Lecture 14 (Monday, 18-Feb-13)

Matrix models (cont.). Approach to equilibrium.

Return to Main Index pageGo back to notes for Lecture 13, 13-Feb Go forward to notes for Lecture 15, 20-FebGo to notes on matrix algebra (e.g., how to multiply matrices)

(Software that calculates the eigenvalues and eigenvectors is now available as an Excel add-in at http://www.cse.csiro.au/poptools/)

(I use the programming language/mathematical software *Mathematica*)

**Population's approach to equilibrium (stable age distribution).**

Now let's try a variant of a matrix/life cycle graph but with real (preliminary) data for endangered black-footed ferrets.

Reproduction starts at the end of the first year. The main change from the matrix and life cycle graph at
the end of Lecture 12 is, therefore, that our starter matrix now has a
"**self-loop**" in the upper left hand corner (an arc that points back to the node it came from).

Remember that the *F*_{i} fertility terms include a survival term, *P*_{i}, for the mothers.
We counted mothers just **after** a breeding pulse (post-breeding census).
In order for the mothers to reproduce at the end of our one-year census interval,
they must survive through the interval (with probability *P*_{i}).

Figure 14.1 (compare to Fig. 12.1) shows the change we are introducing, which involves the self-loop on Node 1:

Fig. 14.1.Life cycle graph of black-footed ferrets, for showing approach to stable age distribution (SSD) and other matrix phenomena.

The difference between this graph and that of Fig. 12.1 is the addition of aself-loopfor reproduction

at the end of the first year (production of more beginning-of-first-year animals by end-of-first-year animals).

Question to ponder: Since we know thatP_{3}= 0.346, what ism_{3}?

Crunch the revised matrix:

0.691 0.898 0.606 0.464 0.534 0.346

Fig. 14.2. Projection matrix for black-footed ferrets used for the 'age wave'

demonstration. The matrix has the same number of rows (4) and columns (4)

as the life cycle graph (Fig. 14.1) has nodes (4). The fertilities (F_{i}) are in the top row,

while the annual survival rates (P_{i}) are in the subdiagonal. Each cell in the matrix

corresponds to an arc in the life cycle graph. Where no arrow occurs, the corresponding

matrix cell is empty or zero.and we get:

l = 1.16

The reproductive value(RV) vector is:{1, 1.01, 0.52, 0}

The stable age distribution (SSD) is:

0.61(i.e., at the time of the census, 61% of the population are 1st-year individuals)

0.24

0.11

0.03The scalar product (

SP) is 0.91 (given theRV, SSD and SPyou can calculate the sensitivities yourself, using Eqn 12.1)Now, let's say we start with a census vector that is NOT at the stable age distribution (i.e., it is not at

equilibrium) and project it for 10 years. Say, for example that we start with 25% of the population in each of the 4 age classes at timet= 0 (second bar in Fig. 13.3). Here's what happens to the proportions :

Fig. 14.3.Age distribution for the black-footed population with four age classes projected over nine time intervals. At timet= 0, the four age classes account for the given proportions of the population (20% 1st-year, 20% 2nd-year, 40% third-year, and 20% 4th-year; each of the colors is the proportion in the age-class). Byt= 9, the population has essentially converged on the stable (st)age distribution (SSD) shown at the far right in darker colors. Notice that the over-representation of certain age classes (e.g., the third-year class shown in white) is 'squeezed out' in a wave-like pattern.SSD valuesfrom bottom to top in stacked bar at right: Gray (bottom): equilibrium percentage of 1st-year individuals; black: equilibrium percentage of 2nd-year individuals; yellow: equilibrium percentage of 3rd-year individuals; dark blue (top): equilibrium percentage of 4th-year individuals.Observed values(t= 0 to 9): Light blue: observed proportion of 1st-year individuals; purple: observed proportion of 2nd-year individuals; white: observed proportion of 3rd-year individuals; light green: observed proportion of 4th-year individuals. The proportions were calculated by sequentially right-multiplying an initial census vector (e.g., 20, 40, 20, 20) against the Leslie matrix shown in Fig. 14.2.Let's try the same thing but starting with all the individuals in the 3rd age-class.

Fig. 14.4.Approach to the stable age distribution by a population that starts att=0with

all 100 individuals in the third age-class. Notice the dramatic fluctuations on the way to time step 4.

Compared to the projection series in Fig. 14.3. the proportions here approach the SSD values slightly

more slowly and are slightly further from the SSD att= 10.

All bars have the same meanings as in Fig. 14.3.

The "waves" in the age structure disappear as we approach

equilibriumat the stable (st)age distribution, SSD. Note that the equilibrium for the matrix model is defined in terms of the age structure. Thatcontrasts with the stable equilibrium in the logistic model, which was defined in terms of population size (N=K).

Questions to ponder.

If I started with 0 1^{st}-, 2^{nd}-, and 3^{rd}-year individuals and 100 4^{th}-year individuals and projected to the next census interval, how many would I expect to have at the next census?Alternative question.What is the reproductive value (value as a seed for future population growth) of a 4^{th}-year individual?

Does that help answer the first question?

If a life cycle graph has five nodes, how many possible arcs does it have? Is a life cycle likely actually to have that many arcs (indeed,canit, if it is an age-classified model)?Here's the matrix of sensitivity values (remember, each number corresponds to an arc in the life cycle graph)

0.670.270.12 0.680.14 0

Fig. 14.5. Matrix of sensitivity values for the black-footed ferret values. l is most sensitive to a change in first-year survival. First-year survival is

s_{21}= 0.68, corresponding to theP_{1}arc. l is not at all sensitive to a change in third-year survival.

The highest-sensitivity arc comes from the node with the highest SSD (0.61) and goes to the node with the highest RV (1.01). Again, "size of source (SSD) times reproductive value of target (RV)" tells us about the sensitivity.

The matrix of elasticities is:

0.40.210.06 0.270.06 0

Fig. 14.6. Matrix,

E, of elasticity values, with cellse_{ij}, for the black-footed ferret model. Note that the arc to which l is mostelasticisF_{1}. That's because the elasticities (impact on l of a proportional change in a vital rate) are computed as the sensitivities weighted by the value of the original arcs. BecauseF_{1}>>P_{1}, it follows thate_{11}>e_{21}.

Overview of matrix-based analyses:

Purpose:Analysis of complex age- or stage-structured populations

(vs. 'simpler' dynamics of populations with non-overlapping generations).Analysis of which life cycle transitions (e.g., adult survival, growth probability for yearling trout, offspring production by yearlings) have a large impact on population dynamics.

Measures produced:Sensitivity: effect on lambda of a small

absolutechange in a given vital rate

Mathematically, the sensitivity is the product of the stable (st)age distribution value of the node from which the arc originates, times the reproductive value of the node to which it points (that is it weighs the reproductive value of the destination, by how much flows into it along that arc).I suggest memorizing this verbal definition.The sensitivities reflect

howvaluablethe destination node is (RV)

times

howlargethe source node is (SSD).

(Valueof target) times (sizeof source)[See Caswell, 20019, p. 208, if you wish to pursue this].

Elasticity: effect on lambda (l) of a small

proportionalchange in a given vital rate. Calculated by weighting the sensitivities by the value of the original arc/matrix cell.Population growth rate (l, the dominant eigenvalue)

Stable (st)age distribution (SSD, right eigenvector)

Reproductive values (RV, left eigenvector): reproductive value curves tend

to peak atage of first reproduction(unless survival or fertility prospects continue to increase dramatically)

Reproductive value is the value of an individual of a given stage as a(in first stage equivalents).

seed for future population growth

Reproductive value represents an individual'scurrent and expected future offspring production,

weighted by its survival prospects.That is, the reproductive value weighs all the pathways along which we can move from a given node

back to the first node (new recruits).

Post-reproductive individuals have a reproductive value of zeroGeneration time

Average age of a stage (for life cycles analyzed by size, social status or some other way that differs from classic age-classified Leslie matrix).

Guidance for field effort and management:

• Which vital rates require the most careful measurements (the most sensitive)

• If we want to manage population dynamics where could we make changes that would have the largest impact?

• Example: let's say we find that cowbirds are a major contributor to the decline of a passerine bird.

By doing a sensitivity analysis of the life cycle of the cowbirds we could find out where to put our efforts into cowbird control.Assumptions:

Vital rates

do not change over time(no effect of environmental stochasticity)Evenly spaced census intervals at a particular time in the cycle (usually an

annual post-breeding censusfor most vertebrates)

Birth-pulse reproduction(as opposed to birth-flow)We have measured the vital rates reasonably accurately

(Accuracy of measurement of sensitive transitions is the most critical)We have accounted for all possible transitions in the life cycle

No density-dependence{We

canadd stochasticity or density-dependence to matrix models, we just haven't done that here}For reproductive values and stable age distribution, sensitivities and elasticities, we are also assuming the population is

at equilibrium. That is, the population has gone through any transient dynamics caused by not starting at the stable age distribution.

{Look back at the changes in the age structure in Figs. 13.3 and 13.4 for examples of how the transient dynamics gradually smooth out and the population reachesequilibrium -- in a matrix model equilibrium means the population has achieved the stable age or stage distribution}.

Things the analysis won't do for us:Even if we know which are the most sensitive transitions, we may not be able to do anything about them. For example a manager who deals with a migratory population only on the breeding grounds may not be able to affect overwinter survival even if that is the most sensitive transition. (Nevertheless, it may be important that she or he know what is most important).

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The material below is not anything I will test directly, but studying it, at least to the point where it makes sense (even if you couldn't do it on your own) is likely to help make the matrix models more comprehensible.

Some notes on matrix algebra and graph theory:Go to web page for definitions of terms in matrix algebra.

Matrix

A(printed in bold, written by hand with a bar or rightward arrow over it)

Ahas elementsa_{ij }where the first subscript (i) refers to the row number and the second (j) refers to the column number. I sometimes use the subscriptsr(for rows) andc(for columns. That yieldsa_{rc}, which is a reminder that the matrix cells correspond to the arcs (arrows) connecting the nodes. In a demographic matrix, thea_{ij }refers to a transition from thej^{th}column to thei^{th}row. In the life cycle graph, this would mean thearcfrom thej^{th}node to thei^{th}node(representing an age class or stage). For example a transitionfromthe 4^{th}nodetothe 2^{nd}node would go in cell a_{24}(2^{nd}row, 4^{th}column).A

self-loop(used more often in stage-classified analyses) is an arc that returns to the node from which it originated (it therefore occurs on the diagonal of the corresponding matrix).

Matrix multiplication:The dimensionality of a matrix is give by

rowsXcolumns(rXc). To multiply two matrices they must becompatible, meaning that the number of columns in the first (left-side) matrix is the same as the number of rows in the second matrix. The result matrix (product matrix) will have the number ofrowsof the left matrix or vector and the number ofcolumnsof the right matrix or vector. For example, a 2X3 times a 3X4 is compatible, because the threes match. The product will be a 2X4 matrix (rows of the first by columns of the second). You can't multiply a 3X4 by a 2X4, because the four columns of the first matrix aren't compatible with the 2 rows of the second matrix.Let's walk through a 3X2 * 2X3 (the left matrix is 3 rows by two columns, while the right matrix is two rows by three columns; they are compatible but the result will be a different size than either 'parent' matrix -- a 3X3).

1 4 14 10 24

2 6 4

3 5 X = 21 23 37

3 1 5

2 2 10 14 18To get the element in the first row and first column of the product matrix we take the first row on the left and the first column on the right and calculate the dot product.

1*2 + 4*3 = 14

Now first row, 2nd column. 1*6 + 4*1 = 10

1st row, 3rd col. 1*4 + 4*5 = 24

2nd row, 1st col. 3*2 + 5*3 = 21

2nd row, 2nd col. 3*6 + 5* 1 = 23

2nd row, 3rd col. 3*4 + 5*5 = 37

3rd row, 1st col. 2*2 + 2*3 = 10

3rd row, 2nd col. 2*6 + 2*1 = 14

3rd row, 3rd col. 2*4 + 2*5 = 18

A little practice goes a long way toward making this seem easier.

You should try multiplying the matrices the other way round. Do you get the same answer? NO!! What you should get is:

28 46

16 27

Here is projection of an initial census vector that consists of just 30 3rd-year individuals. On the left is the 4X4 projection matrix,

A. (It is the matrix of Fig. 12.1). To the right ofAis the census vector, N(ø). We are therefore multiplying a 4 X 4 * 4 X 1. The result will be a 4 X 1 (i.e., another column vector containing the projected number of individuals in each of the age classes). Within about 12 census intervals this census vector will settle down to very nearly stable age distribution proportions and grow each year (or whatever the census interval is) by a factor l.

N(ø) N(1) 0.45 0.7 2.1 0 0 63 0.45 X 0 = 0 0.7 30 0 0.7 0 21 N(1) N(2) 0.45 0.7 2.1 0 63 28.4 0.45 X 0 = 28.4 0.7 0 0 0.7 21 0 N(2) N(3) 0.45 0.7 2.1 0 28.4 32.6 0.45 X 28.4 = 12.8 0.7 0 19.8 0.7 0 0 N(3) N(4) 0.45 0.7 2.1 0 32.6 65.3 0.45 X 12.8 = 14.7 0.7 19.8 8.93 0.7 0 13.9

[If I had started with 30 4th-year individuals in

In calculating the above projections, the only very complex term is the number of individuals in the first age-class.

For the transition from

References:

Caswell, H. 2001. Matrix Population Models (2nd Edn.). Sinauer Associates, Sunderland, MA.

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.

McDonald, D.B., and H. Caswell. 1993. Matrix models for avian demography. Chapter 10

InCurrent Ornithology Vol. 10 (D. Powers, ed.). Plenum Publishing, NYvan 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. [in WyoWeb folder]

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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