AP Calculus AB: 11.2.2 Logistic Growth

• Distinguish between exponential growth models for short-term population growth and logistic growth models for long-term population growth that take into account the carrying capacity of the environment.

• Analyze solutions to logistic growth differential equations using direction fields.

• Solve logistic growth equations numerically using Euler’s Method.

• Use the separable nature of logistic growth differential equations to solve them analytically.

• A simple population model could assume exponential growth. This is a reasonable assumption if the population is small, because there are no constraints restricting
  its growth.

• But in the long run, exponential population growth cannot be sustained due to naturally occurring constraints on resources.

• The logistic growth model takes this into account. When the population is small, the population growth rate is nearly exponential.

• When the population nears its carrying capacity, or the
  maximum population that the environment can sustain in the long run, the growth rate approaches zero.

• The solutions all approach the carrying capacity of 1200.

• When the population is below 1200, the population increases to the carrying capacity. When the population is above 1200, the population decreases to the carrying capacity.

• When the population is 600, the population is increasing the most rapidly. That is, the slopes are the steepest.

• Recall that Euler’s Method is a numerical method for solving differential equations by approximating the solution by a piecewise linear function.

• Given an initial condition, y 0 = y(x 0 ), the next point is given by x 1 = x 0 + h and y 1 = y 0 + hF(x 0 , y 0 ), where h is the step size. This process then repeats with (x 1 , y 1 ), the new starting point.

• With a step size of 25, the population after 50 time units is computed to be approximately 872. The population after 100 time units is computed to be approximately 1184.

• Notice that after 75 time units, the population exceeded its carrying capacity and, as would be expected by the logistic growth model, the population decreased at the next iteration.

• The logistic differential equation can also be solved
  analytically by relying on the fact that it is separable. A
  differential equation is separable if it can be written in the form N(y)dy = M(x)dx. The solution can be obtained by integrating both sides of the equation.

• The analytical solution to the logistic differential equation reveals that the population approaches its carrying capacity in the long run.

• The exact solution can be used to find the actual populations (based on the model) after 50 time units and after 100 time units.

• Substitute 50 for t in the solution to obtain a population of 961 after 50 time units.

• Substitute 100 for t in the solution to obtain a population of 1,185 after 100 time units.

• Notice that the solutions obtained with Euler’s method are reasonable approximations. In fact, at t = 100, the
  approximate solution and the exact solutions only differ by one.

• To find when the population reaches 1,100, substitute 1,100 for P in the exact solution. Then solve for t.

• To solve for t after isolating the exponential expression that contains t, it is necessary to take the natural log of both sides of the equation.

• Solving for t reveals that 67 time units must pass before the population reaches 1,100.

dP/dt=0.8P(1−P/400)

dP/dt=0.8P(1−P/400), P(0)=600

P(10)≈487

dP/dt=0.5P(1−P/500)

P(10)≈600

t=5.55

P(4)≈406

dP/dt=0.5P(1−P/500), P(0)=100

P(4)≈254

t=0.64