On continuous time models in genetic and Bernstein algebras

We discuss the long-time behavior of Andreoli's differential equation for genetic algebras and for Bernstein algebras and show convergence to an equilibrium in both cases. For a class of Bernstein algebras this equilibrium is determined explicitly.     

Long-time behavior of continuous time models in genetic algebras

In [2] the solutions of Andreoli's differential equation in genetic algebras with genetic realization were shown to converge to equilibria. Here we derive an explicit formula for these limits.     

Dispersion population models discrete in time and continuous in space

We analyze a discrete-time model of populations that grow and disperse in separate phases. The growth phase is a nonlinear process that allows for the effects of local crowding. The dispersion phase is a linear process that distributes the population throughout its spatial habitat. Our study quantifies the issues of survival and extinction, the existence and stability of nontrivial steady states, and the comparison of various dispersion strategies. Our results show that all of these issues are tied to the global nature of various model parameters. The extreme strategies of staying-in place and going-everywhere-uniformly are compared numerically to diffusion strategies in various contexts. We approach the mathematical analysis of our model from a functional analysis and an operator theory point of view. We use recent results from the theory of positive operators in Banach lattices.     

Stochastic continuous time neurite branching models with tree and segment dependent rates

In this paper we introduce a continuous time stochastic neurite branching model closely related to the discrete time stochastic BES-model. The discrete time BES-model is underlying current attempts to simulate cortical development, but is difficult to analyze. The new continuous time formulation facilitates analytical treatment thus allowing us to examine the structure of the model more closely. We derive explicit expressions for the time dependent probabilities p(γ,t) for finding a tree γ at time t, valid for arbitrary continuous time branching models with tree and segment dependent branching rates. We show, for the specific case of the continuous time BES-model, that as expected from our model formulation, the sums needed to evaluate expectation values of functions of the terminal segment number μ(f(n),t) do not depend on the distribution of the total branching probability over the terminal segments. In addition, we derive a system of differential equations for the probabilities p(n,t) of finding n terminal segments at time t. For the continuous BES-model, this system of differential equations gives direct numerical access to functions only depending on the number of terminal segments, and we use this to evaluate the development of the mean and standard deviation of the number of terminal segments at a time t. For comparison we discuss two cases where mean and variance of the number of terminal segments are exactly solvable. Then we discuss the numerical evaluation of the S-dependence of the solutions for the continuous time BES-model. The numerical results show clearly that higher S values, i.e. values such that more proximal terminal segments have higher branching rates than more distal terminal segments, lead to more symmetrical trees as measured by three tree symmetry indicators.     

Uninvadability in N-species frequency models for resident-mutant systems with discrete or continuous time

The uninvadability concept, that was originally introduced through static comparisons of individual fitness in resident-mutant systems for a single species, is developed for multi-species models with frequency-dependent fitness by extending its equivalent single-species dynamic characterization. This multi-species definition is then reinterpreted in terms of individual fitness functions based on intra and interspecific interactions. The resultant concept is discussed in relation to that of an N-species ESS (evolutionarily stable strategy) and to dynamic stability of monomorphic and polymorphic evolutionary systems.     

A graph-theoretic method for the basic reproduction number in continuous time epidemiological models

In epidemiological models of infectious diseases the basic reproduction number is used as a threshold parameter to determine the threshold between disease extinction and outbreak. A graph-theoretic form of Gaussian elimination using digraph reduction is derived and an algorithm given for calculating the basic reproduction number in continuous time epidemiological models. Examples illustrate how this method can be applied to compartmental models of infectious diseases modelled by a system of ordinary differential equations. We also show with these examples how lower bounds for can be obtained from the digraphs in the reduction process.     

Heritability of longevity in Large White and Landrace sows using continuous time and grouped data models

Background

Using conventional measurements of lifetime, it is not possible to differentiate between productive and non-productive days during a sow''s lifetime and this can lead to estimated breeding values favoring less productive animals. By rescaling the time axis from continuous to several discrete classes, grouped survival data (discrete survival time) models can be used instead.

Methods

The productive life length of 12319 Large White and 9833 Landrace sows was analyzed with continuous scale and grouped data models. Random effect of herd*year, fixed effects of interaction between parity and relative number of piglets, age at first farrowing and annual herd size change were included in the analysis. The genetic component was estimated from sire, sire-maternal grandsire, sire-dam, sire-maternal grandsire and animal models, and the heritabilities computed for each model type in both breeds.

Results

If age at first farrowing was under 43 weeks or above 60 weeks, the risk of culling sows increased. An interaction between parity and relative litter size was observed, expressed by limited culling during first parity and severe risk increase of culling sows having small litters later in life. In the Landrace breed, heritabilities ranged between 0.05 and 0.08 (s.e. 0.014-0.020) for the continuous and between 0.07 and 0.11 (s.e. 0.016-0.023) for the grouped data models, and in the Large White breed, they ranged between 0.08 and 0.14 (s.e. 0.012-0.026) for the continuous and between 0.08 and 0.13 (s.e. 0.012-0.025) for the grouped data models.

Conclusions

Heritabilities for length of productive life were similar with continuous time and grouped data models in both breeds. Based on these results and because grouped data models better reflect the economical needs in meat animals, we conclude that grouped data models are more appropriate in pig.     

Existence of a globally asymptotically stable equilibrium in Volterra models with continuous time delay

A sufficient condition for the existence of a globally asymptotically stable equilibrium in Volterra models with continuous time delay is obtained, and some properties of the stable equilibrium are proven. Furthermore, some applications in which asymptotic stability only depends on the sign of the coefficients are considered.     

Genetic algebras for tetraploidy with several loci

The algebra of gamete multiplication corresponding to a population of autotetraploids which differ at n loci, each with an arbitrary number of alleres, is proven to be a genetic algebra, irrespective of the mode of segregation of the n loci.     

Survival in continuous structured populations models

The extended McKendrick-von Foerster structured population model is employed to derive a nonautonomous ordinary differential equation model of a population. The derivation assumes that the individual life history can be delineated into several physiological stages. We study the persistence of the population when the model is autonomous and base the nonautonomous survival analysis on the autonomous case and a comparison principle. A brief excursion into alternate life history strategies is presented.This work was supported in part by the U.S. Environmental Protection Agency under cooperative agreement CR 813353010     

LFT models of continuous biotechnological processes

The aim of the paper is to obtain suitable state-space models of continuous biotechnological processes (CBTP) in the framework of Linear Fractional Transformation (LFT). The LFT models are starting point in most of the advanced robust control design and analysis methods. Therefore, a linearized process model in the state-space is used whose elements are supposed to vary within certain bounds to represent the nonlinear behaviour of the real plant. The performance specifications are defined in the frequency domain through weighting functions. Two LFT models of CBTP are obtained ready for controller design aimed to optimize robust stability margins and robust performance, respectively.     

On density and extinction in continuous population models

Survival analyses, investigations of extinction and persistence, are executed for populations represented by a nonautonomous differential equation model. The population is assumed governed by density dependent and time varying density independent demographic parameters. While traditional approaches to extinction postulate extinction on an infinite time horizon and at zero abundance level, survival analysis is developed not only for this traditional setting but also on a finite time horizon and at a nonzero threshold level. A main conclusion is that extinction of a temporally stressed population is determined by a totality of density independent and density dependent factors.     

Genetic models of Parkinson disease

To date, a truly representative animal model of Parkinson disease (PD) remains a critical unmet need. Although toxin-induced PD models have served many useful purposes, they have generally failed to recapitulate accurately the progressive process as well as the nature and distribution of the human pathology. During the last decade or so, the identification of several genes whose mutations are causative of rare familial forms of PD has heralded in a new dawn for PD modelling. Numerous mammalian as well as non mammalian models of genetically-linked PD have since been created. However, despite initial optimism, none of these models turned out to be a perfect replica of PD. Meanwhile, genetic and toxin-induced models alike continue to evolve towards mimicking the disease more faithfully. Notwithstanding this, current genetic models have collectively illuminated several important pathways relevant to PD pathogenesis. Here, we have attempted to provide a comprehensive discussion on existing genetic models of PD.     

Genetic and phenotypic models of natural selection

The following theorem is proposed: when two phenotypes differ in attributes affecting their relative fitness, selection will cease to cause further evolutionary change when the two phenotypes have the same fitness, provided that certain modes of inheritance apply; in particular, all genotypes specifying the same phenotype must have the same average fitness. If these conditions of “uniform fitness” patterns of inheritance are not met, particular genetic models of natural selection should replace an analysis of phenotypes. If the conditions are met, an analysis of the stationary conditions when the phenotypes have equal fitnesses permits quantitative statements about the outcome of selection without recourse to genetic models. Phenotypic analyses of natural selection are illustrated by models of sex ratios in plants, sexual versus asexual reproduction in plants, and parental investment by animals.     

Genetic models for CNS inflammation

The use of transgenic technology to over-express or prevent expression of genes encoding molecules related to inflammation has allowed direct examination of their role in experimental disease. This article reviews transgenic and knockout models of CNS demyelinating disease, focusing primarily on the autoimmune disease multiple sclerosis, as well as conditions in which an inflammatory response makes a secondary contribution to tissue injury or repair, such as neurodegeneration, ischemia and trauma.     

Odefy - From discrete to continuous models

Background  

Phenomenological information about regulatory interactions is frequently available and can be readily converted to Boolean models. Fully quantitative models, on the other hand, provide detailed insights into the precise dynamics of the underlying system. In order to connect discrete and continuous modeling approaches, methods for the conversion of Boolean systems into systems of ordinary differential equations have been developed recently. As biological interaction networks have steadily grown in size and complexity, a fully automated framework for the conversion process is desirable.     

Discrete and continuous mathematical models of DNA branch migration

DNA junctions, known as Holliday junctions, are intermediates in genetic recombination between DNAs. In this structure, two double-stranded DNA helices with similar sequence are joined at a branch point. The branch point can move along these helices when strands with the same sequence are exchanged. Such branch migration is modeled as a random walk. First, we model this process discretely, such that the motion of the branch is represented as transfer between discrete compartments. This is useful in analysing the results of DNA branch migration on junction comprised of synthetic oligonucleotides. The limit in which larger numbers of smaller steps go to continuous motion of the branch is also considered. We show that the behavior of the continuous system is very similar to that of the discrete system when there are more than just a few compartments. Thus, even branch migration on oligonucleotides can be viewed as a continuous process. One consequence of this is that a step size must be assumed when determining rate constants of branch migration.We compare migration where forward and backward movements of the branch are equally probable to biased migration where one direction is favored over the other. In the latter case larger differences between the discrete and continuous cases are predicted, but the differences are still small relative to the experimental error associated with experiments to measure branch migration in oligonucleotides.     

Bifurcation analysis of continuous biochemical reactor models

The validity of a biochemical reactor model often is evaluated by comparing transient responses to experimental data. Dynamic simulation can be a rather inefficient and ineffective tool for analyzing bioreactor models that exhibit complex nonlinear behavior. Bifurcation analysis is a powerful tool for obtaining a more efficient and complete characterization of the model behavior. To illustrate the power of bifurcation analysis, the steady-state and transient behavior of three continuous bioreactor models consisting of a small number of ordinary differential equations are investigated. Several important features, as well as potential limitations, that are difficult to ascertain via dynamic simulation are disclosed through the bifurcation analysis. The results motivate the use of dynamic simulation and bifurcation analysis as complementary tools for analyzing the nonlinear behavior of bioreactor models.