Question

There is considerable evidence to support the theory that for some species there is a minimum population m such that the species will become extinct if the size of the population falls below m. This condition can be incorporated into the logistic equation by introducing the factor (1−m/P). Thus the modified logistic model is given by the differential equation
dPdt=kP(1−PK)(1−mP)
where k is a constant and K is the carrying capacity.
Suppose that the carrying capacity K=40000, the minimum population m=900, and the constant k=0.1. Answer the following questions.
1. Assuming P≥0 for what values of P is the population increasing. Answer (in interval notation):
2. Assuming P≥0 for what values of P is the population decreasing. Answer (in interval notation):

Answer 1

Answer:

1. (0, 1/40000) and (1/900, +∞)

2. (1/40000, 1/900).

Step-by-step explanation:

The population is increasing when dP/dt is positive and the population is decreasing when dP/dt is negative.

First, replacing the values of K, k and m, we get that dP/dt are equal to:

dP/dt=0.1P(1-40000P)(1-900P)

To find the intervals when dP/dt is positive, we need to make dP/dt greater than zero as:

dP/dt=0.1P(1-40000P)(1-900P) > 0

So, we have 3 terms in the equation, so:

0.1P > 0 if P > 0

1 - 40000 P > 0   if P < 1/40000

1 - 900 P > 0   if   P < 1/900

We can said that 0.1P is positive if P is positive,  (1 - 40000P) is positive is P is lower than 1/40000 and (1 -900P) is positive is P is lower than 1/900

Therefore, we have the following intervals: (0, 1/40000), (1/40000, 1/900) and (1/900, +∞)

For the interval (0, 1/40000): 0.1P is positive, (1 - 40000P) is positive and (1 -900P) is positive. So we can said that dP/dt is positive.

For the interval (1/40000, 1/900): 0.1P is positive, (1 - 40000P) is negative and (1 -900P) is positive. So we can said that dP/dt is negative.

For the interval (1/900, +∞): 0.1P is positive, (1 - 40000P) is negative and (1 -900P) is negative. So we can said that dP/dt is positive.

Finally, the population increase at values of P between: (0, 1/40000) and (1/900, +∞) and the population decrease at values of P between: (1/40000, 1/900).