7.1: Finding Equilibrium Points

Finding equilibrium points of a continuous-time model $\frac{dx}{dt} = G(x)$ can be done in the same way as for a discrete-time model, i.e., by replacing all $x$’s with $x_{eq}$’s (again, note that these could be vectors). This actually makes the left hand side zero, because $x_{eq}$ is no longer a dynamical variable but just a static constant. Therefore, things come down to just solving the following equation

$$0=G(x_{eq}) \nonumber$$

with regard to $x_{eq}$. For example, consider the following logistic growth model:

$$\frac{dx}{dt} =rx \left(1-\dfrac{x}{K} \right) \label{7.1}$$

Replacing all the $x$’s with $x_{eq}$’s, we obtain

$$0 =rx_{eq} \left(1-\dfrac{x}{K} \right) \label{7.3}$$

$$x_{eq} =0, K \label{7.4}$$

It turns out that the result is the same as that of its discrete-time counterpart(see Eq.(5.1.6)).

Exercise $\PageIndex{1}$

Find the equilibrium points of the following model:

$$\frac{dx}{dt} =x^{2} -rx +1 \label{7.5}$$

Exercise $\PageIndex{2}$: Simple Pendulum

Find the equilibrium points of the following model of a simple pendulum:

$$\frac{d^{2} \theta}{dt^{2}} = -\frac{g}{L} \sin{\theta} \nonumber$$

Exercise $\PageIndex{3}$: Susceptible-Infected-Recovered model

The following model is called a Susceptible-Infected-Recovered (SIR) model, a mathematical model of epidemiological dynamics. $S$ is the number of susceptible individuals, $I$ is the number of infected ones, and $R$ is the number of recovered ones. Find the equilibrium points of this model.

$$\begin{align} \frac{dS}{dt} &= -aSI \label{7.7} \\[4pt] \frac{dI}{dt} &= aSI -bI \label{7.8} \\[4pt] \frac{dR}{dt} &=bI \label{7.9} \end{align}$$