Theor Ecol (2018) 11:129–140

ORIGINAL PAPER

Demography when history matters: constructionand analysis of second-order matrix population models

Charlotte de Vries1 ·Hal Caswell1

Received: 28 July 2017 / Accepted: 24 October 2017 / Published online: 8 December 2017© The Author(s) 2017. This article is an open access publication

Abstract History matters when individual prior conditionscontain important information about the fate of individ-uals. We present a general framework for demographicmodels which incorporates the effects of history on popula-tion dynamics. The framework incorporates prior conditioninto the i-state variable and includes an algorithm for con-structing the population projection matrix from informationon current state dynamics as a function of prior condi-tion. Three biologically motivated classes of prior conditionare included: prior stages, linear functions of current andprior stages, and equivalence classes of prior stages. Tak-ing advantage of the matrix formulation of the model, weshow how to calculate sensitivity and elasticity of any demo-graphic outcome. Prior condition effects are a source ofinter-individual variation in vital rates, i.e., individual het-erogeneity. As an example, we construct and analyze asecond-order model of Lathyrus vernus, a long-lived herb.We present population growth rate, the stable populationdistribution, the reproductive value vector, and the elastic-ity of λ to changes in the second-order transition rates. Wequantify the contribution of prior conditions to the total het-erogeneity in the stable population of Lathyrus using theentropy of the stable distribution.

Electronic supplementary material The online version of thisarticle (https://doi.org/10.1007/s12080-017-0353-0) contains sup-plementary material, which is available to authorized users.

� Charlotte de Vriesc.devries@uva.nl

Hal Caswellh.caswell@uva.nl

1 Institute for Biodiversity and Ecosystem Dynamics,University of Amsterdam, Amsterdam, The Netherlands

Keywords Second-order matrix population model ·Individual heterogeneity · Prior stage dependence ·Carry over effects · Sensitivity analysis

Introduction

Every demographic analysis requires a classification of indi-viduals by age, size, developmental stage, physiologicalcondition, or some other variable. These variables describeindividual states (i-states) such that the fate of an individualdepends only on its current state and the environment (Metz1977; Caswell and John 1992; Metz and Diekmann 1986).This requires the state variable to capture all the aspects ofthe individual’s history that are relevant to its future fate(Caswell et al. 1972; Caswell 2001). The task of the popu-lation modeler is to find an i-state variable that successfullycaptures past history. This is not easy; apparently reasonableand frequently used choices of i-states may fail to captureall the relevant information about individual history.

Confronted with this problem, the modeler might choosea completely different i-state variable (as plant ecologistsdid when giving up age-classified demography in favorof size-classified models), or might add a dimension tothe state space (as in extending stage-classified models toinclude both age and stage). Sometimes, however, it mightbe difficult or impossible to measure the relevant currentcharacteristic, but a proxy for that characteristic might befound in some function of the prior condition of an individ-ual. For example, resource storage can influence vital ratesin plants but resource storage can be difficult to measure.However, reproduction in the prior year can be used as aproxy for resource storage in species where reproduction inone year may deplete resource storage and reduce fertility inthe following year. If it is not possible to measure resource

https://doi.org/10.1007/s12080-017-0353-0

130 Theor Ecol (2018) 11:129–140

storage, one might therefore incorporate prior reproductivestatus into the state variable to improve the model.

A variety of prior conditions which affect vital rates havebeen found empirically. Reynolds and Burke (2011) foundthat chestnuts with fast early growth died younger thanchestnuts with slow early growth. Warren et al. (2014) foundthat previous breeding success and current body conditionmay be among the most important determinants of breedingpropensity in female lesser scaup. Rouan et al. (2009) foundthat choice of next breeding site is affected by both currentand prior breeding site in Branta canadensis. These exam-ples show that prior condition can be a source of individualvariation in vital parameters, i.e., a source of heterogene-ity. Some attempts have been made to include this sourceof heterogeneity into population models, see Pfister andWang (2005), Ehrlen (2000), and Rouan et al. (2009), buta framework for incorporating general prior conditions intodemographic models does not exist and will be presented inthis paper.

When constructing a structured population model, the i-state variables are used to classify individuals into states ina population vector n(t) whose entries give the densities ofeach state. The population vector is projected forward by apopulation projection matrix A

n(t + 1) = An(t). (1)

The matrix A can be decomposed into a matrix U, contain-ing transition probabilities for existing individuals, and amatrix F, describing the generation of new individuals byreproduction:

A = U + F. (2)

If prior conditions influence present dynamics, the vector nand the matrices U, F, and A must be modified to accountfor these influences. Our goal in this paper is to present asystematic method for constructing such models in whichindividuals are classified by current stage and (very gener-ally defined) prior condition. Because we are consideringeffects of individual condition at just one prior time, werefer to these as second-order matrix population models. Wewill present the demographic analysis of such models at thelevel of the individual, the cohort, and the population, andshow how to carry out sensitivity analyses of model results

to changes in parameters. As an example, we develop amodel and calculate the elasticity of the population growthrate of the herbaceous perennial plant Lathyrus vernus.

Terminology Our discussion requires careful definitions ofterms in order to clarify the way that historical effects areincorporated. We will say that the life cycle is described interms of stages (e.g., size classes). The prior condition of anindividual is some function of its stage at the prior time andits stage at the current time, and thus incorporates histori-cal information. The combination of current stage and priorcondition serves as the individual state variable for the anal-ysis. We give an overview of the terminology used in thispaper in Table 1.

The prior condition can be any arbitrary function of priorand current stage. Prior condition might be the stage at theprior time, it might be defined by membership in a set ofstages at the prior time, or it might be defined as the differ-ence between the current and the prior stages. Suppose forexample that stages are defined by size, in the hope that asize classification would be a satisfactory i-state variable. Itmight turn out that historical effects require including sizeat the prior time in the i-state variable. Alternatively, thei-state might require information only on membership ina class of sizes (e.g., larger than average or smaller thanaverage); we will refer to these classes of prior stages asequivalence classes. Or, the i-state might require informa-tion on the change in size between the previous time and thecurrent time, and individuals might be classified by whetherthey have grown, shrunk, or remained in the same size class.

Models based on prior stage are described in section“Full prior stage dependence,” models based on general func-tions of prior and current stages are described in section“Prior condition models,” and models based on equivalenceclasses of prior stage are described in section “Equivalenceclasses of prior stages.” The matrices, vectors, and mathe-matical operations used in this paper are listed in Table 2.

Model construction

We will begin by constructing the model in which the priorcondition is defined by full information on the prior stage,which we refer to as full prior stage dependence.

Table 1 Terminology used to distinguish the state of an individual, the stage of the life cycle, and the prior condition of the individual

• The state of an individual is the information necessary to predict the response of an individual to its environment.

• The stage of an individual refers to a biologically defined category, usually a life cycle stage, which is usedto define a (possibly unsuccessful) state variable.

• The condition of an individual is a flexible term that refers to some function of the prior stage and currentstage of the individual; this historical information may be combined with the present stage to obtain a statevariable based on current stage and prior condition.

Theor Ecol (2018) 11:129–140 131

Table 2 Mathematical notation used in this paper

Quantity Description Dimension

U Prior stage dependent cohort projection matrix s(s + 1) × s(s + 1)Ui Matrix with transition rates (from current to future state) for individuals with prior stage i s × s

V Prior condition dependent cohort projection matrix sr × sr

F Prior stage dependent fertility matrix s(s + 1) × s(s + 1)Fi Matrix with fertility rates (from current to future state) for individuals with prior stage i s × s

G Prior condition dependent fertility matrix sr × sr

n Prior stage model population vector n = vec(N) s(s + 1) × 1m Prior condition model population vector m = vec(M) sr × 1C Matrix relating the prior stage dependent vector n to the prior condition dependent population vector m sr × s(s + 1)φ(i, j) Matrix whose (i, j)th entry indicates the prior condition for an individual that makes an i → j transition s × s

φk Matrix whose (i, j)th entry is one if φ(i, j) = k and zero otherwise, in MATLAB notation φk(i, j) = (φ == k) s × s

Is Identity matrix s × s

1s Vector of ones s × 1ei The ith unit vector, with a 1 in the ith entry and zeros elsewhere VariousEij A matrix with a 1 in the (i,j) position and zeros elsewhere Various⊗ Kronecker productX(:, i) Column i of matrix X

vecX The vec operator, which stacks the columns of an m × n matrix X into a mn × 1 vector

Dimensions of vectors and matrices are given where relevant; s denotes the number of classes in the full second-order model and r denotes thenumber of classes in the reduced second-order model

Full prior stage dependence

Creating a prior stage model requires transition and fertil-ity rates to be measured for each possible prior stage. Sincenewborns do not have a well-defined prior stage in general,an extra prior stage must be added for newborns. If there ares current stages, we will label the prior condition of new-borns as stage s + 1. Individuals are thus classified by theircurrent stage 1, 2, . . . , s and their prior stage 1, 2, . . . , s+1.The transition and fertility rates, conditional on prior stage,are given by the matrices Uk and Fk:

Uk s × s Transitions among stages for individuals (3)

with prior stage k = 1, . . . , s + 1,

Fk s × s Reproduction by individuals (4)

with prior stage k = 1, . . . , s + 1.

The entries of Uk , denoted by ukij , are defined as

ukij = P

[j → i | prior stage = k

], (5)

f kij = E

[offspring of stage i | prior stage = k

]. (6)

It is useful to think of the state of the population asdescribed by a two-dimensional array N of size s × (s +1)

where nij is the number of individuals whose currentstage is i and whose prior stage was j . Ehrlen (2000)suggested that this array could be updated by multipli-cation with a three-dimensional matrix. However, matrixmultiplication can not be used directly to project such atwo-dimensional array (this would require tensors). Instead,the two-dimensional array is transformed into a vector bystacking the columns on top of each other:

n(t) = vecN(t). (8)

We use the tilde notation to denote vectors and matrices thatrelate to the prior condition model. The entries in the popu-lation vector n are now ordered first by prior stage and thenby current stage:

The vector n is projected by a transition matrix U anda fertility matrix F. The U and F matrices are constructed,respectively, from the set of matrices Uk and Fk . The transi-tion matrix U projects individuals into their next stage whilekeeping track of their prior stage. The matrix U is writtenin terms of block matrices corresponding to the blocks in n

132 Theor Ecol (2018) 11:129–140

in Eq. 9. In MATLAB notation, the transition matrix for amodel with s = 2 is

where Uk(:, i) refers to the ith column of the matrix Uk and0 is a column vector of size s×1. To understand the structureof U, consider the (1,1) block in the upper left corner. Thisblock projects individuals with prior stage 1 at t to priorstage 1 at t +1. The only such individuals have current stage1 at time t . They are projected by column 1 of U1. All othercolumns of the (1,1) block of U are zero. The other blocksof U are filled similarly.

The blocks in the last row of U are zero because transi-tions into the newborn stage are impossible; the matrix Fwill fill up this row block. In general, for any number ofstages s,

U =s+1∑

j=1

s∑

i=1

Eij ⊗(e�i ⊗ Uj ei

), (11)

where ei has size s × 1 and Eij has size (s + 1) × (s + 1)and has a one in position (i,j) and zeros elsewhere.

The fertility matrix F is constructed from the Fi . Becauseindividuals are always born into stage s + 1, the Fi appearin the last row of blocks in F. For the case with s = 2, thefertility matrix is

In general, for any number of stages s,

F =s+1∑

k=1

E(s+1)k ⊗ Fk, (13)

where E(s+1)k has size (s + 1) × (s + 1) and has a one inposition (s + 1, k) and zeros everywhere else.

Given a set of transition matricesUi and fertility matricesFi , Eqs. 11 and 13 define the second-order matrices U and F,and the population projection matrix

A = U + F. (14)

These matrices are subject to all the usual demographicanalyses, including population growth, population structure,and sensitivity analysis (see section “Case study” for anexample).

Prior condition models

A more general second-order model structure allows transi-tions and fertility to depend on some function of current and

prior stages. We consider the case where this function canbe defined as a linear transformation of the full prior stagedependent model, where the population vector is written asm, defined by

m(t) = Cn(t), (15)

for some matrix C. An example of such a prior condition ishaving previously grown, shrunk, or stayed the same size.The matrix C maps i-states defined by the combinationof (current stage × prior stage) to i-states defined by thecombination of (current stage × prior condition).

Suppose there are r distinct prior conditions. The state ofthe population is now given by a s × rarray M

where mij represents the number of individuals whose cur-rent stage is i and whose prior condition is j . As in Eq. 8,the population vector m is

m(t) = vecM(t). (17)

The key to the construction of the prior condition modelis to derive C from the rule defining the prior condition. Todo so, define a matrix φ

φ(i, j) = prior condition for an individual that makes an

j → i transition. (18)

For example, if the r = 3 prior conditions are shrinking,stasis, and growth

φ(i, j) =⎧⎨

⎩

1 for i < j shrinking,2 for i = j stasis,3 for i > j growth.

(19)

Next, we define a set of matrices φk(i, j), for k = 1, . . . , r ,given by

φk(i, j) ={1 if φ(i, j) = k,

0 otherwise .(20)

Given the matrices φk , the matrix C is given by

C =r∑

k=1

(1Ts+1 ⊗ ek ⊗ Is

)diag

(vecφk

), (21)

see Appendix A. To project the new population vector m,we replace the matrices U and F with new matrices V andG, respectively, so that

m(t + 1) =(V + G

)m(t) (22)

The matrix V describes transitions of extant individualsbetween the different i-states of the prior condition model

Theor Ecol (2018) 11:129–140 133

and the matrix G describes the production of new individu-als in the prior condition model. Substituting (15) into bothsides of Eq. 22 yields

C(U + F

)n(t) =

(V + G

)Cn(t). (23)

Equation 23 is satisfied if V and G satisfy the followingequations:

VC = CU (24)

GC = CF. (25)

In general, the matrix C is not square and does not havean inverse. So we use the Moore-Penrose pseudo-inverseC†

(see Abadir and Magnus (2005)) to solve for V and G:

V = CUC†, (26)

G = CFC†. (27)

If C is square and non-singular, the pseudo-inverse is theordinary inverse:

C† = C−1. (28)

IfC is not square but has linearly independent rows (i.e., hasfull row rank),

C† = CT(CCT)−1. (29)

If C has rows of zeroes (this happens if some combina-tions of current stage and prior conditions are impossible),then C will not have full rank. In this case, C† is com-puted from the singular value decomposition, implementedin MATLAB with the function pinv(C) and in R with thefunction Ginv(C). For example, in a size-classified model,it is impossible to be in the smallest size class and to havegrown into it from a smaller size class. In such cases, C†

and thus also V and G have rows and columns of zeroscorresponding to the impossible combinations.

Equations 26 and 27 define the prior condition matricesV and G and the population projection matrix

A = V + G. (30)

As in Eq. 14, the usual demographic results can be obtainedfrom these matrices.

Equivalence classes of prior stages

Equivalence classes of prior stages are a special case offunctions of current and prior stage. Equivalence classes aresubsets of prior conditions that depend only on the priorstage. For example, individuals in a size-classified modelmight be categorized into two equivalence classes depend-ing on whether they were previously above or below somethreshold size.

The machinery described in the previous section, i.e.,Eqs. 21, 26, and 27, can be used to write down the equiva-lence class model. However, there is an easier way to find

the C matrix that transforms the full prior stage dependentpopulation vector n to the equivalence class population vec-tor m. We transform the population matrix N in Eq. 7 intothe equivalence class population matrix M in Eq. 16 by amatrix B;

M = NB. (31)

The matrix B has size (s + 1) × r and its entries are

B(i, j) ={1 if stage i ∈ equivalence class j,

0 otherwise.(32)

Applying the vec operator to Eq. 31 gives

m(t) = (B� ⊗ Is)n(t), (33)

where we have used the following result from Roth (1934)that vecABC = (

CT ⊗ A)vecB. Equation 33 is a special

case of Eq. 15 with

C = (B� ⊗ Is). (34)

Since the rows of B� are orthogonal, the matrix B� and thematrix (B� ⊗ Is) both have full row rank and therefore C†

can be calculated from Eq. 29.

Sensitivity analysis

Sensitivity analysis provides the effect of changes in anyparameter on any model outcome. In general, these com-putations require derivatives of scalar-, vector-, or matrix-valued functions with respect to scalar-, vector-, or matrix-valued arguments. Matrix calculus is a formalism whichenables us to consistently calculate such derivates. For anintroduction to matrix calculus, see Abadir and Magnus(2005); for details see Magnus and Neudecker (1988). Eco-logical applications of sensitivity analysis appear in Caswell(2007, 2009, 2012).

Consider some scalar or vector output of the model, ξ ,which is computed from U and F, or from V and G. Thematrices U, F, V, and G are in turn computed from thematrices Ui and Fi . Suppose Ui and Fi depend on a p × 1vector of parameters θ , i.e., Ui = Ui[θ ], then we use thechain rule to write

dξ

dθ� = dξ

dvec�UdvecU

dθ� + dξ

dvec�FdvecF

dθ� , (35)

or

dξ

dθ� = dξ

dvec�VdvecV

dθ� + dξ

dvec�GdvecG

dθ� . (36)

The first terms in Eqs. 35 and 36 capture effects through sur-vival and transitions, and the second terms capture effects

134 Theor Ecol (2018) 11:129–140

through fertility. The elasticities, or proportional sensitivi-ties, are given by

εξ

εθ� = diag−1(ξ)dξ

dθ� diag(θ). (37)

The derivatives of U, F, V, and G with respect to θ

depend on how U and F depend on the Ui and Fi matrices.Differentiating (11), we obtain

dvecU

dθ� =s+1∑

j=1

QUj

dvecUj

dθ� . (38)

Each term in the summation captures the effect of θ throughone of the Ui . The matrix QU

j is given by

QUj =

s∑

i=1

(Is+1⊗ Ks,s+1⊗ Is

)(vec

(Eij

)⊗ Is2)(Eii ⊗ Is) .

(39)

Similarly, differentiating (13) gives

dvecF

dθ� =s+1∑

i=1

QFi

dvecFi

dθ� , (40)

where each term captures the effect of θ through one of theFi . The matrix QF

i is given by

QFi = (

Is+1 ⊗ Ks,s+1 ⊗ Is) (vec

(Es+1,i

) ⊗ Is2). (41)

To calculate the derivatives of the prior condition matri-ces V and G, we use the chain rule to write

dvecV

dθ� = dvecV

dvecU

dvecU

dθ� , (42)

dvecG

dθ� = dvecG

dvecF

dvecF

dθ� . (43)

Differentiating Eqs. 26 and 27, we obtain

dvecV

dvecU=

(C†

)� ⊗ C, (44)

dvecG

dvecF=

(C†

)� ⊗ C. (45)

Thus,

dvecV

dθ� =[(

C†)� ⊗ C

]dvecU

dθ� , (46)

dvecG

dθ� =[(

C†)� ⊗ C

]dvecF

dθ� . (47)

Equations 38 and 40 are substituted into Eq. 35.Equations 46 and 47 are substituted into Eq. 36. All thatremains is to calculate the dependence of ξ on U and F (or V

and G), and the dependence ofUi and Fi on the parameter θ .These items are specific to the question under consideration,so we provide an example in the next section.

Case study

As an example, we apply the second-order formalism toa fully prior stage dependent model of a perennial plant.We will construct the model and calculate the populationgrowth rate, the stable population vector, the reproductivevalue vector, and the elasticity of the population growth rateλ to proportional changes in the demographic rates.

Our analysis is based on a study by Ehrlen (2000) ofLathyrus vernus, a long-lived herb native to forest mar-gins and woodlands in central and northern Europe andSiberia. Ehrlen classified individuals into seven stages: seed(SD), seedling (SL), very small (VS), small (SM), largevegetative (VL), flowering (FL), and dormant (DO). Ehrlenconstructed two matrix models: a first-order model with nohistorical effects and a second-order model. We constructour second-order model from the transition and fertilityrates calculated by Ehrlen (2000) (their Table 2).

We introduce a special prior stage for newborns; thesecond-order model therefore has a total of 7 × 8 = 56states. We used Ehrlen (2000)’s demographic rates to con-struct a transition matrixUi and a fertility matrix Fi for eachof the eight prior stages. In Table 3, the transition matrixfor individuals who were previously in stage SM is shownas an example. The first two columns of this matrix arezero because small plants can not go back to being seedsor seedlings. All eight transition and fertility matrices are inthe Supplementary Material.

We constructed the projection matrix, A, from the Ui

and Fi matrices using Eqs. 11 and 13. The resulting 56 ×56 matrices are available in the Supplementary Material.

Table 3 The transition matrix for Lathyrus vernus individuals withprior stage small (SM)

The columns in this matrix represent transitions out of the followingcurrent stages from left to right: seed (SD), seedling (SL), very small(VS), small (SM), vegetative large (VL), flowering (FL), and dormant(DO)

Theor Ecol (2018) 11:129–140 135

The population growth rate λ is given by the dominanteigenvalue of A,

λ = 0.985. (48)

This agrees with the value reported for the second-ordermodel by Ehrlen (2000). Ehrlen also fitted a first-ordermodel to the same data and found a value just above one(λ = 1.010). The stable population vector, w, and the repro-ductive value vector, v, which are displayed in Figs. 1 and 2,are the right and left eigenvectors of A, respectively. Theentries of the stable population vector are denoted by wij

for individuals with current stage i and prior stage j , analo-gously to the entries of the population vector n in Eq. 9. Theentries of the marginal stable current stage distribution, wc,are given by

wci =

r∑

j=1

wij (49)

and are shown in Fig. 1b. Similarly, the entries of the marginalstable prior stage distribution, wp, are given by

wpj =

s∑

i=1

wij (50)

and are shown in Fig. 1c.Individuals with the same current stage have different

vital rates if they differ in prior stage and this heterogeneitydue to prior stage affects the population growth rate λ. Toquantify the relative effect of the different prior stage depen-dent transition matrices on the population growth rate, wecalculate the elasticity of the population growth rate, λ, tochanges in the Ui matrices

ελ

εvec�Ui

, i = 1, . . . , 8. (52)

Substituting λ for ξ in Eq. 37 yields

ελ

εvec�Ui

= 1

λ

dλ

dvec�Ui

diag (vecUi ) . (53)

As shown in Caswell (2001),

dλ

dvec�Ui

= wTi ⊗ vTi , (54)

where wi and vi are the right and left eigenvectors of Ui ,respectively, scaled so that

vTi wi = 1 for all i, (55)

1Twi = 1 for all i. (56)

We sum the entries of Eq. 53 to get the elasticity of λ

to a proportional change in all of the entries of Ui . Theresults are shown in Fig. 3. We note that there are largedifferences among prior stages. Proportional changes in the

NBDO

FL

Prior stage

VLSM

VSSL

SDDO

FLCurrent stage

VLSM

VSSL

SD

0

0.05

0.15

0.25

0.2

0.1

Frac

tion

Marginal distribution over current stagesSD SL VS SM VL FL DO

Frac

tion

0

0.05

0.1

0.15

0.2

0.25

0.3

Marginal distribution over prior stagesSD SL VS SM VL FL DO NB

Frac

tion

0

0.05

0.1

0.15

0.2

0.25

0.3

Fig. 1 a Stable current stage × prior stage distribution. b Marginalcurrent stage distribution. cMarginal prior stage distribution. Notation:seed (SD), seedling (SL), very small (VS), small (SM), vegetative large(VL), flowering (FL), dormant (DO), and newborn prior (NB)

demography of individuals with prior stages seed, seedling,or newborn have little effect on λ. Proportional changes inindividuals who were small vegetative at the prior time haveeffects an order of magnitude larger.

136 Theor Ecol (2018) 11:129–140

SDSLVSSM

Prior stage

VLFLDONBSD

SLVS

Current stage

SMVL

FLDO

1

0

0.8

0.6

0.4

0.2

Rep

rodu

ctiv

e va

lue

Current stageSD SL VS SM VL FL DO

Mar

gina

l rep

rodu

ctiv

e va

lue

0

0.5

1

1.5

2

2.5

3

3.5

4

Prior stageSD SL VS SM VL FL DO NB

Mar

gina

l rep

rodu

ctiv

e va

lue

0

0.5

1

1.5

2

2.5

3

3.5

Fig. 2 a Reproductive value vector for each current stage × priorstage combination. bMarginal current stage reproductive value vector.c Marginal prior stage reproductive value vector. Notation: seed (SD),seedling (SL), very small (VS), small (SM), vegetative large (VL),flowering (FL), dormant (DO), and newborn prior (NB)

Discussion

When does history matter? Population models are basedon i-state variables chosen by some mixture of biological

intuition, tradition, practical limitations, and formal statisti-cal analyses. Even after a careful choice of i-state, it mayhappen that individual prior conditions contain importantinformation about the fate of individuals. In such cases,history matters, and the methods presented here solve theproblem of how to incorporate information about it intomatrix models.

Why stop at second-order effects, what about the effectof the prior condition at t − 2, t − 3, etc.? It is theo-retically possible to extend the framework presented hereto include dependence on higher-order effects. However, ifthe i-state variable is such that third- or even fourth-orderhistorical effects are important, it might better to recon-sider the choice of i-state variable instead of including everincreasing historical dependence.

Statistical tests for second-order effects in longitudinaldata using log-linear models have been developed by Bishopet al. (1975) and Usher (1979). Time series of longitudinaldata are needed to perform these tests as well as for the sub-sequent estimation of a prior condition dependent model.Capture–mark–recapture analyses for prior stage dependentmodels have been developed by Pradel (2005) and Coleet al. (2014).

Incorporating individual history requires a decision aboutwhat aspects of the prior condition are important. We havepresented three biologically motivated cases: prior condi-tion as the prior stage, prior condition as an arbitrary linearfunction of current and prior stages, and prior condition asan equivalence class of prior stages. In each case, the nec-essary information is a set of fertility and survival/transitionmatrices for each prior stage. The resulting models useblock-structured matrices to project a vector of stages withinprior conditions. These matrices can be used to calculateall the usual demographic outcomes. Because the matri-ces are carefully constructed from Ui and Fi , they can besubjected to sensitivity analysis. It is straightforward to cal-culate the sensitivity and elasticity of any model outcome toany parameters affecting the vital rates.

Prior condition effects are more than just a convenienttool in constructing i-states for population models. Theyare a biologically real source of inter-individual variation.The importance of individual variation in vital parametersto ecological processes has become increasingly evidentin recent years (Bolnick et al. 2011; Valpine et al. 2014;Caswell 2014; Vindenes and Langangen 2015; Steiner andTuljapurkar 2012; Cam et al. 2016). Ignoring individualvariation in vital parameters, also referred to as individ-ual heterogeneity, can have important consequences for thedemographic outcomes and subsequent conclusions; see forexample Vaupel et al. (1979), Rees et al. (2000), Fujiwaraand Caswell (2001), Vindenes and Langangen (2015), andCam et al. (2016). In his study of Lathyrus, Ehrlen (2000)found that including heterogeneity due to prior stage had

Theor Ecol (2018) 11:129–140 137

Fig. 3 The elasticity of λ to aproportional change in all of theUi

Prior stageSeed Seedling Very small Small Vegetative Flowering Dormant Newborn

Ela

stic

ity o

f to

cha

nge

in U

i0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

only a small effect on λ, although it was sufficient to causethe population to decrease rather than increase.

How much heterogeneity is introduced by the prior con-dition effects in the Lathyrus example? Observations ofindividual plants can identify their current stage, but nottheir prior condition. The stable structure, w (Fig. 1), showsthe joint probability distribution over current and priorstages, and the amount of heterogeneity in the stable popu-lation can be calculated from the entropy of this distribution.The entropy of the joint current × prior stage distribution is

H(p, c) = −∑

i,j

wij ln(wij

) = 2.61. (57)

This measures the overall heterogeneity in the stable pop-ulation structure. The heterogeneity in the marginal currentstage distribution, wc [Eq. 49], is

H(c) = −s∑

i=1

wci ln

(wc

i

) = 1.69. (58)

This is the observable heterogeneity in current stage. Theheterogeneity contributed by the unobservable prior stage,taking into account the relationship between current andprior stage, is

H (p|c) = H(p, c) − H(c) = 0.92, (59)

Khinchin (1957). Thus, in this example, the prior stage con-tributes about 35% of the total heterogeneity in the stablepopulation.

In this paper, we have considered only linear models inconstant environments. However, models could easily beconstructed to include density effects, by making theUi andFi functions of density. Periodic or stochastic models couldbe constructed by making the Ui and/or the Fi appropriatefunctions of time. The analysis of the resulting models may

pose interesting challenges and is an open research problemfor population ecology.

Acknowledgements We thank Gregory Roth for helpful discussionsand Johan Ehrlen for providing the data for the Lathyrus case study.

Funding information This work was supported by EuropeanResearch Council Advanced Grant 322989.

Appendix A: Derivation of the matrix C

In this appendix, we will derive the matrix C in Eq. 21, whichtransforms the fully prior stage dependent population vector,n, into the prior condition dependent population vector, m,

m(t) = Cn(t), (60)

as in Eq. 15. Recall that we defined the matrix φ withelements

φ(i, j) = prior condition for an individual that makes an

j → itransition, (61)

and we defined a set of indicator matrices φk , for k =1, . . . , r , given by

φk(i, j) ={1 if φ(i, j) = k,

0 otherwise .(62)

In Appendix B, we show a few examples of φ matrices.Recall furthermore the matrix

138 Theor Ecol (2018) 11:129–140

where nij is the number of individuals whose current stageis i and whose prior stage was j , and the matrix

where mij is the number of individuals whose current stageis i and whose prior condition was j . Finally, recall that thepopulation vectors are given by

n(t) = vecN(t), (65)

m(t) = vecM(t). (66)

Using the above defined matrices φk and N, the matrixM can be written as

M =r∑

k=1

(φk ◦ N

) (1s+1 ⊗ eTk

). (67)

The first term in the product is a matrix containing the den-sities of individuals in all entries of N that correspond toprior condition k, and zeros elsewhere. The second term inthe product adds the densities across each row of the matrixφk ◦ N and puts these elements in the kth column of M.Taking the vec of both sides of this equation gives

m =r∑

k=1

(1Ts+1 ⊗ ek ⊗ Is

)vec

(φk ◦ N

), (68)

=r∑

k=1

(1Ts+1 ⊗ ek ⊗ Is

)diag

(vecφk

)n, (69)

where we have used vecABC = (CT ⊗ A

)vecB (Roth

1934) and vec (A ◦ B) = diag (vecA) vec (B). Comparisonwith Eq. 60 shows that

C =r∑

k=1

(1Ts+1 ⊗ ek ⊗ Is

)diag

(vecφk

). (70)

Appendix B: Examples of φ and C matrices

B.1 Examples of φ matrices

In this appendix, we show a few examples of the matrix φ.

(A) When prior condition = prior stage, then

φ(i, j) = j for all i. (71)

For example, when s = 3, φ is

(B) When prior condition = equivalence classes of priorstages, then

φ(i, j) = function(j) (74)

For example, for s = 3 and for the case of theequivalence classes j = 1 and j > 1, we get

φ(i, j) ={1 for j = 12 for j = 2, 3

(75)

(C) When prior condition = shrinkage, stasis, or growth,then

φ(i, j) =⎧⎨

⎩

1 for i < j shrinkage2 for i = j stasis3 for i > j growth

(76)

For example, for s = 3, φ is

(D) General more complicated growth conditions. Forexample, suppose the prior condition reflects theamount of growth, so that

φ(i, j) = i − j + s. (78)

Theor Ecol (2018) 11:129–140 139

For s = 4, this results in the following φ

B.2 An example C matrix

We consider the example in Eqs. 76 and 77 with three sizeclasses and three growth classes corresponding to growth,shrinkage, and stasis. For simplicity, we consider a cohortmodel with no fertility and therefore no special prior condi-tion for newborns. The matrices φk are

φ1 =⎛

⎝0 1 10 0 10 0 0

⎞

⎠ , (80)

φ2 =⎛

⎝1 0 00 1 00 0 1

⎞

⎠ , (81)

φ3 =⎛

⎝0 0 01 0 01 1 0

⎞

⎠ . (82)

Substituting these into Eq. 70, and letting the sum rangefrom 1 to 3 only, gives the matrix C as

The transition matrix for the prior condition model, V,is given by Eq. 26 and requires the pseudo-inverse of thematrix C, which can be found using pinv(C) in MATLAB

or Ginv(C) in R. Substitution of the pseudo-inverse of Cinto Eq. 26 and a few lines of algebra result in

This matrix captures the complicated dependence of transi-tions on both current stage and prior condition. The place-ment of the weighted sums of transition probabilities fromthe Ui matrices would be challenging without the metho-dology presented here.

Open Access This article is distributed under the terms of the Crea-tive Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distri-bution, and reproduction in any medium, provided you give appropri-ate credit to the original author(s) and the source, provide a link to theCreative Commons license, and indicate if changes were made.

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