Numerical equilibrium analysis for structured consumer resource models.

In this paper, we present methods for a numerical equilibrium and stability analysis for models of a size structured population competing for an unstructured resource. We concentrate on cases where two model parameters are free, and thus existence boundaries for equilibria and stability boundaries can be defined in the (two-parameter) plane. We numerically trace these implicitly defined curves using alternatingly tangent prediction and Newton correction. Evaluation of the maps defining the curves involves integration over individual size and individual survival probability (and their derivatives) as functions of individual age. Such ingredients are often defined as solutions of ODE, i.e., in general only implicitly. In our case, the right-hand sides of these ODE feature discontinuities that are caused by an abrupt change of behavior at the size where juveniles are assumed to turn adult. So, we combine the numerical solution of these ODE with curve tracing methods. We have implemented the algorithms for "Daphnia consuming algae" models in C-code. The results obtained by way of this implementation are shown in the form of graphs.

Geometric quantification of the plant endoplasmic reticulum.

One of the most difficult obstacles to make biological sciences more quantitative is the lack of understanding the interplay of form and function. Each cell is full of complex-shaped objects, which moreover change their form over time. To tackle this problem, we suggest the use of geometric invariants that are able to produce precise reference points to compare a cell's different functional elements such as organelles under fixed and varying physiological conditions. In this paper, we look at the topology of an almost static sample of the plant cortical endoplasmic reticulum (ER) under close-to-normal physiological conditions using a multi-disciplinary approach combining confocal microscopy, image processing techniques, visualization, computational geometry and graph theory. Data collected from a series of optical sections taken at short, regular intervals along the optical axis are used to reconstruct the ER in three dimensions. A graph structure of the ER network is obtained after thinning the ER geometry to its essential features. The graph is the final and most abstract quantification of the ER and serves very well as a geometrical invariant, even and very importantly, in cases in which the ER sample is moving or slightly changing shape during image acquisition. Moreover, graph theoretic features, such as the number of nodes and their degrees and the number of edges and their lengths, are very robust against different kinds of small perturbations that should not change the ER function. We will also attach surface areas and volumes estimated for the plant ER network as weights to the graph, allowing an even more precise quantitative characterization of this organelle. In total, we have compared 28 different samples under similar experimental conditions. The methods used in this paper should also be applicable to the quantification of other organelles in which geometric abstraction is possible to analyse function. Finally, by the use of confocal microscopy, our techniques will be transferable to situations in which protein markers can move inside the organelle's lumen and/or on the membrane surface to test further aspects of protein distribution.

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

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

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.

Ring vaccination.

Based on the description of an outbreak of foot-and-mouth disease (FMD), a particle model is developed describing the most important properties of this epidemic. Also control measures (mass and ring vaccination) are implemented. This model shows the expected behavior in simulations. Since it is impossible to treat this model analytically, we use ideas of branching processes on two levels to derive a caricature of the particle model. In simulations it is shown that this caricature exhibits similar behavior as the particle system. It is possible to analyze the caricature and, in this way, to obtain expressions for the most important quantities like the reproduction number or the expected final number of infected individuals etc. In this way mass vaccination and ring vaccination can be compared and control strategies can be optimized.