be seen reflected in the fact that the wolves’ curve reaches its asymptote slightly later than the rabbits’ curve.

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