Question

A special type of autonomous differential equation, called the logistic equation, is typically written as Cape = rP(1- ), and is used to model ecological populations P>0. The constant r is called the intrinstic growth rate. The constant K is the greater of two equilibrium solutions called the environmental carrying capacity, and is the maximum sustainable population that the environment can support. Suppose that a population develops according to the logistic equation: dP -= 0.15P - 0.0002P2, where t is measured in weeks. dt a. Rewrite the equation in the typical form to determine the intrinsiſ growth rate and environmental carrying capacity. Note: Since K is an equilibrium solution, it is can also be found by setting the right-hand-side of the equation equal to zero. b. For what values of P is the population increasing and decreasing? Write your answers in interval notation

Answer 1

Answer:

a) dP / dt = 0.15*P ( 1  - P/750)   ,  r = 0.15 , K = 750

b) decreasing : ( - inf , 0 ) & ( 750 , inf )

    increasing : ( 0 , 750 )

Step-by-step explanation:

Given:

- The standard for of logistic equation:

                       dP / dt = r*P( 1 - P/K)

Where, r and K are constants.

- The given experimental relation is:

                       dP / dt = 0.15*P - 0.0002*P^2

Find:

Rewrite the equation in the typical form to determine the intrinsic growth rate(r) and environmental carrying capacity (K).

For what values of P is the population increasing and decreasing? Write your answers in interval notation

Solution:

- First we will convert the given relation into standard form as follows:

                         dP / dt = 0.15*P - 0.0002*P^2

Factor out 0.15*P:

                         dP / dt = 0.15*P ( 1  - P/750)

- Hence, our constants are:

                         r = 0.15 , K = 750

- We will set up and inequality for what values of P is dP/dt > 0 and dP/dt <0

                          dP / dt = 0.15*P - 0.0002*P^2 < 0

= Solve for P:

                          P < 0 , 1-P/750 = 0 -----> P > 750

So when when P is decreasing the intervals are:

                          ( - inf , 0 ) & ( 750 , inf )

And when P is increasing the intervals are:

                              ( 0 , 750)