Ecological Modelling 188 (2005) 15–21 The use of the vec-permutation matrix in spatial matrix population models Christine M. Hunter∗, Hal Caswell Biology Department MS-34, Woods Hole Oceanographic Institution, Woods Hole, MA 02543, USA Abstract Matrix models for a metapopulation can be formulated in two ways: in terms of the stage distribution within each spatial patch, or in terms of the spatial distribution within each stage. In either case, the entries of the projection matrix combine demographic and dispersal information in potentially complicated ways. We show how to construct such models from a simple block-diagonal formulation of the demographic and dispersal processes, using a special permutation matrix called the vec-permutation matrix. This formulation makes it easy to calculate and interpret the sensitivity and elasticity of λ to changes in stage- and patch-specific demographic and dispersal parameters. © 2005 Christine M. Hunter and Hal Caswell. Published by Elsevier B.V. All rights reserved. Keywords: Matrix population model; Spatial model; Metapopulation; Multiregional model; Dispersal; Movement; Sensitivity; Elasticity 1. Introduction If each local population is described by a stage- classified matrix population model (see Caswell, A spatial population model describes a finite set of 2001), a spatial matrix model describing the metapop- discrete local populations coupled by the movement ulation can be written as or dispersal of individuals. Human demographers call these multiregional populations (Rogers, 1968, 1995), n(t + 1) = An(t) (1) ecologists call them multisite populations (Lebreton, 1996) or metapopulations (Hanski, 1999). The dynam- The population vector, n, includes the densities of ics of metapopulations are determined by the patterns each stage in each local population (which we will of dispersal among the local populations and the de- refer to here as a patch). The projection matrix, A, mographic conditions experienced by the local popu- includes both demographic processes, (which gener- lations. ally differ among patches) and dispersal processes (which generally differ among stages). The asymp- λ ∗ Corresponding author. Tel.: +1 508 289 3245; totic population growth rate, , is the dominant fax: +1 508 457 2134. eigenvalue of A, and the stable stage × patch dis- E-mail address: [email protected] (C.M. Hunter). tribution and the reproductive values are given by 0304-3800/$ – see front matter © 2005 Christine M. Hunter and Hal Caswell. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.ecolmodel.2005.05.002 16 C.M. Hunter, H. Caswell / Ecological Modelling 188 (2005) 15–21 the corresponding right and left eigenvectors w and ond arrangement (nstages) the subvectors give the spa- v. tial distribution of each stage. Note that in general it is The elements of A are combinations of demographic not possible to write a model in which the matrix N(t) rates and dispersal probabilities. It can be challenging replaces the vector n(t)inEq.(1) and is projected by to formulate these elements, and difficult to analyze multiplication by a matrix (Logofet, 2002). the role of particular parameters. This can be espe- To model demography without dispersal, it would cially problematic for perturbation analyses. The sen- be convenient to organize the population by patches. sitivity and elasticity of λ to the elements of A are Let Bi be the s × s demographic projection matrix for of little interest, because a change in aij has no sim- patch i, such that n·i(t + 1) = Bin·i(t). Then, without ple biological interpretation. Instead, attention focuses dispersal, the metapopulation would be projected by a on the effects of changes in demography and dispersal block diagonal matrix B separately. Here we describe a method for constructing spa- B1 0 ··· 0 tial matrix population models that keeps demographic n· n· 1 ··· 1 and dispersal parameters clearly distinguished and sep- 0 B2 0 . t + = . t arates their effects in perturbation analysis. It permits . ( 1) . . ( ) .. the inclusion of complicated seasonal patterns of dis- n·p n·p 00··· Bp persal. We describe model construction and the formu- lation of sensitivity and elasticity analyses, and apply B the method to two simple examples. (4) To model dispersal without demography, it would be 2. Spatial model construction more convenient to organize the population by stages. Let Mh be the p × p dispersal matrix for stage h,so Constructing a spatial matrix model requires speci- (h) that m is the probability that an individual of stage fying the state of the metapopulation, the demographic ij h moves from patch j to patch i. Then dispersal would characteristics of each patch, and the dispersal of indi- be described by a block diagonal matrix M viduals among the patches. To describe the state of the metapopulation let p be the number of patches and s be ··· T M1 0 0 T the number of stages. The state of the metapopulation n1· n1· 0 M2 ··· 0 can then be described by the matrix . t + = . t . ( 1) . . ( ) .. n11 n12 ··· n1p T T ns· ns· n n ··· n 00··· Ms 21 22 2p t = t N( ) . ( ) (2) M . .. (5) ns1 ns2 ··· nsp Constructing a spatial model that incorporates both de- where nij (t) is the density of stage i in patch j at time t. The population vector n in (1) can be written in two mography and dispersal has previously required sacri- ficing one or both of these block-diagonal forms. Our ways. Let ni· =rowi and n·j = column j of N, then goal is a method of model construction that maintains T them both. n·1 n1· As many other authors have done, we assume that = . = . npatches . or nstages . (3) demography and dispersal can be treated as operat- ing sequentially within the projection interval (e.g., n·p nT s· Hastings, 1992; Bravo de la Parra et al., 1995, 1997; In the first arrangement (npatches), the subvectors give Sanz and Bravo de la Parra, 1999, 2000; Lebreton and the stage distribution within each patch. In the sec- Gonzalez-Davila, 1993), although either process may C.M. Hunter, H. Caswell / Ecological Modelling 188 (2005) 15–21 17 be seasonal and they may occur in any order. Then, Table 1 constructing a model that maintains the block diagonal The projection matrix A and the sensitivity matrices SB and SM as forms of the demography and dispersal matrices in (4) a function of the arrangement of the population vector (by patches or by stages) and the order of demography and dispersal within the and (5) requires converting a population vector orga- projection interval nized by patches into a population vector organized by Population vector arrangement stages and vice versa. This conversion is accomplished By patches By stages by noting that Demography then dispersal T T = (a) A = P MPB (b) A = M P B P npatches vec(N) (6) T T T T SB = P M PSA SB = P M SA P = BT T = BT T T SM PSA P SM SAP P nstages = vec(N ) (7) Dispersal then demography where the vec operator, vec(·), stacks the columns of (c) A = B PT M P (d) A = P B PT M = T MT = T MT a matrix one on top of the other (this is implemented SB SA P PSB P SA P SM = P BT S PT SM = P BT PT S with the command N(:) in MATLAB). The vectors A A (6) and (7) are related by a special permutation matrix, In all cases, the elasticity matrices satisfy EB = B ◦ SB/λ and EM = M ◦ /λ P, called the vec-permutation matrix; i.e., SM . vec(NT) = P vec(N) (8) If dispersal is followed by demography and we ar- range n by stages, we would have Henderson and Searle (1981) reviewed the properties and derivation of the vec-permutation matrix. It has T T n1· n1· dimension (sp × sp) and is given by . T . . (t + 1) = P B P M . (t) (11) s p T T T P(s, p) = Eij ⊗ Eij (9) ns· ns· i=1 j=1 In this case, the projection matrix is A = P B PT M. s × p i, j where Eij is an matrix witha1inthe( ) The four possible combinations of arrangements of ⊗ position and zeros elsewhere and denotes the Kro- n and sequences of a single demography and a sin- necker matrix product. As with any permutation matrix, gle dispersal event lead to the four projection matrices T = −1 P P . shown in Table 1. Each of these projection matrices can Suppose that, within each projection interval, de- be obtained from any of the others by a cyclic permu- mographic change occurs within each patch and then tation of the matrices making up the product. Hence dispersal redistributes individuals among patches. If we their eigenvalue spectra are identical (Horn and John- organize n by patches we can use the vec-permutation son, 1990, Theorem 1.3.20; see Caswell, 2001 p. 350), matrix to write λ as are the sensitivities and elasticities of to the entries B M n·1 n·1 of and obtained from each. . t + = T M B . t . ( 1) P P . ( ) (10) n·p n·p 3. Sensitivity analysis The projection matrix A in (1) is then A = PT M P B. Eqs. (10) and (11) describe population growth in a In this case, A first applies the block-demography periodic environment that alternates between episodes matrix B to the population vector organized by of demography and dispersal. The sensitivity and elas- patches, then permutes the result to reorganize the ticity of λ to changes in demography or dispersal at any population vector by stages, applies the block- point in the projection cycle can be calculated using re- dispersal matrix M, and then permutes the result sults of Caswell and Trevisan (1994) and Lesnoff et al.