Well-posedness of physiologically structured population models for Daphnia magna

In this paper we heuristically discuss the well-posedness of three variants of the Kooijman/Metz model. Shortcomings concerning the uniqueness and continuous dependence on data of the solutions to one of the variants are traced back to an inconsistency in the biological concept of energy allocation in this model version. The conceptional consequences are discussed and an open question concerning energy allocation is pin-pointed.     

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.      

The application of mass and energy conservation laws in physiologically structured population models of heterotrophic organisms

Rules for energy uptake, and subsequent utilization, form the basis of population dynamics and, therefore, explain the dynamics of the ecosystem structure in terms of changes in standing crops and size distributions of individuals. Mass fluxes are concomitant with energy flows and delineate functional aspects of ecosystems by defining the roles of individuals and populations. The assumption of homeostasis of body components, and an assumption about the general structure of energy budgets, imply that mass fluxes can be written as weighted sums of three organizing energy fluxes with the weight coefficients determined by the conservation law of mass. These energy fluxes are assimilation, maintenance and growth, and provide a theoretical underpinning of the widely applied empirical method of indirect calorimetry, which relates dissipating heat linearly to three mass fluxes: carbon dioxide production, oxygen consumption and N-waste production. A generic approach to the stoichiometry of population energetics from the perspective of the individual organism is proposed and illustrated for heterotrophic organisms. This approach indicates that mass transformations can be identified by accounting for maintenance requirements and overhead costs for the various metabolic processes at the population level. The theoretical background for coupling the dynamics of the structure of communities to nutrient cycles, including the water balance, as well as explicit expressions for the dissipating heat at the population level are obtained based on the conservation law of energy. Specifications of the general theory employ the Dynamic Energy Budget model for individuals. Copyright 1999 Academic Press.     

Stress tolerance and population stability of rock pool Daphnia in relation to local conditions and population isolation

Small fragmented populations can lose genetic variability, which reduces population viability through inbreeding and loss of adaptability. Current and previous environmental conditions can also alter the viability of populations, by creating local adaptations that determine responses to stress. Yet, most studies on stress tolerance usually consider either the effect of genetic diversity or the local environment, missing a more holistic perspective of the factors contributing to stress tolerance among natural populations. Here, we studied how salinity stress affects population growth of Daphnia longispina, Daphnia magna, and Daphnia pulex from rock pools with varying degrees of population isolation and salinity conditions. Standing variation of in situ rock pool salinity conditions explained more variation in salt tolerance than the standing variation of population isolation or genetic diversity, in both a pulse and a press disturbance experiment. This indicates that the level of stress, which these natural populations experience, influences their response to that stress, which may have important consequences for the conservation of fragmented populations. However, long-term population stability in the field decreased with population isolation, indicating that natural populations experience a variety of stresses; thus, population isolation and genetic diversity may stabilize population dynamics over larger spatiotemporal scales.     

Phenotypic structure and bifurcation behavior of population models

Discrete time models for density-regulated populations have been shown to exhibit periodic and chaotic motion in the absence of any external signal. We show how the genetic structure of a population can initiate bifurcations to periodic and chaotic trajectories. We investigate by simulation the dependence of this phenomenon on the strength of assortative mating, the level of heterozygosity, and the intensity of selection. The implications of internally generated chaos for population modeling are discussed.     

Local and global stability for population models

In general, local stability does not imply global stability. We show that this is true even if one only considers population models.We show that a population model is globally stable if and only if it has no cycle of period 2. We also derive easy to test sufficient conditions for global stability. We demonstrate that these sufficient conditions are useful by showing that for a number of population models from the literature, local and global stability coincide.We suggest that the models from the literature are in some sense simple, and that this simplicity causes local and global stability to coincide.     

Approximation of a physiologically structured population model with seasonal reproduction by a stage-structured biomass model

Seasonal reproduction causes, due to the periodic inflow of young small individuals in the population, seasonal fluctuations in population size distributions. Seasonal reproduction furthermore implies that the energetic body condition of reproducing individuals varies over time. Through these mechanisms, seasonal reproduction likely affects population and community dynamics. While seasonal reproduction is often incorporated in population models using discrete time equations, these are not suitable for size-structured populations in which individuals grow continuously between reproductive events. Size-structured population models that consider seasonal reproduction, an explicit growing season and individual-level energetic processes exist in the form of physiologically structured population models. However, modeling large species ensembles with these models is virtually impossible. In this study, we therefore develop a simpler model framework by approximating a cohort-based size-structured population model with seasonal reproduction to a stage-structured biomass model of four ODEs. The model translates individual-level assumptions about food ingestion, bioenergetics, growth, investment in reproduction, storage of reproductive energy, and seasonal reproduction in stage-based processes at the population level. Numerical analysis of the two models shows similar values for the average biomass of juveniles, adults, and resource unless large-amplitude cycles with a single cohort dominating the population occur. The model framework can be extended by adding species or multiple juvenile and/or adult stages. This opens up possibilities to investigate population dynamics of interacting species while incorporating ontogenetic development and complex life histories in combination with seasonal reproduction.     

Dynamic energy budget theory and population ecology: lessons from Daphnia

Dynamic energy budget (DEB) theory offers a perspective on population ecology whose starting point is energy utilization by, and homeostasis within, individual organisms. It is natural to ask what it adds to the existing large body of individual-based ecological theory. We approach this question pragmatically--through detailed study of the individual physiology and population dynamics of the zooplankter Daphnia and its algal food. Standard DEB theory uses several state variables to characterize the state of an individual organism, thereby making the transition to population dynamics technically challenging, while ecologists demand maximally simple models that can be used in multi-scale modelling. We demonstrate that simpler representations of individual bioenergetics with a single state variable (size), and two life stages (juveniles and adults), contain sufficient detail on mass and energy budgets to yield good fits to data on growth, maturation and reproduction of individual Daphnia in response to food availability. The same simple representations of bioenergetics describe some features of Daphnia mortality, including enhanced mortality at low food that is not explicitly incorporated in the standard DEB model. Size-structured, population models incorporating this additional mortality component resolve some long-standing questions on stability and population cycles in Daphnia. We conclude that a bioenergetic model serving solely as a 'regression' connecting organismal performance to the history of its environment can rest on simpler representations than those of standard DEB. But there are associated costs with such pragmatism, notably loss of connection to theory describing interspecific variation in physiological rates. The latter is an important issue, as the type of detailed study reported here can only be performed for a handful of species.     

Application of individual growth and population models of Daphnia pulex to Daphnia magna, Daphnia galeata and Bosmina longirostris

Daphnia models for individual growth and population dynamics have been developed in the manner of models developed by Gurney, McCauley, Andersen and others. All or most of the earlier models were parameterized for Daphnia pulex; we have used the D. pulex model as a baseline model for other species of Daphnia such as magna, galeata and also Bosmina longirostris. Because of the lack of ample data for D. magna, D. galeata and B. longirostris, some of the physiological data had to be relied on the other species whose data were available and in some case calibrated. We were able to produce reasonable results for individual growth as well as population dynamics under the controlled laboratory conditions. Most of the results were compared with the available laboratory data for population as well as growth. All the simulations have been done under high and low food concentrations. The animals are assumed to be feeding on green algae (Chlamydomonas reinhardtti) under the laboratory conditions of 18–20°C. The continuous growth until the end of the life was observed in smaller B. longirostris, whereas rapid growth in the beginning and slower after the start of the reproduction was observed in Daphnia species. The smaller species matured earlier than larger species. B. longirostris population sustained better than Daphnia species in medium food concentrations.     

Spatially structured metapopulation models: global and local assessment of metapopulation capacity

We model metapopulation dynamics in finite networks of discrete habitat patches with given areas and spatial locations. We define and analyze two simple and ecologically intuitive measures of the capacity of the habitat patch network to support a viable metapopulation. Metapopulation persistence capacity lambda(M) defines the threshold condition for long-term metapopulation persistence as lambda(M)>delta, where delta is defined by the extinction and colonization rate parameters of the focal species. Metapopulation invasion capacity lambda(I) sets the condition for successful invasion of an empty network from one small local population as lambda(I)>delta. The metapopulation capacities lambda(M) and lambda(I) are defined as the leading eigenvalue or a comparable quantity of an appropriate "landscape" matrix. Based on these definitions, we present a classification of a very general class of deterministic, continuous-time and discrete-time metapopulation models. Two specific models are analyzed in greater detail: a spatially realistic version of the continuous-time Levins model and the discrete-time incidence function model with propagule size-dependent colonization rate and a rescue effect. In both models we assume that the extinction rate increases with decreasing patch area and that the colonization rate increases with patch connectivity. In the spatially realistic Levins model, the two types of metapopulation capacities coincide, whereas the incidence function model possesses a strong Allee effect characterized by lambda(I)=0. For these two models, we show that the metapopulation capacities can be considered as simple sums of contributions from individual habitat patches, given by the elements of the leading eigenvector or comparable quantities. We may therefore assess the significance of particular habitat patches, including new patches that might be added to the network, for the metapopulation capacities of the network as a whole. We derive useful approximations for both the threshold conditions and the equilibrium states in the two models. The metapopulation capacities and the measures of the dynamic significance of particular patches can be calculated for real patch networks for applications in metapopulation ecology, landscape ecology, and conservation biology.     

Hopf bifurcation in a structured population model for the sexual phase of monogonont rotifers

 We are studying a population of monogonont rotifers in the context of non-linear age-dependent models. In the sexual phase of their reproductive cycle we consider the population structured by age, and composed of three subclasses: virgin mictic females, mated mictic females, and haploid males. The model system has a unique stationary population density which is stable as long as a parameter, related to male-female encounter rate, remains below a critical value. When the parameter increases beyond this critical value, the stationary solution becomes unstable and a stable limit cycle (isolated periodic orbit) appears. The occurrence of this supercritical Hopf bifurcation is shown analytically. Received: 2 August 2001 / Revised version: 3 January 2002 / Published online: 26 June 2002     

Consistency and fluctuation theorems for discrete time structured population models having demographic stochasticity

In this paper we prove a consistency theorem (law of large numbers) and a fluctuation theorem (central limit theorem) for structured population processes. The basic assumptions for these theorems are that the individuals have no statistically distinguishing features beyond their class and that the interaction between any two individuals is not too high. We apply these results to density dependent models of Leslie type and to a model for flour beetle dynamics. Received: 24 February 1999 / Revised version: 23 July 1999 / Published online: 14 September 2000     

Persistence of viral infections on the population level explained by an immunoepidemiological model

We consider a mathematical model of viral spread in a population based on an immune response model embedded in an epidemic network model. The immune response model includes virus load and effector and memory T cells with two possible outcomes depending on parameters: (a) virus clearance and establishment of immune memory and (b) establishment of a non-zero viral presence characterized with increased T-cell concentrations. Isolated individuals can have different immune system parameters and, after a primary infection, can either return to the infection-free state or develop persistent or chronic infection. When individuals are connected in the network, they can reinfect each other. We show that the virus can persist in the epidemic network for indefinite time even if the whole population consists of individuals that are able to clear the virus when isolated from the network. In this case a few individuals with a relatively weak immune response can maintain the infection in the whole population. These results are in contrast to implications of classical epidemiological models that a viral epidemic will end if there is no influx of new susceptibles and if individuals can become immune after infection.     

Daphnia lumholtzi and Daphnia ambigua: population comparisons of an exotica and a native cladoceran in Lake Okeechobee, Florida

The population dynamics of an exotic cladoceran (Daphnialumholtzi Sars) and a native cladoceran (Daphniaambigua) were studied over a 12 month period in subtropical LakeOkeechobee, Florida (USA), to quantify the extent of invasion of the exoticspecies and compare ecological niches. Daphnialumholtzi accounted for up to 70% of theDaphnia assemblage during the summer months(June-August), while D.ambigua accounted for up to 97%of the Daphnia assemblage from fall to spring(October-April). The densities of the two species were inversely corelated.The exotic species was most concentrated in the shallower, warmer, northand south ends of the lake during the summer. It also was present, but atmuch lower densities, in the central lake region during the fall. Thenative species displayed a ubiquitous distribution throughout the lakeduring spring and winter, but was concentrated in the deeper, cooler,central region during the summer. Relationships of the two species withenvironmental conditions indicate that water column temperature mightaffect the seasonal and spatial distribution of the twoDaphnia species. The results also indicate thatD.lumholtzi may be filling a 'vacant' seasonal orspatial niche when conditions are unfavorable forD.ambigua.     

Locking life-cycles onto seasons: Circle-map models of population dynamics and local adaptation

We have formulated a model describing the timing of maturity and reproduction in briefly semelparous organisms whose development rate is primarily controlled by environmental factors. The model is expressed as a circle-map relating time of year at maturation in successive generations. The properties of this map enable us to determine the degree of synchrony to be expected between the life-cycles of members of a population exposed to a regular seasonal environment. We have proved that organisms with a life-history composed of a contiguous series of stages, all with development driven by the same seasonal function, cannot phase-lock their life-cycles to the seasons. However if the organism exhibits facultative diapause induced by a critical time/critical development mechanism of the type proposed by Norling (1984a,b,c) then it will always succeed in phase-locking to a perfectly periodic driving function. Within the context of this circle-map model we have examined population extinctions caused by attempting to over-winter in an inappropriate life-history stage, or by attempting to reproduce at a time of year when this is impossible. We have shown that the possibility of such extinctions limits both the shortness of the post-critical stage, and the lateness of the critical time. We have examined the fitness of persistent cohorts as a function of critical time and development. We find that if the post-critical stage is riskier than the pre-critical then natural selection favors a short post-critical stage and a late critical time; the limitation of this process being dependent on the proportion of the growing season over which successful reproduction is possible. We have determined the variation with life-cycle length (and hence latitude or altitude) of the maturation pattern corresponding to optimal life-history parameters. We find that for organisms which can mature only over a small part of the growing season the majority of any latitudinal gradient exhibits a unimodal maturation pattern. Organisms which can mature and reproduce over the majority of the growing season exhibit more complex patterns, but still exhibit substantial ranges of latitude over which unimodal or bimodal patterns are optimal.     

The frequency spectrum of structured discrete time population models: its properties and their ecological implications

Much research effort has been devoted to the study of the interaction between environmental noise and discrete time nonlinear dynamical systems. A large part of this effort has involved numerical simulation of simple unstructured models for particular ranges of parameter values. While such research is important in encouraging discussion of important ecological issues it is often unclear how general are the conclusions reached. However, by restricting attention to weak noise it is possible to obtain analytical results that hold for essentially all discrete time models and still provide considerable insight into the properties of the noise-dynamics interface. We follow this approach, focusing on the autocorrelation properties of the population fluctuations using the power (frequency) spectrum matrix as the analytic framework. We study the relationship between the spectral peak structure and the dynamical behaviour of the system and the modulation of this relationship by its internal structure, acting as an intrinsic filter and by colour in the noise acting as an extrinsic filter. These filters redistribute power between frequency components in the spectrum. The analysis emphasises the importance of eigenvalues in the identification of resonance, both in the system itself and in its subsystems, and the importance of noise configuration in defining which paths are followed on the network. The analysis highlights the complexity of the inverse problem (in finding, for example, the source of long term fluctuations) and the role of factors other than colour in the persistence of populations.