Regarding the population, what does imply ? What does imply ?
Now look at the right-hand side. If is it positive or negative ? If is it positive or negative ?
Just put the two conclusions together.
The population growth rate dP/dt = rP(1-P/k), where r and k are an positive constant. Suppose that the initial population is P_0. Discuss, by considering the sign of dP/dt, the relationship between P and k if P_0 is less than k and if P_0 is greater than k.
How to show their relationship? Can I consider P as P_0 ? And by what condition? Somebody please guide me and explain it to me. Please take your time and THANK YOU VERY MUCH~~!!!
You should start (as they suggest) by determining the sign of dP/dt. It's reasonable to assume P is positive, and we are given that r is positive. So it all depends on 1-P/k.
If P_0 is less than k, will P ever be greater than k?
P is a function of t, and P_0 is just the value of the function at t=0.
If P is less than k at any time, then dP/dt is positive so P increases as t increases. But since dP/dt would be zero if P were equal to k, P can never be greater than k.
This is the type of thing you do to analyze differential equations qualitatively.