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Using a Logistic Differential Equation the logistic differential equation models the growth rate of a population. Use the equation to (a) find the value of k, (b) find the carrying capacity, (c) graph a slope field using a computer algebra system, and (d) determine the value of P at which the population growth rate is the greatest.

 

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Jyotsana Prakash
Jyotsana PrakashLv10
14 Mar 2021

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Related questions

The logistic function in form of differential equation is given by,

where, P is population growth as function of time, ‘r’ is the growth rate and K is the carrying capacity.

If the carrying capacity is the also function of time K(t) and changing with time given as ,

How the population growth will follow the varying carrying capacity?

 

Endangered Species A conservation organization releases 25 Florida panthers into a game preserve. After 2 years, there are 39 panthers in the preserve. The Florida preserve has a carrying capacity of 200 panthers.

(a) Write a logistic equation that models the population of panthers in the preserve.

(b) Find the population after 5 years.

(c) When will the population reach 100?

(d) Write a logistic differential equation that models the growth rate of the panther population. Then repeat part (b) using Euler’s Method with a step size of . Compare the approximation with the exact answer.

(e) At what time is the panther population growing most rapidly? Explain.

Gompertz Equation Another model that can be used to represent population growth is the Gompertz equation, which is the solution of the differential equation

where k is a constant and L is the carrying capacity.

(a) Find the general solution of the differential equation.

(b) Use a graphing utility to graph the slope field for the differential equation when k = 0.05 and L = 1000.

(c) Describe the behavior of the graph in part (b) as .

(d) Use a graphing utility to graph the equation you found in part (a) for L = 5000, y(0) = 500, and k = 0.02. Determine the concavity of the graph and how it compares with the general solution of the logistic differential equation.

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