7.1: Finding Equilibrium Points

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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.

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