Lecture notes for ZOO 4400/5400 Population Ecology

Lecture 12,  (Monday, 11-Feb-13)
Matrix models (cont.). Formulation, life cycle graph, outputs.

This spreadsheet also illustrates how to calculate sensitivities from the reproductive value vector (RV, left eigenvector) and the stable (st)age distribution (SSD, right eigenvector).

Approach III: The life cycle graph  [The best place to start formulating a model]

The problem with the two formulations in Eqns 11.1 and Eqn 11.2 (equations and matrix) is that they are difficult to visualize in terms of the natural history of the organism of interest.  Eqn 11.2 is great for computers, but not easy to think about when trying to put the life history of the organism one is interested in into a matrix model setup.
The best way to play with possible ways to set up a matrix-based analysis is by the use of a life cycle graph, such as the one shown below.

Fig. 12.1. A life cycle graph corresponding to Eqns 11.1 and the matrix formulation of Eqn 11.2 and Fig. 11.1
(redone as a figure in Fig 12.3).
The four nodes (yellow circles) depict four age classes.
The red arcs (arrows) depict transition probabilities between age classes (in the case of a strictly age-classified matrix, these will be age-specific survival rates).
The green arcs depict fertilities, the production of new recruits. Note that the green arcs (Fi) are compound terms including the survival of the parents (Pi), as well as offspring production (mi).
This is a result of a post-breeding census approach that I will explain below.
Although we are using single subscripts here (Fi, Pi) the more general way to refer to the arcs is as the aij (arc from Node j to Node i).
Thus F3 is equivalent to a13.  The P1 = l1/l0 is to show how one converts the entries in an lx table into the matrix survival entries.

How and when births occur makes a big difference:

All of the above (Eqns 11.1, Eqn 11.2, Fig. 11.1 and Fig. 12.1) represent analysis of a discrete birth pulse (once a year pulse of breeding) post-breeding census formulation. That is, immediately after the annual birth pulse (spring or summer for many birds and mammals in North America), we census the population. It will consist of newborns (first-year individuals), second-year individuals (that have just celebrated their first birthdays), third-year individuals (just celebrated their second birthdays) and fourth-year individuals. The fourth-year will not survive to the next birth pulse (hence the lack of an arc from the fourth node back to the first).  Compare the matrix of Eqn 11.2 or Fig. 12.3 care fully with the life cycle graph of Fig. 12.1.

Fig. 12.2. Difference between discrete age class (subscripted i) used in matrix projection models and continuous age (x) used, for example in lxmx life table analyses. Age classes are discrete (time goes in steps or "chunks" as opposed to flowing continuously).  That is, age classes are first, second, third, etc.  Individuals are "first-year" from the time they are born (and censused) until just before their first birthday. From Caswell, (2000), which is the best single source for learning matrix model approaches. Jenkins (1988) focused especially on the problems of indexing that have confused even very good ecologists. Gotelli’s 1st edition had an error in computing reproductive values that was due to an indexing problem (Steve Jenkins is acknowledged in the Preface to the 2nd edition for finding the error).

The formulations above all represent an annual census taken just after the birth pulse. Strictly speaking, we make the assumption that all members of the population reproduce on the same day and that all the newborns and birthday celebrants are then immediately censused (in practice one would try to have the census at the end of a breeding season that is presumably short compared to the census interval (i.e., census takes a few days within a 1-yr census interval). We can use census intervals other than years but all census intervals must be equal (we can't assess some transitions that occur with a 6-month interval and others that are annual).

If we conduct a pre-breeding census then instead of following the survival of the parents along the reproductive arcs, we must follow the survival of the newborns from birth until we actually census them almost a year later. The life cycle graph's terms then look like the formulation below. [For most vertebrate populations it makes more sense to do a post-breeding census].  See Caswell (2000) for more details on formulating a pre-breeding matrix model. [Personally, I almost always work with a post-breeding census].

Here's a symbolic matrix corresponding to the life cycle graph of Fig. 12.1.

 F2 F3 0 P1 P2 P3 0

Fig. 12.3.  Projection matrix corresponding to the life cycle graph (LCG) of Fig. 12.1.  The top row elements are the fertilities (Fi = Pi * mi) and the subdiagonal elements are survival rates ( Pi).   Note that, for a strictly age-classified analysis, fertilities will be in the top row and annual survival rates in the subdiagonal.  All the other cells are empty (zero).  As we will see in the Greater Sage-Grouse example, however, stage-classified life cycles can have non-zero cells in other places (they will almost always still have the fertilities in the top row, though).

Here are some specific numerical values for the vital rates:
m2 = 1  (1 female offspring per second-year female breeder)
m3 = 3   (3 female offspring per second-year female breeder)
P1 =  0.45 (45% annual survival from birth census to end of first year)
P2 =  0.7 (70% annual survival through second year)
P3 =  0.7 (70% annual survival through third year)

No animals survive to breed at the end of their fourth year

Plugging in the values from above we get the following numeric values:

 0 0.70 2.10 0 0.45 0.70 0.70

Now we crunch that with a matrix demographic analysis package and come up with:

l = 0.99 (the population is almost stationary; stationary means it is neither growing nor shrinking; we use this term to avoid confusion with the term "stable", which has a special meaning referring to the equilibrium reached when the transient dynamics yield to a population that grows with a constant proportion of individuals in each age-class or stage -- the stable (st)age distribution).  At that equilibrium the "stable" population may still be growing very rapidly or it may be declining rapidly.  What isn't changing is the proportion of individuals in each of the age-classes or stages.

Stable age distribution: (because it is a right eigenvector, this one is a vertical 4X1 vector)

0.499
0.227
0.160
0.113

Reproductive values: (a left eigenvector will be a horizontal 1X4 vector) {1., 2.20, 2.12, 0} Note that the reproductive value of the fourth-year individuals is 0.
That's because they will not survive to reproduce again.
Typically reproductive value curves peak at the age of first reproduction because they measure current reproduction plus all future reproduction discounted by the probability of surviving to reproduce in the future. Note also that reproductive values and stable age distribution are properties of an age-class or stage (the nodes), while sensitivities and elasticities are properties of a transition (the arcs).

Scalar product (SP):   1.3389.   {Not required: We can calculate the scalar product as the sum of the cross-products of the corresponding elements of the two eigenvectors.  That is, 1 * 0.499 + 2. 2 * 0.227 + 2.12 * 0.16 + 0 * 0.113 = 1.3389}.

Suggestion for building on skills worked on in Homework 3: use the formula

sij = (RVi*SADj) / SP                                              [Eqn 12.1]

to be sure you can calculate sensitivities correctly. RV is the reproductive value vector, SSD is the stable stage distribution vector. {Remember, aij is the transition from Node j to Node i. Also, arc is the element in the rth row and cth column of the matrix.   I like to use the correspondence P1 = a21 = survival from first-year to second-year to remind me how keep the subscripts straight. For example, s43 would be the sensitivity of lambda to a change in the vital rate that goes from Node 3 to Node 4}.

Sensitivities shown below: (only the "possible" values are listed).   The sensitivities tell us the effect on l of an additive change in a vital rate.

 0 0.169 0.12 0 0.822 0.359 0

Small changes in first-year survival would have the biggest impact on l, followed by second-year survival and then second-year reproduction.

As shown by Eqn 12.1, the sensitivities are cross-products of a right eigenvector element (which has as many elements as the graph has nodes/stages) and a left eigenvector element.  Qualitatively, the sensitivity of l to each arc is the product of the SSD (Stable Stage Distribution = how many) of the source node, times the RV (reproductive value = how valuable) of the target node.

Sensitivity of arc to Node i from Node j = SSD of Node j times RV of Node i
= “how many” in the source times “ how valuable” is the target?

If we have lots of individuals in the age-class (node) that the arrow is coming from (big SSD value), and the arrow is pointing to a very valuable node (high RV), then changing that arrow (aij) will have a big impact on l.  The whole point of eigenanalysis is to tell us how a matrix (of vital rates) affects a vector (a count of the animals in the age-classes or stages) – the matrix is a multi-dimensional object that transforms what it multiplies.  For us, the matrix is a demographic projection matrix (set of vital rates) and the vector it is affecting is a census vector (the population count, categorized by age or stage).  The dominant eigenvalue (l) tells us how fast the projection matrix grows or shrinks the population – the growth rate.  The two eigenvectors tell us two different but important things about how the projection matrix acts.  The left eigenvector (reproductive values, RV) tells us which ages or stages have the biggest per capita role in producing that growing or shrinking – sort of the power of that stage (do we have pawns, knights or queens in our forces?).  The right eigenvector (the stable stage distribution, SSD) tells us how many of each of the age classes or stages we will have (how many pawns, knights or queens are in our army?).  Multiplying importance (left eigenvector elements) by how many (right eigenvector elements) tells us how much of an effect that arc (vital rate, or transition between stages) will have on l.  If our chess move (the arc or vital rate) involves a large cohort of queens, it will have much more effect than if it involves a small force of pawns.  It will take many pawns to have the same effect as one queen.  One or two knights may not be a match for a large force of pawns.  We can trade off number (SSD of source) against importance (RV of target) in determining sensitivity.

Elasticities:

The elasticities are calculated from the formula:

eij = (sij * aij)/l                                  [Eqn 12.2]

That is, they are the sensitivities (sij) times the original matrix entries (aij, arc values) all divided by lambda.  (See also the table at the bottom of the notes for Lecture 11).  The elasticities will always sum to 1.0.  Putting the elasticities into a matrix we get:

 0 0.120 0.254 0 0.373 0.254 0

Notice that the second-year survival and third-year reproduction are now much more nearly equal in importance to first-year survival.
The elasticities tell us the effect on l of a proportional change in a vital rate.

m1 (cohort generation time) = 2.68 (years). The pace of evolution is largely determined by generation time (remember back to my contrast between the growth rates of Iceland and Ireland in Lecture 8 -- that difference is largely due to the shorter generation time in Iceland). Bacteria can evolve much more quickly than can elephants or humans because of their much shorter generation time. m1 is the mean of the discrete equivalent of the lxmx schedule (or net maternity function). The skewness and kurtosis of the lxmx  distribution are also of interest. The skewness is the third moment of the distribution (first is the mean, second is the variance), while the kurtosis is the fourth moment. Skewness tells us how drawn-out the tails of a distribution are, kurtosis tells us how peaked (leptokurtic) or flat (platykurtic) a curve is.  Together these moments determine the shape of the curve, and can tell us a great deal about the life history of the organism.  For example, if reproduction is spread out over the lifespan, the curve (plot of lxmx against age) will be flat (platykurtic).

Hint: Make sure to distinguish between arc-related measures and node-related measures.
Look at several examples of life cycle graphs (or matrices).  Each has a certain number of stages or age-classes (equal to the number of rows and to the number of columns in the projection matrix).  The eigenvectors have exactly that many elements – they relate directly to nodes/stages rather than to arcs.  Each of the elements of either the SSD or the RV vectors has a one-to-one correspondence to one of the stages.  Now look at the arcs (transitions, vital rates) in a variety of life cycle graphs.  The graph will have at least as many arcs as it has nodes/stages.  The number of arcs is the same as the number of non-zero elements in the projection matrix.  Any particular node may have several incoming and several outgoing arcs.  Most matrices will have many more non-zero cells than they do rows or columns.  The sensitivities and elasticities relate to arcs/transitions rather than to nodes.  Put differently, they involve the relationship (connecting arrow) between two different stages (or back to the same stage in the case of self-loops).  In fact, we can compute sensitivities for transitions that don't even occur on the life cycle graph.  That is, we can ask a sort of “what if” question for the importance of non-existent transitions (this is why I often restrict my attention to the “possible” sensitivities – those that correspond to non-zero cells or existing arcs in the life cycle graph.  Often the non-existent arcs are meaningless (transition from age four to age two is unlikely to yield much insight), but occasionally a non-existent arc is plausible, even if not yet observed (e.g. reproduction at earlier stages by organisms with delayed reproduction).

References:

Caswell, H. 2000. Matrix Population Models (2nd Edn.). Sinauer Associates, Sunderland, MA

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

Cochran, M.E., and Ellner, S. 1992. Simple methods for calculating age-based life history parameters for stage-structured populations, Ecol. Monogr. 62: 345-364.

van Groenendael, J., H. de Kroon, S. Kalisz, and S. Tuljapurkar. 1994. Loop analysis: evaluating life history pathways in population projection matrices. Ecol. 75: 2410-2415.

Wisdom, M.J., and L.S. Mills, 1997. Sensitivity analysis to guide population recovery: prairie-chickens as an example. J. Wildlife Mgmnt 61: 302-312.

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