4.4 The logistic equation  (Page 3/12)

Solving the logistic differential equation

The logistic differential equation is an autonomous differential equation, so we can use separation of variables to find the general solution, as we just did in [link] .

Step 1: Setting the right-hand side equal to zero leads to $P=0$ and $P=K$ as constant solutions. The first solution indicates that when there are no organisms present, the population will never grow. The second solution indicates that when the population starts at the carrying capacity, it will never change.

Step 2: Rewrite the differential equation in the form

Then multiply both sides by $dt$ and divide both sides by $P\left(K-P\right).$ This leads to

Multiply both sides of the equation by $K$ and integrate:

The left-hand side of this equation can be integrated using partial fraction decomposition. We leave it to you to verify that

Then the equation becomes

Now exponentiate both sides of the equation to eliminate the natural logarithm:

We define $C_1=e^c$ so that the equation becomes

To solve this equation for $P\left(t\right),$ first multiply both sides by $K-P$ and collect the terms containing $P$ on the left-hand side of the equation:

Next, factor $P$ from the left-hand side and divide both sides by the other factor:

The last step is to determine the value of $C_1.$ The easiest way to do this is to substitute $t=0$ and $P_0$ in place of $P$ in [link] and solve for $C_1\text{:}$

Finally, substitute the expression for $C_1$ into [link] :

Now multiply the numerator and denominator of the right-hand side by $\left(K-P_0\right)$ and simplify:

We state this result as a theorem.

Now that we have the solution to the initial-value problem, we can choose values for $P_0,r,$ and $K$ and study the solution curve. For example, in [link] we used the values $r=0.2311,K=\mathrm{1,072,764},$ and an initial population of $\mathrm{900,000}$ deer. This leads to the solution

Dividing top and bottom by $\mathrm{900,000}$ gives

This is the same as the original solution. The graph of this solution is shown again in blue in [link] , superimposed over the graph of the exponential growth model with initial population $\mathrm{900,000}$ and growth rate $0.2311$ (appearing in green). The red dashed line represents the carrying capacity, and is a horizontal asymptote for the solution to the logistic equation.