Persistent age-distributions for a pair-formation model

Conditions on the vital rates of a two-sex population are presented which imply the existence or nonexistence of exponentially growing persistent age-distributions.     

Chaos and population disappearances in simple ecological models

A class of truncated unimodal discrete-time single species models for which low or high densities result in extinction in the following generation are considered. A classification of the dynamics of these maps into five types is proven: (i) extinction in finite time for all initial densities, (ii) semistability in which all orbits tend toward the origin or a semi-stable fixed point, (iii) bistability for which the origin and an interval bounded away from the origin are attracting, (iv) chaotic semistability in which there is an interval of chaotic dynamics whose compliment lies in the origin’s basin of attraction and (v) essential extinction in which almost every (but not every) initial population density leads to extinction in finite time. Applying these results to the Logistic, Ricker and generalized Beverton-Holt maps with constant harvesting rates, two birfurcations are shown to lead to sudden population disappearances: a saddle node bifurcation corresponding to a transition from bistability to extinction and a chaotic blue sky catastrophe corresponding to a transition from bistability to essential extinction. Received: 14 February 2000 / Revised version: 15 August 2000 / Published online: 16 February 2001     

Periodic solutions in a model of competition between plasmid-bearing and plasmid-free organisms in a chemostat with an inhibitor

We obtain necessary and sufficient conditions on the existence of a unique positive equilibrium point and a set of sufficient conditions on the existence of periodic solutions for a 3-dimensional system which arises from a model of competition between plasmid-bearing and plasmid-free organisms in a chemostat with an inhibitor. Our results improve the corresponding results obtained by Hsu, Luo, and Waltman [1]. Received: 20 November 1997 / Revised version: 12 February 1999 / Published online: 20 December 2000     

Parametric analysis of the ratio-dependent predator–prey model

We present a complete parametric analysis of stability properties and dynamic regimes of an ODE model in which the functional response is a function of the ratio of prey and predator abundances. We show the existence of eight qualitatively different types of system behaviors realized for various parameter values. In particular, there exist areas of coexistence (which may be steady or oscillating), areas in which both populations become extinct, and areas of "conditional coexistence" depending on the initial values. One of the main mathematical features of ratio-dependent models, distinguishing this class from other predator-prey models, is that the Origin is a complicated equilibrium point, whose characteristics crucially determine the main properties of the model. This is the first demonstration of this phenomenon in an ecological model. The model is investigated with methods of the qualitative theory of ODEs and the theory of bifurcations. The biological relevance of the mathematical results is discussed both regarding conservation issues (for which coexistence is desired) and biological control (for which extinction is desired).     

Models of central pattern generators for quadruped locomotion I. Primary gaits

In this paper we continue the analysis of a network of symmetrically coupled cells modeling central pattern generators for quadruped locomotion proposed by Golubitsky, Stewart, Buono, and Collins. By a cell we mean a system of ordinary differential equations and by a coupled cell system we mean a network of identical cells with coupling terms. We have three main results in this paper. First, we show that the proposed network is the simplest one modeling the common quadruped gaits of walk, trot, and pace. In doing so we prove a general theorem classifying spatio-temporal symmetries of periodic solutions to equivariant systems of differential equations. We also specialize this theorem to coupled cell systems. Second, this paper focuses on primary gaits; that is, gaits that are modeled by output signals from the central pattern generator where each cell emits the same waveform along with exact phase shifts between cells. Our previous work showed that the network is capable of producing six primary gaits. Here, we show that under mild assumptions on the cells and the coupling of the network, primary gaits can be produced from Hopf bifurcation by varying only coupling strengths of the network. Third, we discuss the stability of primary gaits and exhibit these solutions by performing numerical simulations using the dimensionless Morris-Lecar equations for the cell dynamics.     

Inverse bifurcation problem, singular Wiener-Hopf equations, and mathematical models in ecology

A single-species population dynamics with dispersal in a spatially heterogeneous environment is modeled by a nonlinear reaction-diffusion equation with a potential term. To each nonlinear kinetics there corresponds a bifurcation curve that describes the relation between the growth rate and the central density of a steady-state population distribution. Our main concern is an inverse problem for this correspondence. The existence of nonlinear kinetics realizing a prescribed bifurcation curve is established. It is shown that the freedom of such kinetics is of degree finite and even, depending only on the heterogeneity of the environment, and conversely that any nonnegative even integer occurs as the degree of freedom in some environments. A discussion is also made on under what kind of environment the degree is equal to zero or is positive. The mathematical analysis involves the development of a general theory for singular multiplicative Wiener-Hopf integral equations.     

Coevolutionary interactions between a haploid species and a diploid species

We investigate a general model describing coevolutionary interaction between a haploid population and a diploid population, each with two alleles at a single locus. Both species are allowed to evolve, with the fitness of the genotypes of each species assumed to depend linearly on the frequencies of the genotypes of the other species. We explore the resulting outcomes of these interactions, in particular determining the location of equilibria under various conditions. The coevolution here is much more complex than that between two haploid populations and allows for the possibility of two polymorphic equilibria. To allow for further analysis, we construct a semi-symmetric model. The variety of outcomes possible even in this second model provides support for the geographic mosaic theory of coevolution by suggesting the possibility of small local populations coevolving to very different outcomes, leading to a shifting geographic mosaic as neighboring populations interact with each other through migration.     

The effect of dispersal on single-species nonautonomous dispersal models with delays

In this paper, single-species nonautonomous dispersal models with delays are considered. An interesting result on the effect of dispersal for persistence and extinction is obtained. That is, if the species is persistent in a patch then it is also persistent in all other patches; if the species is permanent in a patch then it is also permanent in all other patches; if the species is extinct in a patch then it is also extinct in all other patches. Furthermore, some new sufficient conditions for the permanence and extinction of the species in a patch are established. The existence of positive periodic solutions is obtained in the periodic case by employing Teng and Chen's results on the existence of positive periodic solutions for functional differential equations. Received: 26 June 2000 / Revised version: 6 October 2000 / Published online: 10 April 2001     

Oscillations in a refractory neural net

A functional differential equation that arises from the classic theory of neural networks is considered. As the length of the absolute refractory period is varied, there is, as shown here, a super-critical Hopf bifurcation. As the ratio of the refractory period to the time constant of the network increases, a novel relaxation oscillation occurs. Some approximations are made and the period of this oscillation is computed.     

Numerical bifurcation analysis of delay differential equations arising from physiological modeling

This paper has a dual purpose. First, we describe numerical methods for continuation and bifurcation analysis of steady state solutions and periodic solutions of systems of delay differential equations with an arbitrary number of fixed, discrete delays. Second, we demonstrate how these methods can be used to obtain insight into complex biological regulatory systems in which interactions occur with time delays: for this, we consider a system of two equations for the plasma glucose and insulin concentrations in a diabetic patient subject to a system of external assistance. The model has two delays: the technological delay of the external system, and the physiological delay of the patient's liver. We compute stability of the steady state solution as a function of two parameters, compare with analytical results and compute several branches of periodic solutions and their stability. These numerical results allow to infer two categories of diabetic patients for which the external system has different efficiency.     

Approximating the long-term behaviour of a model for parasitic infection

In a companion paper two stochastic models, useful for the initial behaviour of a parasitic infection, were introduced. Now we analyse the long term behaviour. First a law of large numbers is proved which allows us to analyse the deterministic analogues of the stochastic models. The behaviour of the deterministic models is analogous to the stochastic models in that again three basic reproduction ratios are necessary to fully describe the information needed to separate growth from extinction. The existence of stationary solutions is shown in the deterministic models, which can be used as a justification for simulation of quasi-equilibria in the stochastic models. Host-mortality is included in all models. The proofs involve martingale and coupling methods.     

Mutation and recombination with tight linkage

An exact solution of the mutation-recombination equation in continuous time is presented, with linear ordering of the sites and at most one mutation or crossover event taking place at every instant of time. The differential equation may be obtained from a mutation-recombination model with discrete generations, in the limit of short generations, or weak mutation and recombination. The solution relies on the multilinear structure of the dynamical system, and on the commuting properties of the mutation and recombination operators. It is obtained through diagonalization of the mutation term, followed by a transformation to certain measures of linkage disequilibrium that simultaneously linearize and diagonalize the recombination dynamics. The collection of linkage disequilibria, as well as their decay rates, are given in closed form. Received: 26 January 1999 / Revised version: 20 October 2000 / Published online: 10 April 2001     

Stochastic models of a parasitic infection, exhibiting three basic reproduction ratios

Two closely related stochastic models of parasitic infection are investigated: a non-linear model, where density dependent constraints are included, and a linear model appropriate to the initial behaviour of an epidemic. Host-mortality is included in both models. These models are appropriate to transmission between homogeneously mixing hosts, where the amount of infection which is transferred from one host to another at a single contact depends on the number of parasites in the infecting host. In both models, the basic reproduction ratio R0 can be defined to be the lifetime expected number of offspring of an adult parasite under ideal conditions, but it does not necessarily contain the information needed to separate growth from extinction of infection. In fact we find three regions for a certain parameter where different combinations of parameters determine the behavior of the models. The proofs involve martingale and coupling methods.     

A dynamical model for plant cell wall architecture formation

We discuss a dynamical mathematical model to explain cell wall architecture in plant cells. The highly regular textures observed in cell walls reflect the spatial organisation of the cellulose microfibrils (CMFs), the most important structural component of cell walls. Based on a geometrical theory proposed earlier [A. M. C. Emons, Plant, Cell and Environment 17, 3–14 (1994)], the present model describes the space-time evolution of the density of the so-called rosettes, the CMF synthesizing complexes. The motion of these rosettes in the plasma membrane is assumed to be governed by an optimal packing constraint on the CMFs plus adherent matrix material, that couples the direction of motion, and hence the orientation of the CMF being deposited, to the local density of rosettes. The rosettes are created inside the cell in the endoplasmatic reticulum and reach the cell-membrane via vesicles derived from Golgi-bodies. After being inserted into the plasma membrane they are assumed to be operative for a fixed, finite lifetime. The plasma membrane domains within which rosettes are activated are themselves also supposed to be mobile. We propose a feedback mechanism that precludes the density of rosettes to rise beyond a maximum dictated by the geometry of the cell. The above ingredients lead to a quasi-linear first order PDE for the rosette-density. Using the method of characteristics this equation can be cast into a set of first order ODEs, one of which is retarded. We discuss the analytic solutions of the model that give rise to helicoidal, crossed polylamellate, helical, axial and random textures, since all cell walls are composed of (or combinations of) these textures. Received: 10 July 1999 / Revised version: 7 June 2000 / Published online: 16 February 2001     

An alternative stochastic model of generation of oligodendrocytes in cell culture

According to our previous model, oligodendrocyte – type 2 (O-2A) astrocyte progenitor cells become competent for differentiation in vitro after they complete a certain number of critical mitotic cycles. After attaining the competency to differentiate, progenitor cells divide with fixed probability p in subsequent cycles. The number of critical cycles is random; analysis of data suggests that it varies from zero to two. The present paper presents an alternative model in which there are no critical cycles, and the probability that a progenitor cell will divide again decreases gradually to a plateau value as the number of completed mitotic cycles increases. In particular all progenitor cells have the ability to differentiate from the time of plating. The Kiefer-Wolfowitz procedure is used to fit the new model to experimental data on the clonal growth of purified O-2A progenitor cells obtained from the optic nerves of 7 day old rats. The new model is shown to fit the experimental data well, indicating that it is not possible to determine whether critical cycles exist on the basis of these experimental data. In contrast to the fit of the previous model, which suggested that the addition of thyroid hormone increased the limiting probability of differentiation as the number of mitotic cycles increases, the fit of the new model suggests that the addition of thyroid hormone has almost no effect on the limiting probability of differentiation. Received: 6 March 2000 / Revised version: 18 September 2000 / Published online: 30 April 2001     

Dynamical consequences of harvest in discrete age-structured population models

The role of harvest in discrete age-structured one-population models has been explored. Considering a few age classes only, together with the overcompensatory Ricker recruitment function, we show that harvest acts as a weak destabilizing effect in case of small values of the year-to-year survival probability P and as a strong stabilizing effect whenever the survival probability approaches unity. In the latter case, assuming n=2 age classes, we find that harvest may transfer a population from the chaotic regime to a state where the equilibrium point (x₁*, x₂*) becomes stable. However, as the number of age classes increases (which acts as a stabilizing effect in non-exploited models), we find that harvest acts more and more destabilizing, in fact, when the number of age classes has been increased to n=10, our finding is that in case of large values of the survival probabilities, harvest may transfer a population from a state where the equilibrium is stable to the chaotic regime, thus exactly the opposite of what was found in case of n=2. On the other hand, if we replace the Ricker relation with the generalized Beverton and Holt recruitment function with abruptness parameter larger than 2, several of the conclusions derived above are changed. For example, when n is large and the survival probabilities exceed a certain threshold, the equilibrium will always be stable.Revised version: 18 September 2003     

The effect of density-dependent treatment and behavior change on the dynamics of HIV transmission

In this work, we propose a model for heterosexual transmission of HIV/AIDS in a population of varying size with an intervention program in which treatment and/or behavior change of the infecteds occur as an increasing function of the density of the infected class in the population. This assumption has socio-economic implications which is important for public health considerations since density-dependent treatment/behavior change may be more cost-saving than a program where treatment/behavior change occurs linearly with respect to the number of infecteds. We will make use of the conservation law of total sexual contacts which enables us to reduce the two-sex model to a simpler one-sex formulation. Analytical results will be given. Unlike a similar model with linear treatment/behavior change in Hsieh (1996) where conditions were obtained for the eradication of disease, we will show that density-dependent treatment/behavior change cannot eradicate the disease if the disease is able to persist without any treatment/behavior change. This work demonstrates the need to further understand how treatment/behavior change occurs in a society with varying population.     

A unifying framework for chaos and stochastic stability in discrete population models

 In this paper we propose a general framework for discrete time one-dimensional Markov population models which is based on two fundamental premises in population dynamics. We show that this framework incorporates both earlier population models, like the Ricker and Hassell models, and experimental observations concerning the structure of density dependence. The two fundamental premises of population dynamics are sufficient to guarantee that the model will exhibit chaotic behaviour for high values of the natural growth and the density-dependent feedback, and this observation is independent of the particular structure of the model. We also study these models when the environment of the population varies stochastically and address the question under what conditions we can find an invariant probability distribution for the population under consideration. The sufficient conditions for this stochastic stability that we derive are of some interest, since studying certain statistical characteristics of these stochastic population processes may only be possible if the process converges to such an invariant distribution. Received 15 May 1995; received in revised form 17 April 1996     

Overcompensatory recruitment and generation delay in discrete age-structured population models

 The effect of overcompensatory recruitment and the combined effect of overcompensatory recruitment and generation delay in discrete nonlinear age-structured population models is studied. Considering overcompensatory recruitment alone, we present formal proofs of the supercritical nature of bifurcations (both flip and Hopf) as well as an extensive analysis of dynamics in unstable parameter regions. One important finding here is that in case of small and moderate year to year survival probabilities there are large regions in parameter space where the qualitative behaviour found in a general n+1 dimensional model is retained already in a one-dimensional model. Another result is that the dynamics at or near the boundary of parameter space may be very complicated. Generally, generation delay is found to act as a destabilizing effect but its effect on dynamics is by no means unique. The most profound effect occurs in the n-generation delay cases. In these cases there is no stable equilibrium X ^* at all, but whenever X ^* small, a stable cycle of period n+1 where the periodic points in the cycle are on a very special form. In other cases generation delay does not alter the dynamics in any substantial way. Received 25 April 1995; received in revised form 21 November 1995