AiS Challenge 2003 STI :: Predator-Prey Model

  1. Real Problem: In a lake there are two species of fish: X (prey), which lives on plants of which there is a plentiful supply, and Y (predators) which subsists by eating the X (prey). Study the interaction between these two fish populations.

  2. Working Problem: We make the following assumptions. X, the prey, is relatively long-lived and rapidly breeding if left alone; birth rate is larger than natural death rate). In the absence of prey, the starvation (death) rate of Y, the predators, exceeds their birth rate. Both the death rate of the prey and the growth rate of the predators are proportional to the number of encounters with the other species. X and Y are the only fish in our model environment, and there is no immigration into or emigration out of the environment.

  3. Math Model: Let x(t) be the number of prey at time t and y(t) be the number of predators at time t. Then in an time interval dt, x(t) increases by an amount a*x(t)*dt, where the factor a > 0 results from births and "natural" deaths. In addition, x(t) increases negatively (decreases) in dt by the amount -c*x(t)*y(t)*dt where the factor c > 0 results from prey being eaten by predators. So, the net population increase of x(t) in the time interval dt is
    x(t + dt) - x(t) = a*x(t)dt - c*x(t)*y(t)*dt
    In a similar manner, we find that in the interval dt the predators increase by an amount
    y(t + dt) - y(t) = -b*y(t) + d*x(t)*y(t)
    where b > 0 is the predator starvation rate and d > 0 is the predator birth rate resulting from predators eating prey.

  4. Computational Model: There are many ways to implement a model for predator-prey interactions on a computer; e.g., Java, C++, or Excel. A "for loop" or similar construct will be useful.

  5. Results/Conclusions: Since it will not be easy to obtain data to validate your model, you will perform "what if" scenarios. Chose values for a, b, c, and d that you think are reasonable. Plot a time history of both the predator population and the prey population on the same graph. Comment on your results. Finally, plot your results in the "phase plane": number of prey on the x-axis and number of predators on the y-axis. Again, comment on what you see. Can you think of ways to refine your model?