   Next: Numerical Solution using MATLAB Up: Solving the Logistic Equation Previous: Solving the Logistic Equation

Analytic Solution

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?

 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?   Next: Numerical Solution using MATLAB Up: Solving the Logistic Equation Previous: Solving the Logistic Equation
James Powell
2002-02-15