The logistic equation for the population (in thousands) of a certain species is given by:

dp/dt= 3p- 2p²

a. sketch the direction field by using either a computer software package or the method of isoclines.
b. if the initial population is 3000 [that is, p(0) = 3], what can you say about the limiting population lim t →+[infinity] p(t)?
c. if p(0) = 0.8, what is lim t →+[infinity] p(t)?
d. can a population of 2000 ever decline to 800?

a.

b. 1.5

c. 1.5

d. No

Step-by-step explanation:

a. First, let's solve the differential equation:

Divide both sides by  and multiply both sides by dt:

Integrate both sides:

Evaluate the integrals and simplify:

Where C1 is an arbitrary constant

I sketched the direction field using a computer software. You can see it in the picture that I attached you.

b. First let's find the constant C1 for the initial condition given:

Solving for C1:

Now, let's evaluate the limit:

The expression tends to zero as x approaches ∞ . Hence:

c. As we did before, let's find the constant C1 for the initial condition given:

Solving for C1:

Now, let's evaluate the limit:

The expression tends to zero as x approaches ∞ . Hence:

d. To figure out that, we need to do the same procedure as we did before. So,  let's find the constant C1 for the initial condition given:

Solving for C1:

Can a population of 2000 ever decline to 800? well, let's find the limit of the function when it approaches to ∞:

The expression tends to zero as x approaches ∞ . Hence:

Therefore, a population of 2000 never will decline to 800.

2160ft3

step-by-step explanation: