As it turns out the logistic equation can be solved analytically, using
separation of variables. First, separate the and :
First,
Solving now for , we first cross-multiply to arrive at
EXERCISE 2: Compare this exact solution with the Euler-type numerical solution you computed above. How does the accuracy change as is decreased? | ||
EXERCISE 3: In class we discussed how direct conversion of resources to offspring
and a finite resource base results in the logistic equation. Suppose now
we want to investigate the effect of per-capita `harvesting' of the
population, that is, to see what happens when we subtract a given fraction,
, of
the population per time. This changes the logistic equation to
Solve this new equation using the separation of variables technique. What is the effective carrying capacity in this situation? |
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