Suppose That a Population Grows According to a Logistic Model

question 135

Essay

Suppose that a population grows according to a logistic model.(a) Write the differential equation for this situation with k = 0.01 and carrying capacity of 60 thousand.(b) Solve the differential equation in part (a) with the initial condition t = 0 (hours) and population P = 1 thousand.(c) Find the population for t = 10 hours, t = 100 hours, and t = 1000 hours.(d) After how many hours does the population reach 2 thousand? 30 thousand? 55 thousand?
(e) As the time t increases without bound, what happens to the population?
(f) Sketch the graph of the solution of the differential equation.

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Definitions:

Disposable Income

The pool of funds households have for saving and spending pursuits after income taxes are factored out.

Consumption

The use of goods and services by households, constituting a major part of the economy's total output.

Permanent Income Hypothesis

A theory proposed by Milton Friedman suggesting that an individual's consumer behavior is determined by their long-term income expectations rather than their current income.

Services

Intangible products offered to consumers and other businesses, such as banking, education, and healthcare, which unlike goods, are not physical objects.