Logistic Growth, Part 2

The logistic growth function f(t)= The interactive figure below shows a direction field for the logistic differential equation. as well as a graph of the slope function, f(P) = r P (1 - P/K). Click on the left-hand figure to generate solutions of the logistic equation for various starting populations P(0) . [Note: The vertical coordinate of the point at which you click is considered to be P(0) . The horizontal (time) coordinate is ignored.] Explain why P(t) = 0 is a solution. A constant solution is called an equilibrium . The logistic equation has another equilibrium, i.e., a 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 is carrying capacity an appropriate name for K ? 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. 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.