Research Interests: Mathematics, Modal Logic, Computer Science, Chemistry, Epistemic Logic, and 11 moreTheoretical Computer Science, Common Knowledge, Information Flow, Compositional Analysis, Finite Automata, Dynamic Epistemic Logic, Axiom, Dynamic Logic, Propositional dynamic logic, Mathematical Proof, and Agent Communication
In Part I of this paper we discussed new methods for the numerical continuation of point-tocycle connecting orbits in 3-dimensional autonomous ODE’s using projection boundary conditions. In this second part we extend the method to the... more
In Part I of this paper we discussed new methods for the numerical continuation of point-tocycle connecting orbits in 3-dimensional autonomous ODE’s using projection boundary conditions. In this second part we extend the method to the numerical continuation of cycle-to-cycle connecting orbits. In our approach, the projection boundary conditions near the cycles are formulated using eigenfunctions of the associated adjoint variational equations, avoiding costly and numerically unstable computations of the monodromy matrices. The equations for the eigenfunctions are included in the defining boundary-value problem, allowing a straightforward implementation in auto, in which only the standard features of the software are employed. Homotopy methods to find the connecting orbits are discussed in general and illustrated with an example from population dynamics. Complete auto demos, which can be easily adapted to any autonomous 3-dimensional ODE system, are freely available.
The motivation for the research reported in this paper comes from modeling the spread of vector-borne virus diseases. To study the role of the host versus vector dynamics and their interaction we use the susceptible-infected-removed (SIR)... more
The motivation for the research reported in this paper comes from modeling the spread of vector-borne virus diseases. To study the role of the host versus vector dynamics and their interaction we use the susceptible-infected-removed (SIR) host model and the susceptible-infected (SI) vector model. When the vector dynamical processes occur at a faster scale than those in the host-epidemics dynamics, we can use a time-scale argument to reduce the dimension of the model. This is often implemented as a quasi steady-state assumption (qssa) where the slow varying variable is set at equilibrium and an ode equation is replaced by an algebraic equation. Singular perturbation theory will appear to be a useful tool to perform this derivation. An asymptotic expansion in the small parameter that represents the ratio of the two time scales for the dynamics of the host and vector is obtained using an invariant manifold equation. In the case of a susceptible-infected-susceptible (SIS) host model this algebraic equation is a hyperbolic relationship modeling a saturated incidence rate. This is similar to the Holling type II functional response (Ecology) and the Michaelis-Menten kinetics (Biochemistry). We calculate the value for the force of infection leading to an endemic situation by performing a bifurcation analysis. The effect of seasonality is studied where the force of infection changes sinusoidally to model the annual fluctuations of the vector population. The resulting non-autonomous system is studied in the same way as the autonomous system using bifurcation analysis.
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Multiple attractors and boundary crises in a tri-trophic food chain
A bow is a mechanical device where energy is stored in parts of the limbs that is transferred as kinetic energy to the arrow supported at the middle of the string attached to both limb ends. The energy storage capacity of the material of... more
A bow is a mechanical device where energy is stored in parts of the limbs that is transferred as kinetic energy to the arrow supported at the middle of the string attached to both limb ends. The energy storage capacity of the material of the limbs is crucial to get a high efficiency of this energy transmission. Also the strength of the string is important to make it as light as possible, as well as the quality and reliability the arrow. In this chapter an overview of the materials used in archery equipment is given. Also quality performance criteria for use in shooting at archery tournaments such as Olympic Games, are reviewed and obtained by mathematical modeling of the mechanical function of the bow and arrow. These quality coefficients for the modern bow are only slightly better than those of the historical bows. Materials used in modern working-recurve bows can store more deformation energy per unit of mass than materials used in the past. Moreover the mechanical properties of t...
Research Interests:
Abstract The need to follow structured populations, as opposed to unstructured ones, is well-recognized. The most detailed category of population models are the individual-based population models (IBMs), also called agent-based population... more
Abstract The need to follow structured populations, as opposed to unstructured ones, is well-recognized. The most detailed category of population models are the individual-based population models (IBMs), also called agent-based population models (ABMs). Their analysis is generally by simulation in time followed by statistical analysis of the numerical results. A less detailed method is the physiologically structured population model approach (PSPMs) leading originally to continuous-time partial differential equations ( pde s) for the p(opulation)-states such as number(-density) with respect to time and i(ndividual)-state such as age and/or size and later to a delay equation formulation. Their mathematical analysis and computational methods are generally complex. Discrete-time matrix population models (MPMs) are much simpler to analyse in all respects, but the applicability is limited due to stringent modelling assumptions made. We discuss here a class of models we call the Cohort Projection Models (CPMs), which were formerly introduced as a special case of PSPMs with pulsed reproduction. CPMs follow cohorts of identical individuals in a Lagrangian way of which the changes of their i-states such as, size, energy reserves and maturity, are described by age dependent ordinary differential equations ( ode )s from DEB theory. Simultaneously the p-states, such as number of individuals are described by time dependent ode s obeying conservation laws. The population is subdivided in generations on the assumption that seasonal cycles synchronize reproduction events among cohorts and all eggs that are produced by different generations are the same. Feedback from the environment can be included via specification of food dynamics that accommodates competition. Temperature follows a specified periodic trajectory in time. This allows for the definition of a projection map of i-states and p-states, from one reproduction event to the next. The projection interval is typically one year for seasonal variability. The properties of the map can be studied using nonlinear dynamical system theory, such as existence and stability of fixed points and, thereby, the long-term dynamics of the food-population system. We demonstrate this using deb parameter values from the Add-my-Pet (AmP) collection for over 2000 animal species, which were estimated from empirical data. CPMs are meant to match the relative simplicity of the analysis of MPMs with the realism of the deb models for the dynamics of the population individuals.
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Research Interests: Algorithms, Kinetics, Biomass, Biology, Biological Sciences, and 15 moreFood web, Animals, Lotka Volterra, Jacobian Matrix, Ecosystem, Growth rate, Dynamic behaviour of materials, Basin of Attraction, Food Chain, Bifurcation Analysis, Mass Balance, Bifurcation diagram, Hopf Bifurcation, Limit Cycle, and Dynamic Behaviour
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Research Interests: Modeling, Biology, Multidisciplinary, Ecological Modelling, Ecosystem Modelling, and 15 moreFood web, Predation, Nutrient Enrichment, Trophic Level, Theoretical Analysis, Nutrient, Intraguild Predation, Spectrum, Ecosystem, Trophic Interaction, Guild, Food Chain, Bifurcation Analysis, Interaction Strength, and Omnivore
Research Interests:
Research Interests:
Research Interests:
We study the dynamics of a predator-prey system where predators fight for captured prey besides searching for and handling (and digestion) of the prey. Fighting for prey is modelled by a continuous time hawk-dove game dynamics where the... more
We study the dynamics of a predator-prey system where predators fight for captured prey besides searching for and handling (and digestion) of the prey. Fighting for prey is modelled by a continuous time hawk-dove game dynamics where the gain depends on the amount of disputed prey while the costs for fighting is constant per fighting event. The strategy of the predator-population is quantified by a trait being the proportion of the number of predator-individuals playing hawk tactics. The dynamics of the trait is described by two models of adaptation: the replicator dynamics (RD) and the adaptive dynamics (AD). In the RD-approach a variant individual with an adapted trait value changes the population's strategy, and consequently its trait value, only when its payoff is larger than the population average. In the AD-approach successful replacement of the resident population after invasion of a rare variant population with an adapted trait value is a step in a sequence changing the p...
Research Interests:
Many current issues in ecology require predictions made by mathematical models, which are built on somewhat arbitrary choices. Their consequences are quantified by sensitivity analysis to quantify how changes in model parameters propagate... more
Many current issues in ecology require predictions made by mathematical models, which are built on somewhat arbitrary choices. Their consequences are quantified by sensitivity analysis to quantify how changes in model parameters propagate into an uncertainty in model predictions. An extension called structural sensitivity analysis deals with changes in the mathematical description of complex processes like predation. Such processes are described at the population scale by a specific mathematical function taken among similar ones, a choice that can strongly drive model predictions. However, it has only been studied in simple theoretical models. Here, we ask whether structural sensitivity is a problem of oversimplified models. We found in predator-prey models describing chemostat experiments that these models are less structurally sensitive to the choice of a specific functional response if they include mass balance resource dynamics and individual maintenance. Neglecting these proces...
Research Interests:
Research Interests:
Research Interests:
Generally a predator-prey system is modelled by two ordinary differential equations which describe the rate of changes of the biomasses. Since such a system is two-dimensional no chaotic behaviour can occur. In the popular... more
Generally a predator-prey system is modelled by two ordinary differential equations which describe the rate of changes of the biomasses. Since such a system is two-dimensional no chaotic behaviour can occur. In the popular Rosenzweig-MacArthur model, which replaced the Lotka-Volterra model, a stable equilibrium or a stable limit cycle exist. In this paper the prey consumes a non-viable nutrient whose. dynamics is modelled explicitly and this gives an extra ordinary differential equation. For a predator-prey system under chemostat conditions where all parameter values are biologically meaningful, coexistence of multiple chaotic attractors is possible in a narrow region of the two-parameter bifurcation diagram with respect to the chemostat control parameters. Crisis-limited chaotic behaviour and a bifurcation point where two coexisting chaotic attractors merge will be discussed. The interior and boundary crises of this continuous-time predator-prey system look similar to those found f...
Research Interests:
Research Interests:
Research Interests: Plant Biology and Ecology
Research Interests:
The invention of the bow and arrow probably ranks for social impact with the inven-tion of the art of kindling a fire and the invention of the wheel. It must have been in prehistoric times that the first missile was launched with a bow,... more
The invention of the bow and arrow probably ranks for social impact with the inven-tion of the art of kindling a fire and the invention of the wheel. It must have been in prehistoric times that the first missile was launched with a bow, we do not know where and when. The event may well have occurred in different parts of the world at about the same time or at widely differing times. Numerous kinds of bows are known, they may have long limbs or short limbs, upper and lower limbs may be equal or unequal in length whilst cross-sections of the limbs may take various shapes. Wood or steel may be used, singly as in 'self' bows, or mixed when different layers are glued together. There are 'composite' bows with layers of several kinds of organic material, wood, sinew and horn, and, in modern forms, layers of wood and synthetic plastics reinforced with glassfibre or carbon. The shape of the bow when relaxed, may be straight or recurved, where the curvature of the parts of the...
Research Interests:
Accompanying manuscript to the demonstration of the detection and contin-uation of a homoclinic cycle-to-cycle connection in the 3D food chain model by Rosenzweig–MacArthur with the bifurcation software package AUTO, by use of the... more
Accompanying manuscript to the demonstration of the detection and contin-uation of a homoclinic cycle-to-cycle connection in the 3D food chain model by Rosenzweig–MacArthur with the bifurcation software package AUTO, by use of the homotopy method described in the paper. The files are downloadable from http://www.bio.vu.nl/thb/research/project/globif.
ABSTRACT In classical population dynamic models all the individuals in a population are assumed to be identical or only population averages are considered. The state of the population is therefore the number of individuals or the total... more
ABSTRACT In classical population dynamic models all the individuals in a population are assumed to be identical or only population averages are considered. The state of the population is therefore the number of individuals or the total amount of biomass in the population. The mathematical model is a first-order ordinary differential equation (ODE) that specifies the time derivative of this population state. Subsequently, the dynamics of an ecosystem where populations interact with each other and with their abiotic environment is usually described by a system of ODEs. The classical models are sometimes called unstructured population models, as opposed to structured population models, which take into account the differences between individuals and characterize an individual by some state. The development of this state is modeled from birth until death, often using a first-order ODE that specifies the time derivative of the individual state. The model is complemented with models for the birth and death of individuals. The structured population model is derived straightforwardly from the individual model using balance laws. The state of a population is no longer a single number; it is the distribution of the individuals over their possible states. If the number of individuals is large, this distribution is continuous instead of discrete. If, in addition, the distribution is a density with respect to the individual state, the population model can be written as a partial differential equation (PDE). The first structured population models use age for the state of the individuals. For many organisms, however, a difference in age alone does not explain the differences in individual behavior, and other state variables such as size or energy reserves are more suitable. This has led to physiologically structured population models (PSPs). The life cycle of the individuals may consist of a number of life stages, such as egg, juvenile, and adult. The distribution of the number of individuals with respect to the state variable may be irregular. The introduction of these two extra components, namely, several life stages and nonsmooth distributions, gave mathematical difficulties in the PDE formulation. Recently, an alternative cumulative formulation in terms of renewal integral equations has been proposed that deals with these difficulties. In this review we describe the various model formulations from a modeling perspective. The dynamics of a population of worms that propagate by division into two unequal daughters serves as an example. We give an overview of numerical methods for structured population models.
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... more
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. Consum...
Research Interests:
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.... more
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.