Tutorial 6: Stability (linear multivariate)

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Model

Records from the Hudson's Bay Company famously show ~10-year population size cycles in hare and lynx. Moran (1953) built a model to fit to the lynx data collected from the Mackenzie River district from 1821 to 1934.

Let \(Y(t)\) be the number of lynx caught in year \(t\). Following Moran, we will work with the logarithm, \(X(t) = \log_{10} Y(t)\). The expected deviation from the mean, \(X(t) - m\) (where \(m\) is the mean value of \(X\) over years) is predicted to change over time like

\[ x(t+1) = a x(t) + b x(t-1). \]

Here, \(a\) and \(b\) are constants that Moran estimated from data but we'll consider more generally: \(a\) describes how much next year's population size is determined by the current year and \(b\) describes how much next year's population size is determined by the previous year (a time-lag). These parameters could be negative (eg, big population sizes in the past mean less resources for the future). Linear models with such time-lags are often called autoregressive.

Consider \(x(t+1)\) and \(x(t)\) as our two variables, with equations

\[ \begin{align} x(t+1) = a x(t) + b x_(t-1)\\ x(t) = x(t). \end{align} \]

This can be written in matrix form,

\[ \begin{pmatrix} x(t+1)\\ x(t) \end{pmatrix} = \begin{pmatrix} a & b\\ 1 & 0 \end{pmatrix} \begin{pmatrix} x(t)\\ x(t-1) \end{pmatrix}. \]

If we define \(\vec{x}(t)= \begin{pmatrix} x(t)\\ x(t-1) \end{pmatrix}\), this matrix equation is simply

\[ \vec{x}(t+1) = \mathbf{M} \vec{x}(t). \]

Questions

Determine the equilibrium.

Calculate the eigenvalues of \(\mathbf{M}\).

Determine what needs to be true about the parameters for there to be oscillations/cycles. Interpret the condition biologically.

Given there are cycles, determine when the equilibrium is stable.