Basic theory and concepts

Richard P. Shefferson, Shun Kurokawa, and Johan Ehrlén

This document was built in Markdown in R 4.1.0 and compiled on 29 June 2021. It covers package lefko3 version 3.5.0.


R package lefko3 is devoted to the analysis of demographic data through matrix projection models (MPMs) (Shefferson, Kurokawa, and Ehrlen 2021). It is intended to serve as a one-stop destination for the construction of size-classified Lefkovitch MPMs, which divide the life of an individual into different life stages rather than ages. It was originally developed specifically to estimate and analyze historical matrix projection models (hMPMs), which are matrices designed to include not only current state but also past history in terms of previous states. Such matrices are large, typically having dimensions higher than their standard, ahistorical counterparts (the latter will be hereafter referred to as ahistorical MPMs, or ahMPMs, while the acronym MPM will be used to refer to all matrix projection models, whether historical or not). As this package has developed, we have prioritized the development of core algorithms and methods to construct these matrices quickly, efficiently, and at least relatively painlessly. To make comparison possible and straightforward, the package contains methods covering the estimation of both ahistorical and historical MPMs, using both raw matrix and function-based formats. It can also be used to create integral projection models (IPMs) and age × stage matrices, both of which are available in function-based formats.

Package lefko3 introduces a complete suite of functions covering the MPM workflow, from dataset management to the construction of MPMs to their analysis. Dataset management functions standardize demographic datasets from the dominant formats into a format that facilitates hMPM estimation while accounting for individual identity and other parameters. Demographic vital rate models may be estimated using demographic datasets with this standardized format, and these models take the form of generalized linear or mixed linear models using a variety of response distributions. Matrix estimation functions produce all necessary matrices from a single dataset, including all times, patches, and populations in a single shot, and do so quickly through core binaries engineered for speed and accuracy.

This vignette was written as a basic introduction to the concepts and methods underlying lefko3. The target audience is everyone from beginners with little knowledge of population ecology and even less of R, to experienced ecologists with advanced working knowledge of R and the analysis of population dynamics. In addition to this vignette, which covers the concepts, background, and workflow underlying the package, we have created a series of vignettes focused on specific practical examples using the package to run suites of comprehensive analyses. These vignettes show the development of raw and function-based MPMs, IPMs, and age-by-stage MPMs using two datasets included in lefko3: a dataset from a Swedish population of the perennial herb Lathyrus vernus, and a dataset from an American population of the terrestrial orchid Cypripedium candidum. In addition, we have created a number of other vignettes and tutorials that are available on the projects page of the Shefferson lab website.


Life history models and population projection matrices

Matrix projection models (MPMs) are representations of the dynamics of a population across all life history stages deemed relevant, across the most relevant time interval (typically one year, but sometimes different, and assumed to be consistent within each analysis). They require a complete model of the organism’s life history prior to construction, and this model must explicitly show all life stages and all life history transitions to be modeled. Each stage is mutually exclusive, meaning that an individual can only be in a single stage at a given time. Each stage is represented in the matrix by a single column and a single row. Each transition takes exactly one full time step, which needs to be defined consistently. Matrix elements (akj) show either the probability of transition for an individual in stage j at time t (along the columns), to stage k at time t+1 (along the rows), or the mean rate of production of new recruits into stage k at time t+1 (along the rows) by individuals in reproductive stage j in time t (along the columns). Conceptually, each individual is in a particular stage in the instance of monitoring or observation, and then transitions in the interval between that time’s observation and the next time’s observation. Death is not an explicit life stage and so is not modeled as such, instead becoming a potential endpoint of each transition.

Stages are generally defined under the assumption that they are unique moments in an organisms life. This uniqueness is assumed to extend to the demography of the organism during its time within that stage. In some cases, stages are defined as developmental stages, as happens with insect instars. In other cases, stages are defined almost purely on the basis of size, as occurs with perennial herbs that may exhibit a different number of stems, leaves, and flowers in each growing season. In still other cases, stages may be defined via other characteristcs. For example, the vegetative dormancy stage of some perennial herbs is defined by a lack of aboveground size, and so is characteristically not observable (Shefferson et al. 2001).

The timing of monitoring relative to the reproductive season impacts the structure of life history models. Life history models are typically categorized as either pre-breeding or post-breeding. Here, “breeding” refers to the production of offspring, and so a pre-breeding model assumes that monitoring is conducted immediately before the new recruits are born, while a post-breeding model assumes that the monitoring is conducted just after new recruits are born (in both cases, breeding is assumed to be seasonal). In a pre-breeding model, fecundity equals the production of newborns multiplied by the survival of newborns to age 1, since fecundity must take place over a full time step and the timing of the census misses both the birth event itself and the time in which the organism is a newborn (Kendall et al. 2019). In a post-breeding model, fecundity equals the survival of the parent from the stage/age preceding reproduction to reproduction itself, multiplied by the number of newborns produced (Kendall et al. 2019).

In monitoring studies of plant populations, the typical strategy taken is that of a pre-breeding model, in which fecundity is estimated as the production of seeds in a given year multiplied by their over-winter survival probability as seeds and their germination probability in the following year. If the life history includes seed dormancy, then it is modeled by multiplying seed production by seed survival (which is given as the probability of maintaining seed viability from one year to the next), in the first instance, and then as seed survival for seeds already produced and dormant (seed survival may be modeled to decrease with increasing age of dormant seed). Added complexity can arise if there are multiple fecundity pathways, for example when clonal reproduction is also possible or if multiple propagule stages exist. We urge users to be careful with this step, as properly defining the life history model has important implications for all analyses of population dynamics (Kendall et al. 2019).

Figure 1 is a simple example of a stage-classified model and matrix for a terrestrial orchid species, Cypripedium candidum (Shefferson et al. 2001). Here, we show each stage as a node, and each transition as a uni-directional arrow (a). The rates and probabilities are shown as mathematical symbols in (b), with Skj denoting survival-transition probability from stage j in time t to stage k in time t+1, and Fkj denoting the fecundity of reproductive stage j into recruit stage k in this life history.

Figure 1. Simple life history model (a) and ahistorical MPM (b) for Cypripedium candidum, a North American herbaceous plant species. Here, S is the dormant seed stage, J is the seedling stage, D is adult vegetative dormancy, V is adult vegetative sprouting, and F is flowering.

Ahistorical vs. historical matrix models

Figure 1 is an example of an ahistorical MPM (ahMPM), which is a matrix projection model in which the future stage of an individual is dependent only on its current stage, and not on previous stages. In other words, individual history is not incorporated into ahMPMs. It may seem odd that individual history is not incorporated into the matrices given that demographic datasets are composed of records of individual histories spanning several or even many observation events. However, construction of a typical ahMPM breaks up these individual histories into pairs of consecutive stages across time, with each pair treated as independent of every other pair of consecutive stages. For example, if an individual is in stage A in time 1, stage B in time 2, stage E in time 3, and dead in time 4, then its individual history is broken up into A-B, B-E, and E-Dead, with each pair assumed to be independent. The resulting projection matrix would be the same if these transitions originated from different individuals or the same one - their order and relationships do not have any impacts in ahMPM construction.

The matrix shown in figure 1 can be used to project population dynamics by predicting the future numbers of individuals in each stage. For this example, we can project forward by multiplying the projection matrix by a vector of the numbers of individuals in each stage, given in the same order as the order of stages corresponding to the rows and columns of the matrix, in a particular time, as in

\[\begin{equation} \tag{1} \left[\begin{array} {rrr} n _{2,S} \\ n _{2,J} \\ n _{2,V} \\ n _{2,D} \\ n _{2,F} \end{array}\right] = \left[\begin{array} {rrr} S _{SS} & 0 & 0 & 0 & F _{SF} \\ S _{JS} & S _{JJ} & 0 & 0 & F _{JF} \\ 0 & S _{VJ} & S _{VV} & S _{VD} & S _{VF} \\ 0 & S _{DJ} & S _{DV} & S _{DD} & S _{DF} \\ 0 & 0 & S _{FV} & S _{FD} & S _{FF} \end{array}\right] \left[\begin{array} {rrr} n _{1,S} \\ n _{1,J} \\ n _{1,V} \\ n _{1,D} \\ n _{1,F} \end{array}\right] \end{equation}\]

where \(n _{1,S}\) is the number of individuals in stage \(S\) in time 1. The total population size in any given time is the sum of the numbers of individuals in that time, and the asymptotic growth rate \(\lambda\) can be estimated as the dominant eigenvalue of this matrix.

The independence of consecutive stage-pairs reflects a central assumption in ahMPM analysis: the stage of an individual in the next time is influenced only by its current stage. Conceptually, if an organism’s stage in the next time is entirely determined by its current stage, then its previous states do not influence these transitions. Thus, standard ahMPMs are two dimensional and reflect only the current and next immediate stage of individuals, given by the columns and rows, respectively. This is ultimately an extension of the iid assumption in statistics - that the states of individuals are independent and originate from identically-distributed random variables.

MPMs have been estimated since the 1940s (2001). We have never done a meta-analysis of all of these studies, but nonetheless it is safe to say that studies considering individual history are rare. The typical MPM study uses ahMPMs, and so assumes independence of stage transitions across time even from the same individual. In fact, at the time of writing, we are aware of only 5 examples of studies breaking these assumptions and using a historical approach, meaning that they incorporated some degree of individual history into matrix estimation and analysis (Ehrlen 2000; Shefferson, Warren II, and Pulliam 2014; Shefferson, Mizuta, and Hutchings 2017; Shefferson et al. 2018; Vries and Caswell 2018).

The historical MPM (hMPM) is an extension of the matrix projection model that incorporates information on one previous time into the determination of vital rates. Thus, the expected survival-transition probability of an individual in stage j at time t to stage k at time t+1 depends not only on its stage in time t but also on its stage in time t-1. Population ecologists considering this problem analytically might be inclined to add an extra dimension to the matrix to deal with this, thus creating a 3d array or cube. However, this approach is computationally intensive and makes analysis challenging, particularly some projection analyses impossible (Vries and Caswell 2018). Instead, we utilize the approach developed by Ehrlén (2000), in which rows and columns represent life history stages paired in consecutive times. Thus, columns now represent the From pair of stages (stage in times t and t-1), and rows now represent the To pair of stages (stage in times t and t+1), as in Figure 2. This model is effectively equivalent to the second-order model proposed by de Vries and Caswell (2018), although it organizes the elements differently and incorporates more history into transitions involving individuals just born.

Figure 2. Historical MPM for Cypripedium candidum, a North American herbaceous plant.

Here, we can refer to matrix elements as akjl, where the element represents the rate at which individuals transition to stage k in time t+1 after having been stage j in time t and stage l in time t-1. If nkjl is the number of individuals making this transition, n.jl is the total number of individuals in stage j in time t and stage l in time t-1 regardless of stage (or even status as alive or dead) in time t+1, m is the number of stages in the life history model, and d is the number of stages plus death in the life history model, then we have

\[\begin{equation} \tag{2} a _{kjl} = \frac{n _{kjl}}{n _{.jl}} = \frac{n _{kjl}}{\sum_{i=1}^{d} n _{ijl}} \end{equation}\]

These values can be used to compute the elements of matrices in ahMPMs, as follows:

\[\begin{equation} \tag{3} a _{kj} = \frac{\sum_{l=1}^{m} n _{kjl}}{\sum_{l=1}^{m} \sum_{i=1}^{d} n _{ijl}} \end{equation}\]

Note that one cannot use the matrix elements in a historical MPM to calculate the associated ahistorical MPM elements, because the historical transitions must be weighted by the numbers of individuals corresponding to each transition from time t to t+1 that were in each previous stage to produce the proper weighted average rate.

Historical MPMs can be projected forward in the same way that ahMPMs can. Here we see an example of a projection going forward one time step, using the hMPM shown in figure 2.

\[\begin{equation} \tag{4} \tiny \left[\begin{array} {rrr} n _{2,SS} \\ n _{2,JS} \\ n _{2,JJ} \\ n _{2,VJ} \\ n _{2,DJ} \\ n _{2,FJ} \\ n _{2,VV} \\ n _{2,DV} \\ n _{2,FV} \\ n _{2,VD} \\ n _{2,DD} \\ n _{2,FD} \\ n _{2,VF} \\ n _{2,DF} \\ n _{2,FF} \\ n _{2,SF} \\ n _{2,JF} \end{array}\right] = \left[\begin{array} {rrr} S _{SSS} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & F _{SSF} & 0 \\ S _{JSS} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & F _{JSF} & 0 \\ 0 & S _{JJS} & S _{JJJ} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & F _{JJF} \\ 0 & S _{VJS} & S _{VJJ} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & F _{VJF} \\ 0 & S _{DJS} & S _{DJJ} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & F _{DJF} \\ 0 & S _{FJS} & S _{FJJ} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & F _{FJF} \\ 0 & 0 & 0 & S _{VVJ} & 0 & 0 & S _{VVV} & 0 & 0 & S _{VVD} & 0 & 0 & S _{VVF} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & S _{DVJ} & 0 & 0 & S _{DVV} & 0 & 0 & S _{DVD} & 0 & 0 & S _{DVF} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & S _{FVJ} & 0 & 0 & S _{FVV} & 0 & 0 & S _{FVD} & 0 & 0 & S _{FVF} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & S _{VDJ} & 0 & 0 & S _{VDV} & 0 & 0 & S _{VDD} & 0 & 0 & S _{VDF} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & S _{DDJ} & 0 & 0 & S _{DDV} & 0 & 0 & S _{DDD} & 0 & 0 & S _{DDF} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & S _{FDJ} & 0 & 0 & S _{FDV} & 0 & 0 & S _{FDD} & 0 & 0 & S _{FDF} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & S _{VFV} & 0 & 0 & S _{VFD} & 0 & 0 & S _{VFF} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & S _{DFV} & 0 & 0 & S _{DFD} & 0 & 0 & S _{DFF} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & S _{FFV} & 0 & 0 & S _{FFD} & 0 & 0 & S _{FFF} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & F _{SFJ} & 0 & 0 & F _{SFV} & 0 & 0 & F _{SFD} & 0 & 0 & F _{SFF} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & F _{JFJ} & 0 & 0 & F _{JFV} & 0 & 0 & F _{JFD} & 0 & 0 & F _{JFF} & 0 & 0 \\ \end{array}\right] \left[\begin{array} {rrr} n _{1,SS} \\ n _{1,JS} \\ n _{1,JJ} \\ n _{1,VJ} \\ n _{1,DJ} \\ n _{1,FJ} \\ n _{1,VV} \\ n _{1,DV} \\ n _{1,FV} \\ n _{1,VD} \\ n _{1,DD} \\ n _{1,FD} \\ n _{1,VF} \\ n _{1,DF} \\ n _{1,FF} \\ n _{1,SF} \\ n _{1,JF} \end{array}\right] \end{equation}\]

Here, \(n _{1,SS}\) is the number of individuals that were in stage \(S\) in both times 0 and 1, and \(n _{2,DF}\) is the number of individuals that were in stage \(D\) in time 2 and stage \(F\) in time 1.

Historical MPMs normally require a much larger number of elements to be parameterized than ahMPMs do. Figure 2 illustrates this issue, but the function-based matrix analysis in our Cypripedium candidum vignette provides a more realistic example. In that case study, we use a life history with 54 life stages, of which the first is a dormant seed stage, the next four are immature stages, the sixth is adult vegetative dormancy, the next 24 stages are size-classified non-flowering adults, and the final 24 stages are size-classified flowering adults. A life history with 54 stages yields an ahistorical matrix with 54 columns and 54 rows, and so 2,916 elements. A full historical matrix should have 2,916 columns and 2,916 rows, consisting of 8,503,056 elements, because the column and row stages represent pairs of stages.

In an ahistorical matrix, the only true 0s will be those elements corresponding to biologically impossible transitions (although, in the raw matrix case, there may be 0s caused by a lack of individuals taking a particular transition in a given time step). However, most elements in a historical MPM are structural 0s, and increasing the number of stages causes these structural 0s to increase at a faster rate than the rate at which the number of truly estimable elements increases. In fact, if there are m stages in a life history, yielding m2 elements in an ahistorical matrix, then although there will be m4 elements in the historical matrix, only m3 will be potentially estimable while (m-1)m3 are structural 0s (we say potentially because some of the logically possible transitions may still be biologically impossible). This happens because transition elements in hMPMs are only estimable if stage at time t is equal in the column and row stage pairs. For example, the transition probability between stage A in time t-1 and stage B in time t (column stage), to stage C in time t and stage D in time t+1 (row stage) equals 0, because an organism cannot be in both stages B and C in time t. Thus, in the Cypripedium candidum example, all 2,916 elements of the ahMPM are potentially estimable (although some are biologically impossible), while only 157,464 of the 8,503,056 elements of the associated hMPM are potentially estimable and the rest are structural 0s.

Raw vs. function-based matrix models, including integral projection models

Matrix projection models, whether historical or ahistorical, can also be categorized as either raw or function-based. The oldest and most common in the literature is the raw MPM, in which transition probabilities in the matrix are estimated by counting all of the individuals alive in a particular stage at time t, and dividing that number into the number of individuals from that set that are still alive in each possible stage in time t+1 (equations 2 and 3). For example, if 100 individuals are alive in stage D in year 2010, and 20 of these are alive in stage D in year 2011, as are a further 40 in stage V and a further 25 in stage F, then the associated transition probabilities are 0.20, 0.40, and 0.25, respectively. The overall survival probability for stage D is the sum of these transitions, or 0.85. Methods for estimating fecundity in raw matrices vary from system to system. In the Cypripedium candidum vignette, we used counts of the number of fruits produced by each individual in a given year, and multiplied by the mean number of seeds per fruit and the mean germination probability in the next year. This study was conducted as a pre-breeding census - fecundity was estimated using a count of the seeds or other propagules that the plant produces (or a proxy thereof, such as the number of fruits multiplied by the mean number of seeds per fruit) multiplied by the probability of that propagule surviving to the next year in order to make sure that fecundity reflects demographic processes across a full time step (Kendall et al. 2019). In other systems in which offspring may be observed and tracked, counts of actual recruits may be possible, such as in studies of nesting birds, and as in Lathyrus vernus, an herbaceous perennial plant that is the subject of our second vignette.

In function-based MPMs, most matrix elements are populated by kernels that link together functions representing key demographic processes governing each transition. Typically, demographic datasets are analyzed via linear models to estimate demographic parameters such as survival probability, the probability of becoming a certain size assuming survival, and fecundity. These linear models are developed using nested subsets of the demographic data used in a study in order to estimate conditional probabilities reflecting all of the demographic processes that must occur over the span of a single time step. Matrix elements are then estimated as products of responses from these linear models set to particular inputs corresponding to stage at time t and time t+1, as well as any other parameters governing the construction of the matrix. The most common examples of function-based MPMs are integral projection projection models (IPMs), which break up a life history into many fine-scale size classes using a continuous measure of size, and then estimate survival-transitions and fecundity according to these fine-scale size classes (Ellner and Rees 2006). Function-based MPMs are more recent inventions than raw MPMs, and their strengths are making them increasingly common in the literature.

The example from Figures 1 and 2 may help to illustrate the procedure of creating a function-based MPM. In Cypripedium candidum, the simple life history model shown in Figure 1 includes one adult stage that is not observable (D - vegetative dormancy), one adult stage that is observable but not reproductive (V - vegetative sprouting), and one adult stage that is both observable and reproductive (F - flowering). If \(\sigma _{.jl}\) is the probability of survival from stage j in time t and stage l in time t-1 to any stage in time t+1, \(\psi _{.jl}\) is the probability of sprouting in time t+1 after being stage j in time t and stage l in time t-1 and surviving to time t+1, and \(\rho _{.jl}\) is the probability of flowering in time t+1 after being stage j in time t and stage l in time t-1 and surviving to and sprouting in time t+1, then

\[\begin{equation} \tag{5} a _{DDD} = \sigma _{.DD} (1 - \psi _{.DD}) \end{equation}\]

\[\begin{equation} \tag{6} a _{VDD} = \sigma _{.DD} \psi _{.DD} (1 - \rho _{.DD}) \end{equation}\]

\[\begin{equation} \tag{7} a _{FDD} = \sigma _{.DD} \psi _{.DD} \rho _{.DD} \end{equation}\]

In both raw and function-based MPMs, most estimated elements are survival-transition probabilities. In ahMPMs, these give the probability of an individual in stage j at time t surviving and transitioning to stage k at time t+1. Fewer elements are devoted to fecundity, although life history models with more reproductive stages and more recruit stages will have more fecundity elements. In Figure 1, fecundity is shown in the top-right of the matrix, as the mean production of seeds either dormant or germinating in the next time interval. Ignoring the fecundity elements, we have a survival-transition matrix (symbolized as either U or T), in which the column sums correspond to the expected survival probabilities of individuals in each stage from time t to time t+1. Ignoring the survival-transition terms, we have the fecundity matrix (symbolized as F), in which the column sums correspond to the expected overall fecundity of individuals in these respective stages. The full MPM is the matrix sum of the survival-transition matrix and the fecundity matrix.

In principle, everything just noted about function-based ahMPMs also applies to function-based hMPMs. However, there is a key difference in terms of the distribution of elements within the matrix. The inclusion of stage or condition in time t-1 increases both the number of survival-transition elements and the number of fecundity elements, and also has the effect of dispersing the locations of these elements across the matrix. Whereas estimated transitions in ahMPMs will generally occur in dense patches of estimated elements, in hMPMs, most elements are structural 0s, with elements occurring either individually surrounded by 0s, or in small patches of estimated elements surrounded by 0s. This makes error-checking more challenging, because the user needs to be aware of where elements are supposed to occur in order to check their values properly.

Different parameterizations of the historical MPM

de Vries and Caswell (2018) detail different approaches to the development of hMPMs, differing in how state or stage in time t-1 is incorporated into the matrix. Full prior stage dependence models deal with history by incorporating prior condition as the exact stage of an organism in time t-1, yielding a 2d matrix showing stage pairs in time t-1 and t along the columns of the matrix, and stage pairs in time t and t+1 along the rows of the matrix. Prior condition models deal with history by making the transition from stage t to stage t+1 a function of stage at time t and condition in time t-1. Prior condition can be determined in the same way that current stage is determined (e.g. size classification), or a different measure of condition can be used, such as growth (i.e. the change in size between times t-1 and t).

One key feature proposed by de Vries and Caswell (2018) is the addition of a new stage to account for the prior status of newborn individuals. This reflects a different interpretation from Ehrlén (2000) of the historical transition. In Ehrlén (2000), all matrix simply reflect the order of the events in a single transition, including both fecundity and survival-transition events. However, in de Vries and Caswell (2018), these matrix elements must also reflect the history of specific individuals. In the latter case, the fact that a newborn in time t did not exist in time t-1 means that a new prior stage is constructed and used in the matrix. This new stage only exists in the prior time, and is only used for newborns, particularly in fecundity transitions, and in the first survival transition from birth. So, for example, in a 3 stage MPM where the 1st stage is the newborn stage and only the 3rd stage is reproductive, we add a 4th stage to the prior portion of the row and column. Thus, we may start with the following hMPM in Ehrlén (2000) format:

\[\begin{equation} \tag{8} \tiny \left[\begin{array} {rrr} n _{2,AA} \\ n _{2,BA} \\ n _{2,CA} \\ n _{2,AB} \\ n _{2,BB} \\ n _{2,CB} \\ n _{2,AC} \\ n _{2,BC} \\ n _{2,CC} \end{array}\right] = \left[\begin{array} {rrr} S _{AAA} & 0 & 0 & S _{AAB} & 0 & 0 & S _{AAC} & 0 & 0 \\ S _{BAA} & 0 & 0 & S _{BAB} & 0 & 0 & S _{BAC} & 0 & 0 \\ S _{CAA} & 0 & 0 & S _{CAB} & 0 & 0 & S _{CAC} & 0 & 0 \\ 0 & S _{ABA} & 0 & 0 & S _{ABB} & 0 & 0 & S _{ABC} & 0 \\ 0 & S _{BBA} & 0 & 0 & S _{BBB} & 0 & 0 & S _{BBC} & 0 \\ 0 & S _{CBA} & 0 & 0 & S _{CBB} & 0 & 0 & S _{CBC} & 0 \\ 0 & 0 & S _{ACA} + F _{ACA} & 0 & 0 & S _{ACB} + F _{ACB} & 0 & 0 & S _{ACC} + F _{ACC} \\ 0 & 0 & S _{BCA} & 0 & 0 & S _{BCB} & 0 & 0 & S _{BCC} \\ 0 & 0 & S _{CCA} & 0 & 0 & S _{CCB} & 0 & 0 & S _{CCC} \\ \end{array}\right] \left[\begin{array} {rrr} n _{1,AA} \\ n _{1,BA} \\ n _{1,CA} \\ n _{1,AB} \\ n _{1,BB} \\ n _{1,CB} \\ n _{1,AC} \\ n _{1,BC} \\ n _{1,CC} \end{array}\right] \end{equation}\]

This hMPM becomes as follows in de Vries and Caswell (2018) format:

\[\begin{equation} \tag{9} \tiny \left[\begin{array} {rrr} n _{2,AA} \\ n _{2,BA} \\ n _{2,CA} \\ n _{2,AB} \\ n _{2,BB} \\ n _{2,CB} \\ n _{2,AC} \\ n _{2,BC} \\ n _{2,CC} \\ n _{2,AP} \\ n _{2,BP} \\ n _{2,CP} \end{array}\right] = \left[\begin{array} {rrr} S _{AAA} & 0 & 0 & S _{AAB} & 0 & 0 & S _{AAC} & 0 & 0 & S _{AA,AP} & 0 & 0 \\ S _{BAA} & 0 & 0 & S _{BAB} & 0 & 0 & S _{BAC} & 0 & 0 & S _{BA,AP} & 0 & 0 \\ S _{CAA} & 0 & 0 & S _{CAB} & 0 & 0 & S _{CAC} & 0 & 0 & S _{CA,AP} & 0 & 0 \\ 0 & S _{ABA} & 0 & 0 & S _{ABB} & 0 & 0 & S _{ABC} & 0 & 0 & 0 & 0 \\ 0 & S _{BBA} & 0 & 0 & S _{BBB} & 0 & 0 & S _{BBC} & 0 & 0 & 0 & 0 \\ 0 & S _{CBA} & 0 & 0 & S _{CBB} & 0 & 0 & S _{CBC} & 0 & 0 & 0 & 0 \\ 0 & 0 & S _{ACA} & 0 & 0 & S _{ACB} & 0 & 0 & S _{ACC} & 0 & 0 & 0 \\ 0 & 0 & S _{BCA} & 0 & 0 & S _{BCB} & 0 & 0 & S _{BCC} & 0 & 0 & 0 \\ 0 & 0 & S _{CCA} & 0 & 0 & S _{CCB} & 0 & 0 & S _{CCC} & 0 & 0 & 0 \\ 0 & 0 & F _{AP,CA} & 0 & 0 & F _{AP,CB} & 0 & 0 & F _{AP,CC} & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array}\right] \left[\begin{array} {rrr} n _{1,AA} \\ n _{1,BA} \\ n _{1,CA} \\ n _{1,AB} \\ n _{1,BB} \\ n _{1,CB} \\ n _{1,AC} \\ n _{1,BC} \\ n _{1,CC} \\ n _{1,AP} \\ n _{1,BP} \\ n _{1,CP} \end{array}\right] \end{equation}\]

Note that matrix 9 above can be reduced by two rows and 2 columns, since transitions from the newborn prior stage (P) can only be to the newborn stage (A).

Package lefko3 generally implements the full prior dependence model approach, particularly in the development of raw matrices. However, in principle, function-based matrices developed with lefko3 can be seen as falling within the prior condition model approach where prior condition is determined in the same way as current stage. We also implement both the Ehrlén (2000) format and the de Vries and Caswell (2018) format, and use the former as the default. We leave it to the user to decide which approach to use, and point out only that the literature suggests that maternal condition can have long-term effects on offspring survival, for example through maternal care and epigenetic influences (Descamps et al. 2008; Beamonte-Barrientos et al. 2010; Dogra and Dani 2019).

de Vries and Caswell (2018) also suggest that hMPMs can be organized differently to yield more intuitive models. In particular, matrices may be as organized as block matrices where the component blocks are essentially matrices showing the transitions from all stages in time t to stages in time t+1 conditioned on individuals having been in the same stage in time t-1. We refer to these block component matrices as conditional matrices, since they show transitions from time t to time t+1 conditional on the same previous stage in time t-1. We have developed function cond_hmpm() to take full hMPMs and decompose them into their associated conditional matrices.


Step 1: Life history model development

The basic workflow to analysis with package lefko3 starts with the development of a life history model that encapsulates all of the appropriate life stages relevant to population dynamics. We do not present a complete discussion on this step, instead focusing on a sketch of the process focused on the life cycle graph and noting key issues related to this package. We encourage users to explore the literature on this subject, but particularly note chapters 3 and 4 in Caswell (2001) for a good treatment useful for beginners. Stages need to be defined carefully, and should strongly account for variability in vital rates. We also note Kendall et al. (2019) as a good reference detailing common problems in MPM construction and how to avoid them through proper development and operationalization of life history models. Beissinger and Westphal (1998), Wardle (1998), and Salguero-Gómez and Casper (2010) provide good discussions of the proper application of life history models to understand population dynamics through the MPM approach.

Here we utilize a life cycle graph approach to build a life history model. A life cycle graph shows the life history of an organism as a series of nodes and arrows. The nodes represent unique stages that the individual may go through during development, and individuals are noted as occurring in these stages during monitoring sessions conducted over a roughly or strictly regular frequency, such as at the start of the growing season every year, or over specific dates each year. Arrows represent transitions between stages across consecutive times. These may be interpreted either as probabilities of survival when reflecting the individual’s stage across time, or as rates when reflecting the production of offspring. A good life cycle graph, and in general a good life history model, must explicitly include representations of all stages possible for the organism to enter, and all biologically plausible transitions between these transitions. Figure 1a shows an example of a life cycle graph.

The construction of a good life history model is not as easy as it might appear, particularly because it can be influenced by factors seemingly external to the life history of the organism. Some of the most important considerations outside of the species’ biology include the size of the dataset, and the decision of whether to create a raw MPM or a function-based MPM. All else being equal, a larger dataset and a function-based MPM will allow more stages to be used in the life history. Assuming a dataset of some fixed size, raw MPMs will include more and more 0s as the number of stages in the life history model increases. This happens because the chance of having any individuals actually move through a specific transition decreases as the number of possible transitions increases. These functional 0s that become increasingly common with increasing stages can cause problems in analysis, because they suggest that some transitions are impossible when, in fact, individuals moving through them were missing just by chance. Even a transition with extremely high probability may be missing just by chance. One example of an impact of this is the reduction of the mean transition within an element-wise mean matrix, causing the predicted population growth rate to be artificially low.

Function-based MPMs are better able to handle large numbers of matrix elements because overparameterization is prevented by parsimonious model selection when vital rate models are determined. However, the size of the dataset influences the statistical power of these vital rate models, with smaller datasets yielding larger error associated with matrix elements. Loss of statistical power in vital rate models can lead to the loss of important process variance, and can make the population appear more static than it really is. Chapter 6 in Morris and Doak (2002) provides a good description of techniques that may be used to define stages and to determine the exact number to use. Ellner and Rees (2006) provides some discussion of this issue within the context of integral projection model development.

Once a life history model is developed, we need to characterize the main life stages in a way that is relevant to the dataset. We do this with the sf_create() function, which creates a stageframe object. This term is a combination of the terms ‘life history stage’ and ‘data frame,’ and the object itself is a data frame that describes all of the life history stages. This object is created in a way that shows how stage relates to size, reproductive status, observation status, propagule status, maturity status, and presence in the dataset, and in which the description of each life stage is unique. A good stageframe describes stages in ways that matrix-estimating functions utilize to compute elements accurately, and also allows the user to stipulate extra descriptive information on each stage as text. If the user wishes to create an IPM, then this function also allows that to be specified in a way that simplifies calculations.

Step 2a: Dataset organization

Once the stageframe has been created, the demographic data needs to be organized into a format that lefko3 can use. The proper format is one that we term historically-formatted vertical format, or hfv for short. We have developed two functions for this purpose, depending on the format of the data. If the data is arranged horizontally, meaning that individual life histories are recorded in a spreadsheet with rows corresponding to unique individuals and columns corresponding to their condition at different times, then the verticalize3() function can turn this dataset into the proper format. If the dataset is in vertical, ahistorical format, in which an individual’s condition across time is recorded across rows in a spreadsheet, with a single row corresponding to a single observation, then the historicalize3() function can organize the data properly. Both functions are powered by binaries built in C++ using single control loops to propagate each data frame properly and efficiently, and so run very quickly (typically under 1s on a 2019 MacBook Pro). The case studies included in lefko3 illustrate the usage of these functions on real data.

Step 2b: Provide supplemental information for matrix estimation

MPMs are often estimated only partially from available demographic datasets. Some transitions are parameterized using information gathered from other studies, whether through direct input in the matrix or through the development of kernels contingent on external information. Other transitions might also be estimated via proxy transitions elsewhere in the matrix. In lefko3, this information can be provided in one of two ways. The preferred, and most recently developed, method is through the supplement table, which can be developed using the supplemental() function. This function allows users to create a data frame detailing:

  1. specific transitions to set as constants,
  2. specific transitions to estimate via proxy transitions, and
  3. specific multipliers for fecundity in cases in which fecundity estimates resulting from linear modeling must be modified to characterize the full transition.

Examples might illustrate where this approach is useful. If I lack my own data on subterranean juvenile stages in a plant species, but I have estimates of survival for those stages form another study, then I might use those estimates as constants in the MPM. Further, if I lack demographic data on the development of these stages to the seedling or earliest adult stage, but I have reason to believe that the survival probabilities should be similar to the survival within the same stage of seedlings or the smallest adults, then I can use the latter survival-transitions as proxies. Finally, if fecundity is a function of seed production, survival to the next year, and germination probability, then germination probability might be estimated via a separate field germination study, and then used to develop the fecundity kernel in the MPM by way of a multiplier term. Supplement tables provide a means to include all of this information.

Function supplemental() was recently developed to provide a robust, general solution to the addition of these data to MPMs. An alternative, older method exists that is composed of two parts. First, the user may specify specific rates and probabilities to incorporate directly into the matrix via overwrite tables, which can be created using the overwrite() function. This function provides an opportunity to show the specific transition, in either historical or ahistorical format, along with the exact value to use. It also allows the user to specify whether a specific transition should be equal to another proxy transition to be estimated within the matrix estimation procedure. This makes it functionally similar to supplemental(), although it lacks the latter’s ability to incorporate fecundity multipliers, as well as some of its error-checking power. Users choosing to use the overwrite() approach may also need to incorporate fecundity multipliers, in which case they can do so by creating a numeric matrix equal in both dimension and row/column designation to the ahistorical matrix associated with the study. We refer to this as a reproductive matrix. Assuming the same order of stages as in the associated stageframe, the user can create a zero matrix and add fecundity elements set to the appropriate multipliers. Changing all fecundity elements to 1 simply yields fecundity estimated as equal to whatever variable corresponds to fecundity in the dataset. Multiplying by, say, 0.25 will yield estimates 1/4 of that magnitude. The default behavior of lefko3 if neither a reproductive matrix nor a supplemental() table is supplied is to assume that all fecundity is to the earliest stage supplied in the stage frame, where the earliest stage would be the first propagule stage, or the first immature stage if no propagule stage is noted, or the first overall stage if only mature stages are provided.

Step 3: Assess whether historical MPM is justified, and develop models of demographic parameters

Historical MPMs are not necessarily justified in all cases. We urge the reader to use the principle of parsimony in deciding this. For example, users may analyze whether lagged effects of previous stage appear to influence vital rates. Package lefko3 includes one method to do this based on linear modeling. A number of other methods exist, mostly focused on what are known as memory models (Brownie et al. 1993; Pradel, Wintrebert, and Gimenez 2003; Cole et al. 2014). Here, we focus only on the lefko3 method, and leave it to the reader to explore other options.

In lefko3, function modelsearch() allows users to explore whether history influences component vital rates, and to develop linear models of these vital rates. The results can be used to decide not only whether a historical MPM is justified, but also to develop function-based hMPMs and ahMPMs themselves. Package lefko3 can estimate linear models to estimate up to nine different vital rates:

  1. Survival probability - This is the probability of surviving from time t to time t+1, given that the individual is in stage j in time t (and, if historical, in stage l in time t-1). In lefko3, this parameter may be modeled as a function of size, reproductive status, patch, year, age, a number of individual or environmental covariates, and individual identity in times t and t-1. This parameter is required in all function-based matrices.

  2. Observation probability - This is an optional parameter denoting the probability of observation in time t+1 of an individual in stage k given survival from time t to time t+1. This parameter is only used when at least one stage is technically not observable. For example, some plants are capable of vegetative dormancy, in which case they are alive but do not necessarily sprout in all years. In these cases, the probability of sprouting may be estimated as the observation probability. Note that this probability does not refer to observer effort, and so should only be used to differentiate completely unobservable stages where the observation status refers to an important biological phenomenon, such as when individuals may be alive but have a size of 0. In lefko3, this parameter may be modeled as a function of size, reproductive status, patch, year, age, a number of individual or environmental covariates, and individual identity in times t and t-1.

  3. Size transition probability - This is the probability of becoming size k in time t+1 assuming survival from time t to time t+1 and observation in that time. In lefko3, this parameter may be modeled as a function of size, reproductive status, patch, year, age, a number of individual or environmental covariates, and individual identity in times t and t-1. This parameter is required in all function-based size-classified matrices.

  4. Reproduction probability - This is an optional parameter denoting the probability of reproducing in time t+1 given survival from time t to time t+1, and observation in that time. Note that this should be used only if the researcher wishes to separate breeding from non-breeding mature stages. If all adult stages are potentially reproductive and no separation of reproducing from non-reproducing adults is desired, then this parameter is not needed. In lefko3, this parameter may be modeled as a function of size, reproductive status, patch, year, age, a number of individual or environmental covariates, and individual identity in times t and t-1.

  5. Fecundity rate - Under the default setting, this is the rate of successful production of offspring in time t by individuals alive, observable, and reproductive in that time, and, if provided in the dataset, the survival of those offspring into time t+1 in whatever juvenile class is possible. Thus, the fecundity rate of seed-producing plants might be split into seedlings, which are plants that germinated within a year of seed production, and dormant seeds. Alternatively, it may be given only as produced fruits or seeds, with the survival and germination of seeds provided elsewhere in the MPM development process, such as within a supplement table. An additional setting allows fecundity rate to be estimated using data provided for time t+1. In lefko3, this parameter may be modeled as a function of size, reproductive status, patch, year, age, a number of individual or environmental covariates, and individual identity in times t and t-1.

  6. Juvenile survival probability - This is an optional parameter that is used to model the probability of surviving from juvenile stage j in time t to a mature stage in time t+1. It is used only when the user wishes to model juvenile vital rates separately from adults. In lefko3, this parameter may be modeled as a function of size, patch, year, a number of individual or environmental covariates, and individual identity in times t and t-1.

  7. Juvenile observation probability - This is an optional parameter denoting the probability of observation in time t+1 of an individual in mature stage k given survival from a juvenile stage in time t to time t+1. It is used only when the user wishes to model juvenile vital rates separately from adults. In lefko3, this parameter may be modeled as a function of size, patch, year, a number of individual or environmental covariates, and individual identity in times t and t-1, and all other caveats noted in (2) above apply.

  8. Juvenile size transition probability - This is an optional parameter denoting the probability of becoming mature size k in time t+1 assuming survival from juvenile stage j in time t to time t+1 and observation in that time. It is used only when the user wishes to model juvenile vital rates separately from adults. In lefko3, this parameter may be modeled as a function of size, patch, year, a number of individual or environmental covariates, and individual identity in times t and t-1.

  9. Juvenile reproduction probability - This is an optional parameter denoting the probability of reproducing in mature stage k in time t+1 given survival from juvenile stage j in time t to time t+1, and observation in that time. It is used only when the user wishes to model juvenile vital rates separately from adults. In lefko3, this parameter may be modeled as a function of size, patch, year, a number of individual or environmental covariates, and individual identity in times t and t-1, and all other caveats in (4) apply. Note that this parameter denotes transition to maturity.

Of these nine vital rates, most users will estimate at least parameters (1) survival probability, (3) size transition probability, and (5) fecundity. These three are the default set for function modelsearch(). Parameters (2) observation probability and (4) reproduction probability may be used when some stages are included that are completely unobservable (and so do not have any size), or that are mature but non-reproductive, respectively. Parameters (6) through (9) should only be added if the dataset contains juvenile individuals transitioning to maturity.

Function modelsearch() handles the entire modeling process, including the development of global models, exhaustive model building, and the selection of best-fit models. Users should provide this function with information about the following:

  1. Individual history - Are the matrices to be built historical or ahistorical? If the former, then the state of the individual in time t-1 will be included in modeling.

  2. Modeling approach - Should the models be estimated as generalized linear models (GLMs) or mixed linear models? While most function-based matrix models are estimated as the former, the latter approach can account for repeated observations of the same individual by including individual identity as a random factor. Mixed models also allow time and patch to be treated as random variables.

  3. Suite of factors - Should both size and reproductive status be tested as causal factors? Should two-way interactions be included? Alternatively, should models be estimated as constants?

  4. Suite of vital rates - Which adult demographic parameters should be estimated? The defaults are (1) survival, (3) size, and (5) fecundity. Should (2) observation status or (4) reproductive status also be modeled?

  5. Juvenile vital rate estimation - Should juvenile parameters (6) through (9) also be modeled?

  6. Best-fit criterion - If a model with fewer parameters exists within 2.0 AICc units of the model with the lowest AICc, then should this model be used as the best-fit model (the default), or should the model with the lowest AICc always be chosen?

  7. Size distribution - Should size be modeled as a continuous variable under a Gaussian distribution, or as a count variable under either a Poisson or negative binomial distribution? If a count variable, then should the distribution be zero-inflated to account for excess zeroes, zero-truncated to account for a lack of zeroes, or left unaltered?

  8. Fecundity distribution - Should fecundity be modeled as a continuous variable under a Gaussian distribution, or as a count variable under either a Poisson or negative binomial distribution? If a count variable, then should the distribution be zero-inflated to account for excess zeroes, zero-truncated to account for a lack of zeroes, or left unaltered?

  9. Timing of fecundity - Function modelsearch() assumes that linear models of fecundity use a metric counted or measured in time t as the response. This applies well with most herbaceous plant datasets, where flowers or seeds produced in one year might be the fecundity response measured. However, users not wishing to follow this default behavior can use the fectime option to stipulate a fecundity metric measured in time t+1.

  10. Age - Is an age x stage MPM the main goal? Currently only ahistorical age x stage MPMs are offered in lefko3.

  11. Individual covariates and environmental state variables - Should individual or environmental covariates be tested as causal factors on vital rates?

  12. Missing values for times and patches - Should times or patches with coefficients that drop to 0 be estimated as random values from the corresponding distributions of time or patch coefficients?

  13. Censoring - Should data points marked as questionable be used or eliminated?

  14. Variable names - The names of all relevant variables in the dataset need to be specified. Note that the default behavior assumes variable names produced via the historicalize3() or verticalize3() functions, which produce standardized historical vertical datasets.

Once all inputs are provided, modelsearch() goes to work. The result is a lefkoMod object, which is a list in which the first several elements are the best-fit models developed for each vital rate. These are followed by an equivalent number of elements showing the full model tables developed and tested, followed by an element detailing the best-fit criterion used, and ending on quality control data showing the number of individuals and the number of unique transitions used in the estimation of each model. Depending on user choices, linear modeling is handled through the lm() and glm() functions in the stats package, the lmer() and glmer() functions in lme4, the glmmTMB() function in glmmTMB, the glm.nb() function in package MASS, the vglm() function in package VGAM, or the zeroinfl() function in package pscl. Exhaustive model building proceeds through the dredge() function in package MuMIn (Barton 2014). Model selection is handled through assessment of AICc and the number of parameters estimated per model (see 6. Best-fit criterion above).

If modelsearch() is set for historical analysis (historical = TRUE, the default), then the decision of whether to develop a historical MPM can be made on the basis of whether any best-fit vital rate model includes size or reproductive status in time t-1. If at least one vital rate does, then a historical MPM is justified. Otherwise, it is not. Regardless, the output can be used to create a function-based MPM in the next step.

Advanced users of lefko3 may wish to create their own models without the package’s automated model selection function. In that case, lefko3’s matrix functions can accommodate single models developed using base R (including packages stats and MASS), package lme4 (Bates et al. 2015), package glmmTMB (Brooks et al. 2017), package pscl (Zeileis, Kleiber, and Jackman 2008), and package VGAM (Yee and Wild 1996; Yee 2015). Plans are in the works to add nlme compatibility (Pinheiro et al. 2020), as well as non-linear models such as general additive models (GAMs).

Step 4: Estimate matrices

MPM creation can be accomplished with five different functions:

  1. rlefko2() - Creates raw ahistorical MPMs given a dataset, a stageframe, and either a supplement table or a reproductive matrix and an overwrite table.

  2. rlefko3() - Creates raw historical MPMs given a dataset, a stageframe, and either a supplement table or a reproductive matrix and an overwrite table.

  3. flefko2() - Creates function-based ahistorical MPMs given a dataset, a set of models, a stageframe, and either a supplement table or a reproductive matrix and an overwrite table.

  4. flefko3() - Creates function-based historical MPMs given a dataset, a set of models, a stageframe, and either a supplement table or a reproductive matrix and an overwrite table.

  5. aflefko3() - Creates function-based ahistorical age x stage MPMs given a dataset, a set of models, a stageframe, and either a supplement table or a reproductive matrix and an overwrite table.

These functions incorporate binary kernels developed to handle the estimation of matrix elements quickly and efficiently. A single run of flefko3(), for example, should be able to yield all annual matrices for all patches for the Cypripedium candidum dataset provided with lefko3 in under a minute on most machines (14s or so on RPS’ 2019 MacBook Pro with 2.3 GHz 8-Core Intel Core i9). Parallel computing should not be necessary, even with the slowest of current machines, provided that the machine is current enough to handle at least R 3.6.3.

Users wishing to check the output hMPMs for errors or simply understand more about them may utilize function cond_hmpm(). This function decomposes hMPMs into MPMs showing transitions from time t to time t+1, but conditioned on the same stage at time t-1. Thus, a single hMPM with 5 stages and 25 stage pairs would be broken down into 5 conditional MPMs, one for each stage at time t-1, with 5 columns and 5 rows denoting stage at times t and t+1, respectively. These can be examined individually, and the survival of stages from time t to time t+1 can be assessed as a function of stage in time t-1 by taking the associated column sums of these conditional matrices.

Step 5. MPM analyses

Package lefko3 includes a number of functions to aid analysis once matrices are created. These may be of greater utility in some circumstances than functions such as eigen(), because our functions are made to handle even extremely large, sparse matrices. Currently, we include functions to estimate element-wise arithmetic mean matrices, discrete and stochastic population growth rate, stable stage structure, reproductive value, and deterministic and stochastic sensitivity and elasticity. Plans are in the works for further analysis functions in the future.

Population growth rate

Package lefko3 allows the estimation of both the deterministic and stochastic population growth rates. The deterministic population growth rate is estimated with the function lambda3(), and estimates the population growth rate (\(\lambda\)) as the dominant eigenvalue of each matrix provided (technically, the largest real part of the estimated eigenvalues). Where matrices are large, as in most historical or age-by-stage cases, lambda3() first converts matrices to sparse format in order to speed up estimation.

The function slambda3() estimates the log stochastic population growth rate in its instantaneous form (\(a = \text{log} \lambda _{S}\)). This is estimated as the mean log discrete population growth rate across a user-specified number of random draws of the supplied annual matrices, where the default number of draws is 10,000:

\[\begin{equation} \tag{10} a = \text{log} \lambda _{S} = \frac{1}{T} \sum _{t=1}^{T} \text{log} \frac{N _{t}}{N _{t-1}} \end{equation}\]

where \(N _t\) is the population size in time \(t\), and \(T\) is the number of time steps projected, set to 10,000 by default. Function slambda3() does not shuffle across patches or populations, instead shuffling within patches, or shuffling annual matrices calculated as element-wise means of patch matrices within the same population and the same time. The methodology is based on Morris and Doak (2002), though accounts for spatial averaging of patches and corrects for the geometric nature of growth rates.

Stable stage distribution and reproductive value

In ahistorical analyses, the stable stage distribution and the reproductive value of stages are estimated as standardized right and left eigenvectors associated with the dominant eigenvalue of the matrix. Standardization of the stable stage distribution is handled by dividing each respective element of the right eigenvector by the sum of the elements in that eigenvector. Standardization of the reproductive value vector is handled by dividing each element in the left eigenvector by the value of the first non-zero element in that eigenvector.

The methods mentioned above apply to historical matrices as well. However, as described, they only provide the stable stage distribution and reproductive values of stage pairs. We provide two functions, stablestage3() and repvalue3() to allow the estimation of these vectors for both ahistorical and historical MPMs. When provided with a historical MPM, these functions also estimate historically-corrected stable stage distributions and reproductive value vectors, which correspond to the original stages in the associated life history model rather than the stage pairs, and so are comparable against the ahistorical stable stage distributions and reproductive value vectors. The historically-corrected stable stage proportion for stage j is estimated as the sum of all stable stage proportions for stage j in time t across all stages in time t-1, as in:

\[\begin{equation} \tag{11} SS _j = \sum _{l=1}^{m} w _{jl} \end{equation}\]

where \(l\) is stage in time t-1, \(m\) is the number of stages, \(SS_j\) is the stable stage proportion of stage j, and \(w _{jl}\) is the stable stage proportion of stage pair \(jl\) given as the standardized right eigenvector corresponding to the dominant eigenvalue of the hMPM. The historically-corrected reproductive value of stage j is calculated as the sum of all reproductive values for stage j in time t across all stages in time t-1, weighted by the stable stage proportion of the respective stage pair divided by the stable stage proportion of stage \(j\) at time t (Ehrlen 2000), as in:

\[\begin{equation} \tag{12} RV _j = \sum _{l=1}^{m} v _{jl} (w _{jl} / SS _{j}) \end{equation}\]

where \(RV _j\) refers to reproductive value of stage \(j\), and \(v _{jl}\) refers to the reproductive value of the stage pair as given by the standardized left eigenvector of the dominant eigenvalue of the historical MPM. The influence of history can make these numbers differ quite dramatically from those produced by ahistorical matrices.

We do not yet offer stochastic stable stage distribution and reproductive value estimation.

Sensitivity and elasticity

Package lefko3 contains functions allowing users to conduct deterministic and stochastic sensitivity and elasticity analyses. Sensitivity and elasticity analysis are forms of perturbation analysis, and we urge readers to consult Caswell (2001) and Caswell (2019) to become fully acquainted with the topic. Here, we discuss just the most important aspects to understand these analyses as conducted using lefko3.

The sensitivity of \(\lambda\), the deterministic population growth rate, to a specific element in a projection matrix can be calculated using eigen analysis as

\[\begin{equation} \tag{13} \frac{\partial \lambda}{\partial a _{kj}} = \frac{\bar{v} _{k} w _{j}}{<\mathbf{w}, \mathbf{v}>} \end{equation}\]

Here, \(\mathbf{w}\) is the stable stage distribution vector calculated as the right eigenvector of the dominant eigenvalue of the matrix (standardized to sum to 1.0), \(\mathbf{v}\) is the reproductive value vector calculated as the associated left eigenvector (standardized by dividing by the value of the first non-zero value), and \(\bar{v} _{k}\) is the complex conjugate of element \(k\) of \(\mathbf{v}\). \(k\) is also the stage in time t+1 and in the ahMPM refers to the corresponding row, while \(j\) refers to stage in time t and in an ahMPM refers to the corresponding column. The term \(<\mathbf{w}, \mathbf{v}>\) refers to the scalar product of these vectors.

In the hMPM case, the basic method is essentially the same as in equation 13, although the equation itself changes somewhat, with the sensitivity of \(\lambda\) to each matrix element given as

\[\begin{equation} \tag{14} \frac{\partial \lambda}{\partial a _{kjl}} = \frac{\bar{v} _{kj} w _{jl}}{<\mathbf{w}, \mathbf{v}>} \end{equation}\]

Here, \(k\) is stage in time t+1, \(j\) is stage in time t, \(l\) is stage in time t-1, \(kj\) refers both to the stage pair in times t+1 and t as well as the corresponding row in the historical matrix, and \(jl\) refers both to the stage pair in times t and t-1 as well as the corresponding column in the historical matrix. Historically-corrected sensitivities may also be estimated for the basic stages of the life history model using the historically-corrected stable stage distributions and reproductive values given in equations 11 and 12 as input in equation 13.

Sensitivities show how much a small change in a matrix element can alter \(\lambda\), but do so by estimating the local slope of \(\lambda\) given in terms of the matrix element (Caswell 2001). This makes it difficult to compare the sensitivities of transition probabilities to fecundity rates, because they are given on different scales. Sensitivities are also typically non-zero even for impossible transitions represented by 0s in the matrix. Elasticities avoid these problems by showing the influence of a proportional change in a matrix element on \(\lambda\). In the ahistorical case, the elasticity, \(e _{kj}\), of \(\lambda\) to change in a matrix element \(a _{kj}\) is given as

\[\begin{equation} \tag{15} e _{kj} = \frac{a _{kj}}{\lambda} \frac{\partial \lambda}{\partial a _{kj}} \end{equation}\]

The historical case is just an extension of the above.

\[\begin{equation} \tag{16} e _{kjl} = \frac{a _{kjl}}{\lambda} \frac{\partial \lambda}{\partial a _{kjl}} \end{equation}\]

Here, we use row and column definitions equivalent to those used in the historical sensitivity case. This makes the elasticity a function of the sensitivity of \(\lambda\) to the matrix element, as well as of the value of the matrix element itself. Thus, structural 0s in the hMPM must have elasticity equal to 0, although it is entirely possible that they have non-zero sensitivities.

Elasticities calculated for hMPMs can be compared to those calculated in ahMPMs easily because elasticities are additive, with all elasticities corresponding to a specific matrix necessarily summing to 1.0. Thus, we can calculate the stage pair elasticities resulting from a historical MPM analysis as

\[\begin{equation} \tag{17} e _{kj} = \sum_{i=1}^{m} e _{kji} \end{equation}\]

with all symbol definitions as before. We have provided a summary() function for elasticity output in lefko3 that automatically groups the resulting elasticities by the kind of ahistorical or historical transition.

Stochastic sensitivity and elasticity analysis are also available. Per Caswell (2001), the sensitivity of \(a = \text{log} \lambda _{S}\) to changes in a specific element is given as:

\[\begin{equation} \tag{18} \frac{\partial \text{log} \lambda _S}{\partial a _{kj}} = \lim_{T \to \infty} \frac{1}{T} \sum_{t=0}^{T-1} \frac{\mathbf{v}(t+1) \mathbf{w}^\top(t)}{\mathbf{v}^\top(t+1) \mathbf{w}(t+1)} \end{equation}\]

where \(t\) refers to a specific time and \(T\) refers to the number of times simulated. Similarly, the stochastic elasticity of \(a = \text{log} \lambda _{S}\) to changes in a specific element is given as:

\[\begin{equation} \tag{19} \frac{\partial \text{log} \lambda _S}{\partial \text{log} a _{kj}} = \lim_{T \to \infty} \frac{1}{T} \sum_{t=0}^{T-1} \frac{\bigl(\mathbf{v}(t+1) \mathbf{w}^\top(t) \bigr) \circ \mathbf{A_t}}{\mathbf{v}^\top(t+1) \mathbf{w}(t+1)} \end{equation}\]

where \(\mathbf{A_t}\) refers to the A matrix corresponding to time t. Stochastic sensitivities for hMPMs may be converted to historically-corrected format as in the deterministic case, and stochastic elasticities may be summed as before.

Users working with elasticities, whether deterministic or stochastic, may find the summary.lefkoElas function useful. If an elasticity analysis is conducted on a lefkoMat object, then the information in that object can be used to summarize summed elasticity according to different sorts of transitions. In the ahistorical case, elasticities are sorted by growth, stasis, shrinkage, and fecundity. In the historical case, they are sorted by transition pairs across three times (e.g. stasis followed growth vs. shrinkage followed by stasis vs. growth followed by reproduction).

Further analyses

Users can take the MPMs produced by MPM creation functions in package lefko3 and plug them into MPM or matrix analysis functions in other packages. Matrices produced by lefko3 are stored within lefkoMat objects, which are S3 objects structured as lists. These lists include a number of elements, but among the most important are elements $A, $U, and $F, which are themselves lists of complete projection matrices, survival-transition matrices, and fecundity matrices, respectively. For example, code such as matobject$A[[1]] would access the first complete projection matrix in a lefkoMat object named matobject. The $labels element is a data frame giving a description of each matrix in the order in which it occurs within the $A element, including its population, patch, and year designations. This allows the use of all functions that work with matrices, including functions in base R such as eigen(), as well as in packages such as popbio (Stubben and Milligan 2007) and popdemo (Stott, Hodgson, and Townley 2012). We encourage users to explore whether the packages and functions they wish to use can handle sparse matrices, as well as large matrices - some were not designed to and can fail or yield unpredictable behavior when applied particularly to historical matrices produced by lefko3.


We are grateful to two anonymous reviewers whose scrutiny improved the quality of this vignette. The project resulting in this package and this tutorial was funded by Grant-In-Aid 19H03298 from the Japan Society for the Promotion of Science.

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