How should I go about obtaining the explicit solution to this logistic first-order nonlinear ordinary differential equation?

Question

I have to find the explicit solution to this harvesting problem in a population model where $\frac{dN}{dt}=rN(1-\frac{N}{K})-H(N)$ such that $H=qEN$, subject to initial condition $N(0)=N_0$. Here $N=N(t)$ and $r$,$K$,$q$ and $E$ are positive constants. I also have to deduce the two limiting behaviors as $t$ approaches infinity. So far I have broken down the equation to $\frac{1}{N}\frac{dN}{dt}+\frac{r}{k}N=(r-qE)$. Any hints on how to do the problem would be very much appreciated. I'm having trouble understanding what they mean by finding the explicit solution and deducing two limiting behaviors.

Answer 1

First of all, "finding the explicit solution" means finding an equation of the form $N=N(t)$ that satisfies your differential equation. "Deducing the limiting behaviors" (I think) means determining what happens to the population/harvest if the initial amount of plants (in your case) is below vs. above the "carrying capacity".

The logistic differential equation is separable, so you just separate the variables, use partial fraction decomposition, and solve.