Concept explainers
There is considerable evidence to support the theory that for some species there is a minimum population m such that the species will become extinct if the size of the population falls below m. This condition can be incorporated into the logistic equation by introducing the factor (1 − m/P).
Thus the modified logistic model is given by the differential equation
- (a) Use the differential equation to show that any solution is increasing if m < P < M and decreasing if 0 < P < m.
- (b) For the case where k = 0.08, M = 1000, and m = 200, draw a direction field and use it to sketch several solution curves. Describe what happens to the population for various initial populations. What are the equilibrium solutions?
- (c) Solve the differential equation explicitly, either by using partial fractions or with a computer algebra system. Use the initial population P0.
- (d) Use the solution in part (c) to show that if P0 < m, then the species will become extinct. [Hint: Show that the numerator in your expression for P(t) is 0 for some value of t.]
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Chapter 9 Solutions
Single Variable Calculus: Early Transcendentals
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