Crash course on equilibrium and stability

Feedback from self-reflection 2

  • Two-thirds of responses mentioned confusion about Equilibrium, Stability, Carrying capacities, or Allee effects.

  • Nearly one-third mentioned confusing about graphing in general.

These themes continue to be important throughout ecology, so let’s take some time to review

Let us analyze the Logistic growth model:

\[\frac{dN}{dt} = rN*\bigg(1-\frac{N}{K}\bigg)\]

  • What are the equilibrium points?
    • i.e. “At what point does the population size stop shrinking or growing?”
    • We can set these by setting the equation equal to zero:

\[\frac{dN}{dt} = rN*\bigg(1-\frac{N}{K}\bigg) = 0\]

  • Two possibilities
    • First possibility: \(N = 0\)
    • Second possibility: \(N = K\)

\[\frac{dN}{dt} = rN*\bigg(1-\frac{N}{K}\bigg) = 0\]

  • Two equilibrium points
    • First equilibrium point: \(N = 0\)
    • Second equilibrium point: \(N = K\)
  • This means at if at some point \(N = 0\), then \(N = 0\) will continue – unless something happens!
  • This means at if at some point \(N = K\), then \(N = K\) will continue – unless something happens!

Stability of an equilibrium

  • We can then ask if each equilibrium point is stable or not stable.
    • What happens if \(N = 0\), and the system gets “pushed” slightly?
    • What happens if \(N = K\), and the system gets “pushed” slightly?

Graphical representation

  • Two equilibrium points
    • First equilibrium point: \(N = 0\)
    • Second equilibrium point: \(N = K\)
  • We can visualize population dynamics along a number line:

  • What happens if the system is pushed just a bit off?

Stability of equilibria in Allee effects model

  • The logistic growth model assumes declining fitness with population size
  • But in nature, fitness may increase with population size

\[\frac{dN}{dt} = - rN \bigg( 1-\frac{N}{T} \bigg) \bigg( 1-\frac{N}{K} \bigg)\]

  • What are the equilibrium points?
    (Cases where \(\frac{dN}{dt} = 0\))

  • \(N = 0\), \(N = T\), \(N = K\): If this is true, the population isn’t growing or shrinking

  • The model has three equilibrium points: \(N = 0\), \(N = T\), \(N = K\). Let’s evaluate their stability.