Ecologists are monitoring the proliferation of a non-native water plant in a 50-acre…
MATHEMATICS · Multi-part · AP · Calculus BC · Differential Equations · Differential Equations
Problem
Ecologists are monitoring the proliferation of a non-native water plant in a 50-acre reservoir. Historical data indicate that nutrient levels and shading from existing vegetation place an upper limit on the plant coverage. Let A(t) be the area, in acres, covered by the plant t months after it is first observed, and suppose A satisfies a differential equation of the form dtdA=f(A).
Answer the following in complete sentences, using precise mathematical language.
(a) Describe two specific characteristics of this situation that suggest a logistic, rather than an exponential, growth model for A(t). Relate each characteristic explicitly to features of the reservoir described above.
(b) In a logistic model, the carrying capacity K is the limiting value of A(t) as t→∞. Explain the practical meaning of K in this context and identify one physical or environmental factor from the description that determines its value.
(c) For a logistic model, the quantity A1dtdA represents the relative growth rate. Without solving a differential equation, explain at which plant coverage (in terms of K) this relative growth rate is greatest, and justify your answer using properties of logistic functions.