The Use of the Vec-Permutation Matrix in Spatial Matrix Population Models Christine M

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 formulation 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. back to the original organization of the population by (2003). In general, let A = Dm ···D1 be any matrix patches. product. The sensitivity matrix of A (i.e., the matrix 18 C.M. Hunter, H. Caswell / Ecological Modelling 188 (2005) 15–21 whose (i, j) entry is ∂λ/∂aij )is 4.1. The peregrine falcon

vwT Wootton and Bell (1992) presented a matrix model S = (12) A w, v for the peregrine falcon, Falco peregrinus anatum, in California. The model describes a population in λ To calculate the sensitivity of to the entries of one of two patches (northern and southern California) with = the matrices in the product, say Dk, write A FkDkGk two stages (juveniles and adults). This case is simple where enough that the projection matrix can be written down directly (as Wootton and Bell did), but shows clearly Dm ···Dk+1 k = m Fk = how the block-demography and block-dispersal matri- k = m I ces can be used in such analyses. Wootton and Bell arranged the population by Dk−1 ···D1 k = 1 Gk = (13) patches. They assumed dispersal happens before de- k = I 1 mography, so the survival of dispersing birds depends on their destination. Only juveniles disperse, and they Then the sensitivity and elasticity matrices for Dk assume that the probability of juvenile dispersal is d re- are gardless of direction. Thus, the dispersal matrices for ∂λ juveniles and adults are T T S = = Fk S Gk (14) Dk ∂d(k) A ij 1 − dd M = (18) 1 d − d (k) 1 dij ∂λ Dk E = = ◦ S (15) Dk λ (k) λ Dk = ∂dij M2 I (19) where ◦ denotes the Hadamard product (Caswell and Demography in patch i is described by Trevisan, 1994; Lesnoff et al., 2003). Table 1 shows 0 fi the sensitivity matrices resulting from applying (14) to Bi = (20) each of the four projection matrices. In each case, the pi qi elasticity matrices are given by where fi is fertility, pi is juvenile survival, and qi is 1 adult survival in patch i (1 = north, 2 = south). From EB = B ◦ SB (16) λ (9), the vec-permutation matrix is   1 EM = M ◦ SM (17) 1000 λ    0010 P(2, 2) =   (21) The sensitivities and elasticities relevant to demo-  0100 graphic and dispersal transitions appear in the s × s 0001 diagonal blocks of SB and EB, and the p × p diagonal blocks of SM and EM. This makes the interpretation of Constructing the block-diagonal matrices B and M and the results particularly clear. combining these with P gives the metapopulation projection matrix   4. Examples 0 f1 00    (1 − d)p q dp 0  = B TM =  1 1 1  In this section we present two examples of the A P P   000f2 use of the vec-permutation matrix in constructing p − d p q and analyzing spatial matrix models for metapopula- d 2 0(1 ) 2 2 tions. (22) C.M. Hunter, H. Caswell / Ecological Modelling 188 (2005) 15–21 19

Table 2 The off-diagonal blocks of SB and EB are of no in- Parameter values for the northern and southern peregrine falcon pop- terest because they show the results of perturbations ulations (from Wootton and Bell, 1992) to elements of B that are inherently zero. The elastic- Parameter Northern Southern ity matrix shows that λ is most elastic to changes in f 0.26 0.19 adult survival. The sums of the diagonal blocks of EB p 0.72 0.72 give the proportional contributions of the northern and q 0.77 0.77 southern populations to the metapopulation growth rate d 0.27 0.27 (0.68 and 0.32, respectively). The sensitivity and elasticity of λ to the dispersal Parameter values from Wootton and Bell (1992) are matrices are shown in Table 2. Only fertility differed between the two populations. The resulting metapopulation projec-   tion matrix is 0.1128 0.0662 ––     .  0.0662 0.0388 – –  002556 00 SM =   (27)    . .   0.5256 0.77 0.1944 0  ––05454 0 4287 A =   (23) . .  0000.1908  ––03200 0 2515 . . . 0 1944 0 0 5256 0 77   0.0873 0.0189 –– In the absence of dispersal, the northern population,  . .  which has higher fertility, would have a higher growth  0 0189 0 0301 – –  EM =   (28) λ = . λ =  .  rate ( B1 0 961) than the southern population ( B2 ––05781 0 . λ = 0 919). The growth rate of the metapopulation is A ––00.2666 0.943. Although all these growth rates are less than 1, the precision of the parameter estimates is unknown (and probably not high); and for the purposes of this where again the off-diagonal blocks are irrelevant. The example only the relative values of λ are relevant. columns of the M1 must sum to 1. Thus we are inter- ested only in perturbations in which a change in any From (12), the sensitivity matrix for λA is   entry is compensated by changes elsewhere in that col- 0.1062 0.392 0.0623 0.3082 umn. We do this by calculating the sensitivity to the    0.1567 0.5781 0.0919 0.4544  dispersal probability d, treating d as a lower-level pa- SA =   (24)  0.0835 0.3082 0.049 0.2422  rameter (Caswell, 2001). 0.0919 0.3392 0.0539 0.2666 ∂λ ∂λ ∂m(1) The sensitivity and elasticity matrices for the demogra- = ij ∂d (1) ∂d phy and dispersal components are given in Table 1(c). i,j ∂mij λ The sensitivity and elasticity of to the demographic ∂λ ∂λ ∂λ ∂λ matrices are = + − −   ∂m(1) ∂m(1) ∂m(1) ∂m(1) 0.0944 0.392 –– 12 21 11 22   =− .  0.1392 0.5781 – –  0 02 (29) SB =   (25)  ––0.0583 0.2422  m(1) i, j ––0.0642 0.2666 where ij is the element of M1. Thus, an increase   in d, which would increase dispersal from the high qual- 00.1062 –– ity patch to the low quality patch and vice-versa, would    0.1062 0.4719 – –  reduce λ. Numerical calculations show that ∂λ/∂d be- EB =   (26) comes more negative as the difference in fertility be-  ––00.049  tween the northern and southern populations increases, . . ––0049 0 2177 as intuition would suggest. 20 C.M. Hunter, H. Caswell / Ecological Modelling 188 (2005) 15–21

4.2. The black-headed gull with λ = 0.997. The non-zero blocks in A can be la- belled as Many formulations of spatial matrix models in the   F1 F2 F3 F4 F5 literature can be written using the vec-permutation ma-     trix. For example, Lebreton (1996) presents a spatial P1 0000 model for the black-headed gull (Larus ridibundus)   A =  0 P2 000 (34) in two patches, one of good quality and one of poor    00P 00 quality, in central France. Five age classes are distin- 3 guished, with demographic matrices and population 000P4 P5 growth rates   which replicates, in block form, the generalized Leslie 00.096 0.160 0.224 0.320 matrix format of B1 and B2.    .  The sensitivity and elasticity of λ to B and M are  0 800000   obtained from Table 1(d). As in the peregrine falcon ex- B1 =  00.82000 ,   ample, the elasticities of λ to B show that most (77%) of  000.82 0 0  λ is contributed by the high quality patch. The columns 00 00.82 0.82 of M1 must sum to 1 so we calculate the sensitivity of λ to p and q directly λ = 1.011 (30) ∂λ ∂λ ∂λ   = − =−0.043 (35) 00.100 0.160 0.200 0.200 ∂p ∂m(1) ∂m(1)   21 11  0.800000   ∂λ ∂λ ∂λ   = − = . B2 =  00.82000 , 0 0173 (36)   ∂q ∂m(1) ∂m(1)  000.82 0 0  12 22 00 00.82 0.82 This formulation has useful theoretical properties, because A is a block matrix version of the age-classiﬁed λ = 0.968 (31) Leslie matrix. Lebreton (1996), extending the earlier Only age class 1 disperses, so work of Le Bras (1971) and Rogers (1974), showed this form can be analyzed with a block-matrix generaliza- 0.75 0.375 1 − pq tion of the age-classiﬁed renewal equation, and applied M1 = = (32) 0.25 0.625 p 1 − q the approach to the evolution of dispersal (Lebreton et al., 2000; Khaladi et al., 2000). and M2 = M3 = M4 = M5 = I. The metapopulation projection matrix, given by Table 1(d),is   000.096 0 0.160 0 0.224 0 0.320 0    00 00.10000.16000.20000.200    . .  0 60 0 3000000000   0.20 0.5000000000    .   000820000000 A =   (33)  00 00.82000000      00 0 00.8200000    00 0 0 00.820000    0000000.82 0 0.82 0  00000000.82 0 0.82 C.M. Hunter, H. Caswell / Ecological Modelling 188 (2005) 15–21 21

5. Discussion Bravo de la Parra, R., Auger, P., Sanchez, E., 1997. Time scales in density dependent discrete models. J. Biol. Syst. 5, 111–129. The vec-permutation matrix provides a framework Caswell, H., 2001. Matrix Population Models: Construction, Anal- ysis and Interpretation, 2nd ed. Sinauer Associates, Sunderland, for constructing spatial matrix models that incorporate Massachusetts. demography and dispersal. Because it yields a matrix Caswell, H., Trevisan, M.C., 1994. The sensitivity analysis of peri- product in which dispersal and demography appear as odic matrix models. Ecology 75, 1299–1303. block-diagonal matrices, it makes perturbation analysis Hanski, I., 1999. Metapopulation Ecology. Oxford University Press. particularly simple, permitting straightforward calcu- Hastings, A., 1992. Age dependent dispersal is not a simple process: λ density dependence, stability, and chaos. Theor. Popul. Biol. 41, lation of the sensitivity and elasticity of to changes 388–400. in demography and dispersal at any point in the annual Henderson, H.V., Searle, S.R., 1981. The vec-permutation, the vec cycle and for any stage. operator and Kronecker products: a review. Linear Multilinear Although the cases we have examined here include Algebra 9, 271–288. only a single episode each of dispersal and demogra- Horn, R.A., Johnson, C.R., 1990. Matrix Analysis. Cambridge Uni- versity Press, Cambridge, Massachusetts. phy, it also applies to situations that include seasonal- Khaladi, M., Grosbois, V., Lebreton, J.-D., 2000. An explicit ity of demography or dispersal, multiple episodes of approach to evolutionarily stable dispersal strategies with a dispersal, and dispersal of different stages at different cost of dispersal. Nonlinear Anal.: Real World Appl. 1, 137– times of year. Any number and any order of demogra- 144. phy and dispersal events, connected by the appropriate Le Bras, H., 1971. Equilibre et croissance des populations soumises a des migrations. Theor. Popul. Biol. 2, 100–121. vec-permutation and reverse permutation matrices, can Lebreton, J.-D., 1996. Demographic models for subdivided popu- be included in the projection matrix A. This construc- lations: the renewal equation approach. Theor. Popul. Biol. 49, tion is particularly apt for species in which dispersal 291–313. occurs at particular seasons within the year, but is not Lebreton, J.-D., Gonzalez-Davila, G., 1993. An introduction to mod- restricted to this case. Nor is it restricted to popula- els of subdivided populations. J. Biol. Syst. 1, 389–423. Lebreton, J.-D., Khaladi, M., Grosbois, V., 2000. An explicit ap- tions classiﬁed by stage and location; it can be applied proach to evolutionarily stable dispersal strategies: no cost of to populations stratiﬁed by other pairs of factors, such dispersal. Math. Biosci. 165, 163–176. as age and size. In a subsequent paper we will exam- Lesnoff, M., Ezanno, P., Caswell, H., 2003. Sensitivity analysis in ine such cases that require more complex structures to periodic matrix models: a postscript to Caswell and Trevisan. incorporate the biology of dispersal. Appl. Math. Lett.: Math. Comput. Model. 37, 945–948. Logofet, D.O., 2002. Three sources and three constituents of the formalism for a population with discrete age and stage structures. Matematicheskoe Modelirovanie (Math. Model.) 14, 11–22 (in Acknowledgements Russian, with English summary). Rogers, A., 1968. Matrix Analysis of International Population We thank Jean-Dominique Lebreton and Brian Growth and Distribution. University of California Press, Berke- McArdle for discussion, two reviewers for comments ley, California. on an earlier version of the paper, and the Statistics De- Rogers, A., 1974. The multiregional net maternity function and multiregional stable growth. Demography 11, 473–481. partment of the University of Auckland for hospitality. Rogers, A., 1995. Multiregional Demography: Principles, Methods Financial support provided by EPA Grant R-82908901, and Extensions. Wiley, New York, New York. a grant from the David and Lucile Packard Foundation, Sanz, L., Bravo de la Parra, R., 1999. Variables aggregation in a time and a Maclaurin Fellowship from the New Zealand discrete linear model. Math. Biosci. 157, 111–146. Institute of Mathematics and its Applications to HC. Sanz, L., Bravo de la Parra, R., 2000. Time scales in stochastic multiregional models. Nonlinear Anal.: Real World Appl. 1, 89– Woods Hole Oceanographic Institution Contribution 122. 11200. Wootton, J.T., Bell, D.A., 1992. A metapopulation model of the peregrine falcon in California: viability and management strategies. Ecol. Appl. 2, 307–321. References

Bravo de la Parra, R., Auger, P., Sanchez, E., 1995. Aggregation methods in discrete models. J. Biol. Syst. 3, 603–612.