You would be able to substitute values into your DE if you knew the rate at any point in time, but you don't.
Besides, it's much easier to work with functions of only one variable.
I was looking at this excellent video
YouTube - patrickJMT's Channel
but I still don't understand why I need to first find the analytic solution to this Logistic Model
before I can answer this question
In other words, why can't I just plug in the values for Population (P), Carrying Capacity (K) and the constant (k) into the Logistic Model equation? Why do I have to first find an explicit solution for the Logistic Model equation before I can answer population growth problems?Biologists stock a lake with 500 fish and estimate the carrying capacity to be 10,000. The number of fish tripled during the first year.
a) Assuming that the size of the fish population satisfies the logistic equation, find an expression for the size of the population after 't' years.
b) How long will it take the population to reach 4,000?