A population of buffalo grows exponentially (the rate of growth is determined by the population itself) but has a carrying capacity. Its population (in tens of thousands) at a time t ( in years ) is governed by the differential equation $\frac{dP}{dt} = kP(1-P)$ where $k$ the growth rate is yet to be determined.
What is the carrying capacity?
Solve the differential equation for $P(t)$ if we have that $P(0) = \frac23$
Suppose that $P(1) = \frac{8}{10}$ Find k.
How many buffalo will be alive when $t = 2\text{ years}$
I dont know how to solve for the carrying capacity first of all because what I think needs to be done is solve the equation for 0 but then I get P = 0 or 1 for an answer and I don't know if that really makes sense.... unless its 10,000 buffalo
And solving the differential is confusing because I don't know how to separate this equation, so some help on that would be appreciated.