Population Growth Models

Part 4.4: Symbolic Solutions

Now we will use the standard technique for separable differential equations to find symbolic descriptions of the solutions of the logistic equation. See Part 5 of the module Introduction to Differential Equations for a general discussion of separable equations.

1. Use "pencil and paper" to separate the variables in the logistic differential equation
   

   Then integrate both sides of the resulting equation. (This is easy for the "t" side -- you will need to use partial fractions for the "P" side.)

2. After calculating both integrals, set the results equal. (Don't forget the constant of integration!) Then solve the equation for P as a function of t.

3. Now use your computer algebra system's differential equation solver to solve the logistic equation directly.

4. The results from steps 2 and 3 are -- or should be -- formulas for the same family of functions. If the formulas do not look alike, reconcile any differences that you see. Show that one formula can be put in the form of the other. If the formulas have "arbitrary constants" in different places, show how the arbitrary constant in one formula is related to the arbitrary constant in the other formula.

5. Consider the representation you obtained in step 2. Does the representation include all solutions of the differential equation -- in particular, both equilibrium solutions? If not, explain why your calculation did not include all solutions.

6. Suppose the starting population P(0) is a specific number P₀ (which may be either smaller or larger than K). Choose whichever solution form you prefer, and determine the value of the "arbitrary" constant (in terms of K and P₀) such that the solution P(t) satisfies the initial condition P(0) = P₀. Simplify as much as possible.

7. Explain why your solution function P(t) in step 6 approaches K as t becomes large.

8. For the same solution form as you used in step 6, describe what choices of the arbitrary constant correspond to solutions between 0 and K. What choices of the arbitrary constant correspond to solutions greather than K? What choices of the arbitrary constant correspond to negative solutions?

9. For specific values of r, K, and P₀, plot the direction field and your solution function to verify visually that your formula is correct. Repeat with two different combinations of the parameters so that the graphs of the three solution functions have three distinct shapes.