On the concept of individual in ecology and evolution.

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 dynamics. By appropriately delimiting individuals, followed by defining their states as well as their environment, it become possible to construct the general population equations that were introduced and studied by Odo Diekmann and his collaborators. In this essay I argue for taking the properties that led to these successes as the defining characteristics of the concept of individual, delegating the properties classically invoked by philosophers to the secondary role of possible empirical indicators for the presence of those characteristics. The essay starts with putting in place as rule for effective concept engineering that one should go for relations that can be used as basis for deductive structure building rather than for perceived ontological essence. By analysing how we want to use it in the mathematical arguments I then build up a concept of individual, first for use in population dynamical considerations and then for use in evolutionary ones. These two concepts do not coincide, and neither do they on all occasions agree with common intuition-based usage.

Daphnias: from the individual based model to the large population equation.

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 deterministic models. The models treat the interaction between a general size-structured consumer population ('Daphnia') and an unstructured resource ('algae'). The discrete, size and age-structured Daphnia population changes through births and deaths of its individuals and through their aging and growth. The birth and death rates depend on the sizes of the individuals and on the concentration of the algae. The latter is supposed to be a continuous variable with a deterministic dynamics that depends on the Daphnia population. In this model setting we prove that when the Daphnia population is large, the stochastic differential equation describing the IBM can be approximated by the delay equation featured in (Diekmann et al., loc. cit.).

A simple fitness proxy for structured populations with continuous traits, with case studies on the evolution of haplo-diploids and genetic dimorphisms.

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 Mendelian diploids. When individuals have variable birth states, as is, for example, the case in spatial models, R (0) is itself an eigenvalue, which usually cannot be expressed explicitly in the trait vectors under consideration. In that case, Q(Y| X):=-det (I-L(Y| X)) can often be used as fitness proxy, with L the next-generation matrix for a potential mutant characterized by the trait vector Y in the (constant) environment engendered by a resident characterized by X. If the trait space is connected, global uninvadability can be determined from it. Moreover, it can be used in all the usual local calculations like the determination of evolutionarily singular trait vectors and their local invadability and attractivity. We conclude with three extended case studies demonstrating the usefulness of Q: the calculation of ESSs under haplo-diploid genetics (I), of evolutionarily steady genetic dimorphisms (ESDs) with a priori proportionality of macro- and micro-gametic outputs (an assumption that is generally made but the fulfilment of which is a priori highly exceptional) (II), and of ESDs without such proportionality (III). These case studies should also have some interest in their own right for the spelled out calculation recipes and their underlying modelling methodology.

How to lift a model for individual behaviour to the population level?

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 immature. Alternatively, one may describe a population in terms of the history of the population birth rate together with the history of any environmental variables affecting i-state changes, reproduction and survival. Thus, a population model leads to delay equations. This delay formulation corresponds to a restriction of the p-dynamics to a forward invariant attracting set, so that no information is lost that is relevant for long-term dynamics. For such equations there exists a well-developed theory. In particular, numerical bifurcation tools work essentially the same as for ordinary differential equations. However, the available tools still need considerable adaptation before they can be practically applied to the dynamic energy budget (DEB) model. For the time being we recommend simplifying the i-dynamics before embarking on a systematic mathematical exploration of the associated p-behaviour. The long-term aim is to extend the tools, with the DEB model as a relevant goal post.

Daphnia revisited: local stability and bifurcation theory for physiologically structured population models explained by way of an example.

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 the resource concentration). As many results for such systems are available (Diekmann et al. in SIAM J Math Anal 39:1023-1069, 2007), we can draw rigorous conclusions concerning dynamical behaviour from an analysis of a characteristic equation. We derive the characteristic equation for a fairly general class of population models, including those based on the Kooijman-Metz Daphnia model (Kooijman and Metz in Ecotox Env Saf 8:254-274, 1984; de Roos et al. in J Math Biol 28:609-643, 1990) and a model introduced by Gurney-Nisbet (Theor Popul Biol 28:150-180, 1985) and Jones et al. (J Math Anal Appl 135:354-368, 1988), and next obtain various ecological insights by analytical or numerical studies of special cases.

Adaptive dynamics for physiologically structured population models.

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), an approximation for the rate of evolutionary change in characters under directional selection, can be extended so as to apply to general physiologically structured population models with multiple birth states. Secondly, we show that the invasion fitness function (up to and including second order terms, in the distances of the trait vectors to the singularity) for a community of N coexisting types near an evolutionarily singular point has a rational form, which is model-independent in the following sense: the form depends on the strategies of the residents and the invader, and on the second order partial derivatives of the one-resident fitness function at the singular point. This normal form holds for Lotka-Volterra models as well as for physiologically structured population models with multiple birth states, in discrete as well as continuous time and can thus be considered universal for the evolutionary dynamics in the neighbourhood of singular points. Only in the case of one-dimensional trait spaces or when N = 1 can the normal form be reduced to a Taylor polynomial. Lastly we show, in the form of a stylized recipe, how these results can be combined into a systematic approach for the analysis of the (large) class of evolutionary models that satisfy the above restrictions.

Speciation: more likely through a genetic or through a learned habitat preference?

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 evolve either by means of a genetic mechanism or by means of learning. Employing an adaptive-dynamical analysis, we show that evolution proceeds either to a single population of specialists with a genetic preference for their optimal habitat, or to a population of generalists without a habitat preference. The generalist population subsequently experiences disruptive selection. Learning promotes speciation because it increases the intensity of disruptive selection. An individual-based version of the model shows that, when loci are completely unlinked and learning confers little cost, the presence of disruptive selection most probably leads to speciation via the simultaneous evolution of a learned habitat preference. For high costs of learning, speciation is most likely to occur via the evolution of a genetic habitat preference. However, the latter only happens when the effect of mutations is large, or when there is linkage between genes coding for the different traits.

Adaptive walks on changing landscapes: Levins' approach extended.

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 selection. Here we extend Levins method towards models, which include these selection regimes and where therefore fitness landscapes change with population state. We employ the same kind of curves Levins used: trade-off curves and fitness contours. However, fitness contours are not fixed but a function of the resident traits and we only consider those that divide the trait space into potentially successful mutants and mutants which are not able to invade ('invasion boundaries'). The developed approach allows to make a priori predictions about evolutionary endpoints and about their bifurcations. This is illustrated by applying the approach to several examples from the recent literature.

Evolution of dispersal in metapopulations with local density dependence and demographic stochasticity.

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 between patches is explicitly modelled by a disperser pool. We show, first, that evolutionarily stable dispersal rates do not necessarily increase with rates for the local extinction of populations due to external disturbances in habitable patches. Second, we describe how demographic stochasticity affects the evolution of dispersal rates: evolutionarily stable dispersal rates remain high even when disturbance-related rates of local extinction are low, and a variety of qualitatively different responses of adapted dispersal rates to varied levels of disturbance become possible. This paper shows, for the first time, that evolution of dispersal rates may give rise to monotonically increasing or decreasing responses, as well as to intermediate maxima or minima.

On the concept of attractor for community-dynamical processes I: the case of unstructured populations.

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 densities.

On the concept of attractor for community-dynamical processes II: the case of structured populations.

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 extinct species can not be rescued by noise. In this part we extend the theory to communities of physiologically structured populations. One difference between the structured and unstructured cases is that a structured population may be doomed to extinction and not rescuable by any biologically relevant noise before actual extinction has taken place. Another difference is that in the structured case we have to use different topologies to define continuity of orbits and to measure noise. Biologically meaningful noise is furthermore related to the linear structure of the community state space. The construction of extinction preserving chain attractors developed in this paper takes all these points into account.

Steady-state analysis of structured population models.

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 on steady states we shall restrict to constant functions). The steady-state problem has two components: (i). the environmental condition should be such that the existing populations do neither grow nor decline; (ii). a feedback consistency condition relating the environmental condition to the community/population size and composition should hold. In this paper we develop, justify and analyse basic formalism under the assumption that individuals can be born in only finitely many possible states and that the environmental condition is fully characterized by finitely many numbers. The theory is illustrated by many examples. In addition to various simple toy models introduced for explanation purposes, these include a detailed elaboration of a cannibalism model and a general treatment of how genetic and physiological structure should be combined in a single model.

Predictors and outcomes of stent thrombosis: an intravascular ultrasound registry.

AIMS: To investigate whether intravascular ultrasound provides additional information regarding the prediction of stent thrombosis, a retrospective multicentre registry was designed to enrol patients with stent thrombosis following stent deployment under ultrasound guidance. METHODS AND RESULTS: A total of 53 patients were enrolled (mean age 61+/-9 years) with stable angina (43%), unstable angina (36%), and post-infarct angina (21%) who underwent intracoronary stenting. The majority had balloon angioplasty alone prior to stenting (94%) with 6% also undergoing rotational atherectomy. The indication for stenting was elective (53%), suboptimal result (32%) and bailout (15%). There were 1.6+/-0.8 stents/artery with 87% undergoing high-pressure dilatation (> or =14 atmospheres). The minimum stent area was 7.7+/-2.8 mm(2)with a mean stent expansion of 81.5+/-21.9%. Overall, 94% of cases demonstrated one abnormal ultrasound finding (stent under-expansion, malapposition, inflow/outflow disease, dissection, or thrombus). Angiography demonstrated an abnormality in only 32% of cases (chi-square=30.0, P<0.001). Stent thrombosis occurred at 132+/-125 h after deployment. Myocardial infarction occurred in 67% and there was an overall mortality of 15%. CONCLUSION: On comparison with angiography, the vast majority of stents associated with subsequent thrombosis have at least one abnormal feature by intravascular ultrasound at the time of stent deployment.

On the formulation and analysis of general deterministic structured population models. II. Nonlinear theory.

This paper is as much about a certain modelling methodology, as it is about the constructive definition of future population states from a description of individual behaviour and an initial population state. The key idea is to build a nonlinear model in two steps, by explicitly introducing the environmental condition via the requirement that individuals are independent from one another (and hence equations are linear) when this condition is prescribed as a function of time. A linear physiologically structured population model is defined by two rules, one for reproduction and one for development and survival, both depending on the initial individual state and the prevailing environmental condition. In Part I we showed how one can constructively define future population state operators from these two ingredients. A nonlinear model is a linear model together with a feedback law that describes how the environmental condition at any particular time depends on the population size and composition at that time. When applied to the solution of the linear problem, the feedback law yields a fixed point problem. This we solve constructively by means of the contraction mapping principle, for any given initial population state. Using subsequently this fixed point as input in the linear population model, we obtain a population semiflow. We then say that we solved the nonlinear problem.

Testing the vulnerability of the phylotypic stage: on modularity and evolutionary conservation.

The phylotypic stage is the developmental stage at which vertebrates most resemble each other. In this study we test the plausibility of the hypotheses of Sander [1983, Development and Evolution, Cambridge University Press] and Raff [1994, Early Life on Earth, Columbia University Press; 1996, The Shape of Life, University of Chicago Press] that the phylotypic stage is conserved due to the intense and global interactivity occurring during that stage. First, we test the prediction that the phylotypic stage is much more vulnerable than any other stage. A search of the teratological literature shows that disturbances at this stage lead to a much higher mortality than in other stages, in accordance with the prediction. Second, we test whether that vulnerability is indeed caused by the interactiveness and lack of modularity of the inductions or, alternatively, is caused by some particularly vulnerable process going on at that time. From the pattern of multiple induced anomalies we conclude that it is indeed the interactiveness that is the root cause of the vulnerability. Together these results support the hypotheses of Sander and Raff. We end by presenting an argument on why the absence of modularity in the inductive interactions may also be the root cause of the conservation of the much discussed temporal and spatial colinearity of the Hox genes. J. Exp. Zool. (Mol. Dev. Evol.) 291:195-204, 2001.

Butterfly eyespot patterns: evidence for specification by a morphogen diffusion gradient.

In this paper we describe a test for Nijhout's (1978, 1980a) hypothesis that the eyespot patterns on butterfly wings are the result of a threshold reaction of the epidermal cells to a concentration gradient of a diffusing degradable morphogen produced by focal cells at the centre of the future eyespot. The wings of the nymphalid butterfly, Bicyclus anynana, have a series of eyespots. each composed of a white pupil, a black disc and a gold outer ring. In earlier extirpation and transplantation experiments (Nijhout 1980a; French and Brakefield, 1995) it has been established that these eyespots are indeed organised around groups of signalling cells active during the first hours of pupal development. If these cells were to supply the positional information for eyespot formation in accordance with Nijhout's diffusion-degradation gradient model, then, when two foci are close together. the signals should sum, and this effect should be apparent in the detailed shape of the resulting pigment pattern. We give an equation for the form of the contours that would be obtained in this manner. We use this to test the morphogen gradient hypothesis on measurements of the outlines of fused eyespots obtained either by grafting focal cells close together, or by using a mutation (Spotty) that produces adjacent fused eyespots. The contours of the fused patterns were found to satisfy our equation, thus corroborating Nijhout's hypothesis to the extent possible with this particular type of experiment.

How should we define fitness in structured metapopulation models? Including an application to the calculation of evolutionarily stable dispersal strategies.

We define a fitness concept applicable to structured metapopulations consisting of infinitely many equally coupled patches. In addition, we introduce a more easily calculated quantity Rm that relates to fitness in the same manner as R0 relates to fitness in ordinary population dynamics: the Rm of a mutant is only defined when the resident population dynamics converges to a point equilibrium and Rm is larger (smaller) than 1 if and only if mutant fitness is positive (negative). Rm corresponds to the average number of newborn dispersers resulting from the (on average less than one) local colony founded by a newborn disperser. Efficient algorithms for calculating its numerical value are provided. As an example of the usefulness of these concepts we calculate the evolutionarily stable conditional dispersal strategy for individuals that can account for the local population density in their dispersal decisions. Below a threshold density x, at which staying and leaving are equality profitable, everybody should stay and above x everybody should leave, where profitability is measured as the mean number of dispersers produced through lines of descent consisting of non-dispersers.

Desynchronization rate in cell populations: mathematical modeling and experimental data.

We characterize the kinetics of two cancer cell lines: IGROV1 (ovarian carcinoma) and MOLT4 (leukemia). By means of flow cytometry, we selected two populations from exponentially growing in vitro cell lines, depending on the cells' DNA synthesis activity during a preceding labeling period. For these populations we determined the time course of the percentages of cells in different phases of the cycles, sampling every 3 hr for 60 hr. Initially, semi-synchronous populations quickly converged to a stable age distribution, which is typical of the cell line (at equilibrium); this desynchronization reflects the intercell variability in cell cycle duration. By matching these experimental observations to mathematical modelling, we related the convergence rate toward the asymptotic distribution (R) and the period of the phase-percentage oscillations (T), to the mean cell cycle duration and its coefficient of variation. We give two formulas involving the above-mentioned parameters. Since T and R can be drawn by fitting our data to an asymptotic formula obtained from the model, we can estimate the other two kinetic parameters. IGROV1 cells have a shorter mean cell cycle time, but higher intercell variability than the leukemia line, which takes longer to lose synchrony.