Using Phase planes, how do I find graphically the equilibria and their stability of a logistic growth model??

Question

I'm having trouble understanding the concept of phase portrait which I never learned in my applied differential equations class. The question is asking to study the logistic growth model, $$ \frac{dN}{dt}=rN\left(1-\frac{N}{K}\right)-H, $$ for a constant harvesting rate $H$ by using phase planes to find graphically the equilibria and their stability.

I've looked up phase planes in my Diff Eq book but it only talks about using phase planes for a linear system of first order ordinary differential equations, here the equation is a first order nonlinear differential equation. Here $N$ stands for the population, $K$ stands for the carrying capacity of the population, and $r$ is the net growth rate per unit of population.

Answer 1

Your ODE is one-dimensional, thus also the phase portrait is one-dimensional. You can of course embed this one axis into the graph of the system function.

The essential information is to determine the roots of the system function, those are the stationary points, and the sign of the system function on the intervals generated by the root, those determine the stability.

If you want to do it nicely, you can take the graph of the parable $y=rx(1-x/K)$ and the horizontal lines $y=H$ and mark on each region of those horizontal lines the direction of the vector field. Each of those decorated lines is a phase portrait for its value of $H$.