Mass Balance Equation Versus Logistic Equation in Food Chains

The dynamic behavior of tri-trophic food chains consisting of resources, prey, predator and top-predator is dealt with. We compare a formulation whereby the prey growth is logistic, with a mass balance formulation. In the case of the mass balance formulation both the linear and the hyperbolic functional response are discussed. The consequences of the different formulations on the dynamics of a microbial food chain in chemostat situation are described. Bifurcation diagrams for the nonlinear dynamic systems are given. When the prey grows logistically there is no coexistence of the three species for biologically realistic parameter values for a microbial food chain. The same holds for the mass balance equations with a linear functional response for the prey. For a hyperbolic functional response, however, there is a stable equilibrium for the whole food chain in a rather large region of the parameter space. Furthermore, this model shows more complex dynamic behaviors; besides point attractors, limit cycles and chaotic attractors.

Sublethal toxic effects in a generic aquatic ecosystem

The dynamical behaviour of an aquatic ecosystem stressed by limiting nutrients and exposure to a conservative toxicant is investigated. The ecosystem downstream of a pollution source consists of: nutrients, biotic pelagic and benthic communities, and detritus pools in the water body and on the sediment. The long-term dynamic behaviour of this system is analysed using bifurcation theory. A reference state is defined and our aim is to quantify the effects of toxicological (toxic exposure), ecological (feeding, predation, competition) and environmental stressors (nutrient supply, dilution rate). To that end we calculate the ranges of stress levels where the long-term dynamics (equilibrium, oscillatory or chaotic behaviour) is qualitatively the same. In this way we obtain levels of toxicological loading where the abundances of all populations are the same as in the reference case, the no-effect region. We will also calculate toxic exposure levels that do not lead to a change in the composition of the ecosystem, and therefore its structure, with respect to the reference unexposed situation, but where population abundances and internal toxicant concentrations may have been changed quantitatively. The model predicts that due to indirect effects even low sublethal toxic exposure can lead to catastrophic changes in the ecosystem functioning and structure, and that the long-term sensitivities of oligotrophic and eutrophic systems to toxic stress are different.

Bifurcation theory, adaptive dynamics and dynamic energy budget-structured populations of iteroparous species

In this paper, we describe a technique to evaluate the evolutionary dynamics of the timing of spawning for iteroparous species. The life cycle of the species consists of three life stages, embryonic, juvenile and adult whereby the transitions of life stages (gametogenesis, birth and maturation) occur at species-specific sizes. The dynamics of the population is studied in a semi-chemostat environment where the inflowing food concentration is periodic (annual). A dynamic energy budget-based continuous-time model is used to describe the uptake of the food, storage in reserves and allocation of the energy to growth, maintenance, development (embryos, juveniles) and reproduction (adults). A discrete-event process is used for modelling reproduction. At a fixed spawning date of the year, the reproduction buffer is emptied and a new cohort is formed by eggs with a fixed size and energy content. The population consists of cohorts: for each year one consisting of individuals with the same age which die after their last reproduction event. The resulting mathematical model is a finite-dimensional set of ordinary differential equations with fixed 1-year periodic boundary conditions yielding a stroboscopic map. We will study the evolutionary development of the population using the adaptive dynamics approach. The trait is the timing of spawning. Pairwise and mutual invasibility plots are calculated using bifurcation analysis of the stroboscopic map. The evolutionary singular strategy value belonging to the evolutionary endpoint for the trait allows for an interpretation of the reproduction strategy of the population. In a case study, parameter values from the literature for the bivalve Macoma balthica are used.

Modelling long-term ecotoxicological effects on an algal population under dynamic nutrient stress

We study the effects of toxicants on the functioning of phototrophic unicellular organism (an algae) in a simple aquatic microcosm by applying a parameter-sparse model. The model allows us to study the interaction between ecological and toxicological effects. Nutrient stress and toxicant stress, together or alone, can cause extinction of the algal population. The modelled algae consume dissolved inorganic nitrogen (DIN) under surplus light and use it for growth and maintenance. Dead algal biomass is mineralized by bacterial activity, leading to nutrient recycling. The ecological model is coupled with a toxicity-module that describes the dependency of the algal growth and death rate on the toxicant concentration. Model parameter fitting is performed on experimental data from Liebig, M., Schmidt, G., Bontje, D., Kooi, B.W., Streck, G., Traunspurger, W., Knacker, T. [2008. Direct and indirect effects of pollutants on algae and algivorous ciliates in an aquatic indoor microcosm. Aquatic Toxicology 88, 102-110]. These experiments were especially designed to include nutrient limitation, nutrient recycling and long-term exposure to toxicants. The flagellate species Cryptomonas sp. was exposed to the herbicide prometryn and insecticide methyl parathion in semi-closed Erlenmeyers. Given the total limiting amount of nitrogen in the system, the estimated toxicant concentration at which a long-term steady population of algae goes extinct will be derived. We intend to use the results of this study to investigate the effects of ecological (environmental) and toxicological stresses on more realistic ecosystem structure and functioning.

Model analysis of a simple aquatic ecosystems with sublethal toxic effects

The dynamic behaviour of simple aquatic ecosystems with nutrient recycling in a chemostat, stressed by limited food availability and a toxicant, is analysed. The aim is to find effects of toxicants on the structure and functioning of the ecosystem. The starting point is an unstressed ecosystem model for nutrients, populations, detritus and their intra- and interspecific interactions, as well as the interaction with the physical environment. The fate of the toxicant includes transport and exchange between the water and the populations via two routes, directly from water via diffusion over the outer membrane of the organism and via consumption of contaminated food. These processes are modelled using mass-balance formulations and diffusion equations. At the population level the toxicant affects different biotic processes such as assimilation, growth, maintenance, reproduction, and survival, thereby changing their biological functioning. This is modelled by taking the parameters that described these processes to be dependent on the internal toxicant concentration. As a consequence, the structure of the ecosystem, that is its species composition, persistence, extinction or invasion of species and dynamics behaviour, steady state oscillatory and chaotic, can change. To analyse the long-term dynamics we use the bifurcation analysis approach. In ecotoxicological studies the concentration of the toxicant in the environment can be taken as the bifurcation parameter. The value of the concentration at a bifurcation point marks a structural change of the ecosystem. This indicates that chemical stressors are analysed mathematically in the same way as environmental (e.g. temperature) and ecological (e.g. predation) stressors. Hence, this allows an integrated approach where different type of stressors are analysed simultaneously. Environmental regimes and toxic stress levels at which no toxic effects occur and where the ecosystem is resistant will be derived. A numerical continuation technique to calculate the boundaries of these regions will be given.

Stabilization due to predator interference: comparison of different analysis approaches

We study the influence of the particular form of the functional response in two-dimensional predator-prey models with respect to the stability of the nontrivial equilibrium. This equilibrium is stable between its appearance at a transcritical bifurcation and its destabilization at a Hopf bifurcation, giving rise to periodic behavior. Based on local bifurcation analysis, we introduce a classification of stabilizing effects. The classical Rosenzweig-MacArthur model can be classified as weakly stabilizing, undergoing the paradox of enrichment, while the well known Beddington-DeAngelis model can be classified as strongly stabilizing. Under certain conditions we obtain a complete stabilization, resulting in an avoidance of limit cycles. Both models, in their conventional formulation, are compared to a generalized, steady-state independent two-dimensional version of these models, based on a previously developed normalization method. We show explicitly how conventional and generalized models are related and how to interpret the results from the rather abstract stability analysis of generalized models.

A new class of non-linear stochastic population models with mass conservation

We study the effects of random feeding, growing and dying in a closed nutrient-limited producer/consumer system, in which nutrient is fully conserved, not only in the mean, but, most importantly, also across random events. More specifically, we relate these random effects to the closest deterministic models, and evaluate the importance of the various times scales that are involved. These stochastic models differ from deterministic ones not only in stochasticity, but they also have more details that involve shorter times scales. We tried to separate the effects of more detail from that of stochasticity. The producers have (nutrient) reserve and (body) structure, and so a variable chemical composition. The consumers have only structure, so a constant chemical composition. The conversion efficiency from producer to consumer, therefore, varies. The consumers use reserve and structure of the producers as complementary compounds, following the rules of Dynamic Energy Budget theory. Consumers die at constant specific rate and decompose instantaneously. Stochasticity is incorporated in the behaviour of the consumers, where the switches to handling and searching, as well as dying are Poissonian point events. We show that the stochastic model has one parameter more than the deterministic formulation without time scale separation for conversions between searching and handling consumers, which itself has one parameter more than the deterministic formulation with time scale separation for these conversions. These extra parameters are the contributions of a single individual producer and consumer to their densities, and the ratio of the two, respectively. The tendency to oscillate increases with the number of parameters. The focus bifurcation point has more relevance for the asymptotic behaviour of the stochastic model than the Hopf bifurcation point, since a randomly perturbed damped oscillation exhibits a behaviour similar to that of the stochastic limit cycle particularly near this bifurcation point. For total nutrient values below the focus bifurcation point, the system gradually becomes more confined to the direct neighbourhood of the isocline for which the producers do not change.

Advantage of storage in a fluctuating environment

We will elaborate the evolutionary course of an ecosystem consisting of a population in a chemostat environment with periodically fluctuating nutrient supply. The organisms that make up the population consist of structural biomass and energy storage compartments. In a constant chemostat environment a species without energy storage always out-competes a species with energy reserves. This hinders evolution of species with storage from those without storage. Using the adaptive dynamics approach for non-equilibrium ecological systems we will show that in a fluctuating environment there are multiple stable evolutionary singular strategies (ss's): one for a species without, and one for a species with energy storage. The evolutionary end-point depends on the initial evolutionary state. We will formulate the invasion fitness in terms of Floquet multipliers for the oscillating non-autonomous system. Bifurcation theory is used to study points where due to evolutionary development by mutational steps, the long-term dynamics of the ecological system changes qualitatively. To that end, at the ecological time scale, the trait value at which invasion of a mutant into a resident population becomes possible can be calculated using numerical bifurcation analysis where the trait is used as the free parameter, because it is just a bifurcation point. In a constant environment there is a unique stable equilibrium for one species following the "competitive exclusion" principle. In contrast, due to the oscillatory dynamics on the ecological time scale two species may coexist. That is, non-equilibrium dynamics enhances biodiversity. However, we will show that this coexistence is not stable on the evolutionary time scale and always one single species survives.

The relationship between elimination rates and partition coefficients

Rate constants for uptake and elimination of chemicals in organisms are often related to partition coefficients (typically the octanol-water partition coefficient). We show that the well-mixed one-compartment model for toxico-kinetics implies that the elimination rate is inversely proportional to the square root of the partition coefficient. When chemical exchange is limited by diffusion in the boundary layers adjacent to the interface, two-film models are appropriate, which have more complex implications for the relationships between the exchange rates and the partition coefficient. We also show that the popular steady-flux approximation of the two-film model is not a conceptual generalization of the one-compartment model, although it shares the first-order kinetics. We compare the kinetics of a series of models with an increasing number of well-mixed compartments for exchange, such that the two-film model results for an infinite number of compartments. The latter model formulation in terms of partial differential equations, and more in particular its boundary condition at the interface of the two media, is believed to be new. In the steady-flux approximation and in the model with single well-mixed boundary layers and low diffusivities, the elimination rate depends hyperbolically on the partition coefficient. The available data for abiotic systems (SPME fibers) supports a hyperbolic relationship, whereas the data for aquatic biota are less discriminating between a hyperbolic or a square root relationship with the partition coefficient. The daphnia data showed less scatter than the fish data, possibly due to the small variance in body sizes, since elimination rates are inversely proportional to body length. The square root relationship fitted these data best.

Consequences of symbiosis for food web dynamics

Basic Lotka-Volterra type models in which mutualism (a type of symbiosis where the two populations benefit both) is taken into account, may give unbounded solutions. We exclude such behaviour using explicit mass balances and study the consequences of symbiosis for the long-term dynamic behaviour of a three species system, two prey and one predator species in the chemostat. We compose a theoretical food web where a predator feeds on two prey species that have a symbiotic relationships. In addition to a species-specific resource, the two prey populations consume the products of the partner population as well. In turn, a common predator forages on these prey populations. The temporal change in the biomass and the nutrient densities in the reactor is described by ordinary differential equations (ODE). Since products are recycled, the dynamics of these abiotic materials must be taken into account as well, and they are described by odes in a similar way as the abiotic nutrients. We use numerical bifurcation analysis to assess the long-term dynamic behaviour for varying degrees of symbiosis. Attractors can be equilibria, limit cycles and chaotic attractors depending on the control parameters of the chemostat reactor. These control parameters that can be experimentally manipulated are the nutrient density of the inflow medium and the dilution rate. Bifurcation diagrams for the three species web with a facultative symbiotic association between the two prey populations, are similar to that of a bi-trophic food chain; nutrient enrichment leads to oscillatory behaviour. Predation combined with obligatory symbiotic prey-interactions has a stabilizing effect, that is, there is stable coexistence in a larger part of the parameter space than for a bi-trophic food chain. However, combined with a large growth rate of the predator, the food web can persist only in a relatively small region of the parameter space. Then, two zero-pair bifurcation points are the organizing centers. In each of these points, in addition to a tangent, transcritical and Hopf bifurcation a global heteroclinic bifurcation is emanating. This heteroclinic cycle connects two saddle equilibria where the predator is absent. Under parameter variation the period of the stable limit cycle goes to infinity and the cycle tends to the heteroclinic cycle. At this global bifurcation point this cycle breaks and the boundary of the basin of attraction disappears abruptly because the separatrix disappears together with the cycle. As a result, it becomes possible that a stable two-nutrient-two-prey population system becomes unstable by invasion of a predator and eventually the predator goes extinct together with the two prey populations, that is, the complete food web is destroyed. This is a form of over-exploitation by the predator population of the two symbiotic prey populations. When obligatory symbiotic prey-interactions are modelled with Liebig's minimum law, where growth is limited by the most limiting resource, more complicated types of bifurcations are found. This results from the fact that the Jacobian matrix changes discontinuously with respect to a varying parameter when another resource becomes most limiting.

Numerical bifurcation analysis of ecosystems in a spatially homogeneous environment

The dynamics of single populations up to ecosystems, are often described by one or a set of non-linear ordinary differential equations. In this paper we review the use of bifurcation theory to analyse these non-linear dynamical systems. Bifurcation analysis gives regimes in the parameter space with quantitatively different asymptotic dynamic behaviour of the system. In small-scale systems the underlying models for the populations and their interaction are simple Lotka-Volterra models or more elaborated models with more biological detail. The latter ones are more difficult to analyse, especially when the number of populations is large. Therefore for large-scale systems the Lotka-Volterra equations are still popular despite the limited realism. Various approaches are discussed in which the different time-scale of ecological and evolutionary biological processes are considered together.

Quantitative steps in symbiogenesis and the evolution of homeostasis

The merging of two independent populations of heterotrophs and autotrophs into a single population of mixotrophs has occurred frequently in evolutionary history. It is an example of a wide class of related phenomena, known as symbiogenesis. The physiological basis is almost always (reciprocal) syntrophy, where each species uses the products of the other species. Symbiogenesis can repeat itself after specialization on particular assimilatory substrates. We discuss quantitative aspects and delineate eight steps from two free-living interacting populations to a single fully integrated endosymbiotic one. The whole process of gradual interlocking of the two populations could be mimicked by incremental changes of particular parameter values. The role of products gradually changes from an ecological to a physiological one. We found conditions where the free-living, epibiotic and endobiotic populations of symbionts can co-exist, as well as conditions where the endobiotic symbionts outcompete other symbionts. Our population dynamical analyses give new insights into the evolution of cellular homeostasis. We show how structural biomass with a constant chemical composition can evolve in a chemically varying environment if the parameters for the formation of products satisfy simple constraints. No additional regulation mechanisms are required for homeostasis within the context of the dynamic energy budget (DEB) theory for the uptake and use of substrates by organisms. The DEB model appears to be dosed under endosymbiosis. This means that when each free-living partner follows DEB rules for substrate uptake and use, and they become engaged in an endosymbiotic relationship, a gradual transition to a single fully integrated system is possible that again follows DEB rules for substrate uptake and use.

Numerical bifurcation analysis of a tri-trophic food web with omnivory

We study the consequences of omnivory on the dynamic behaviour of a three species food web under chemostat conditions. The food web consists of a prey consuming a nutrient, a predator consuming a prey and an omnivore which preys on the predator and the prey. For each trophic level an ordinary differential equation describes the biomass density in the reactor. The hyperbolic functional response for single and multi prey species figures in the description of the trophic interactions. There are two limiting cases where the omnivore is a specialist; a food chain where the omnivore does not consume the prey and competition where the omnivore does not prey on the predator. We use bifurcation analysis to study the long-term dynamic behaviour for various degrees of omnivory. Attractors can be equilibria, limit cycles or chaotic behaviour depending on the control parameters of the chemostat. Often multiple attractor occur. In this paper we will discuss community assembly. That is, we analyze how the trophic structure of the food web evolves following invasion where a new invader is introduced one at the time. Generally, with an invasion, the invader settles itself and persists with all other species, however, the invader may also replace another species. We will show that the food web model has a global bifurcation, being a heteroclinic connection from a saddle equilibrium to a limit cycle of saddle type. This global bifurcation separates regions in the bifurcation diagram with different attractors to which the system evolves after invasion. To investigate the consequences of omnivory we will focus on invasion of the omnivore. This simplifies the analysis considerably, for the end-point of the assembly sequence is then unique. A weak interaction of the omnivore with the prey combined with a stronger interaction with the predator seems advantageous.

Light-induced mass turnover in a mono-species community of mixotrophs

We formulate a simple model for growth of a facultative photoautotroph with chemoheterotrophic capabilities. The organism is described by zero, one or three reserve components, and one structural component, all taken to be generalized compounds. The rules of synthesizing units are used for interactions among the uptake processes of the various nutrients and light (parallel processing), and for the merging of autotrophic and heterotrophic activities (sequential processing). For simplicity, we focus on the assimilation of inorganic carbon, inorganic nitrogen and light, and of two organic compounds (dead reserves and dead structure) that originate from aging. The process of resource recycling in a closed environment, as driven by light, and its links with community's structure (amount of biomass) is analysed in this simplest of all communities. Explicit analytical expressions for the steady states show how structure and function depend on the system parameters light, total carbon and total nitrogen. The behaviour resembles the Monod model for the Canonical Community, a three-species ecosystem consisting of producers, consumers and decomposers. If trophic preferences of a mixotroph are allowed to follow a random walk across generations, a trophic structure evolves where mixotrophs coexist with auto- and heterotrophs. Depth profiles are presented for the implied steady-state concentrations of dissolved inorganic carbon and nitrogen.

Iteroparous reproduction strategies and population dynamics

Asymptotic relationships between a class of continuous partial differential equation population models and a class of discrete matrix equations are derived for iteroparous populations. First, the governing equations are presented for the dynamics of an individual with juvenile and adult life stages. The organisms reproduce after maturation, as determined by the juvenile period, and at specific equidistant ages, which are determined by the iteroparous reproductive period. A discrete population matrix model is constructed that utilizes the reproductive information and a density-dependent mortality function. Mortality in the period between two reproductive events is assumed to be a continuous process where the death rate for the adults is a function of the number of adults and environmental conditions. The asymptotic dynamic behaviour of the discrete population model is related to the steady-state solution of the continuous-time formulation. Conclusions include that there can be a lack of convergence to the steady-state age distribution in discrete event reproduction models. The iteroparous vital ratio (the ratio between the maximal age and the reproductive period) is fundamental to determining this convergence. When the vital ratio is rational, an equivalent discrete-time model for the population can be derived whose asymptotic dynamics are periodic and when there are a finite number of founder cohorts, the number of cohorts remains finite. When the ratio is an irrational number, effectively there is convergence to the steady-state age distribution. With a finite number of founder cohorts, the number of cohorts becomes countably infinite. The matrix model is useful to clarify numerical results for population models with continuous densities as well as delta measure age distribution. The applicability in ecotoxicology of the population matrix model formulation for iteroparous populations is discussed.

Bi-trophic food chain dynamics with multiple component populations

Food web models describe the patterns of material and energy flow in communities. In classical food web models the state of each population is described by a single variable which represents, for instance, the biomass or the number of individuals that make up the population. However, in a number of models proposed recently in the literature the individual organisms consist of two components. In addition to the structural component there is an internal pool of nutrients, lipids or reserves. Consequently the population model for each trophic level is described by two state variables instead of one. As a result the classical predator-prey interaction formalisms have to be revised. In our model time budgets with actions as searching and handling provide the formulation of the functional response for both components. In the model, assimilation of the ingested two prey components is done in parallel and the extracted energy is added to a predators reserve pool. The reserves are used for vital processes; growth, reproduction and maintenance. We will explore the top-down modelling approach where the perspective is from the community. We will demonstrate that this approach facilitates a check on the balance equations for mass and energy at this level of organization. Here it will be shown that, if the individual is allowed to shrink when the energy reserves are in short to pay the maintenance costs, the growth process has to be 100% effective. This is unrealistic and some alternative model formulations are discussed. The long-term dynamics of a microbial food chain in the chemostat are studied using bifurcation analysis. The dilution rate and the concentration of nutrients in the reservoir are the bifurcation parameters. The studied microbial bi-trophic food chain with two-component populations shows chaotic behaviour.

Multiple attractors and boundary crises in a tri-trophic food chain

The asymptotic behaviour of a model of a tri-trophic food chain in the chemostat is analysed in detail. The Monod growth model is used for all trophic levels, yielding a non-linear dynamical system of four ordinary differential equations. Mass conservation makes it possible to reduce the dimension by 1 for the study of the asymptotic dynamic behaviour. The intersections of the orbits with a Poincaré plane, after the transient has died out, yield a two-dimensional Poincaré next-return map. When chaotic behaviour occurs, all image points of this next-return map appear to lie close to a single curve in the intersection plane. This motivated the study of a one-dimensional bi-modal, non-invertible map of which the graph resembles this curve. We will show that the bifurcation structure of the food chain model can be understood in terms of the local and global bifurcations of this one-dimensional map. Homoclinic and heteroclinic connecting orbits and their global bifurcations are discussed also by relating them to their counterparts for a two-dimensional map which is invertible like the next-return map. In the global bifurcations two homoclinic or two heteroclinic orbits collide and disappear. In the food chain model two attractors coexist; a stable limit cycle where the top-predator is absent and an interior attractor. In addition there is a saddle cycle. The stable manifold of this limit cycle forms the basin boundary of the interior attractor. We will show that this boundary has a complicated structure when there are heteroclinic orbits from a saddle equilibrium to this saddle limit cycle. A homoclinic bifurcation to a saddle limit cycle will be associated with a boundary crisis where the chaotic attractor disappears suddenly when a bifurcation parameter is varied. Thus, similar to a tangent local bifurcation for equilibria or limit cycles, this homoclinic global bifurcation marks a region in the parameter space where the top-predator goes extinct. The 'Paradox of Enrichment' says that increasing the concentration of nutrient input can cause destabilization of the otherwise stable interior equilibrium of a bi-trophic food chain. For a tri-trophic food chain enrichment of the environment can even lead to extinction of the highest trophic level.

Numerical methods and parameter estimation of a structured population model with discrete events in the life history

We consider two numerical methods for the solution of a physiologically structured population (PSP) model with multiple life stages and discrete event reproduction. The model describes the dynamic behaviour of a predator-prey system consisting of rotifers predating on algae. The nitrate limited algal prey population is modelled unstructured and described by an ordinary differential equation (ODE). The formulation of the rotifer dynamics is based on a simple physiological model for their two life stages, the egg and the adult stage. An egg is produced when an energy buffer reaches a threshold value. The governing equations are coupled partial differential equations (PDE) with initial and boundary conditions. The population models together with the equation for the dynamics of the nutrient result in a chemostat model. Experimental data are used to estimate the model parameters. The results obtained with the explicit finite difference (FD) technique compare well with those of the Escalator Boxcar Train (EBT) method. This justifies the use of the fast FD method for the parameter estimation, a procedure which involves repeated solution of the model equations.

The role of intracellular components in food chain dynamics

The dynamics of a simple food web, including multiple substrates and predator-prey interactions, is studied. An individual-based model is presented that describes the intracellular composition of the biomass of each population with two components: reserves and structural biomass. The model describes the simultaneous uptake of multiple substrates via specific carriers and their assimilation into reserve energy via multiple assimilation pathways. The available energy is used for maintenance and growth. Parameters are estimated by curve-fitting data from the literature under the condition that the elemental balances and the enthalpy balances are met. The proposed model provides an adequate description of the macromolecular composition of biomass. The model is not too complicated to be of use in the study of food webs. The consequences of the presence of intracellular components in a food web on its long-term dynamics are investigated with bifurcation analysis.

Discrete event versus continuous approach to reproduction in structured population dynamics

The governing equations are derived for the dynamics of a population consisting of organisms which reproduce by laying one egg at the time, on the basis of a simple physiological model for the uptake and use of energy. Two life stages are assumed, the egg and the adult stage where the adults do not grow. These assumptions hold true, for instance, for rotifers. From the model for the life history of the individuals, a physiologically structured population model for a rotifer population is derived. On the basis of this discrete event reproduction population model a continuous reproduction population model is proposed. The population model together with the equation for the food result in chemostat equations which are solved numerically. We show that for the calculation of the transient population dynamic behaviour after a step-wise change of the dilution rate, an age structure suffices, despite the size and energy structure used to describe the dynamics of the individuals. Aggregation of the continuous reproduction population model yields an approximate lumped parameter model in terms of delay differential equations. In order to assess the performance of the models, experimental data from the literature are fitted. The main purpose of this paper is to discuss the consequences of discrete event versus continuous reproduction. In both population models death by starvation is taken into account. Unlike the continuous reproduction model, the discrete model captures the experimentally observed lack of egg production shortly after the step change in the dilution rate of the chemostat.

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.

Resistance of a food chain to invasion by a top predator

We study the invasion of a top predator into a food chain in a chemostat. For each trophic level, a bioenergetic model is used in which maintenance and energy reserves are taken into account. Bifurcation analysis is performed on the set of nonlinear ordinary differential equations which describe the dynamic behaviour of the food chain. In this paper, we analyse how the ability of a top predator to invade the food chain depends on the values of two control parameters: the dilution rate and the concentration of the substrate in the input. We investigate invasion by studying the long-term behaviour after introduction of a small amount of top predator. To that end we look at the stability of the boundary attractors; equilibria, limit cycles as well as chaotic attractors using bifurcation analysis. It will be shown that the invasibility criterion is the positiveness of the Lyapunov exponent associated with the change of the biomass of the top predator. It appears that the region in the control parameter space where a predator can invade increases with its growth rate. The resulting system becomes more resistant to further invasion when the top predator grows faster. This implies that short food chains with moderate growth rate of the top predator are liable to be invaded by fast growing invaders which consume the top predator. There may be, however, biological constraints on the top predator's growth rate. Predators are generally larger than prey while larger organisms commonly grow slower. As a result, the growth rate generally decreases with the trophic level. This may enable short food chains to be resistant to invaders. We will relate these results to ecological community assembly and the debate on the length of food chains in nature.

Diffusion barriers limit the effect of mobile calcium buffers on exocytosis of large dense cored vesicles

Fast exocytosis in melanotropic cells, activated by calcium entry through voltage-gated calcium channels, is very sensitive to mobile calcium buffers (complete block at 800 microM ethylene glycol bis(beta-aminoethyl ether)-N,N,N'N'-tetraacetic acid (EGTA)). This indicates that calcium diffuses a substantial distance from the channel to the vesicle. Surprisingly, 1, 2-bis(2-aminophenoxy)ethane-N,N,N',N'-tetraacetic acid (BAPTA), having a similar KD for calcium as EGTA but a approximately 100 times faster binding rate, blocked exocytosis only twice as effectively as EGTA. Using computer simulations, we demonstrate that this result cannot be explained by free diffusion and buffer binding rates. We hypothesized that local saturation of calcium buffers is involved. A diffusion barrier for both calcium and buffer molecules, located 50-300 nm from the membrane and reducing diffusion 1000 to 10,000 times, generated similar calcium concentrations for specific concentrations of EGTA and BAPTA. With such barriers, calcium rise phase kinetics upon short step depolarizations (2-20 ms) were faster for EGTA than for BAPTA, implying that short depolarizations should allow exocytosis with 50 microM EGTA but not with 25 microM BAPTA. This prediction was confirmed experimentally with capacitance measurements. Coupling exocytosis to calcium dynamics in the model, we found that a barrier with a approximately 3000 times reduced diffusion at approximately 130 nm beneath the membrane best explains the experimentally observed effects of EGTA and BAPTA on block and kinetics of release.

Consequences of population models for the dynamics of food chains

A class of bioenergetic ecological models is studied for the dynamics of food chains with a nutrient at the base. A constant influx rate of the nutrient and a constant efflux rate for all trophic levels is assumed. Starting point is a simple model where prey is converted into predator with a fixed efficiency. This model is extended by the introduction of maintenance and energy reserves at all trophic levels, with two state variables for each trophic level, biomass and reserve energy. Then the dynamics of each population are described by two ordinary differential equations. For all models the bifurcation diagram for the bi-trophic food chain is simple. There are three important regions; a region where the predator goes to extinction, a region where there is a stable equilibrium and a region where a stable limit cycle exists. Bifurcation diagrams for tritrophic food chains are more complicated. Flip bifurcation curves mark regions where complex dynamic behaviour (higher periodic limit cycles as well as chaotic attractors) can occur. We show numerically that Shil'nikov homoclinic orbits to saddle-focus equilibria exists. The codimension 1 continuations of these orbits form a 'skeleton' for a cascade of flip and tangent bifurcations. The bifurcation analysis facilitates the study of the consequences of the population model for the dynamic behaviour of a food chain. Although the predicted transient dynamics of a food chain may depend sensitively on the underlying model for the populations, the global picture of the bifurcation diagram for the different models is about the same.

Bow-arrow interaction in archery

A mathematical model of the flight of the arrow during its discharge from a bow was proposed by Pekalski (1990). His description of the model was incomplete. In this paper, I give a full description of the model. Furthermore, I propose some improvements that make his model more consistent with reality. One achievement is the modelling of contact of the arrow and grip; the pressure button is modelled as a unilateral elastic support. The acceleration force acting upon the arrow during the launch is predicted by an advanced mathematical model of bow dynamics. There is a satisfactory conformity of the simulation and experimental results. The new model predicts that the arrow leaves the pressure button before it leaves the string, as reported previously. The ability to model arrow dynamics can be used to improve the adjustment of the bow-arrow system for optimal performance.

Food chain dynamics in the chemostat

The asymptotic behavior of a tri-trophic food chain model in the chemostat is studied. The Monod-Herbert growth model is used for all trophic levels. The analysis is carried out numerically, by finding both local and global bifurcations of equilibria and of limit cycles with respect to two chemostat control parameters: the dilution rate of the chemostat and the concentration of input substrate. It is shown that the bifurcation structure of the food chain model has much in common with the bifurcation structure of a one-dimensional map with two turning points. This map is used to explain how attractors are created and destroyed under variation of the bifurcation parameters. It is shown that low as well as high concentration of input substrate can lead to extinction of the highest trophic level.

Numerical analysis of pulmonary indicator gas wash-in

Mathematical modelling of gas transport in the human lung yields a diffusion-convection problem. Transport is described by a second-order parabolic partial differential equation with variable coefficients and initial and boundary conditions. A new numerical approach for the solution of these governing equations is proposed in which the convection term is dealt with exactly. This method combines integration along characteristics and the finite difference method. The results obtained are in agreement with those of other investigators but are based on less stringent assumptions. To judge the merit of various finite difference schemes, the results are compared with analytical solutions for a special case in which the lung is modelled as a uniform cylinder. Laplace transform techniques and expansion into Fourier series are used to obtain the analytical solutions.