How to find the equilibrium of differential equation model?

Question

Continuous population model, with the following equation model: $$\frac{\mathrm dN}{\mathrm dt}=rN\left(1-\frac Nk\right)-\frac{aN}{b+N}.$$ To find the equilibrium I make the equation equal to $0$. Then, do I rearrange it to get $N$ on its own?

Answer 1

One obvious solution is $N=0$

To find other equilibrium points, we can do the following: $$0 = rN\left(1-\frac{N}{k}\right)-\frac{aN}{b+N}$$ $$\frac{a}{b+N}=r\left(1-\frac{N}{k}\right)$$ $$\frac{a}{b+N}=r\left(\frac{k-N}{k}\right)$$ $$\frac{ak}{r} = (b+N)(k-N)$$ $$N^2-(k-b)N+\frac{ak}{r}-bk = 0$$ Solve the quadratic equation to get the solutions for $N$.