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 $H$.

Answer 2

The right side of the differential equation can be written as the quadratic polynomial $ \ -\frac{r}{K} \ N^2 + rN - H $ $ = \ -\frac{r}{K} · \left(N^2 - KN + \frac{KH}{r} \right) \ , $ the zeroes (if any) of which give the equilibrium values of $ \ N \ $ for the system, $$ N \ = \ \frac{K \pm \sqrt{K^2 - \frac{4KH}{r}}}{2} \ \ . $$

Alternatively, we may "complete the square" in the polynomial to produce $$ \frac{dN}{dt} \ = \ -\frac{r}{K} \ \left(N - \frac{K}{2} \right)^2 \ - \ \left(H \ - \ \frac{rK}{4} \right) \ \ . $$

These expressions indicate that there are three cases to consider, depending on the "size" of the harvesting term. The parabola alluded to by Lutz Lehmann appears in the graph of $ \ \frac{dN}{dt} \ $ versus $ \ N \ \ . $

From either of the descriptions given above, we can define a "critical" harvesting rate $ \ H_c = \ \frac{rK}{4} \ . $ For $ \ H < H_c \ \ , $ there are two equilibria, which we might call $ \ N_{+} \ $ and $ \ N_{-} \ $ ; since $ \ \frac{dN}{dt} > 0 \ $ for $ \ N_{-} < N < N_{+} \ $ and $ \ \frac{dN}{dt} < 0 \ $ otherwise, the population "runs asymptotically" to $ \ N_{+} \ $ (the stable equilibrium) and away from $ \ N_{-} \ $ (the unstable equilibrium); if the initial population is less than $ \ N_{-} \ $ , then the population drops to zero (actually, to "negative infinity"). If $ \ H > H_c \ $ ("over-harvesting"), any initial population ultimately declines to zero. The threshold rate, $ \ H = H_c \ $ ("critical harvesting"), will have an initial population $ \ N(0) > \frac{K}{2} \ $ tend to $ \ \frac{K}{2} \ $, but an initial population $ \ N(0) < \frac{K}{2} \ $ will fall to zero; this peculiar "one-sided" equilibrium is a merger of the two equilibria of the sub-critical harvesting case and so is sometimes called "semi-stable".

The parabola is useful for characterizing this dynamical system with a single dependent variable, but it is not the "phase-portrait" itself. As Lutz Lehmann remarks, rather than a "portrait" (which is the term used for systems of more than one variable), we produce what is variously called a "slope-", "direction-" or "flow-field", in which we plot curves for $ \ N \ $ as a function of $ \ t \ $ with the tick marks representing values of $ \ \frac{dN}{dt} \ \ , $ as is also done in phase-portraits. (Since $ \ \frac{dN}{dt} \ $ is independent of time, all of the tick-marks at any level of $ \ N \ $ all have the same slope.) Direction fields are displayed below for the three harvesting cases discussed, with curves for various initial populations included.

$ \quad \quad \quad \quad \large \text{sub-critical harvesting} $

$ \quad \quad \quad \quad \large \text{critical harvesting} $

$ \quad \quad \quad \quad \large \text{over-harvesting} $