Logistic growth with a threshold: Most species have a survival threshold level, and populations of fewer individuals than the threshold cannot sustain themselves. If the carrying capacity is K and the...

Logistic growth with a threshold: Most species have a survival threshold level, and populations of fewer individuals than the threshold cannot sustain themselves. If the carrying capacity is K and the threshold level is S, then the logistic equation of change for the population N = N (t) is

For Pacific sardines, we may use K = 2.4 million tons and r = 0.338 per year, as in Example 6.10. Suppose we also know that the survival threshold level for the sardines is S = 0.8 million tons.

a. Write the equation of change for Pacific sardines under these conditions.

b. Make a graph of dN dt versus N and use it to find the equilibrium solutions. How do the equilibrium solutions correspond with S and K?

c. For what values of N is the graph of N versus t increasing, and for what values is it decreasing?

d. Explain what can be expected to happen to a population of 0.7 million tons of sardines.

e. At what population level will the population be growing at its fastest?