Use Maple for much of this problem.
Under the Volterra model of predator prey interaction, we will show later in the course that the population x of prey and y of predators (each a function of time t) are related by
for suitable positive constants a, b, c, and d.
Suppose that , , , and and that at some time .
a) Find a value of the constant K so that the model fits this data. (The Maple command
will evaluate an expression f at the point .)
b) Use Maple to compute the partial derivatives of f at the point . (Recall that the Maple command
computes the partial derivatives symbolically.)
c) Linear approximation along the level curve says that . We can solve this equation to estimate y in terms of x near the point on the level curve. Use this technique to estimate the predator population y as changes from to x=1900.
d) Use the Maple fsolve command to more exactly solve for the value of y that corresponds to x=1900. How do you explain the discrepancy from your answer in part c)? (The Maple command
will look numerically for solutions to in the range y between 10 and 20. For example might be given by
If no value is returned by Maple , this may mean there is no solution in this range, or it may mean that the solution method failed to converge.
e) Use implicit differentiation to calculate when .
f) Repeat parts c) and d) using x=1999 instead of x=1900. Compare the quality of the linear approximation.