CALCULUS FOR MODELLING 08
Equilibria in Models
An equilibrium is a state at which the model predicts no instantaneous change.
The main idea
For an autonomous model
\[\frac{dX}{dt}=f(X),\]an equilibrium \(X^*\) satisfies
1. Why equilibrium matters
If the system starts exactly at an equilibrium, the model predicts that it remains there.
2. Exponential model
For \(X'=rX\), with \(r\ne0\),
\[rX^*=0\]gives the equilibrium
\[\boxed{X^*=0}.\]3. Logistic model
For
\[P'=rP\left(1-\frac{P}{K}\right),\]the equilibria are
\[\boxed{P^*=0,\qquad P^*=K}.\]4. Equilibrium does not mean no processes occur
In a compartment model, inflows and outflows may both be nonzero but balance exactly. Thus a constant state can result from opposing processes.
5. Systems of equations
For
\[X'=f(X,Y),\qquad Y'=g(X,Y),\]an equilibrium \((X^*,Y^*)\) satisfies
6. Nullclines
The curves \(f(X,Y)=0\) and \(g(X,Y)=0\) are nullclines. Their intersections give equilibria.
7. Disease-free equilibrium
In epidemic modelling, an equilibrium with no infectious individuals is often called a disease-free equilibrium.
8. Endemic equilibrium
An equilibrium with a positive infectious level may be called endemic when the model supports persistent infection.
9. Finding versus understanding equilibria
Solving \(f=0\) locates equilibria. Stability analysis is then needed to determine what nearby trajectories do.