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.
\[ \begin{bmatrix} 0 & 1 & 8 & 0.5 \\ 0.3 & 0 & 0 & 0 \\ 0 & 0.8 & 0 & 0 \\ 0 & 0 & 0.5 & 0 \end{bmatrix} \]
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.
For a population with \(n\) stages, we can capture the demographic transitions in an \(n\mathrm{-by-}n\) 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 \(t=1\)?
Matrix product of the transition matrix and the current distrubition:
\[ \begin{bmatrix} 0 & 1 & 8 & 0.5 \\ 0.3 & 0 & 0 & 0 \\ 0 & 0.8 & 0 & 0 \\ 0 & 0 & 0.5 & 0 \end{bmatrix} \times \begin{bmatrix} 52 \\ 10 \\ 16 \\ 30 \end{bmatrix} \]
Matrix product of the transition matrix and the current distrubition:
\[ \begin{bmatrix} 0 & 1 & 8 & 0.5 \\ 0.3 & 0 & 0 & 0 \\ 0 & 0.8 & 0 & 0 \\ 0 & 0 & 0.5 & 0 \end{bmatrix} \times \begin{bmatrix} 52 \\ 10 \\ 16 \\ 30 \end{bmatrix} = \begin{bmatrix} 153 \\ 15 \\ 8 \\ 8 \end{bmatrix} \]
\[ \begin{bmatrix} 0 & 1 & 8 & 0.5 \\ 0.3 & 0 & 0 & 0 \\ 0 & 0.8 & 0 & 0 \\ 0 & 0 & 0.5 & 0 \end{bmatrix} \xrightarrow[]{\text{Eigenvalue}} 1.33 \]
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?
\[ \begin{bmatrix} 0 & 1 & 8 & 0.5 \\ 0.3 & 0 & 0 & 0 \\ 0 & 0.8 & 0 & 0 \\ 0 & 0 & 0.5 & 0 \end{bmatrix} \]
\[ \begin{bmatrix} 0 & 1 & 8 & 0.5 \\ 0.3 \to 0 & 0 & 0 & 0 \\ 0 & 0.8 & 0 & 0 \\ 0 & 0 & 0.5 & 0 \end{bmatrix} \]
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 \(70\%\)?
\[ \begin{bmatrix} 0 & 1 & 8 & 0.5 \\ 0.3 \to 0.2 & 0 & 0 & 0 \\ 0 & 0.8 & 0 & 0 \\ 0 & 0 & 0.5 & 0 \end{bmatrix} \]
\[ \begin{bmatrix} 0 & 1 & 8 & 0.5 \\ 0.3 \to 0.2 & 0 & 0 & 0 \\ 0 & 0.8 & 0 & 0 \\ 0 & 0 & 0.5 & 0 \end{bmatrix} \Rightarrow \boxed{\lambda = 1.33 \to \lambda = 1.15} \]
\[ \begin{bmatrix} 0 & 1 & 8 & 0.5 \\ 0.3 & 0 & 0 & 0 \\ 0 & 0.8 \to 0.7 & 0 & 0 \\ 0 & 0 & 0.5 & 0 \end{bmatrix} \]
\[ \begin{bmatrix} 0 & 1 & 8 & 0.5 \\ 0.3 & 0 & 0 & 0 \\ 0 & 0.8 \to 0.7 & 0 & 0 \\ 0 & 0 & 0.5 & 0 \end{bmatrix} \Rightarrow \boxed{\lambda = 1.33 \to \lambda = 1.28} \]
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)
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…
\[ \begin{bmatrix} 0 & 1 & 8 & 0.5 \\ 0.3 \to 0.4 & 0 & 0 & 0 \\ 0 & 0.8 & 0 & 0 \\ 0 & 0 & 0.5 & 0 \end{bmatrix} \]
\[ \begin{bmatrix} 0 & 1 & 8 & 0.5 \\ 0.3 & 0 & 0 & 0 \\ 0 & 0.8 \to 0.9 & 0 & 0 \\ 0 & 0 & 0.5 & 0 \end{bmatrix} \]
Quantify the effects of proportional changes
\[ \begin{bmatrix} 0 & 1 & 8 & 0.5 \\ 0.3 \to 0.27 & 0 & 0 & 0 \\ 0 & 0.8 & 0 & 0 \\ 0 & 0 & 0.5 & 0 \end{bmatrix} \]
\[ \begin{bmatrix} 0 & 1 & 8 & 0.5 \\ 0.3 & 0 & 0 & 0 \\ 0 & 0.8 \to 0.72 & 0 & 0 \\ 0 & 0 & 0.5 & 0 \end{bmatrix} \]
Quantify the effects of proportional changes
\[ \begin{bmatrix} 0 & 1 & 8 & 0.5 \\ 0.3 \to 0.27 & 0 & 0 & 0 \\ 0 & 0.8 & 0 & 0 \\ 0 & 0 & 0.5 & 0 \end{bmatrix} \Rightarrow \boxed{\lambda = 1.33 \Rightarrow \lambda = 1.28} \]
\[ \begin{bmatrix} 0 & 1 & 8 & 0.5 \\ 0.3 & 0 & 0 & 0 \\ 0 & 0.8 \to 0.72 & 0 & 0 \\ 0 & 0 & 0.5 & 0 \end{bmatrix} \Rightarrow \boxed{\lambda = 1.33 \to \lambda = 1.29} \]
Quantify the effects of proportional changes
\[ \begin{bmatrix} 0 & 1 & 8 & 0.5 \\ 0.3 \to 0.33 & 0 & 0 & 0 \\ 0 & 0.8 & 0 & 0 \\ 0 & 0 & 0.5 & 0 \end{bmatrix} \Rightarrow \lambda= \boxed{\lambda = 1.33 \to \lambda = 1.38} \]
\[ \begin{bmatrix} 0 & 1 & 8 & 0.5 \\ 0.3 & 0 & 0 & 0 \\ 0 & 0.8 \to 0.88 & 0 & 0 \\ 0 & 0 & 0.5 & 0 \end{bmatrix} \Rightarrow \boxed{\lambda = 1.33 \to \lambda = 1.37} \]
If we have limited resources, and we want to develop effective management,
Where should we focus our efforts?
If we could increase juvenile survival by 10% or adult survival by 10%, which would give bigger impact?
Discussion questions for Crouse et al.
As of the time when this paper was written (1987), what was the primary focus of turtle conservation efforts? Why was this the primary focus of conservation activity?
The authors simplify the turtle life cycle, which can extend more than 50 years, into just 7 different stages. Explain in your own words (a) why they chose to do this, (b) what these stages represent, and (c) the reproductive dynamics of these stages.
In your own words, what does the “Elasticity” of a stage represent? What does Figure 4A tell us about the elasticity of different life stages?
If the authors had to pick a single life stage that should be the target of conservation, which one do you think they would advocate for? Why?
According to the authors, what is an uncomfortable possibility that conservation biologists needed to address about turtle conservation efforts? Where else do you think such lessons are applicable in our contemporary lives?