Johan Metz

Leiden University | LEI · Institute of Biology Leiden

Traditionally, population models distinguish individuals on the basis of their current state. Given a distribution, a discrete time model then specifies (precisely in deterministic models, probabilistically in stochastic models) the population distribution at the next time point. The renewal equation alternative concentrates on newborn individ- ual...

The time-honoured paradigm in the theory of virulence evolution assumes a positive relation between infectivity and harmfulness. However, the aetiology of respiratory diseases yields a negative relation, with diseases of the lower respiratory tract being less infective and more harmful. We explore the evolutionary consequences in a simple model inc...

Traditionally, population models distinguish individuals on the basis of their current state. Given a distribution, a discrete time model then specifies (precisely in deterministic models, probabilistically in stochastic models) the population distribution at the next time point. The renewal equation alternative concentrates on newborn individuals...

We review the importance of developmental mechanisms in animals in constraining evolutionary changes. We first discuss the importance of time scales at which such constraints are relevant and after that focus on near absolute constraints that act on macroevolutionary scales. We could find only a few well-underpinned examples of such near absolute c...

In the original publication of the article, the Subsection 2.1.2 was published incorrectly.

A set of axioms is formulated characterizing ecologically plausible community dynamics. Using these axioms, it is proved that the transients following an invasion into a sufficiently stable equilibrium community by a mutant phenotype similar to one of the community's finitely many resident phenotypes can always be approximated by means of an approp...

This article has been withdrawn: please see Elsevier Policy on Article Withdrawal (http://www.elsevier.com/locate/withdrawalpolicy). This article has been withdrawn at the request of the editor and publisher. The publisher regrets that an error occurred which led to the premature publication of this paper. This error bears no reflection on the arti...

Many species are subject to seasonal cycles in resource availability, affecting the timing of their reproduction. Using a stage-structured consumer-resource model in which juvenile development and maturation are resource-dependent, we study how a species’ reproductive schedule evolves dependent on the seasonality of its resource. We find three qual...

Considering the environmental condition as a given function of time, we formulate a physiologically structured population model as a linear non-autonomous integral equation for the, in general distributed, population level birth rate. We take this renewal equation as the starting point for addressing the following question: When does a physiologica...

In a physiologically structured population model (PSPM) individuals are characterised by continuous variables, like age and size, collectively called their i-state. The world in which these individuals live is characterised by another set of variables, collectively called the environmental condition. The model consists of submodels for (i) the dyna...

We consider the question of when delay systems, which are intrinsically infinite dimensional, can be represented by finite dimensional systems. Specifically, we give conditions for when all the information about the solutions of the delay system can be obtained from the solutions of a finite system of ordinary differential equations. For linear aut...

We review the evolutionary importance of developmental mechanisms in constraining evolutionary changes in animals—in other words, developmental constraints. We focus on hard constraints that can act on macroevolutionary timescales. In particular, we discuss the causes and evolutionary consequences of the ancient metazoan constraint that differentia...

A widespread tenet is that evolution of pathogens maximises their basic reproduction ratio, R0. The breakdown of this principle is typically discussed as exception. Here, we argue that a radically different stance is needed, based on evolutionarily stable strategy (ESS) arguments that take account of the 'dimension of the environmental feedback loo...

An organism's life history is closely interlinked with its allocation of energy between growth and reproduction at different life stages. Theoretical models have established that diminishing returns from reproductive investment promote strategies with simultaneous investment into growth and reproduction (indeterminate growth) over strategies with d...

Infectious diseases still generate huge socio-economic costs and many people would like to see them gone entirely. While success stories like with smallpox and rinderpest give hope that this may be possible, many other eradication attempts have failed. Eradication requires huge and costly efforts, which can only be sustained if sufficient progress...

This paper should be read as addendum to Dieckmann et al. (J Theor Biol 241:370-389, 2006) and Parvinen et al. (J Math Biol 67: 509-533, 2013). Our goal is, using little more than high-school calculus, to (1) exhibit the form of the canonical equation of adaptive dynamics for classical life history problems, where the examples in Dieckmann et al. (...

Over the last two decades evolutionary branching has emerged as a possible mathematical paradigm for explaining the origination of phenotypic diversity. Although branching is well understood for one-dimensional trait spaces, a similarly detailed understanding for higher dimensional trait spaces is sadly lacking. This note aims at getting a research...

The fitness concept and perforce the definition of frequency independent fitnesses from population genetics is closely tied to discrete time population models with non-overlapping generations. Evolutionary ecologists generally focus on trait evolution through repeated mutant substitutions in populations with complicated life histories. This goes wi...

Significance Our study explains one of the riddles of mammal evolution: the strong conservation of the number of trunk vertebrae. The vertebral column and its high evolvability are considered to be of central importance for the evolution of vertebrates, which is why the constancy is both puzzling and important. We hypothesize, on biomechanical and...

One of the powerful tools of adaptive dynamics is its so-called canonical equation (CE), a differential equation describing how the prevailing trait vector changes over evolutionary time. The derivation of the CE is based on two simplifying assumptions, separation of population dynamical and mutational time scales and small mutational steps. (It ma...

We consider a continuous time stochastic individual based model for a population structured only by an inherited vector trait and with logistic interactions. We consider its limit in a context from adaptive dynamics: the population is large, the mutations are rare and we view the process in the timescale of mutations. Using averaging techniques due...

The class of deterministic 'Daphnia' models treated by Diekmann et al. (J Math Biol 61:277-318, 2010) has a long history going back to Nisbet and Gurney (Theor Pop Biol 23:114-135, 1983) and Diekmann et al. (Nieuw Archief voor Wiskunde 4:82-109, 1984). In this note, we formulate the individual based models (IBM) supposedly underlying those determin...

Part of the art of theory building is to construct effective basic concepts, with a large reach and yet powerful as tools for getting at conclusions. The most basic concept of population biology is that of individual. An appropriately reengineered form of this concept has become the basis for the theories of structured populations and adaptive dyna...

Recently, de-Camino-Beck and Lewis (Bull Math Biol 69:1341-1354, 2007) have presented a method that under certain restricted conditions allows computing the basic reproduction ratio [Formula: see text] in a simple manner from life cycle graphs, without, however, giving an explicit indication of these conditions. In this paper, we give various sets...

Body size (≡ biomass) is the dominant determinant of population dynamical processes such as giving birth or dying in almost all species, with often drastically different behaviour occurring in different parts of the growth trajectory, while the latter is largely determined by food availability at the different life stages. This leads to the questio...

Adaptive dynamics (AD) so far has been put on a rigorous footing only for clonal inheritance. We extend this to sexually reproducing diploids, although admittedly still under the restriction of an unstructured population with Lotka–Volterra-like dynamics and single locus genetics (as in Kimura’s in Proc Natl Acad Sci USA 54: 731–736, 1965 infinite...

We analyze long-term evolutionary dynamics in a large class of life history models. The model family is characterized by discrete-time population dynamics and a finite number of individual states such that the life cycle can be described in terms of a population projection matrix. We allow an arbitrary number of demographic parameters to be subject...

This book was first published in 2004. Unraveling the origin of biodiversity is fundamental for understanding our biosphere. This book clarifies how adaptive processes, rather than geographic isolation, can cause speciation. Adaptive speciation occurs when biological interactions induce disruptive selection and the evolution of assortative mating,...

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One of the important questions in understanding infectious diseases and their prevention and control is how infectious agents can invade and become endemic in a host population. A ubiquitous feature of natural populations is that they are spatially fragmented, resulting in relatively homogeneous local populations inhabiting patches connected by the...

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The simplest behaviour one can hope for when studying a mathematical model of evolution by natural selection is when evolution always maximises the value of some function of the trait under consideration, thus providing an absolute measure of fitness for the model. We survey the role of such models, known as optimisation models in the literature, a...

Mammals as a rule have seven cervical vertebrae, except for sloths and manatees. Bateson proposed that the change in the number of cervical vertebrae in sloths is due to homeotic transformations. A recent hypothesis proposes that the number of cervical vertebrae in sloths is unchanged and that instead the derived pattern is due to abnormal primaxia...

This paper explores the effect of discontinuous periodic host absence on the evolution of pathogen transmission rates by using R 0 maximisation techniques. The physiological consequence of an increased transmission rate can be either an increased virulence, i.e. there is a transmission-virulence trade-off or ii) a reduced between season survival, i...

For structured populations in equilibrium with everybody born equal, ln(R (0)) is a useful fitness proxy for evolutionarily steady strategy (ESS) and most adaptive dynamics calculations, with R (0) the average lifetime number of offspring in the clonal and haploid cases, and half the average lifetime number of offspring fathered or mothered for Men...

The quick answer to the title question is: by bookkeeping; introduce as p(opulation)-state a measure telling how the individuals are distributed over their common i(ndividual)-state space, and track how the various i-processes change this measure. Unfortunately, this answer leads to a mathematical theory that is technically complicated as well as i...

We consider the interaction between a general size-structured consumer population and an unstructured resource. We show that stability properties and bifurcation phenomena can be understood in terms of solutions of a system of two delay equations (a renewal equation for the consumer population birth rate coupled to a delay differential equation for...

Goal: To provide an alternative to the usual bet-hedging explanation for delayed germination, one that takes account of known facts about germination in stable, fine-grained environments. Context: Small patches with local environmental conditions (microhabitats) such that seedlings can establish themselves are customarily called safe sites. Key ass...

The fitness concept of evolutionary ecology differs from that of population genetics. The former is geared toward dealing with long-term evolution through the repeated invasion of mutants for potentially complicated ecological scenarios and the latter with short-term changes in relative frequencies of types for heavily simplified ecological scenari...

Aim: To elucidate the role of the eco-evolutionary feedback loop in determining evolution- arily stable life histories, with particular reference to the methodological status of the optimization procedures of classical evolutionary ecology. Key assumption: The fitness ρ of a type depends both on its strategy X and on the environ- ment E, ρ = ρ(X, E...

Aim: To elucidate the role of the eco-evolutionary feedback loop in determining evolutionarily stable life histories, with particular reference to the methodological status of the optimization procedures of classical evolutionary ecology. Key assumptions: The fitness ρ of a type depends both on its strategy X and on the environment E, ρ = ρ(X, E),...

Many studies of evolutionarily stable strategies (ESS) for technical reasons make the simplification that reproduction is clonal. A post-hoc justification is that in the simplest eco-evolutionary models more realistic genetic assumptions, such as haploid sexual or diploid sexual cases, yield results compatible with the clonal ones. For metapopulati...

We develop a systematic toolbox for analyzing the adaptive dynamics of multidimensional traits in physiologically structured population models with point equilibria (sensu Dieckmann et al. in Theor. Popul. Biol. 63:309-338, 2003). Firstly, we show how the canonical equation of adaptive dynamics (Dieckmann and Law in J. Math. Biol. 34:579-612, 1996)...

Stochastic models for metapopulations that consist of a finite number of local populations are discussed. A stochastic Levins model with two local slates (metapopulation level: a patch is occupied or empty) is compared with a detailed model at the level of local populations. The European badger is taken as an example to illustrate that the stochast...

Body plans are remarkably well conserved, but on (very) rare occasions important novelties evolve. Such novelties involve changes at the genotypic and phenotypic level affecting both developmental and adult traits. At all levels, duplications play an important role in the evolution of novelties. Mutations for duplications, including mutations for d...

In most animal taxa, longevity increases with body size across species, as predicted by the oxidative stress theory of aging. In contrast, in within-species comparisons of mammals and especially domestic dogs (e.g. Patronek et al., '97; Michell, '99; Egenvall et al., 2000; Speakman et al., 2003), longevity decreases with body size. We explore two d...

We analyze the consequences of diet choice behavior for the evolutionary dynamics of foraging traits by means of a mathematical model. The model is characterized by the following features. Consumers feed on two different substitutable resources that are distributed in a fine-grained manner. On encounter with a resource item, consumers decide whethe...

Why do all mammals, except for sloths and manatees, have exactly seven cervical vertebrae? In other vertebrates and other regions, the vertebral number varies considerably. We investigated whether natural selection constrains the number of cervical vertebrae in humans. To this end, we determined the incidence of cervical ribs and other homeotic ver...

Abstract Why do all mammals, except for sloths and manatees, have exactly seven cervical vertebrae? In other vertebrates and other regions, the vertebral number varies considerably. We investigated whether natural selection constrains the number of cervical vertebrae in humans. To this end, we determined the incidence of cervical ribs and other hom...

Recent theoretical studies have analyzed the evolution of habitat specialization using either the logistic or the Ricker equation. These studies have implemented evolutionary change directly in population-level parameters such as habitat-specific intrinsic growth rates r or carrying capacities K. This approach is a shortcut to a more detailed analy...

A focus on the eco-evolutionary feedback continually operating between a population's evolution and its environment helps to appreciate the generality of ESS theory. Here we illustrate, through a sequence of four examples, how respecting such feedback in the evolutionary dynamics of quantitative traits may result in qualitatively unexpected outcome...

Robustness of coexistence against changes of parameters is investigated in a model-independent manner by analyzing the feedback loop of population regulation. We define coexistence as a fixed point of the community dynamics with no population having zero size. It is demonstrated that the parameter range allowing coexistence shrinks and disappears w...

Levins's fitness set approach has shaped the intuition of many evolutionary ecologists about resource specialization: if the set of possible phenotypes is convex, a generalist is favored, while either of the two specialists is predicted for concave phenotype sets. An important aspect of Levins's approach is that it explicitly excludes frequency-dep...

Levins’s fitness set approach has shaped the intuition of many evolutionary ecologists about resource specialization: if the set of possible phenotypes is convex, a generalist is favored, while either of the two specialists is predicted for concave phenotype sets. An important aspect of Levins’s approach is that it explicitly excludes frequency‐dep...

already,existed before,the abstract theory,took,off. AD creates order on an abstract level, which in turn helps in constructing,new,tools. As far as the use of the newer tools is concerned, AD can be said to have contributed to predictions. Another,class of predictions,from,AD arise from,arguments,on the frequency,with,which,one,may expect,differen...

We provide the link between population dynamics and the dynamics of Darwinian evolution via studying the joint population dynamics of similar populations. Similarity implies that the relative dynamics of the populations is slow compared to, and decoupled from, their aggregated dynamics. The relative dynamics is simple, and captured by a Taylor expa...

A problem in understanding sympatric speciation is establishing how reproductive isolation can arise when there is disruptive selection on an ecological trait. One of the solutions that has been proposed is that a habitat preference evolves, and that mates are chosen within the preferred habitat. We present a model where the habitat preference can...

Biology takes a special place among the other natural sciences because biological units, be they pieces of DNA, cells or organisms, reproduce more or less faithfully. As for any other biological processes, reproduction has a large random component. The theory of branching processes was developed especially as a mathematical counterpart to this most...

Vargas and Fallon (2005. J Exp Zool (Mol Dev Evol) 304B:86-90) propose that Hox gene expression patterns indicate that the most anterior digit in bird wings is homologous to digit 1 rather than to digit 2 in other amniotes. This interpretation is based on the presence of Hoxd13 expression in combination with the absence of Hoxd12 expression in the...

A recent Perspectives article by Gavrilets (2003) on the theory of speciation ignored advances in understanding processes of adaptive speciation, in which the splitting of lineages is an adaptation caused by frequency-dependent selection. Adaptive, or sympatric, speciation has been modeled since the 1960s, but the large amount of attention from bot...

Abstract A recent Perspectives article by Gavrilets (2003) on the theory of speciation ignored advances in understanding processes of adaptive speciation, in which the splitting of lineages is an adaptation caused by frequency-dependent selection. Adaptive, or sympatric, speciation has been modeled since the 1960s, but the large amount of attention...

: Sex allocation theory explains how size-related variations in male and female fitness may favor the evolution of size-dependent sex allocation in hermaphrodites. Although empirical studies show that sex allocation changes gradually with size in many species, the-oretical studies tend to predict an abrupt sex reversal from one sex to the other, th...

Sex allocation theory explains how size‐related variations in male and female fitness may favor the evolution of size‐dependent sex allocation in hermaphrodites. Although empirical studies show that sex allocation changes gradually with size in many species, theoretical studies tend to predict an abrupt sex reversal from one sex to the other, that...

As anthropogenic environmental changes spread and intensify across the planet, conservation biologists have to analyze dynamics at large spatial and temporal scales. Ecological and evolutionary processes are then closely intertwined. In particular, evolutionary responses to anthropogenic environmental change can be so fast and pronounced that conse...

The assumption that trade-offs exist is fundamental in evolutionary theory. Levins (Am. Nat. 96 (1962) 361-372) introduced a widely adopted graphical method for analyzing evolution towards an optimal combination of two quantitative traits, which are traded off. His approach explicitly excluded the possibility of density- and frequency-dependent sel...

Temporal and spatial variations of the environment are important factors favoring the evolution of dispersal. With few exceptions, these variations have been considered to be exclusively fluctuations of habitat quality. However, since the presence of conspecifics forms part of an individual's environment, demographic stochasticity may be a componen...

Recently at least two papers have appeared that look at cancer from an evolutionary perspective. That cancer has a negative effect on fitness needs no argument. However, cancer origination is not an isolated process, but the potential for it is linked in diverse ways to other genetically determined developmental events, complicating the way selecti...

We introduce a notion of attractor adapted to dynamical processes as they are studied in community-ecological models and their computer simulations. This attractor concept is modeled after that of Ruelle as presented in [11] and [12]. It incorporates the fact that in an immigration-free community populations can go extinct at low values of their de...

In Part I of this paper Jacobs and Metz (2003) extended the concept of the Conley-Ruelle, or chain, attractor in a way relevant to unstructured community ecological models. Their modified theory incorporated the facts that certain parts of the boundary of the state space correspond to the situation of at least one species being extinct and that an...

Our systematic formulation of nonlinear population models is based on the notion of the environmental condition. The defining property of the environmental condition is that individuals are independent of one another (and hence equations are linear) when this condition is prescribed (in principle as an arbitrary function of time, but when focussing...

In this paper, we predict the outcome of dispersal evolution in metapopulations based on the following assumptions: (i) population dynamics within patches are density-regulated by realistic growth functions; (ii) demographic stochasticity resulting from finite population sizes within patches is accounted for; and (iii) the transition of individuals...

Gene expression patterns of the segment polarity genes in the extended and segmented germband stage are remarkably conserved among insects. To explain the conservation of these stages, two hypotheses have been proposed. One hypothesis states that the conservation reflects a high interactivity between modules, so that mutations would have several pl...