Ecological Modelling 251 (2013) 307–311 Contents lists available at SciVerse ScienceDirect Ecological Modelling jo ur n al homep ag e: www.elsevier.com/locate/ecolmodel Short communication Projection matrices in variable environments:  1 in theory and practice Dmitrii O. Logofet ∗ Laboratory of Mathematical Ecology, A.M. Obukhov Institute of Atmospheric Physics, RussianAcademy of Sciences, Moscow, Russia a r t i c l e i n f o Article history: Received 15 September 2012 Received in revised form 20 December 2012 Accepted 21 December 2012 Available online 21 February 2013 Keywords: Life cycle graph Strong components Reproductive submatrix False growth rate Carapa guianensis Stochastic growth rate a b s t r a c t Perron–Frobenius theorem for nonnegative matrices, a mathematical foundation of matrix population models, applies when the projection matrix is not decomposable (or equivalently, when it is irreducible), the application yielding the dominant eigenvalue  1 > 0 as a measure of the growth potential that a pop- ulation with given demography possesses in a given environment. In practice, however, the projection matrix often appears to be decomposable (reducible); to calculate  1 in this case, a principal submatrix should rather be used that corresponds to the reproductive core of the life cycle graph. I call it the repro- ductive submatrix and demonstrate that, when the reproductive submatrix does not coincide with the projection matrix and if this discrepancy is neglected in a case study, the resulting  1 may happen to be overestimated. Averaging over a number of annual projection matrices eliminates the false growth rate but raises the problem of choice among the modes of averaging in the estimation of the stochastic growth rate in a stochastic environment. Computer simulation gives a method that avoids the both kinds of problem. © 2013 Elsevier B.V. All rights reserved. 1. Introduction The standard form of a discrete-structured population model is given by a vector-matrix equation x(t + 1) = Lx(t ), t = 0, 1, . . . , (1) where x(t) is an nD vector of the population structure at time moment t and an n × n matrix L is called the projection matrix (Caswell, 1989, 2001). The pattern of nonzero elements allocation in matrix L can be represented as the associated directed graph (Harary et al., 1965; Horn and Johnson, 1990), which coincides with the life cycle graph (LCG, Logofet and Belova, 2008) for individuals in the population. The graph summarizes the knowledge of species biol- ogy in the terms of status groups specified by the components of vector x(t). It shows all the transitions among status groups that are possible for one time step; it also indicates which of the status groups are reproductive, i.e., produce the population recruitment, and in which status groups the recruits may appear at the time t + 1. Given a particular LCG, it defines, up to node enumeration, the pattern of projection matrix L. Once calibrated form data, matrix L determines the dynamics of vector x(t) from its initial state x(0) in accordance with Eq. (1) under the assumption that matrix L remains invariable in time. It also determines  1 (L) > 0, the dominant eigen- value of matrix L, which is recognized as a quantitative measure of ∗ Tel.: +7 916 6286229; fax: +7 495 953 1652. E-mail address: danilal@postman.ru the population growth potential – the analogue to the scalar intrin- sic growth rate for multi-dimensional population growth governed by Eq. (1) (Caswell, 1989, 2001; Logofet, 1993; Logofet and Belova, 2008). Fig. 1 shows an example of the LCG for a size-structured popu- lation of Carapa guianensis, a tropical tree species that is harvested for timber and non-timber products (Klimas et al., 2012). Five size groups produce offspring, which may appear in two groups, ‘Seedlings 1 ’ and ‘Seedlings 2 ’. However, in some habitats and years of observation, none of the final-class trees does reproduce or none of ‘Seedlings 2 ’ does survive. It means that the corresponding arrows (dashed arrows in Fig. 1) disappear from the graph. Does this affect the general algorithm for calculating  1 (L)? In this communication, I demonstrate that it may do so, indeed, the outcome overestimat- ing the true  1 (L). If a population was censured during a number, T + 1, of succes- sive years and the data resulted in T annual projection matrices, L(0), . . ., L(T - 1), which varied from year to year, then Eq. (1) yields to x(T ) = L(T - 1)L(T - 2) . . . L(0)x(0), (1T) where x(0) is the initial population vector. In C. guianensis case study, T = 4 (ibidem), and the authors believed that the four annual projection matrices scoped the range of temporal random varia- tions in the environment. To estimate the growth potential over the entire ensemble of data, a concept of stochastic growth rate,  S (Tuljapurkar, 1986, 1990; Caswell, 2001), was applied. If, in practice, the estimate of  S makes use of each year-specific  1 , then the contribution from an overestimated  1 (L) biases the estimation 0304-3800/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ecolmodel.2012.12.028