Question

Assuming P≥0, suppose that a population develops according to the logistic equation ????P/????????=0.07P−0.00014P2 where ???? is measured in weeks. Answer the following questions.
(1) What is the carrying capacity?
(2) What is the value of ?????
(3) For what values of P is the population increasing?
(4) For what values of P is the population decreasing?

Answer 1

Answer:

A complete description of the problem and solutions to each point are given below.

Explanation:

The Logistic Growth model of a population is defined as follows:

                                   dP/dt = kP (1 - P/M) (1)

Where:

dP/dt is the population growth rate.

k is the maximal individual growth rate for a given population

P is the number of individual in the population

M= carrying capacity of the population.

In the problem this first order derivative ecuation is given by

                               dP/dt= 0,07P - 0,00014P² (2)

So, we can factorize ecuation (1) and compare it with (2) to find the carryng capacity and k value (questions 1 and 2).

                                        kP (1 - P/M) = 0,07P - 0,00014P²

                                        kP - (k/M)P² = 0,07P - 0,00014P²

Here kP = 0,07P ⇒ k = 0,07 (answer to question 2)

Then, (k/M)P² = 0,00014P²⇒ (k/M) = 0,00014 and given that k= 0,07

⇒ 0,07/M = 0,00014 ⇒ M = 0,07/0,00014 ⇒ M=500 (answer to question 1)

Finally to find for which values of P is the population increasing/decreasing, we have to find the values at which dP/dt is zero. Analyzing the equation as a quadratic function (applying Baskara´s equation) we find that the values at which dP/dt is zero are P= 0 and P = -1.

x = [-0,07 ± √(0,07²)]/ 0,14 ⇒ x=  [-0,07 ± 0,07)]/ 0,14 ⇒ x = 0 and x= -1.

Then the vertex here is P = -0,5.

Now we can state that for P greater than - 0,5 the population is decreasimg (question 4), while for values from -∞ to - 0,5 the population is increasing (question 3).

Summarizing, given the first order derivative, the carrying capacity of the population is 500, the value of k is 0,07, the values of P for which dP/dt is positive (incresing population) are those comprised in the range (-∞, -0,5) and those for which the population is decreasing are (-0,5,∞).