A Bayesian approach to modeling phytoplankton population dynamics from size distribution time series

The rates of cell growth, division, and carbon loss of microbial populations are key parameters for understanding how organisms interact with their environment and how they contribute to the carbon cycle. However, the invasive nature of current analytical methods has hindered efforts to reliably quantify these parameters. In recent years, size-structured matrix population models (MPMs) have gained popularity for estimating division rates of microbial populations by mechanistically describing changes in microbial cell size distributions over time. Motivated by the mechanistic structure of these models, we employ a Bayesian approach to extend size-structured MPMs to capture additional biological processes describing the dynamics of a marine phytoplankton population over the day-night cycle. Our Bayesian framework is able to take prior scientific knowledge into account and generate biologically interpretable results. Using data from an exponentially growing laboratory culture of the cyanobacterium Prochlorococcus, we isolate respiratory and exudative carbon losses as critical parameters for the modeling of their population dynamics. The results suggest that this modeling framework can provide deeper insights into microbial population dynamics provided by size distribution time-series data.

Life history adaptations to fluctuating environments: Combined effects of demographic buffering and lability of vital rates

Demographic buffering and lability have both been identified as important adaptive strategies to optimise long-term fitness in variable environments. These strategies are not mutually exclusive, however we lack efficient methods to measure their relative importance. Here, we define a new index to measure the total lability for a given life history, and use stochastic simulations to disentangle relative fitness effects of buffering and lability. The simulations use 81 animal matrix population models, and different scenarios to explore how the strategies vary across life histories. The highest potential for adaptive demographic lability was found for short- to intermediately long-lived species, while demographic buffering was the main response in slow-living species. This study suggests that faster-living species are more responsive to environmental variability, both for positive or negative effects. Our methods and results provide a more comprehensive view of adaptations to variability, of high relevance to predict species responses to climate change.

“Realistic Choice of Annual Matrices Contracts the Range of λS Estimates” under Reproductive Uncertainty Too

Our study is devoted to a subject popular in the field of matrix population models, namely, estimating the stochastic growth rate, λS, a quantitative measure of long-term population viability, for a discrete-stage-structured population monitored during many years. “Reproductive uncertainty” refers to a feature inherent in the data and life cycle graph (LCG) when the LCG has more than one reproductive stage, but when the progeny cannot be associated to a parent stage in a unique way. Reproductive uncertainty complicates the procedure of λS estimation following the defining of λS from the limit of a sequence consisting of population projection matrices (PPMs) chosen randomly from a given set of annual PPMs. To construct a Markov chain that governs the choice of PPMs for a local population of Eritrichium caucasicum, an short-lived perennial alpine plant species, we have found a local weather index that is correlated with the variations in the annual PPMs, and we considered its long time series as a realization of the Markov chain that was to be constructed. Reproductive uncertainty has required a proper modification of how to restore the transition matrix from a long realization of the chain, and the restored matrix has been governing random choice in several series of Monte Carlo simulations of long-enough sequences. The resulting ranges of λS estimates turn out to be more narrow than those obtained by the popular i.i.d. methods of random choice (independent and identically distributed matrices); hence, we receive a more accurate and reliable forecast of population viability.

Integral projection models

Integral projection models (IPMs) allow projecting the behaviour of a population over time using information on the vital processes of individuals, their state, and that of the environment they inhabit. As with matrix population models (MPMs), time is treated as a discrete variable, but in IPMs, state and environmental variables are continuous and are related to the vital rates via generalised linear models. Vital rates in turn integrate into the population dynamics in a mechanistic way. This chapter provides a brief description of the logic behind IPMs and their construction, and, because they share many of the analyses developed for MPMs, it only emphasises how perturbation analyses can be performed with respect to different model elements. The chapter exemplifies the construction of a simple and a more complex IPM structure with an animal and a plant case study, respectively. Finally, inverse modelling in IPMs is presented, a method that allows population projection when some vital rates are not observed.

Introduction to matrix population models

Matrix population models (MPMs) are currently used in a range of fields, from basic research in ecology and evolutionary biology, to applied questions in conservation biology, management, and epidemiology. In MPMs individuals are classified into discrete stages, and the model projects the population over discrete time-steps. A rich analytical theory is available for these models, for both the linear deterministic case and for more complex dynamics including stochasticity and density dependence. This chapter provides a non comprehensive introduction to MPMs and some basic results on asymptotic dynamics, life history parameters, and sensitivities and elasticities of the long-term growth rate to projection matrix elements and to underlying parameters. We assume that readers are familiar with basic matrix calculations. Using examples with different kinds of demographic structure, we demonstrate how the general stage-structured model can be applied to each case.

Demographic Methods across the Tree of Life

Demography is everywhere in our lives: from birth to death. Demography shapes our daily decisions, as well as the decisions that others make on us (e.g. bank loans, retirement age). Demography is everywhere across the Tree of Life. The universal currencies of demography—survival, development, reproduction, and recruitment—shape the performance of all species, from lions to dandelions. The omnipresence of demography in all things alive and dead, and its multiple applications to better understand the ecology, evolution, and conservation/management of species, allows us to—in principle—apply the wide array of quantitative methods to, for example, bacteria or humans. However, demographic methods to date have remained taxonomically siloed, despite the fact that, to a large extent, they are widely applicable across the Tree of Life. In this book, we walk nonexperts through the ABCs of data collection, model construction, analyses, and interpretation across a wide repertoire of demographic artillery. This book introduces the reader to some of the demographic methods, including abundance-based models, life tables, matrix population models, integral projection models, integrated population models, and individual based models, to mention a few. Through the careful integration of data collection methods, analytical approaches, and applications, clearly guided through fully reproducible R scripts, we provide a state-of-the-art thorough representation of many of the most popular tools that any demographer (or demographically inclined mind) should equip themselves with.

A flexible Bayesian approach to estimating size-structured matrix population models

The rates of cell growth, division, and carbon loss of microbial populations are key parameters for understanding how organisms interact with their environment and how they contribute to the carbon cycle. However, the invasive nature of current analytical methods has hindered efforts to reliably quantify these parameters. In recent years, size-structured matrix population models (MPMs) have gained popularity for estimating rate parameters of microbial populations by mechanistically describing changes in microbial cell size distributions over time. And yet, the construction, analysis, and biological interpretation of these models are underdeveloped, as current implementations do not adequately constrain or assess the biological feasibility of parameter values, leading to inference which may provide a good fit to observed size distributions but does not necessarily reflect realistic physiological dynamics. Here we present a flexible Bayesian extension of size-structured MPMs for testing underlying assumptions describing the dynamics of a marine phytoplankton population over the day-night cycle. Our Bayesian framework takes prior scientific knowledge into account and generates biologically interpretable results. Using data from an exponentially growing laboratory culture of the cyanobacterium Prochlorococcus, we herein demonstrate the performance improvements of our approach over current models and isolate previously ignored biological processes, such as respiratory and exudative carbon losses, as critical parameters for the modeling of microbial population dynamics. The results demonstrate that this modeling framework can provide deeper insights into microbial population dynamics provided by flow-cytometry time-series data.

Potential-Growth Indicators Revisited: Higher Generality and Wider Merit of Indication

The notion of a potential-growth indicator came to being in the field of matrix population models long ago, almost simultaneously with the pioneering Leslie model for age-structured population dynamics, although the term has been given and the theory developed only in recent years. The indicator represents an explicit function, R(L), of matrix L elements and indicates the position of the spectral radius of L relative to 1 on the real axis, thus signifying the population growth, or decline, or stabilization. Some indicators turned out to be useful in theoretical layouts and practical applications prior to calculating the spectral radius itself. The most senior (1994) and popular indicator, R0(L), is known as the net reproductive rate, and we consider two others, R1(L) and RRT(A), developed later on. All the three are different in terms of their simplicity and the level of generality, and we illustrate them with a case study of Calamagrostis epigeios, a long-rhizome perennial weed actively colonizing open spaces in the temperate zone. While the R0(L) and R1(L) fail, respectively, because of complexity and insufficient generality, the RRT(L) does succeed, justifying the merit of indication.

Potential-Growth Indicators Revisited: More Merits of Indication

The notion of potential-growth indicator came to being in the field of matrix population models long ago, almost simultaneously with the pioneering Leslie model for age-structured population dynamics, albeit the term has been given and the theory developed only recent years. The indicator represents an explicit function, R(L), of matrix L elements and indicates the position of the spectral radius of L relative to 1 on the real axis, thus signifying the population growth, or decline, or stabilization. Some indicators turned out useful in theoretical layouts and practical applications prior to calculating the spectral radius itself. The most senior (1994) and popular indicator, R0(L), is known as the net reproductive rate, and we consider two more ones, R1(L) and RRT(A), developed later on. All the three are different in what concerns their simplicity and the level of generality, and we illustrate them with a case study of Calamagrostis epigeios, a long-rhizome perennial weed actively colonizing open spaces in the temperate zone. While the R0(L) and R1(L) fail respectively because of complexity and insufficient generality, the RRT(L) does succeed, justifying the merit of indication.