Lecture notes for ZOO 4400/5400 Population Ecology

Lecture 11,  (Friday, 8-Feb-13)

Matrix population models.

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Go to glossary of matrix algebra terms

        Required reading: Gotelli text, pp. 56-79
               Suggested reading:  McDonald, A primer for stage-classified life cycle graphs (on WyoWeb)
               Suggested reading:  McDonald, Demographic analyses of mating systems (on WyoWeb)
               Suggested reading:  van Groenedael, Projection matrices in population biology (on WyoWeb)

Matrix-based approaches to the analysis of single populations.

For the first few lectures I talked about exponential and logistic models for the growth of single populations. Obviously though, populations can't grow or decline (at least in the long term) without filling the universe or going extinct. Most populations therefore have  r = 0  or  l=1.  We will now study a useful method for analyzing single-population dynamics -- matrix-based approaches -- many of whose most powerful insights will have little to do with whether the population is increasing or decreasing.  I will give a brief general introduction and then work through an actual case history.  Once we've seen a little bit about how and why this kind of approach might be useful, we'll delve into more detail on the techniques used.

Outside readings: van Groenendael et al. (1988) is a good reading for an overview of what the approach can do.  It will be a "suggested" but not required reading, and I have put a copy in the WyoWeb folder.  Caswell ( 2001) has written a book on the subject that is the single comprehensive source for this major approach to modeling population dynamics.  I wrote a book chapter with Hal Caswell that some of you may find useful (McDonald and Caswell, 1993), and I have also put in the WyoWeb folder a "primer" I wrote on matrix models.

One of the things we ignored in the exponential and logistic models was the role of age structure. For almost any organism it's reasonably obvious that age (or size) makes a big difference to the role of an individual in a population -- as we saw in looking at some of the broad patterns of natality and survival in natural populations.  Both the birth rates (female offspring per female) and the survival rates will usually change as a function of age.  In some species (e.g., fish, trees) with indeterminate growth, size will be a major determinant of birth and survival rates --- large individuals will produce more offspring and have higher survival rates.  Matrix models are flexible enough to allow us to classify individuals by size as well as age -- this may be very useful, for example, when analyzing the dynamics of fish or insect populations, where size rather than age is the major determinant of reproductive output and sometimes survival.

So let's see about developing the matrix-based approach that allows us to do that.

Matrix projection models--an under-utilized demographic tool.

1. Historical origins--the Leslie matrix (Leslie, 1945

F1 F2  


Fig. 11.1. An (age-classified) Leslie matrix with fertilities in the top row and survival rates along the subdiagonal.  The fertility and survival rates are collectively  called the vital rates.  We can use this matrix to project population growth from a census vector (a count of the individuals in each age-class).  This matrix has 3 age-classes  (the matrix is 3-by-3).
Compare and contrast this matrix with the equations and matrix of Eqn 11.2  and Fig. 11.1 and the corresponding life cycle graph shown in Fig. 11.2.

2. Setting up the model (Caswell 2001; McDonald & Caswell, 1993)

a) Why are we doing a demographic analysis? What won't we get?
b) Deciding what kinds of transitions matter
c) The problem of missing or incomplete data
d) Parameterization -- putting in the survival and reproduction terms (vital rates).  More on this later
e) A guide to future field work. Matrix models tell us about the transitions in the life cycle to which the population growth rate (l) is sensitive or elastic -- these terms mean that changes in those vital rates will have a big impact on the population growth rate.  If a transition is highly sensitive, then we need very accurate measurements of any component terms such as survival rates, because inaccurate values will yield an inaccurate analysis. On the other hand, insensitive arcs suggest places where one should not overdo the effort (many ornithologists spend too much time and effort assessing nesting success and not enough assessing survival).

N.B. Even if a transition is highly sensitive it doesn't mean we can do anything about it. A manager may have to weigh the cost and feasibility of acting on the most sensitive transitions against the practical reality that it may be easier to do something about a relatively insensitive transition.

3. The major outputs:

Lambda (l) the population growth rate (at equilibrium).
         Usually close to 1.0 (1.0 = stationary, neither growing nor shrinking).
Sensitivity--what effect does an absolute change in a vital rate (arc in the life cycle graph or cell in the matrix) have on l?
    For example, if we change first-year survival by 0.01, how much will that affect the population growth rate?
Elasticity--what effect does a proportional change in an arc (vital rate) have on l?
    For example, if we change first-year survival by 1%, how much will that affect population growth?
Stable (st)age distribution--proportion of the population in each (st)age.
    What proportion of the total population are first-year individuals?  Second-year individuals?  Etc.
     At equilibrium, a population's age structure won't change, even if the population is growing or shrinking.
Reproductive value -- value of a given age-class or stage as a seed for population growth
        (newborn or first age class reproductive value = 1.0 by definition).
        Example: an adult female sea turtle is "worth" 588 turtle eggs (Crowder 1994).

4. Comparing life histories (quantitatively!)

a) Using the elasticity vs. the sensitivity.  Elasticity assesses effect of a proportional change in the values,
            whereas the sensitivities assess the effect of an additive change.

b) Matrix output gives us a whole suite of parameters to compare among populations and species.

Some other tools and data requirements:

i. We will usually want robust estimates of survival rates from mark-recapture (e.g., mist netting) data

ii. Basic natural history: if you don't know your organism well, you may overlook critical life history transitions.

iii. Incorporating greater realism by accounting for demographic stochasticity (important in small populations) vs.
        environmental stochasticity (can be important even to large populations)

iv. Another important addition is to incorporate density dependence: how does density affect birth rates, mortality and other vital rates?  Our initial matrix models will ignore density dependence.  That is, the vital rates will be constant through time and over all population sizes.

Real-world examples:

Matrix models of sensitive species in Forest Service Region 2 (including WY).  Assessment of population of black-footed ferrets in Badlands National Park and adjacent Buffalo Gap National Grassland in South Dakota.  Population has grown to over 200 free-living ferrets from the captive-bred founders.  Individuals born in the wild do much better as founders for new introduced populations.  Can the managers of the SD population afford to "give away" some of their ferrets to help establish other populations?  Matrix models provide a quantitative basis to help guide management decisions.  Recently, the population in the Shirley Basin, WY has shown exponential growth highlighted in a short paper in the journal Science (Grenier et al., 2007). 

Black-footed ferrets have thrived in the large black-tailed prairie dog colonies in the Conata Basin, SD.  Matrix population models have helped guide their management. 

A matrix population model is a demographic technique for understanding phenomena at the population level based on information from the individual level (birth, death, and growth rates of individuals; we'll call these demographic rates the vital rates).  The major mathematical links are from equations on the one hand, and graph theory on the other hand, to the matrix formulations that are the way computer programs actually analyze the models.  

Consider a very simple life cycle with four age classes (Eqn 11.1, Fig. 11.1).  We could look at population change over time in three equivalent ways: using difference equations, using a population projection matrix (Eqn 11.2; remember the Leslie matrix from ecology courses?) or with a life cycle graph (Fig. 12.1, in the next lecture).  The three are mathematically equivalent–but their utility is context-dependent.  

Approach I. Equation-based: we could use continuous differential equations (such as those we used for the exponential and logistic) but usually use discrete, difference equations (time comes in chunks or "intervals" as fine as days or as coarse as years).   The reason for using the discrete (vs. continuous) formulation is that many of the organisms we are interested in are pulse breeders. That is, they have a single, relatively well-defined period (pulse) of breeding per interval (often an annual cycle).  Complex analyses of difference equations are very challenging mathematically.  Complex analyses of discrete formulations using matrix algebra are much easier (especially since the advent of the personal computer, which can handle most matrix algebra routines with ease).
A difference equation formulation would look like this: 

Eqn 11.1                                  (Eqns 11.1)

The terms in the equations are ni(t), the number of individuals of age-class i at time t; Pi, the annual survival rate of individuals in age-class i, and mi, the fertilities of individuals in age-class i. The first line tells us that the number of individuals in the first age-class (n1) at time t+1 is equal to the number of individuals (mothers) in the second age-class (n2) times their annual survival (P2) times the number of babies they produce (m2), plus the same quantities for the third-year mothers (fourth-year individuals don't reproduce).  That is, the total babies produced equal the sum of the babies produced by the two reproductive age-classes.  The second line tells us how many second-year individuals we will have at time t+1.  That will be the number of babies at time t, given by n1(t), times their survival rate through the first year (P1).  And so on for the two remaining age classes.  Here we have an organism with four age classes, first-year individuals, second-year individuals, third-year individuals and fourth-year individuals. Fourth year individuals die on their fourth birthday just after we have counted them (or at least before they survive through to the next breeding season). Note also that individuals first reproduce in their second year. Those birth and survival rates are the vital rates we have discussed in the past few lectures.  For size-classified life cycles, the vital rates could also include growth rates (Gi).

Approach II. Matrix-based: [The way we organize a demographic model for number-crunching analysis by a computer].  Below is a Leslie matrix formulation of Eqns 11.1. Matrix algebra is just a way of organizing sets of equations in an orderly way -- computers can easily do lots of things with matrices that would be a lot more awkward if we used sets of equations.  In the first equation of Eqns 11.1, the first-year individuals at time t+1 are a function of the number of second-year individuals at time t times the fertility of those second-year individuals, plus the number of third-year individuals times their fertility.  All of that becomes condensed into a single matrix row-times-vector column calculation in Eqn 11.2.  We will discuss why there are "parental" survival terms in the fertility equation shortly. The matrix algebra formulation of Eqns 11.1 can be written as:

Eqn 11.2                                           (Eqn 11.2)

In Eqn 11.2, a 4X1 census vector (LHS) at time t+1 results from the right-multiplication of a 4X4 projection matrix by a 4X1 census vector at time t.  The (mathematically equivalent to Eqns 11.1) matrix form shown in Eqn 11.2 is a very convenient way to organize Eqns. 11.1 for computer analysis.  With a computer, we can do many kinds of very powerful analyses very easily (e.g., eigenvalue and eigenvector analyses).  The mathematical results will have biologically meaningful interpretations -- for example, the dominant eigenvalue of the matrix turns out to be the growth rate, l, of the population.  We will refer to the projection matrix (the first big bracketed term after the = sign in Eqn 11.2) as A (bold font, or if writing it by hand we can put a bar over the top), with elements aij, referring to the element in the ith row and jth column.  (Mnemonic -- which means a way to help memorize something: R X C = Roman Catholic, rows first then columns; Arc = arc).  Each element (cell) in the matrix represents a transition (often a probability) of moving from the jth age class (or stage) to the ith age class (or stage).  Thus, in the matrix shown above, element a21 (=P1) is the probability of surviving from the first age class to the second.  Figuring out the correct values of the elements (we call this  parameterizing the matrix) may require hard work in the field and always requires care in the formulation.  We refer to these transition terms collectively as the vital rates (demographic processes such as birth, survival, growth {in mass}, change in social status).  Given the projection matrix A, computer programs can easily calculate the eigenvalues, eigenvectors, and other statistics that provide insights into the population dynamics.

{Notation reminder: this is a projection matrix.  Look back to my speedometer example for the distinction between projection and forecasting}. 

Note that in Eqn 11.2 above I have left out lots of empty cells (zeros).  Another way to write/draw that matrix would be: 


Fig. 11.2.  Leslie projection matrix corresponding to the demographic equations of Eqns 11.1 and to the matrix portion on the RHS of the matrix algebra formulation in Eqn 11.2.  If we right multiply this matrix by a 4X1 census vector at time t, the result will be the projection of the population to time t+1.

The only non-zero cells are in the top row --these are the fertilities, and in the subdiagonal (the cells below the top-left to bottom right diagonal) -- these are the annual survival rates.  A few other points about the matrix and its notation:  note that the right-most column is all zeros.  This is necessary in order to generate a fourth age-class (we want to count them because they have just reproduced and produced offspring, but they won't survive to reproduce again or be censused).  A variant of the Leslie matrix has a non-zero element in the bottom right-hand corner (representing "adult" survival).  Other cells may be non-zero when we consider stage-classified life cycles.  For example, the stage-classified analysis of sage-grouse in an upcoming lecture has a 3X3 matrix with all the cells non-zero (meaning that a transition from any age-class to any other age-class is possible).

Before we look at the third (life cycle) approach let's see what kinds of outputs a computer program can give us:

Mathematical definition
l (lambda) Dominant eigenvalue

Population growth rate (l can also be used as a measure
                                        of individual evolutionary fitness)

Stable (st)age distribution
Right eigenvector Proportion of population in each age-class or stage
Reproductive values (RV) Left eigenvector  Value of a (st)age class as a seed for population growth
Sensitivities (sij) sij = dl/daij    (partial derivative of l
                        with respect to aij ).  
      Calculated from the eigenvectors

Sensitivity of l to a (small) change in a vital rate (aij
             {SSD of source times RV of target }

Elasticities (eij) eij = sij*(aij/l)
               sensitivity "weighted" by aij/l
Proportional sensitivity
Cohort generation time (m1) Mean of net fertility schedule
|   (mean of discrete equivalent of lxmx
Generation time (an evolutionary fundamental)
Life expectancy [Complex formulation] Expectancy of life from current (st)age onward

The sensitivities and elasticities are particularly valuable, and we will return to their interpretation repeatedly.

Caswell, H. 2001. Matrix Population Models (2nd Edn.). Sinauer Associates, Sunderland, MA.
Crowder, L.B. 1994. Predicting the impact of turtle excluder devices on loggerhead sea turtle populations. Ecol. Applic. 4: 437-445.
Grenier, M.B., D.B. McDonald, and S.W. Buskirk. 2007. Rapid population growth of a critically endangered carnivore. Science 317: 799.
Jenkins, S.H. 1988. Uses and abuses of demographic models. Ecol. Bull. 69: 201-207.
Leslie, P.H. 1966. The intrinsic rate of increase and the overlap of successive generations in a population of guillemots. J. Anim. Ecol. 25: 291-301.
McDonald, D.B., and H. Caswell. 1993. Matrix models for avian demography.  Chapter 10 In Current Ornithology Vol. 10 (D. Powers, ed.). Plenum Publishing, NY
van Groenendael, J., H. de Kroon and H. Caswell. 1988. Projection matrices in population biology. TREE 3: 264-269.

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