Structured population growth, pt.2

Practice problems

Photo by Joshua J. Cotten on Unsplash

Monarch butterflies are migratory species that spend the North American winter in southwestern Mexico, and return to the US and Canada to breed during the summer months.

  • The dynamics of these butterflies during the summer months is as follows (estimates from Hunt & Tongan 2017):

  • Each female adult butterfly can lay up to \(45\) viable eggs, of which only \(3.4\%\) survive to the chrysalis stage. Eggs that make it to the chrysalis stage survive into the adult stage at rate of \(85\%\).

  • Your task : Draw the life transition diagram and write the transition matrix for the monarch butterfly.

  • Consider a species whose life cycle is described by the following matrix:

\[ \begin{bmatrix} 0 & 1 & 8 & 0.5 \\ 0.3 & 0 & 0 & 0 \\ 0 & 0.8 & 0 & 0 \\ 0 & 0 & 0.5 & 0 \end{bmatrix} \]

Draw a life transition diagram representing this matrix

Consider a species of fish whose individuals live for 3 years (age classes 0, 1, 2, and 3). Newborn fish have a \(30\%\) survival rate to year 1; year 1 fish have a \(80\%\) survival rate to year 2; year 2 fish have a \(50\%\) survival rate to year 3; and all fish die in their third year.

Newborn fish are sexually immature and cannot give birth. Year 1 fish can give birth to \(1\) newborn per year; Year 2 fish can give birth to \(8\) newborns per year; and Year 3 fish can only give birth to \(0.5\) newborns per year.

Structure of transition matrices

For a population with \(n\) stages, we can capture the demographic transitions in an \(n\mathrm{-by-}n\) matrix.

  • Each element captures the contribution from the current column, to the current row.
    • The top row always reflects new births
    • The first (leftmost) column reflects the youngest individuals
    • The last (rightmost) column reflects the oldest individuals

The value of a transition matrix

  • The product of the current population distribution and the transition matrix tells you the future population distribution

The value of a transition matrix

  • From our worked example, recall the transition matrix

\[ \begin{bmatrix} 0 & 1 & 8 & 0.5 \\ 0.3 & 0 & 0 & 0 \\ 0 & 0.8 & 0 & 0 \\ 0 & 0 & 0.5 & 0 \end{bmatrix} \]

  • At time \(t=0\), the population has \(52\) individuals in stage 0, \(10\) in stage 1, \(15\) in stage 2, and \(30\) in stage 3.

  • What is the expected distribution of individuals at \(t=1\)?

The value of a transition matrix

Matrix product of the transition matrix and the current distrubition:

\[ \begin{bmatrix} 0 & 1 & 8 & 0.5 \\ 0.3 & 0 & 0 & 0 \\ 0 & 0.8 & 0 & 0 \\ 0 & 0 & 0.5 & 0 \end{bmatrix} \times \begin{bmatrix} 52 \\ 10 \\ 16 \\ 30 \end{bmatrix} \]

The value of a transition matrix

Matrix product of the transition matrix and the current distrubition:

\[ \begin{bmatrix} 0 & 1 & 8 & 0.5 \\ 0.3 & 0 & 0 & 0 \\ 0 & 0.8 & 0 & 0 \\ 0 & 0 & 0.5 & 0 \end{bmatrix} \times \begin{bmatrix} 52 \\ 10 \\ 16 \\ 30 \end{bmatrix} = \begin{bmatrix} 153 \\ 15 \\ 8 \\ 8 \end{bmatrix} \]

The value of a transition matrix

  • We can keep iterating over and over (and over), or….
  • Calculate the \(\mathrm{eigenvalue}\) of the transition matrix
  • The dominant eigenvalue reflects the long-term growth rate.

\[ \begin{bmatrix} 0 & 1 & 8 & 0.5 \\ 0.3 & 0 & 0 & 0 \\ 0 & 0.8 & 0 & 0 \\ 0 & 0 & 0.5 & 0 \end{bmatrix} \xrightarrow[]{\text{Eigenvalue}} 1.33 \]

The value of a transition matrix

  • The dominant Eigenvalue represents the expected long-term annual growth rate (\(\lambda\))
  • Populations grow when \(\lambda \gt 1\)
  • Populations shrink when \(\lambda \lt 1\)
  • Population size is stable when \(\lambda = 1\)

For more on what Eigenvalues are…

Check out this video.

How does this help?

Application of eigenvalues to management

  • Let’s return to our previous fish example. In normal conditions, newborn fish have a \(30\%\) survival rate to year 1; year 1 fish have a \(80\%\) survival rate to year 2; year 2 fish have a \(50\%\) survival rate to year 3; and all fish die in their third year.

  • Changes to the water chemistry cause newborn survival rates to plummet to \(0%\) What are the consequences for the population?

What happens when transition rates change?

\[ \begin{bmatrix} 0 & 1 & 8 & 0.5 \\ 0.3 & 0 & 0 & 0 \\ 0 & 0.8 & 0 & 0 \\ 0 & 0 & 0.5 & 0 \end{bmatrix} \]

\[ \begin{bmatrix} 0 & 1 & 8 & 0.5 \\ 0.3 \to 0 & 0 & 0 & 0 \\ 0 & 0.8 & 0 & 0 \\ 0 & 0 & 0.5 & 0 \end{bmatrix} \]

eigen() decomposition
$values
[1]  0.90975372+0.0000000i -0.43917558+0.7121757i -0.43917558-0.7121757i
[4] -0.03140256+0.0000000i

$vectors
              [,1]                     [,2]                     [,3]
[1,] 0.98809238+0i -0.98437105+0.000000000i -0.98437105+0.000000000i
[2,] 0.10861097+0i  0.06175270+0.100139377i  0.06175270-0.100139377i
[3,] 0.09550802+0i  0.05050534-0.100512941i  0.05050534+0.100512941i
[4,] 0.05249114+0i -0.06696735+0.005838105i -0.06696735-0.005838105i
                 [,4]
[1,] -0.0007726432+0i
[2,]  0.0024604462+0i
[3,] -0.0626814138+0i
[4,]  0.9980302548+0i

What happens when transition rates change?

  • What if instead, the reduced newborn survival had decreased from \(30%\) to \(20\%\)?

  • Or if newborn rates were unaffected, but survival from year 1 to 2 was reduced from \(80\%\) to \(70\%\)?

\[ \begin{bmatrix} 0 & 1 & 8 & 0.5 \\ 0.3 \to 0.2 & 0 & 0 & 0 \\ 0 & 0.8 & 0 & 0 \\ 0 & 0 & 0.5 & 0 \end{bmatrix} \]

\[ \begin{bmatrix} 0 & 1 & 8 & 0.5 \\ 0.3 \to 0.2 & 0 & 0 & 0 \\ 0 & 0.8 & 0 & 0 \\ 0 & 0 & 0.5 & 0 \end{bmatrix} \Rightarrow \boxed{\lambda = 1.33 \to \lambda = 1.15} \]

\[ \begin{bmatrix} 0 & 1 & 8 & 0.5 \\ 0.3 & 0 & 0 & 0 \\ 0 & 0.8 \to 0.7 & 0 & 0 \\ 0 & 0 & 0.5 & 0 \end{bmatrix} \]

\[ \begin{bmatrix} 0 & 1 & 8 & 0.5 \\ 0.3 & 0 & 0 & 0 \\ 0 & 0.8 \to 0.7 & 0 & 0 \\ 0 & 0 & 0.5 & 0 \end{bmatrix} \Rightarrow \boxed{\lambda = 1.33 \to \lambda = 1.28} \]

The value of a transition matrix

How is the growth rate affected by particular transitions?

\[ \begin{bmatrix} 0 & 1 & 8 & 0.5 \\ 0.3 \to 0.2 & 0 & 0 & 0 \\ 0 & 0.8 & 0 & 0 \\ 0 & 0 & 0.5 & 0 \end{bmatrix} \Rightarrow \boxed{\lambda = 1.33 \to \lambda = 1.15} \]

\[ \begin{bmatrix} 0 & 1 & 8 & 0.5 \\ 0.3 & 0 & 0 & 0 \\ 0 & 0.8 \to 0.7 & 0 & 0 \\ 0 & 0 & 0.5 & 0 \end{bmatrix} \Rightarrow \boxed{\lambda = 1.33 \to \lambda = 1.28} \]

We can go through each entry one-by-one and identify critical transitions (Sensitivity analysis)

The value of a transition matrix

This could also go the other way– increase in \(\lambda\)

\[ \begin{bmatrix} 0 & 1 & 8 & 0.5 \\ 0.3 \to 0.4 & 0 & 0 & 0 \\ 0 & 0.8 & 0 & 0 \\ 0 & 0 & 0.5 & 0 \end{bmatrix} \Rightarrow \boxed{\lambda = 1.33 \to \lambda = 1.47} \]

\[ \begin{bmatrix} 0 & 1 & 8 & 0.5 \\ 0.3 & 0 & 0 & 0 \\ 0 & 0.8 \to 0.9 & 0 & 0 \\ 0 & 0 & 0.5 & 0 \end{bmatrix} \Rightarrow \boxed{\lambda = 1.33 \to \lambda = 1.38} \]

We can go through each entry one-by-one and identify critical transitions (Sensitivity analysis)

But there is a problem…

  • Increasing one parameter by 0.1 might not be equivalent to increasing another by the same amount.

\[ \begin{bmatrix} 0 & 1 & 8 & 0.5 \\ 0.3 \to 0.4 & 0 & 0 & 0 \\ 0 & 0.8 & 0 & 0 \\ 0 & 0 & 0.5 & 0 \end{bmatrix} \]

\[ \begin{bmatrix} 0 & 1 & 8 & 0.5 \\ 0.3 & 0 & 0 & 0 \\ 0 & 0.8 \to 0.9 & 0 & 0 \\ 0 & 0 & 0.5 & 0 \end{bmatrix} \]

Elasticity analysis

Quantify the effects of proportional changes

\[ \begin{bmatrix} 0 & 1 & 8 & 0.5 \\ 0.3 \to 0.27 & 0 & 0 & 0 \\ 0 & 0.8 & 0 & 0 \\ 0 & 0 & 0.5 & 0 \end{bmatrix} \]

\[ \begin{bmatrix} 0 & 1 & 8 & 0.5 \\ 0.3 & 0 & 0 & 0 \\ 0 & 0.8 \to 0.72 & 0 & 0 \\ 0 & 0 & 0.5 & 0 \end{bmatrix} \]

Elasticity analysis

Quantify the effects of proportional changes

\[ \begin{bmatrix} 0 & 1 & 8 & 0.5 \\ 0.3 \to 0.27 & 0 & 0 & 0 \\ 0 & 0.8 & 0 & 0 \\ 0 & 0 & 0.5 & 0 \end{bmatrix} \Rightarrow \boxed{\lambda = 1.33 \Rightarrow \lambda = 1.28} \]

\[ \begin{bmatrix} 0 & 1 & 8 & 0.5 \\ 0.3 & 0 & 0 & 0 \\ 0 & 0.8 \to 0.72 & 0 & 0 \\ 0 & 0 & 0.5 & 0 \end{bmatrix} \Rightarrow \boxed{\lambda = 1.33 \to \lambda = 1.29} \]

Elasticity analysis

Quantify the effects of proportional changes

\[ \begin{bmatrix} 0 & 1 & 8 & 0.5 \\ 0.3 \to 0.33 & 0 & 0 & 0 \\ 0 & 0.8 & 0 & 0 \\ 0 & 0 & 0.5 & 0 \end{bmatrix} \Rightarrow \lambda= \boxed{\lambda = 1.33 \to \lambda = 1.38} \]

\[ \begin{bmatrix} 0 & 1 & 8 & 0.5 \\ 0.3 & 0 & 0 & 0 \\ 0 & 0.8 \to 0.88 & 0 & 0 \\ 0 & 0 & 0.5 & 0 \end{bmatrix} \Rightarrow \boxed{\lambda = 1.33 \to \lambda = 1.37} \]