Math 326 Feb 12 Model

Math 326, Feb 12 - Fox vs Rabbit

We build the model in 2.4, Predator-Prey models. As we know, the rabbits must be kept in check lest their population come to dominate the earth. Behold the foxes. The rabbit population grows exponentially with only the foxes to reduce their numbers. The fox population grows when there are plenty of rabbits to eat, but it drops when rabbits are scarce.

During each time period, each fox eats a number of rabbits equals to the total number of rabbits times the "prey factor". This is a simplification since foxes won't indefinitely increase their feeding as rabbit numbers increase, but it is a reasonable assumption when their are too few rabbits for the foxes to eat their fill.

Change in Rabbits = (Total Rabbits) [(Base Growth Rate) - (Foxes)(Prey Factor)]

Fox growth rate starts negative by subtracting Base Diminishing Rate, which is the rate the foxes will die with no rabbits at all. The fox growth rate is then boosted by the availability of rabbits; more rabbits, healthier foxes and more baby foxes.

Change in Foxes = (Total Foxes) [(- Base Diminishing Rate) + (Feeding rate)(Rabbits)]

Set up the model, starting with 1000 rabbits and 100 foxes. Initial rates are:
Rabbit Base Growth Rate = 0.04
Prey Factor = 0.0004
Fox Base Diminishing Rate = 0.08
Feeding Rate = 0.0001

Make two graphs: one graphing Foxes and Rabbits over time and one with Rabbits on the x-axis and Foxes on the y-axis. See how these change as you adjust the rates. It takes a lot of time to see the patterns. Start with 500 time periods, maybe adding more if you think it would be helpful.

Additional Questions

By analyzing the equations used to update the two populations, deduce the number of Foxes and Rabbits which would result in perfect equilibrium (neither population changes as time passes). Check your answer using the spreadsheet.
Too many foxes scare the farmers. Introduce a hunt which kills off 30 foxes every time the fox population goes over 130. Make these two numbers adjustable by scroll bar.
There's a limit to how many rabbits a fox can eat. Put an adjustable max (start with 0.5) on the number of rabbits eaten per fox each period.
Even in perfect conditions, there are only so many young a fox family can produce. Put an adjustable max (start with 0.02) on the fox growth rate ((feeding rate)*(rabbits)).