Population Growth Models
Part 4.2: Equilibria
We explore in this part and the next what we can say about the family of solutions of the logistic differential equation
without determining a symbolic description for the family. One tool for understanding the solutions will be the applet below. It displays a slope field for the differential equation as well as a graph of the slope function, f(P) = r P (K - P). Click on the left-hand figure to generate solutions of the logistic equation for various starting populations P(0). [Notes: 1. The vertical coordinate of the point at which you click is considered to be P(0). The horizontal (time) coordinate is ignored. 2. The graph of the slope function will not change as you change initial conditions. In Part 5 we will consider changes to the right-hand side of the equation.]
- Experiment with a range of initial conditions to get a "feel" for the family of solutions to the logistic differential equation.
- Use the differential equation itself to explain why P(t) = 0 is a solution.
A constant solution of a differential equation is called an equilibrium. Locating and classifying the equilibrium solutions is an important step in understanding the family of all solutions of the equation.
- The logistic equation has another equilibrium, i.e.,
another solution of the form P(t) = constant. What is the constant? Explain how you know from the differential equation that this function is a solution.
- If the starting population P(0) is greater than K, what can you say about the solution P(t)? What do you see in the differential equation that confirms this behavior?
- If the starting population P(0) is less than than K, what can you say about the solution P(t)? What do you see in the differential equation that confirms this behavior?
- Why are carrying capacity and maximum supportable population appropriate names for K?
Classification of equilibrium solutions
We classify equilibrium solutions according to the behavior of other solutions that start nearby.
- An equilibrium solution P = c is called stable if any solution P(t) that starts near P = c stays near it.
- The equilibrium P = c is called asymptotically stable if any solution P(t) that starts near P = c actually converges to it -- that is,
- If an equilibrium is not stable, it is called unstable. This means there is at least one solution that starts near the equilibrium and runs away from it.
- For the logistic equation, is the equilibrium solution P = 0 stable or unstable? If stable, is it also asymptotically stable? Explain.
- Is the equilibrium solution you found in step 3 stable or unstable? If stable, is it also asymptotically stable? Explain.
