Chapter 1 Introduction

1.1 What are matrix population models?

The population models that are taught in undergraduate ecology courses—the exponential growth model, logistic model, Ricker model, etc.—are useful for explaining and illustrating basic patterns in population dynamics, such as patterns of exponential growth, the effects of density dependence, and phenomena such as population cycles and chaos. Their utility stems from their simplicity, which allows them to be analyzed without recourse to pages of advanced math. In particular, most of these foundational models assume that all individuals in the population can be treated as if they have identical birth and death rates.

As a practicing ecologist or conservation biologist, you know first-hand that all individuals in a population are not identical: they age, develop, and grow; and birth and death rates (which I will collectively call “demography”) often vary systematically with age, developmental stage, and size. Individuals can differ in other important ways as well: for example, individuals of a given age may exhibit morphological variation resulting from genetic differences or heterogeneity in their developmental environment; or they might exhibit different behavioral responses to a given stimulus resulting from variation in prior experience. These phenotypic differences can also affect demography.

Ultimately, an individual’s short-term fitness (its survival probability or mortality risk, as well as its expected birth rate) depends on both its state (age, stage, or size), its phenotype, and the environment (encompassing abiotic conditions and resources as well other organisms with which it interacts) in which it lives. Simple models that characterize the population simply by its abundance, density, or biomass cannot represent this biological richness; thus, they do no provide insight on whether this richness matters for population dynamics, or how it might be exploited to manage the population (e.g., to arrest the decline of a rare species). On the other hand, a model that encompassed all of this, together with all of the feedback loops between the focal species and the ecological community it’s embedded in, would be tremendously complex (browse any issue of the journal Ecological Modelling to see examples) and would require tremendous amounts of data to estimate all the parameters.

Matrix population models (MPMs) are a compromise with intermediate complexity. They assume that individuals can be classified into a finite number of discrete states, and that an individual’s state fully determines the individual’s reproductive, survival, and growth rates in a given environment.

MPMs also use a discrete time step; this is usually one year for models of free-living organisms, but may be shorter or longer. Thus, the model implicitly integrates over events occurring within the time step.

In addition, the standard MPM assumes no density dependence in demography. This allows access to a variety of analytic tools, but means that long-term projections should be taken with a large dash of salt.

The “matrix” in matrix population models does not have any biological meaning. Rather, it comes from use of matrix algebra to both make representation more compact and to give access to analytic and computational tools

The standard models only require information on survival fraction and mean birth rates across individuals within each class, as well as information on the probability that an individual moves from one class to another. This information might be assembled in a life table, especially if the population is classified by age, but this is not a requirement. This information can come from a variety of data sources, each with strengths and weaknesses (see chapter XX). If you have data from a single year, or estimates that are multi-year averages, then you can construct a single matrix. With estimates from multiple years, you can construct a matrix for each year, which allows you to construct a stochastic MPM that allows you to examine the effects of environmental variability.

1.2 What are matrix population models used for?

You may be reading this book because you have collected demographic data on a population for some other objective, and a colleague or reviewer said “You should use that data to construct a matrix population model!” Why would you want to do that?

Matrix population models have been extensively used in studies of life history evolution as well as for addressing management questions for endangered or invasive species (fisheries scientists tend not to use the MPM formalism for their age-structured models). In the literature, we find that most MPMs are used to calculate the populations deterministic or stochastic asymptotic population growth rate, together with sensitivities of that growth rate to underlying model parameters. The asymptotic growth rate, generally denoted \(\lambda\) for deterministic MPMs and \(\log \lambda_S\) for stochastic MPMs, can be viewed as an integrated measure of population performance (for population management problems) or as fitness (for evolutionary questions). The sensitivities (and their cousins, elasticities) of the asymptotic growth rate provides information about the likely effect of a management intervention or response to selection, respectively.

Quite a few other quantities can be calculated from an MPM, and sensitivities of these quantities to various model components can be analyzed ((???)). However, few of these have been taken up to the degree that the analysis of the asymptotic growth rate has. Recent and ongoing comparative analyses of MPMs in the COMADRE and COMPADRE databases may reveal which of these analyses are particularly useful for various purposes.

1.3 A brief history of matrix population models

1.4 Common pitfalls in matrix population model construction

If you know the life history of your organism, it is usually straightforward to draw a conceptual diagram that shows all of the class levels an individual could be in (e.g., ages or stages), and arrows linking them (Fig. X). The arrows are of two kinds: the first describes what happens to existing individuals (survival, aging, maturation, growth) and the other relates to the production of new individuals through reproduction. This diagram is a highly intuitive but qualitative representation of the life cycle. When converting this into a model for projecting the population into the future, we need to be precise in our bookkeeping.

It turns out to be surprisingly difficult to introduce this precision. Kendall et al. (2019) examined the prevalence of three common mistakes in the construction of animal MPMs, and found error rates as high as 62%, with no trend in error rates over time. I won’t belabor the details of those errors here; see the publication if you are interested. Based on teaching MPMs to graduate students for 20 years, I have concluded that there are three general factors leading to high error rates persisting despite the fact that most textbooks describe the issues:

  1. Linguistic ambiguity arises when biologically meaningful terms are applied to the model in ways that don’t quite match up. For example, “yearling” and “young of year” refer to individuals in their first year of life. In an age-structured MPM, individuals need to be classified as “newborn” or “one year old” (i.e., having just reached their first birthday). Which of those does the term “yearling” apply to? As a second example, the terms “fecundity,” “fertility,” and “reproduction” are all used (often interchangeably) to refer to the production of new individuals by parents. However, each of these terms as specific (and sometimes contrasting) meanings in various fields of biology; when this doesn’t match the meaning as applied to the MPM, it can lead to confusion.
  2. The correct bookkeeping can lead to terms in the model that seem to defy biological intuition (this is true in financial bookkeeping as well!), leading non-modelers to discard these terms in favor of ones that match intuition. For example, in some types of MPMs that require a direct link from non-reproductive juveniles to newborns.
  3. Not all processes in the life cycle span the same amount of time, yet the model requires a consistent timestep. For example, birth or egg laying occurs on timescales of seconds to days, maturing from a newly laid egg to an independent fledgling may take a couple of months, aging a year takes a year, and reaching reproductive maturity may take decades. Fitting these into a model with an annual time step requires explicitly integrating across events occurring within a year, and appropriately subdividing processes that take more than a year. Doing these leads to complicated mathematical expressions that even experienced modelers make mistakes with.

If one can “think like a modeler” then the complexities of MPM construction make sense. Most, if not all, existing textbooks try to teach you to think like a modeler. But if you are trained to think like a biologist, or ecologist, or conservationist, then learning to reliably “think like a modeler” is a big task, probably only worth doing if modeling is going to be an important component of your future work.

My goal in this book is to take a different approach. It turns out that most of the bookkeeping that generates complex and non-intuitive terms is entirely rote, and can be attained through simple matrix multiplication (which can be done either by hand or by a computer). I provide you with recipes for shaping your biological information (e.g., age-specific birth and survival rates) into forms from which a turn of the matrix multiplication machinery will produce an appropriate MPM. Furthermore, I try to link the notation as closely as possible to the biology. Finally, I try to provide clear guidance where you need to think about your biological information in a very specific way before putting it into the model.

1.5 Mathematical essentials

1.5.1 Notation

There is no getting around the fact that MPMs are mathematical constructs. However, I will try to make the notation be as closely tied to the biology as possible. One challenge is that we need a consistent way to reference demographic rates of specific age, size, or stage classes. In mathematical notation, \(p_i\) is the survival rate of individuals in the \(i\)’th class. Thus, for example, in an age-structured model with age classes 0, 1, 2, and so on, \(p_2\) is the survival of one-year-olds. In a stage-structured model with stage classes newborn, juvenile, sub-adult and adult, then \(p_2\) is the survival of juveniles. Keeping track of this mapping requires extra cognitive work, and can be especially confusing in age-structured models where the mathematical index and the biological age don’t align.4 Thus I introduce a biological notation, in which the actual age or the name of the class is shown between parentheses in the subscript. Thus, in an age-structured model, \(p_{(2)}\) is the survival of two-year-olds (survival from age 2 to age 3). In a stage-structured model, \(p_{(\text{sub-adult})}\) is the annual survival of individuals who start the year in the sub-adult stage.

This distinction is also useful in referencing the vector of population abundances in year \(t\), \(\mathbf{n}(t)\). As we will see, MPMs can be constructed in different ways that either do or do not include newborn individuals in the population vector. In that case, careful attention to context is required to know what \(n_1(t)\) (the first element of \(\mathbf{n}(t)\)) refers to. In contrast, \(n_{(0)}(t)\) vs. \(n_{(1)}(t)\), or \(n_{(\text{newborn})}(t)\) vs. \(n_{(\text{juvenile})}(t)\), is clear and unambiguous.

Transition notation

1.5.2 Matrix multiplication

1.6 Matrix population models in R

References

Kendall, B. E., M. Fujiwara, J. Diaz-Lopez, S. Schneider, J. Voigt, and S. Wiesner. 2019. “Persistent problems in the construction of matrix population models.” Ecological Modelling 406: 33–43.

  1. One could, of course, start the mathematical index at zero, and you will see this in some textbooks; but this leads to inconsistencies between age-structured and other models. It also doesn’t work in R, where the indexing of a vector always starts at one.↩