As it turns out the logistic equation can be solved analytically, using
separation of variables. First, separate the and :

Our next goal would be to integrate both sides of this equation, but the form of the right hand side doesn't look elementary and will require a partial fractions expansion. That is, we wish to write

where and are unknown constants. If we multiply on the left and right hand sides by (which is equivalent to putting the right hand side over a common denominator) we arrive at the equation

Since there is no term with on the left hand side, we see that

If we set then we are left with , and thus the partial fraction decomposition is

Now we turn to integrating the right hand side of

First,

which is not too bad. For the second term, we must use a substitution , which gives a differential . Thus we may write the second term on the right hand side as

Putting all these terms together gives us

Here we have used the property of logarithms to equate the difference of the logs with the log of the quotient. The additional term, , on the left hand side is the free constant of integration, which will be determined by considering initial conditions to the differential equation. Exponentiating both sides of the equation gives

which must be solved for . However, it is convenient at this point to find out how the constant relates to the initial condition. Noting that at and substituting gives

Solving now for , we first cross-multiply to arrive at

and putting all terms including on one side of the equation,

Solving now for ,

Simplifying this expression by multiplying numerator and denominator by gives

EXERCISE 2: Compare this exact solution with the Euler-type numerical solution you
computed above. How does the accuracy change as is decreased? |
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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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2002-02-15