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Mathematics LibreTexts

7.1: Finding Equilibrium Points

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

0=G(xeq)

with regard to xeq. For example, consider the following logistic growth model:

dxdt=rx(1xK)

Replacing all the x’s with xeq’s, we obtain

0=rxeq(1xK)

xeq=0,K

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

Exercise 7.1.1

Find the equilibrium points of the following model:

dxdt=x2rx+1

Exercise 7.1.2: Simple Pendulum

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

d2θdt2=gLsinθ

Exercise 7.1.3: 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.

dSdt=aSIdIdt=aSIbIdRdt=bI


This page titled 7.1: Finding Equilibrium Points is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by Hiroki Sayama (OpenSUNY) via source content that was edited to the style and standards of the LibreTexts platform.

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