There is one last variable to calculate. Vx , or the reproductive value, measures the relative importance of different age classes to the future reproduction of a population.

If many die before reaching the age at which reproduction begins, those organisms are less important to future growth than are those that have begun to reproduce.

Assume you are a manager trying to determine how much harvesting to allow, while optimizing the catch available to be harvested. MacArthur worked out the math, and suggested the following: The permissible harvest from an age classis inversely related to the reproductive value of that class.

A definition: Vx is defined as the age specific expectation of future offspring (for an equilibrium population) or age specific contribution to future population growth (if population size is changing).

For equilibrium populations (and thus in a stable age distribution) the calculation of Vx is simple: 

Vx = limi/lx 

For populations whose size is changing, the formula and calcuations are more complex:  V

V

e

le l mx

rx

x

rii i

i x0

V0 is 1 in our formulation of a life table and reproductive value, and can be dropped from the equation (but the approach permits comparing other ages).

Reproductive value normally peaks at the age of first reproduction, and will always rise from birth to . It could continue to rise if survivorship during initial bouts of reproduction was high and litter size increased during those years as experience increased, but that rarely, if ever, happens.

The reproductive values for our sample life table are: Age x lx mx app.Vx Vx 0 1.000 0 1.4 1.0 1 .800 0 1.75 1.426 2 .600 1 2.33 2.168 3 .400 2 2.00 1.999 4 .200 0 0 0 5 0 - - -

The reproductive values tell us why young age classes can be harvested more heavily (at least according to theory) – their reproductive values are low, and thus their relative contribution to population growth. The values also indicate why harvesting prime, reproductive animals must be so limited - they are the ones most critical to future population growth.

The Leslie Matrix

The Leslie matrix is a simpler way to organize the life tablecalculations we've done. The arithmetic is really the same, and what we're really doing is setting the numbers up in a form most easily computerized. The matrix (sometimes also called the 'population projection matrix') is simply a catalog of age-dependent fecundities and proportional survivorships.

The first row of the matrix is simply the fecundities F0 --> Fn. In each succeeding row (forming a sub-diagonal) the x+1th column is filled with px (i.e. column 1 for newborns, age 0 and age class 1, has p0 in column 1 of row 2).

Here’s what the matrix looks like in abstract terms…

F0 F1 F2 F3 F4 F5

p0 0 0 0 0 0

0 p1 0 0 0 0

0 0 p2 0 0 0

0 0 0 p3 0 0

0 0 0 0 p4 0

And for the sample life table:   0 0 .66 1 0

.8 0 0 0 0

0 .75 0 0 0

0 0 .66 0 0

0 0 0 .50 0

0 0 0 0 0

Now if we post-multiply this matrix by a matrix (really a column vector) which is the age-class structure at time t, we'll get a new column vector which is the age structure at time t+1. _ _ _

L x Nt = Nt+1

and _ _ _ _ _ _ _ _ Nt+2 = L x Nt+1 = L x (L x Nt) = L2Nt

Using the sample data, let’s project a simple age structure 1 time unit forward: L x Nt = Nt+1

0 0 .66 1 0 100 166

.8 0 0 0 0 100 80

0 .75 0 0 0 100 75

0 0 .66 0 0 100 66

0 0 0 .50 0 100 50

Everything we’ve done thus far has been in a world of exponential growth. How can we make the calculations and/or matrix projection reflect the density dependence we know exists in the real world?

Obviously, we need to find a way to have the mx and px respond to density, rather than remaining fixed.

To add density dependence to the dynamics we construct another matrix to represent density effects. While it is possible to separate density effects on fecundity and survivorship, that's unnecessarily complicated for our purposes.

The matrix we're constructing Qt, recognizing, in principle at

least, that there may be time dependence, as well as pure density dependence.

While the t will remain in the model structure, calculations typically end up assuming a time-invariant density- dependence.

We also recognize that density effects will predominantly be of 2 kinds: effects resulting from crowding (density) at birth (those effects are typically manifest in reduced size, survivorship and fecundity throughout life) and effects due to density at the present time (which may compound birth-density effects).

W e assume that effects are directly proportional to appropriate densities, and that the two sources of effect are additive.

To reduce fecundity and survivorship according to density we define a 'density-dependence‘ term q(x,t): 

q(x,t) = 1 + bNt-x-1 + aNt

b measures the effects of crowding at birth for each age class x, and a measures the effects of current density.

In this simplification of reality, a and b are not only time invariant, but also invariant with age.

The q varies with time and with density. We generally consider the time variation to result solely from density change.

The result is a matrix:Qt = |q(0,t) 0 0 ...|

| 0 q(1,t) 0 ...| | 0 0 q(2,t) ...| |... |

In practice we use the inverse of this matrix, so that each term in the multiplier is not q(x,t), but 1/q(x,t).Values of q(x,t) > 1then indicate density-dependent reductions in fecundity and survivorship. What we produce is a modified Leslie matrix L' for use in population projection.

L' = L x Q-1 = |F0/q(0,t) F1/q(1,t) F2/q(2,t) ...| |p0/q(0,t) 0 0 ...|

| 0 p1/q(1,t) 0 ...| | 0 0 p2/q(2,t) ...| | ... |

Population projection using L’ produces density-dependent, logistic population growth, first leading to a stable age distribution, then leading to the population reaching its carrying capacity, K.

K is related to population growth and density dependence in the following way: K = (- 1)/(a + b)

The Lefkovitch or Stage Projection Matrix

The Leslie matrix is the correct approach for animals, but not for many plants. For them size or stage is more important than age.

Think back to the diagrammatic model for teasel – what determined whether the plants flowered or not was the size of the rosette of basal leaves. And look back at the diagram … some plants did not advance to the next size class during one measurement interval (a year); they remained in the same class. That would not fit into a Leslie matrix model.

Instead, plants (and things like corals) are better described using a transition matrix approach.

Nt+1 = ANt

A is a Lefkovitch or transition matrix

It contains all stage- specific rates of transition and seed production

There are i columns and j (= i) rows for a population with i stages

A This year (t)

t+1

stage 1

stage 2

stage i

stage 1 a11 a12 a1i

stage 2 a21 a22 a2i

stage j aj1 aj2 aj3

If i = j the value indicates the fraction of organisms that remain in the same stage.

The elements of the Lefkovitch matrix are not proportional survivorships, but the probabilities of making transitions from one stage (or size) class to others.

We could use the teasel data to produce a Lefkovitch matrix, but we’ll use a much simpler case of Coryphanthum robbinsorum, an endangered cactus from the Sonoran desert.

This cactus has no seed bank, so that annual census would not detect seeds, and we can use a fecundity measure that incorporates not just seeds produced, but probability of seeds germinating and surviving to the time of the census. The result is this simplified life cycle graph, which is the equivalent of the diagrammatic life tables you saw earlier…

Since this model doesn’t include a seed stage, consider how we determine the number of small juveniles at the ‘next’ census:

n1(t+1) = p11n1 + n3(t)F

This formulation means that, unlike the Leslie matrix, the first row of this Lefkovitch matrix includes both the probability of remaining in the small juvenile class in row1,column1 and the fecundity of adults in row1,column3. Here’s the basic matrix projection equation:

n1(t+1) p11 0 F n1(t) n2(t+1) = p21 p22 0 x n2(t) n3(t+1) 0 p32 p33 n3(t)

Here are the values for the Lefkovitch matrix for one of the three sites collected by Schmalzel (1995):

p11 0 F 0.434 0 0.560 p21 p22 0 = 0.333 0.610 0 0 p32 p33 0 0.304 0.956

From this matrix you can calculate various important things about the population: the stable age distribution – in the figure below this matrix describes the population at site C, the only one where a fairly large proportion of small juveniles grow to large juveniles and mature to reproduce.

That difference in the proportion growing and maturing also produces a larger for site C (1.12, indicating a growing population, versus 0.998 and 0.997, indicating declining populations at the other sites).

is the dominant (largest) eigenvalue of the matrix. There are additional, smaller eigenvalues as solutions to the polynomial equation you saw in the matrix algebra section of the notes.

Even before the population has reached a SAD and a stable growth rate, the growth can be predicted from eigenvalues and eigenvectors (Caswell, 2001) according to the formula:

n t c xi it

ii

N

( )

1

where the c values are constants set by initial population structure.

For site C the eigenvalues and eigenvectors can be plotted over time. The plots show that the observed population growth is explained by the dominant eigenvalue (component 1) after only a very few years. The other components (eigenvalues) converge on 0 contribution…

Remember (from the matrix algebra section) that the left eigenvector gives proportions in each ‘age class’ in a SAD when age classes are appropriate, or here the stable relative stage abundances, and the right eigenvector gives reproductive values. The reproductive values for site C Coryphantha robbinsorum are:

small juveniles 1 large juveniles 2.06 adults 3.44

How sensitive are these results to small changes in survivorship or transition probabilities? The matrices can also be used to determine that. Without going into the means of calculation, sensitivity measures how much population growth changes for a unit change in each measure.

Sensitivity as a measure therefore has the problem that a unit change in fecundity is much different than a unit change in transition probability.

To deal with that problem, experts developed a better measure, called elasticity. Elasticities for a Lefkovitch (or a Leslie) matrix sum to 1. They therefore measure the relative contributions of each element of the matrix to the growth rate (), tested by unit change in each, all other elements held constant. As is evident in the figure, it is survivorship as adults that largely determines growth rate in this cactus at all sites…

Appendix – the Lefkovitch matrix treatment of teasel data

Let’s look at the matrix for teasel, using the same data as in the diagrammatic life table. Remember the classes into which the population was divided:

n1 - dormant seeds in year 1 n2 - dormant seeds in year 2

n3 - small rosettes n4 - medium rosettes n5 - large rosettes n6 - flowering plants

Here’s the projection matrix for that field of teasel:

0 0 0 0 0 322.38 .966 0 0 0 0 0 0.013 0.010 0.125 0 0 3.448 0.07 0 0.125 0.238 0 30.170 0.002 0 0.038 0.245 0.167 0.862 0 0 0 0.023 0.750 0 

And here’s what some of those numbers mean:

a) 0.966 of the seeds which were dormant in the first year remain dormant until at least the beginning of the second year. b) 0.013 of the seeds which were dormant through the winter germinate and become small rosettes in the first year.c) 0.007 of the seeds which were dormant through their first winter grow in the next year to become medium rosettes, and the next number, 0.008, represents the fraction which grew to become large rosettes in their first growth season.

d) Now look at the 3rd column. The two 0.125s represent the fraction which remain in the small rosette category and the fraction which advance to become medium rosettes. The 0.036 represents the fraction which grow so rapidly that they become large rosettes in their first year of growth. e) Similarly, the 4th column includes values for remaining in the medium rosette category, growing to become large rosettes, and the small fraction which reach the large category so early in the season that they proceed to flower.f) The last column may be initially confusing. If we wrote the matrix without correction, there would be only one non-zero value in it: the average number of seeds produced by a flowering plant, since teasel dies after flowering. That number would have been 431. However, correction recognizes that some seeds are not dormant in their first year. The fraction which enter dormancy is 0.749. Thus the proper value for seeds produced to enter the first category, seeds dormant in their first year, is 431 x 0.749 = 322.38.

f) (cont.) The other numbers reflect seeds which germinate and begin growth in their first year, rather than becoming dormant. The fraction germinating and becoming small rosettes is 0.008, which, multiplied by the total seed production (431), gives 3.448. The number which grow to medium rosettes is 30.170, and the number which grow all the way to large rosettes is 0.862. As you might expect of biennials, none flower in their first year.

If you do a standard matrix multiplication L x Nt with the populationvector being numbers in the various stages, the product is the numberin those stages at time t+1.