Population Ecology, pt 2:
Structured populations
Stage structured populations
Think back to the assumptions made in the simple exponential model of population growth (\(\frac{dN}{dt} = rN\)).
- Which assumption(s) are we relaxing when accounting for age/stage structure in populations?
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
For timestep \(2\):
\[ \begin{bmatrix} 0 & 1 & 8 & 0.5 \\ 0.3 & 0 & 0 & 0 \\ 0 & 0.8 & 0 & 0 \\ 0 & 0 & 0.5 & 0 \end{bmatrix} \times \begin{bmatrix} 153 \\ 15 \\ 8 \\ 8 \end{bmatrix} \]
The value of a transition matrix
For timestep \(2\):
\[ \begin{bmatrix} 0 & 1 & 8 & 0.5 \\ 0.3 & 0 & 0 & 0 \\ 0 & 0.8 & 0 & 0 \\ 0 & 0 & 0.5 & 0 \end{bmatrix} \times \begin{bmatrix} 153 \\ 15 \\ 8 \\ 8 \end{bmatrix} = \begin{bmatrix} 83 \\ 45.9 \\ 12 \\ 4 \end{bmatrix} \]
The value of a transition matrix
For timestep \(3\):
\[ \begin{bmatrix} 0 & 1 & 8 & 0.5 \\ 0.3 & 0 & 0 & 0 \\ 0 & 0.8 & 0 & 0 \\ 0 & 0 & 0.5 & 0 \end{bmatrix} \times \begin{bmatrix} 83 \\ 45.9 \\ 12 \\ 4 \end{bmatrix} = \dots \]
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.
What happens when transition rates change?
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.
An extreme heat event causes newborn survival rates to plummet to \(0%\) in a given year. What are the consequences for the population?
What happens when transition rates change?
\[ \begin{bmatrix} 0 & 1 & 8 & 0.5 \\ 0.3 & 0 & 0 & 0 \\ 0 & 0.8 & 0 & 0 \\ 0 & 0 & 0.5 & 0 \end{bmatrix} \]
\[ \begin{bmatrix} 0 & 1 & 8 & 0.5 \\ 0.3 \to 0 & 0 & 0 & 0 \\ 0 & 0.8 & 0 & 0 \\ 0 & 0 & 0.5 & 0 \end{bmatrix} \]
What happens when transition rates change?
What if instead, the extreme heat event only reduced newborn survival from \(30%\) to \(20\%\)?
Or if newborn rates were unaffected, but survival from year 1 to 2 was reduced from \(80\%\) to \(70\%\)?
\[ \begin{bmatrix} 0 & 1 & 8 & 0.5 \\ 0.3 \to 0.2 & 0 & 0 & 0 \\ 0 & 0.8 & 0 & 0 \\ 0 & 0 & 0.5 & 0 \end{bmatrix} \]
\[ \begin{bmatrix} 0 & 1 & 8 & 0.5 \\ 0.3 \to 0.2 & 0 & 0 & 0 \\ 0 & 0.8 & 0 & 0 \\ 0 & 0 & 0.5 & 0 \end{bmatrix} \Rightarrow \lambda = 1.15 \]
\[ \begin{bmatrix} 0 & 1 & 8 & 0.5 \\ 0.3 & 0 & 0 & 0 \\ 0 & 0.8 \to 0.7 & 0 & 0 \\ 0 & 0 & 0.5 & 0 \end{bmatrix} \]
\[ \begin{bmatrix} 0 & 1 & 8 & 0.5 \\ 0.3 & 0 & 0 & 0 \\ 0 & 0.8 \to 0.7 & 0 & 0 \\ 0 & 0 & 0.5 & 0 \end{bmatrix} \Rightarrow \lambda = 1.28 \]
The value of a transition matrix
How is the growth rate affected by particular transitions?
\[ \begin{bmatrix} 0 & 1 & 8 & 0.5 \\ 0.3 \to 0.2 & 0 & 0 & 0 \\ 0 & 0.8 & 0 & 0 \\ 0 & 0 & 0.5 & 0 \end{bmatrix} \Rightarrow \lambda = \boxed{1.33 \Rightarrow 1.15} \]
\[ \begin{bmatrix} 0 & 1 & 8 & 0.5 \\ 0.3 & 0 & 0 & 0 \\ 0 & 0.8 \to 0.7 & 0 & 0 \\ 0 & 0 & 0.5 & 0 \end{bmatrix} \Rightarrow \lambda = \boxed{1.33 \Rightarrow 1.28} \]
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 \(\lambda\)
\[ \begin{bmatrix} 0 & 1 & 8 & 0.5 \\ 0.3 \to 0.4 & 0 & 0 & 0 \\ 0 & 0.8 & 0 & 0 \\ 0 & 0 & 0.5 & 0 \end{bmatrix} \Rightarrow \lambda = \boxed{1.33 \Rightarrow 1.47} \]
\[ \begin{bmatrix} 0 & 1 & 8 & 0.5 \\ 0.3 & 0 & 0 & 0 \\ 0 & 0.8 \to 0.9 & 0 & 0 \\ 0 & 0 & 0.5 & 0 \end{bmatrix} \Rightarrow \lambda = \boxed{1.33 \Rightarrow 1.38} \]
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
\[ \begin{bmatrix} 0 & 1 & 8 & 0.5 \\ 0.3 \to 0.27 & 0 & 0 & 0 \\ 0 & 0.8 & 0 & 0 \\ 0 & 0 & 0.5 & 0 \end{bmatrix} \Rightarrow \lambda= \boxed{1.33 \Rightarrow 1.28} \]
\[ \begin{bmatrix} 0 & 1 & 8 & 0.5 \\ 0.3 & 0 & 0 & 0 \\ 0 & 0.8 \to 0.72 & 0 & 0 \\ 0 & 0 & 0.5 & 0 \end{bmatrix} \Rightarrow \lambda = \boxed{1.33 \Rightarrow 1.29} \]
Elasticity analysis
Quantify the effects of proportional changes
\[ \begin{bmatrix} 0 & 1 & 8 & 0.5 \\ 0.3 \to 0.33 & 0 & 0 & 0 \\ 0 & 0.8 & 0 & 0 \\ 0 & 0 & 0.5 & 0 \end{bmatrix} \Rightarrow \lambda= \boxed{1.33 \Rightarrow 1.38} \]
\[ \begin{bmatrix} 0 & 1 & 8 & 0.5 \\ 0.3 & 0 & 0 & 0 \\ 0 & 0.8 \to 0.88 & 0 & 0 \\ 0 & 0 & 0.5 & 0 \end{bmatrix} \Rightarrow \lambda = \boxed{1.33 \Rightarrow 1.37} \]
Age vs. stage-based matrices
So far, we have conflated age and stage (e.g. “1-year-old” == “Juvenile”, “2-year-old” == “Subadult”, etc.)
But, there’s a potential issue – e.g. what if the juvenile stage extends for multiple year?
Empirical application:
Population dynamics of red cockaded woodpeckers
Red cockaded woodpeckers: an uncommon (near threatened) species
Causes for the decline
From Ziegler & Walter 2014, in Ecological Applications
The Red Cockaded Woodpecker is an endangered species endemic to fire dependent longleaf pine ecosystems in the southeastern United States. Primary threats to the species include a lack of suitable cavity trees, habitat fragmentation, and fire suppression that results in hardwood midstory encroachment and the declining suitability of foraging habitat.
Preferred habitat for the species consists of mature, open pine forest with large trees, sparse midstory, and a lush herbaceous ground cover, conditions primarily maintained through frequent, low-intensity ground fires. Old pines are an especially important part of RCW ecology because RCWs construct cavities in living trees, and heartwood diameter is a function of age. Cavity construction is difficult and time intensive, and RCW groups will defend and use the same cavity trees for many years
Photo by Gaurav Kandlikar near Abita Springs
How to build a matrix population model for RCW?
Question 1: What are the stages?
RCWs are territorial and cooperatively breeding with a complex social structure.
Groups are composed of a monogamous breeding pair and non-breeding helpers.
Fledglings either remain on the natal territory as helpers or disperse
Almost all females disperse in either the fall or spring following fledging and ultimately obtain breeding positions if they survive, although they may act as floaters before acquiring a breeding opportunity.
Substantial numbers of male fledglings stay as helpers, and those that survive ultimately inherit the breeding position in their natal territory or fill breeding vacancies in neighboring territories. Other male fledglings disperse to either become breeders, solitary males, floaters, or (to a lesser extent) helpers in another territory
So… what are the stages? It’s complex!
So… what are the stages? It’s complex!
Banded individuals were classified according to age and stage class each year, with potential stage classes including fledglings, individuals <1 year old; breeders, males and females that occupy a territory and have the potential to produce offspring; helpers, nonbreeding adults that are part of the social group and assist breeders; solitary males, adult males that maintain a territory but are unpaired and do not breed; and floaters, adults without territories that do not breed
Question 2: What are the transition rates and reproduction rates?
Question 3: How to put these together into a matrix?
As in Heppell et al. (1994), matrices were constructed in this study as male-only models because, unlike females, males almost exclusively comprise the helper class and can hold territories despite being unpaired
Simplified example
- Consider the example from last week:
\[ \begin{bmatrix} 0 & 0 & 10 \\ 0.1 & 0 & 0 \\ 0 & 0.5 & 0 \end{bmatrix} \]
Simplified example
- Now, consider what happens when individuals can persist in a given stage.
- Juveniles stay juveniles at a rate of 0.25, adults stay adults at a rate of 0.33
\[ \begin{bmatrix} 0 & 0 & 10 \\ 0.1 & 0 \to 0.25 & 0 \\ 0 & 0.5 & 0 \to 0.33 \end{bmatrix} \]
Many potential options
- What biology to include? What to exclude?
Which model to choose?
Takeaways
Matrix population models as important tools for conservation and restoration
Importance of comparing models that incorporate different aspects of a species’ biology
Importance (and difficulty!) of collecting field data
Integrating data with models enables predictions