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.     

Mixtures of varying coefficient models for longitudinal data with discrete or continuous nonignorable dropout

The analysis of longitudinal repeated measures data is frequently complicated by missing data due to informative dropout. We describe a mixture model for joint distribution for longitudinal repeated measures, where the dropout distribution may be continuous and the dependence between response and dropout is semiparametric. Specifically, we assume that responses follow a varying coefficient random effects model conditional on dropout time, where the regression coefficients depend on dropout time through unspecified nonparametric functions that are estimated using step functions when dropout time is discrete (e.g., for panel data) and using smoothing splines when dropout time is continuous. Inference under the proposed semiparametric model is hence more robust than the parametric conditional linear model. The unconditional distribution of the repeated measures is a mixture over the dropout distribution. We show that estimation in the semiparametric varying coefficient mixture model can proceed by fitting a parametric mixed effects model and can be carried out on standard software platforms such as SAS. The model is used to analyze data from a recent AIDS clinical trial and its performance is evaluated using simulations.     

Stability in N-species coevolutionary systems

Stability criteria have recently been developed for coevolutionary Lotka-Volterra systems where individual fitness functions are assumed to be linear in the population state. We extend these criteria as part of a general theory of coevolution (that combines effects of ecology and evolution) based on arbitrary (i.e. nonlinear) fitness functions and a finite number of individual phenotypes. The central role of the stationary density surface where species' densities are at equilibrium is emphasized. In particular, for monomorphic resident systems, it is shown coevolutionary stability is equivalent to ecological stability combined with evolutionary stability on the stationary density surface. Also discussed is how our theory relates to recent treatments of phenotypic coevolution via adaptive dynamics when there is a continuum of individual phenotypes.     

Linear discrete models with different time scales

Aggregation of variables allows to approximate a large scale dynamical system (the micro-system) involving many variables into a reduced system (the macro-system) described by a few number of global variables. Approximate aggregation can be performed when different time scales are involved in the dynamics of the micro-system. Perturbation methods enable to approximate the large micro-system by a macro-system going on at a slow time scale. Aggregation has been performed for systems of ordinary differential equations in which time is a continuous variable. In this contribution, we extend aggregation methods to time-discrete models of population dynamics. Time discrete micro-models with two time scales are presented. We use perturbation methods to obtain a slow macro-model. The asymptotic behaviours of the micro and macro-systems are characterized by the main eigenvalues and the associated eigenvectors. We compare the asymptotic behaviours of both systems which are shown to be similar to a certain order.     

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.     

Conditions for global stability of two-species population models with discrete time delay

By constructing appropriate Liapunov functionals, asymptotic behaviour of the solutions of various delay differential systems describing prey-predator, competition and symbiosis models has been studied. It has been shown that equilibrium states of these models are globally stable, provided certain conditions in terms of instantaneous and delay interaction coefficients are satisfied.     

On linear stochastic compartmental models in discrete time

This note presents a general time-dependent study of linear stochastic compartmental models in discrete time. The transient distribution of the state of the system is obtained by adapting methods used in the continuous time analysis. Covariance functions with and without a time lag are then deduced by a simple probabilistic argument. Results are derived in the Markov case and are partly extended to the semi-Markov case.     

Complex discrete dynamics from simple continuous population models

Nonoverlapping generations have been classically modelled as difference equations in order to account for the discrete nature of reproductive events. However, other events such as resource consumption or mortality are continuous and take place in the within-generation time. We have realistically assumed a hybrid ODE bidimensional model of resources and consumers with discrete events for reproduction. Numerical and analytical approaches showed that the resulting dynamics resembles a Ricker map, including the doubling route to chaos. Stochastic simulations with a handling-time parameter for indirect competition of juveniles may affect the qualitative behaviour of the model.     

Genetic algebras and time continuous models

It is shown how linear genetic algebras, ordinarily applied in situations with discrete time, will also simplify certain systems of differential equations in time continuous models. These models describe the variation of genotype frequencies in infinite populations in different mating systems. The cases considered include matings between individuals randomly drawn from the population at each moment, a population which is continuously backcrossed to a second, constant population, and a population divided into two age groups, which take part in the matings with different intensities. For the first case the general theory is applied to an example with tetraploids having a mixture of chromatid and chromosome segregation.     

The frequency spectrum of structured discrete time population models: its properties and their ecological implications

Much research effort has been devoted to the study of the interaction between environmental noise and discrete time nonlinear dynamical systems. A large part of this effort has involved numerical simulation of simple unstructured models for particular ranges of parameter values. While such research is important in encouraging discussion of important ecological issues it is often unclear how general are the conclusions reached. However, by restricting attention to weak noise it is possible to obtain analytical results that hold for essentially all discrete time models and still provide considerable insight into the properties of the noise-dynamics interface. We follow this approach, focusing on the autocorrelation properties of the population fluctuations using the power (frequency) spectrum matrix as the analytic framework. We study the relationship between the spectral peak structure and the dynamical behaviour of the system and the modulation of this relationship by its internal structure, acting as an "intrinsic" filter and by colour in the noise acting as an "extrinsic" filter. These filters redistribute "power" between frequency components in the spectrum. The analysis emphasises the importance of eigenvalues in the identification of resonance, both in the system itself and in its subsystems, and the importance of noise configuration in defining which paths are followed on the network. The analysis highlights the complexity of the inverse problem (in finding, for example, the source of long term fluctuations) and the role of factors other than colour in the persistence of populations.     

Linking continuous and discrete intervertebral disc models through homogenisation

At present, there are two main numerical approaches that are frequently used to simulate the mechanical behaviour of the human spine. Researchers with a continuum-mechanical background often utilise the finite-element method (FEM), where the involved biological soft and hard tissues are modelled on a macroscopic (continuum) level. In contrast, groups associated with the science of human movement usually apply discrete multi-body systems (MBS). Herein, the bones are modelled as rigid bodies, which are connected by Hill-type muscles and non-linear rheological spring-dashpot models to represent tendons and cartilaginous connective tissue like intervertebral discs (IVD). A possibility to benefit from both numerical methods is to couple them and use each approach, where it is most appropriate. Herein, the basic idea is to utilise MBS in simulations of the overall body and apply the FEM only to selected regions of interest. In turn, the FEM is used as homogenisation tool, which delivers more accurate non-linear relationships describing the behaviour of the IVD in the multi-body dynamics model. The goal of this contribution is to present an approach to couple both numerical methods without the necessity to apply a gluing algorithm in the context of a co-simulation. Instead, several pre-computations of the intervertebral disc are performed offline to generate an approximation of the homogenised finite-element (FE) result. In particular, the discrete degrees of freedom (DOF) of the MBS, that is, three displacements and three rotations, are applied to the FE model of the IVD, and the resulting homogenised forces and moments are recorded. Moreover, a polynomial function is presented with the discrete DOF of the MBS as variables and the discrete forces an moments as function values. For the sake of a simple verification, the coupling method is applied to a simplified motion segment of the spine. Herein, two stiff cylindrical vertebrae with an interjacent homogeneous cylindrical IVD are examined under the restriction of purely elastic deformations in the sagittal plane.     

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.     

Global stability of single-species diffusion volterra models with continuous time delays

In this paper we consider the stability property of single-species patches connected by diffusion with a within-patch dynamics of Volterra type and with continuous time delays. We prove that this system can only have two kinds of equilibria: the positive and the trivial one. By the assumption that the delay kernels are convex combinations of suitable non-negative and normalized functions, the linear chain trick gives an expanded system of O.D.E. with the same stability properties as the original integro-differential system. Homotopy function techniques provide sufficient conditions for the existence of the positive equilibrium and for its global stability. We also prove the local stability of any positive equilibrium and the local instability both of positive and trivial equilibria. The biological meanings of the results obtained are compared with known results from the literature. This work was performed under the auspices of G.N.F.M., C.N.R. (Italy) and within the activity of the Evolution Equations and Applications group, M.P.I. (Italy). I thank the Department of Applied Mathematics, Shizuoka University, Japan, which enabled me to visit Urbino.     

Stability in a class of discrete time models of interacting populations

Summary Effective Lyapunov and Lyapunov-like functions for a class of discrete time models of interacting populations are presented. These functions are constructed on the biologically meaningful principle that a viable population must absorb energy from external sources when its density is low and it must dissipate energy to the environment when its density is high. These functions can be used to establish that a discrete time model is globally stable or that its solutions are ultimately confined to an acceptable region of the state space. The latter is especially interesting when the model has chaotic solutions. These methods are applied to a single species model and a model of competition between two species.     

Remarks on discrete and continuous large-scale models of DNA dynamics.

We present a comparison of the continuous versus discrete models of large-scale DNA conformation, focusing on issues of relevance to molecular dynamics. Starting from conventional expressions for elastic potential energy, we derive elastic dynamic equations in terms of Cartesian coordinates of the helical axis curve, together with a twist function representing the helical or excess twist. It is noted that the conventional potential energies for the two models are not consistent. In addition, we derive expressions for random Brownian forcing for the nonlinear elastic dynamics and discuss the nature of such forces in a continuous system.