Biomedical subjects

B W Kooi

Publications and source records attributed to B W Kooi.

At least 19 recordsLinked to original sources

PubMed2006

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.

Adaptation, Physiological↗

PubMed2004

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.

Algorithms↗

PubMed2004

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.

Animals↗

PubMed2003

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.

Animals↗

PubMed2003

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.

Animals↗

PubMed2002

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.

Aging↗

PubMed2001

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.

Bacteria↗

PubMed2001

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.

Animals↗

PubMed2001

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.

Animals↗

PubMed2000

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.

Animals↗

PubMed2000

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.

Animals↗

PubMed1999

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.

Animals↗

PubMed1999

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.

Age Factors↗

PubMed1999

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.

Animals↗

PubMed1998

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.

Animals↗

PubMed1998

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.

Animals↗

PubMed1998

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.

Biomechanical Phenomena↗

PubMed1993

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.

Diffusion↗