Mathematical constraints on transition matrix elasticity analysis

It is commonly assumed in matrix models for population dynamics that demographic parameters influence the growth rate (λ). Sensitivity and elasticity analyses are among the general methods for evaluating this influence (de Kroon et al. 1986; de Kroon & van Groenendael 1987; Caswell 1989). There are, however, mathematical consequences that have to be taken into account to avoid misinterpretations, and here we consider those that affect a specific (but important) class of matrix models. Recently a number of works have been published that extend the use of elasticities. For instance, Silvertown et al. (1993) proposed the use of correlation matrices between r (=ln λ) and elasticities as a way of determining trade-offs among life-cycle components. However, Shea et al. (1994) pointed out a number of limitations on this conjecture and formulated artificial models for it where the conjecture fails. van Groenendael et al. (1994) also used elasticities as a way of analysing life histories, but they calculated elasticities for each life-history pathway rather than for each matrix element. Most recently, van Tienderen (1995) introduced ‘integrated elasticities’, in an attempt to incorporate the direct and indirect contributions of a given coefficient to λ. According the Silvertown et al. (1993), the elasticity matrix can be used to ordinate plants into a triangular space representing the contribution of growth (G), survival (L) and fecundity (F) to their finite rate of increase. The elasticity matrix is divided into six regions representing the following transitions: E1, recruitment of seeds to the seed pool; E2, recruitment of seedlings or juveniles from the current seed production; E3, clonal growth; E4, retrogression; E5, stasis in the same state; and E6, growth to the next state. Elasticities for each life-history process were summed into G (E3+E6), L (E4+E5) and F (E1+E2). Although there are no significant correlations between this ordination and the C-S-R classification proposed by Grime (1977), Silvertown et al. (1993) found similar successional trajectories in the G-L-F space to those observed in the C-S-R space. However, it is necessary to be cautious when using elasticity analysis. Enright et al. (1995) showed that the dimension of the transition matrix can be an important cause of variation in the elasticities for G, L and F. In this paper we study a class of matrices where non-zero values are seen for the E1 (or alternatively E2), E5 and E6 transitions. Such a pattern is common for trees and other perennial plants and applies to all age classified models. Elasticities values and the demographic triangle The matrices studied here are of the form: If there is more than one reproductive stage L, G and F are still related, but in a more complex manner (see Appendix). In this case the accessible portion of the G-L-F triangle will be restricted to the region bounded by two lines of the type shown in Fig. 1 (eqns 5 and 8 of Appendix). The demographic triangle proposed by Silvertown et al. (1993) with the lines described by eqn 5 of Appendix with n ranging from 2 to 8. The lines tend to the G-L side with increasing n. Note also that the area between two successive lines decreases with n. The results demonstrate that the mathematical properties of matrix models have important consequences for their use in biological systems. Some direct consequences of these results are as follows: 1. For any model of the form we describe, the elasticities of Gi (growth) decrease with i, implying that, in general, this is not related to the biology of the species. No matter what are the values of Gi we will find that earlier stages contribute more to λ than later ones simply due to the fact that we are using this matrix model. Additionally, for the non-reproductive stages all the elasticities are identical. One consequence is that it will be impossible to measure a trade-off between Gis of non-reproductive stages using correlations between its elasticities. 2. The elasticities of Fi are bound to be smaller than the greatest elasticity of Gi, so they will always contribute less to λ. It is then better to use sensitivities to compare the influence of changes in demographic parameters on λ due to its unmistakable interpretation. 3. The distinct dependence of the accessible region of the triangle on the dimension of the matrix also results in difficulties in correlating elasticities when using data from matrices with different dimensions. Species whose data were obtained using a large number of classes will tend to be positioned towards the G-L side of the demographic triangle while species with a low number of classes will tend to be at the centre of the triangle. In the special case of a single reproductive stage this dependence is more explicit, with the position of the species jumping from one line to another as the number of stages varies. The matrix with a single reproductive stage implies a negative correlation between G and L and a positive one between F and G. These correlations became less strong for other matrices but are none the less possible, due to the form of eqns 6 and 7 of the Appendix. It is therefore impossible or very unlikely that one can use such matrices to measure biological trade-offs between demographic parameters using correlations between elasticities. The ordination of populations into G-L-F space (Silvertown, Franco & McConway 1992; Silvertown & Franco 1993; Silvertown et al. 1993), in the sense of determining the region of the triangle it could access, depends on the type of matrix model used in the study. Consequently, the comparison of demography and life-cycle of woody and herbaceous perennial plants proposed by Silvertown et al. (1993) has to be undertaken with caution. Biological traits could determine the ordination of the plant species in the triangle only within the region between the two predetermined lines. Indeed, Silva Matos (1995) showed that although the position of the tropical palm Euterpe edulis (Mart.) in the triangle varies with the population density, it is always constrained by such lines. It is also important to notice that no matrices of the class studied here can access the region of the triangle below the n = 2 line (F > G, Fig. 1). This result is in accordance with the observation that no published transition matrices for plant species have high elasticities for both F and L relative to G (Enright et al. 1995). The dependence of the position of a plant species in the G-L-F triangle on the dimension of the transition matrix (n) can be qualitatively studied for this class of matrices. The effect of n can be separated in two independent contributions: one associated with changes of the accessible region in the triangle and another due to changes in relative importance of L and G. This latter type of contribution is a consequence of the fact that the mean time spent by the individuals in each stage will decrease with increasing n, so the value of G will necessarily be increased in relation to L. As a result the species will be displaced to a position further from the L vertex of the triangle. With respect to the accessible region, the effects depend on the relation between the total number of stages n and the number of reproductive stages ( f ). When an increase in n is due solely to an increase in f the accessible region is enlarged towards the G-L line. However the effect, particularly when n is large, is small and any change in the position will be mainly determined by biological features of the plant species. When n is increased keeping f fixed, the whole of accessible region is shifted closer to the G-L line and simultaneously shrinks. The result will be a displacement of the position of the plant species towards the G-L line. It is important to stress that these changes of position in the triangle occur without biological changes, and it is thus possible to move a species simply by changing the way we choose the stages. Similar constraints on elasticities are expected to hold for other classes of matrix models, although they are harder to obtain. It has to be noted that the choice of the form of the matrix model to be used in a specific study is always biased by the type of plant life history. It is very unlikely that someone would chose a matrix like the ones studied here to describe a short-lived or a clonal plant. The second author thanks CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico) for the financial support. Here we will make use of the property 1 deduced in the appendix of van Groenendael et al. (1994): We use the following convention for the elasticities: e(Gi) = ei+1,i; e(Pi) = eii; e(Fi) = e1i For the type of matrix studied here, the following relations hold e(Gi-1) = e(Gi)+e(Fi) n > i > n-f e(Gi-1) = e(Gi) n-f+1 > i > 1 e(Gn-1) = e(Fn) These results also have some consequences for the demographic triangle introduced by Silvertown et al. (1993). When f = 1 the e(Gi) and e(Fn) (eqns 2 and 3) are all equal to F. Consequently F and L are related by: L = 1-nF So, the value of F is all we need to specify the point in the triangle. For f > 1 there are (n−f ) values of Gi with equal elasticities (eqn 2) so the following relations are obtained: From eqns 1 and 4 we see that each e(Gi) for reproductive stages is bound to be lower than F and greater than 0 (from its definition). If all these elasticities are equal to F then we get eqn 5. On the other extreme, if all these elasticities are equal to zero we get: L = 1-(n-f+1)F Which is the same as eqn 5 with n replaced by (n−f+1). So these matrices lie between two lines in the demographic triangle: the lines corresponding to n and (n−f+1). Received 2 September 1997revision accepted 23 March 1998