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,∞).
