Typically, we want to know if a population will move towards or away

from a steady state. In other words, we want to know is a steady

state is stable or unstable.

Linear stability analysis allows us to determine what happens close

to a steady state.

Idea: over a sufficiently small scale, nonlinear equations are

approximately linear. We can therefore use a Taylor expansion to

linearise a system around a steady state and see what happens if we

give the system a little nudge. Does the population return to the

steady state or does it move away?

ability of steady states

Let nt = Nt - N* be a small perturbation from a steady state. We

want to know if this perturbation grows or decays over time. The

dynamics of the perturbation are given by:

nt+1 = Nt+1 - N*/nHence, if we can neglect higher order terms, we may summarise the

behaviour of nt, and hence the stability of the steady state, as

follows.

X = f(N*)

x < -1

-1

0 < x < 1

A>1

perturbation dynamics

oscillatory growth

oscillatory decay

monotonic decay

monotonic growth

stability

oscillatory unstable

oscillatory stable

monotonic stable

monotonic unstable

¹It can be shown that we can indeed neglect higher order terms as long as we are not

on the borderline between different behaviours, where the behaviour that changes

may be affected by these terms. For example, if f'(N+) = 0 the steady state N* is

stable for the nonlinear equation, but may be oscillatory or monotonic depending on

the nonlinearity.

Stability of steady states

Q: What is the stability of each of the steady states (N* = 0 and

N* = K) in the logistic growth model, assuming r> 0?

Nt+1= Nt (1+r

1 (1 + r ( 1₁ - ^/^ ) )/nstable iff (N*)| < 1, unstable if |f' (N*)| > 1

oscillatory if f'(N*) < 0, monotonic if f' (N*) > 0

Interpretation

Q: How does the intrinsic growth rate affect the population dynamics?