1. Matrix models

The matrix derived from the transition table in the previous chapter contains more information than just a list of vital rates. In addition to their descriptive use, in order to compare populations for instance, they form the components of a mathematical model that allows us to predict the fate of the population. This is done by matrix "projection", or iterative matrix multiplication, which gives us the population growth rate l, and the stable stage distribution, w.

Because matrices are really sets of linear equations, the mathematical procedure consists of two actions, based on two properties of transitions:

a) multiplication - a portion a_ij of all n_j individuals in stage j at time t make it to stage i at time t+1, or a_ij n_j = n_ij; and

b) addition - the n_i individuals in stage i at time t+1 are composed of individuals from all stages j at time t leading to stage j at time t+1, or n_i = S n_ij = S a_ij n_i for all i.

If this procedure is done once, from census 1 to census 2, we are in fact reversing the derivation of the vital rates. Multiplying the observed nj at time t=0 with the calculated vital rates aij, and adding the n_ij's up for each i, we get the list of the totals per stage observed in census 2, at time t+1=1. In the terms used in matrix (or linear) algebra, such a list is a vector. Thus, we multiplied the initial vector n(0)={n_i (0)} with the matrix A to obtain a new vector n(1)={n_i (1)}=An(0).

2. Matrix projection

Repeating the multiplication many times reveals many new properties of the population being modeled. This is done by multiplying n(1) by matrix A, which yields a new vector n(2) at time t+2=3: n(2)=A n(1)=A² n(0). The vectors n(t)=A^tn(0) describe the distribution of numbers of individuals n_i(t) in stages i at time t. In the beginning of the iterative matrix multiplications, the elements of n(t) usually fluctuate - sometimes quite drastically. This is due to the initial stage distribution n(0). In other words:

The predicted population growth rate, which is one of the reasons we run this mathematical model, is independent of n(0): l= N(t+1)/N(t)= N(t+2)/N(t+1)=N(t+3)/N(t+2) etc. This limiting behavior usually sets in after less than 20 iterations. When l is constant, N(t) still may increase or decrease, but at a constant rate, given by l>1, l-1 or l<1 respectively. If l=1, population density stays constant. l is also known in the form r=lnl, the intrinsic rate of population increase, sometimes called the per capita reproductive rate.

In contrast to transient behavior, during limiting behavior of the population in the model, the vectors n(t) are changing at the same rate, which is equal to l. The condition n(t+1)=ln(t) has two implications:

a) all stages i increase at the same rate l as N(t): n_i(t+1)=l n_i(t); and

b) the proportions of the n_i's are constant, forming a new vector w={w_i}, which represents the stable stage distribution of the population. This quantity is a useful descriptor of the dynamics of a population, in order to compare populations and for testing hypotheses about variation of the environment.

Applying mathematical matrix theory to the behavior leads to the finding that a population, which is subject to a single matrix A, eventually converges to a stable form. This "rule" is called the strong ergodic theorem (Caswell 1989), where ergodicity means independence of initial conditions. The weak ergodic theorem states that if two populations are subject to equal matrices, their stage distributions become equal.

3. Assumptions

Population growth rate l and stable stage distribution w are results of applying matrix projection to matrix A. (They are the dominant eigenvalue l1 and the right eigenvector w of A.) Because the same matrix A is used for many time intervals, these values only correspond to real (future) population dynamics if matrix A does not change in time. This assumption of time-invariance is necessary in order to derive l and w, and is only true if the environment does not change at the year-to-year scale. This does allow seasonal changes, as long as they have the same effect on the vital rates aij of A.

There is one more assumption implied in the repeated use of the same matrix A. The assumption results from the constancy of the vital rates: density N(t) of the population as a whole, and the densities of its stages n(t), do not cause changes in the vital rates. In this case densities of other individuals do not constitute controlling factors in the individuals' environment. In other words, the model assumes that population dynamics are density-independent. This is not such an improbable condition. There is no reason to assume density dependence a priori; it has to be shown by testing against the null-hypothesis of density independence.

In cases where the environment varies, or where density dependence is shown, population dynamics can also be modeled with transition matrices, but differently. In these time-variant models, more than one single matrix A is used, as we will see later.

4. Use of the model

If the predicted population parameters are only theoretical constructs, and are not forecasts of what the population under study is exactly going to do, what is their use? First, in many cases demographic variations in time may indeed be close to time invariance, at least for a while. This may be the case for successional stages between more drastic changes, and may affect many plants and animals living there. Second, in cases that the assumption does not hold, it is informative to know what effect the environment during a particular time period has on population dynamics. There may be merely a tendency toward increase, decrease or constancy, which is replaced the next period by another set of environmental influences resulting in different population dynamics.

Furthermore, the vector w can be used to test whether the found stage distributions n(0) and n(1) are stable already, given the matrix A representing the transition from census 1 to 2. If p(1) is a vector with proportions of stages p_i=n_i/N(1), then rejecting the null hypothesis p(1)=w means that the population has not reached its limiting behavior; this is either a) because matrix A is valid, but dynamics are still transient, or b) because matrix A is only valid for this particular transition, as part of annual environmental variation (including density dependence). The two scenarios (a) and (b) can be distinguished by longer-term monitoring of the population. Not rejecting the hypothesis means that matrix A has been already valid for a while, and that the population has reached a stable state. This implies that the environment has been more or less constant for a at least the time period of transient behavior observed in the model.

5. Elasticity

One of the most useful features of transition matrix models (which they have in common with other dynamic mathematical models) is the possibility to measure the effect that a change in each one of the vital rates aij has on population growth rate l. This is referred to as the elasticity e_ij = (a_ij dl)/(lda_ij) of the vital rate a_ij, or the proportional change in l due to a change in a_ij (Crouse et al. 1987, Boeken and Canham 1995). If one vital rate has greater elasticity than another, a change in the first vital rate, as a proportion of its value, has a greater effect on l than an equivalent change in the second vital rate. Elasticity is derived from what is commonly known as sensitivity, expressing the effects of absolute changes in the values of model components (in our case a_ij's) rather than proportional ones on the outcome of the model (l).

The elasticity of a vital rate depends on aij and l, but also on the proportion of the population that is in stage j under the stable stage distribution, and on the contribution of stage i to l. This is v_i, the reproductive value of stage i, which is one of the terms of v, the left eigenvector of A (the counterpart of the right eigenvector w). The vector v is obtained by doing the same matrix multiplication procedure as for w, but with matrix A^-1, the transpose of A. (transposing is, in graphic terms, rotating the elements of A around an axis formed by the diagonal from top left to bottom right.) In models used for dynamic life tables, reproductive values V_x are calculated and interpreted differently: as the average contribution in numbers of offspring that an individual aged x will make to the next generation before it dies (Fisher 1930).

Elasticities are derived as eij = (a_ij/l) {(v_i w_j)/<v,w>}, where <v,w> is a scalar obtained by multiplication of the two vectors v and w: <v,w> = S v_k w_k, for k=1 to the number of stages (or <v,w> = v₁w₁+ v₂w₂+ v₃w₃+ ...). Note that the equation of e_ij includes the portion of the stable stage distribution of stage j at time t, but the future contribution of stage i at time t+1.

Knowing elasticities of a population is of practical value for management of populations (as conservation or, alternatively, suppression of noxious or undesirable animals or plants).

References

* Boeken, B. and C. D. Canham (1995). Biotic and abiotic control of the dynamics of gray dogwood (Cornus racemosa Lam.) shrub thickets. Journal of Ecology 83: 569-580.

Caswell, H. 1989. Matrix population models. Sinauer, Sunderland, USA.

* Caswell, H., and P. A. Werner. 1978. Transient behavior and life history analysis of teasel (Dipsacus sylvestris Huds.). Ecology 59: 53-66.

* Crouse, D. T., L. B. Crowder, and H. Caswell. 1987. A stage-based population model for loggerhead sea turtles and implications for conservation. Ecology 68: 1412-1423.

Fisher, R. A. 1930. The genetical theory of natural selection. Oxford Univ. Press.