Question

The population of a wildlife habitat is modeled by the equation P(t) = 360/1 + 6.2e−0.35t, where t is given in years. How many animals were originally transported to the habitat? How many years will it take before the habitat reaches half its capacity?

Answer 1

Consider the population of a wildlife habitat is written using the formula;

P(t) = 360/(1 + 6.2e-0.35t)

 

The carrying capacity for the model is determined as follows:

The logistic growth model is approximately exponential at first, but it has a reduced rate of growth as the output approaches the model’s upper bound, called the carrying capacity. For constants a, b, and c the logistic growth of a population over time  is represented by the model;

f(x) = c/(1 + ae-bc)

 

Here, c-is the carrying capacity.

Hence, the carrying capacity of a logistic model is 360.

 

The number of animals originally transported to habitat is:

Put t = 0 as follows;

P(0) = 360/{1 + 6.2-0.35(0)}

        = 360(1 + 1)

       = 360/2

       = 180

 

The number of animals originally transported to habitat 180 animals.

The number of years its take for the habitat to reach half of its capacity is:

                          180 = 360/(1 + 6.2e-0.35t)

180(1 + 6.2e-0.35t) = 360

 180 + 1116e-0.35t = 360

 

Therefore,

e-0.35t = (360 – 180)/1116

e-0.35t = 0.1613

 

Take natural log on both sides;

ln(e-0.35t) = ln(0.1613)

        -0.35t = ln(0.1613)

                 t = ln(0.1613)/-0.35

                   = 5

 Therefore, it takes 5 years its take for the habitat to reach half of its capacity.

 Therefore, it takes 5 years its take for the habitat to reach half of its capacity.