Periodic solutions for a population dynamics problem with age-dependence and spatial structure

Using a linear model with age-dependence and spatial structure we show how a periodical supply of individuals will transform an exponentially decaying distribution of population into a non-trivial asymptotically stable periodic distribution. Next we give an application to an epidemic model.     

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     

On periodic cohort solutions of a size-structured population model

 We consider a size-structured population model with discontinuous reproduction and feedback through the environmental variable ‘substrate’. The model admits solutions with finitely many cohorts and in that case the problem is described by a system of ODEs involving a bifurcation parameter β. Existence of nontrivial periodic n-cohort solutions is investigated. Moreover, we discuss the question whether n cohorts (n≧2) with small size differences will tend to a periodic one-cohort solution as t→∞. Received 16 March 1995; received in revised form 7 January 1997     

Interaction of maturation delay and nonlinear birth in population and epidemic models

 A population with birth rate function B(N) N and linear death rate for the adult stage is assumed to have a maturation delay T>0. Thus the growth equation N′(t)=B(N(t−T)) N(t−T) e⁻ d ₁ T−dN(t) governs the adult population, with the death rate in previous life stages d ₁≧0. Standard assumptions are made on B(N) so that a unique equilibrium N _e exists. When B(N) N is not monotone, the delay T can qualitatively change the dynamics. For some fixed values of the parameters with d ₁>0, as T increases the equilibrium N _e can switch from being stable to unstable (with numerically observed periodic solutions) and then back to stable. When disease that does not cause death is introduced into the population, a threshold parameter R ₀ is identified. When R ₀<1, the disease dies out; when R ₀>1, the disease remains endemic, either tending to an equilibrium value or oscillating about this value. Numerical simulations indicate that oscillations can also be induced by disease related death in a model with maturation delay. Received: 2 November 1998 / Revised version: 26 February 1999     

Multiparametric bifurcations for a model in epidemiology

 In the present paper we make a bifurcation analysis of an SIRS epidemiological model depending on all parameters. In particular we are interested in codimension-2 bifurcations. Received 8 April 1994; received in revised form 29 June 1995     

Analysis of an SEIRS epidemic model with two delays

 A disease transmission model of SEIRS type with exponential demographic structure is formulated. All newborns are assumed susceptible, there is a natural death rate constant, and an excess death rate constant for infective individuals. Latent and immune periods are assumed to be constants, and the force of infection is assumed to be of the standard form, namely proportional to I(t)/N(t) where N(t) is the total (variable) population size and I(t) is the size of the infective population. The model consists of a set of integro-differential equations. Stability of the disease free proportion equilibrium, and existence, uniqueness, and stability of an endemic proportion equilibrium, are investigated. The stability results are stated in terms of a key threshold parameter. More detailed analyses are given for two cases, the SEIS model (with no immune period), and the SIRS model (with no latent period). Several threshold parameters quantify the two ways that the disease can be controlled, by forcing the number or the proportion of infectives to zero. Received 8 May 1995; received in revised form 7 November 1995     

Modeling aggregation and growth processes in an algal population model: analysis and computations

 Aggregation, the formation of large particles through multiple collision of smaller ones is a highly visible phenomena in oceanic waters which can control material flux to the deep sea. Oceanic aggregates more than 1 cm in diameter have been observed and are frequently described to consist of phytoplankton cells as well as other organic matter such as fecel pellets and mucus nets from pteropods. Division of live phytoplankton cells within an aggregate can also increase the size of aggregate (assuming some daughter cells stay in the aggregate) and hence could be a significant factor in speeding up the formation process of larger aggregate. Due to the difficulty of modeling cell division within aggregates, few efforts have been made in this direction. In this paper, we propose a size structured approach that includes growth of aggregate size due to both cell division and aggregation. We first examine some basic mathematical issues associated with the development of a numerical simulation of the resulting algal aggregation model. The numerical algorithm is then used to examine the basic model behavior and present a comparison between aggregate distribution with and without division in aggregates. Results indicate that the inclusion of a growth term in aggregates, due to cell division, results in higher densities of larger aggregates; hence it has the impact to speed clearance of organic matter from the surface layer of the ocean. Received 1 July 1994; received in revised form 23 February 1996     

A spatially discrete model for aggregating populations

 In this paper we introduce a spatially discrete model for aggregating populations described by a system of ODEs. We study the long time behavior of the solutions and we show that the model contains mechanisms by which individuals in the population aggregate at particular points in space. Received: 29 June 1996 / Revised version: 5 August 1997     

Stochastic epidemics: the probability of extinction of an infectious disease at the end of a major outbreak

 The aim of this study is to derive an asymptotic expression for the probability that an infectious disease will disappear from a population at the end of a major outbreak (‘fade-out’). The study deals with a stochastic SIR-model. Local asymptotic expansions are constructed for the deterministic trajectories of the corresponding deterministic system, in particular for the deterministic trajectory starting in the saddle point. The analytical expression for the probability of extinction is derived by asymptotically solving a boundary value problem based on the Fokker-Planck equation for the stochastic system. The asymptotic results are compared with results obtained by random walk simulations. Received 20 July 1995; received in revised form 6 May 1996     

The dynamics of infectious diseases in orchards with roguing and replanting as control strategy

 Roguing and replanting is a widely adopted control strategy of infectious diseases in orchards. Little is known about the effect of this type of management on the dynamics of the infectious disease. In this paper we analyze a structured population model for the dynamics of an S-I-R type epidemic under roguing and replanting management. The model is structured with respect to the total number of infections and the number of post-infectious infections on a tree. Trees are assumed to be rogued, and replaced by uninfected trees, when the total number of infections on the tree reaches a threshold value. Stability analysis and numerical exploration of the model show that for specific parameter combinations the internal equilibrium can become unstable and large amplitude periodic fluctuations arise. Several hypothesis on the mechanism causing the destabilisation of the steady-state are considered. The mechanism leading to the large amplitude fluctuations is identified and biologically interpreted. Received 2 September 1994     

Periodic orbits arising from Hopf bifurcations in a Volterra prey-predator model

 In this paper we derive a formula which enables the stability of periodic solutions to a Volterra integro-differential system to be determined. This system which has been studied by Cushing [1], models a predator-prey interaction with distributed delays. The results are obtained by using the algorithm developed by Kazarinoff, Wan and van den Driessche [2] based on the centre manifold formulas of Hassard and Wan [3]. We discuss an example of the formula for the case of weak kernels and show that under certain conditions stable periodic solutions arising from Hopf bifurcations at different critical values of the parameters can exist together. Received 30 December 1994; received in revised form 12 December 1995     

Identifiability of a stochastic model for cell debris in flow cytometry

 One of the most important problems in recovering DNA distribution from flow cytometric DNA measurements is the presence of background noise. In this paper, we analyse a probabilistic model recently proposed for background debris distribution and based on a specific probabilistic mechanism for the DNA fragmentation process of the cell nucleus. In particular, we carry out some sufficient conditions to uniquely identify the original DNA distribution from the flow cytometric data. Received: 15 June 1997 / Revised version: 18 November 1997     

A non-local model for a swarm

 This paper describes continuum models for swarming behavior based on non-local interactions. The interactions are assumed to influence the velocity of the organisms. The model consists of integro-differential advection-diffusion equations, with convolution terms that describe long range attraction and repulsion. We find that if density dependence in the repulsion term is of a higher order than in the attraction term, then the swarm profile is realistic: i.e. the swarm has a constant interior density, with sharp edges, as observed in biological examples. This is our main result. Linear stability analysis, singular perturbation theory, and numerical experiments reveal that weak, density-independent diffusion leads to disintegration of the swarm, but only on an exponentially large time scale. When density dependence is put into the diffusion term, we find that true, locally stable traveling band solutions occur. We further explore the effects of local and non-local density dependent drift and unequal ranges of attraction and repulsion. We compare our results with results of some local models, and find that such models cannot account for cohesive, finite swarms with realistic density profiles. Received: 17 September 1997 / Revised version: 17 March 1998     

Properties of solutions for a chemotaxis system

 We investigate mathematically the system of equations proposed by Chaplain and Stuart [2], to describe the chemotactic response of endothelial cells under the angiogenesis stimulus. In particular, we characterize the steady state endothelial cell density function, and give conditions on the chemotactic parameter k and cell proliferation parameter b that ensure that migration/ proliferation either does or does not occur in steady state. The time dependent problem is also treated. Received 12 September 1995; received in revised form 6 August 1996     

Analytical solutions of a minimal model of species migration in a bounded domain

 A minimal model of species migration is presented which takes the form of a parabolic equation with boundary conditions and initial data. Solutions to the differential problem are obtained that can be used to describe the small- and large-time evolution of a species distribution within a bounded domain. These expressions are compared with the results of numerical simulations and are found to be satisfactory within appropriate temporal regimes. The solutions presented can be used to describe existing observations of nematode distributions, can be used as the basis for further work on nematode migration, and may also be interpreted more generally. Received: 15 August 1999     

The periodically forced Droop model for phytoplankton growth in a chemostat

 It is proved that the periodically forced Droop model for phytoplankton growth in a chemostat has precisely two dynamic regimes depending on a threshold condition involving the dilution rate. If the dilution rate is such that the sub-threshold condition holds, the phytoplankton population is washed out of the chemostat. If the super-threshold condition holds, then there is a unique periodic solution, having the same period as the forcing, characterized by the presence of the phytoplankton population, to which all solutions approach asymptotically. Furthermore, this result holds for a general class of models with monotone growth rate and monotone uptake rate, the latter possibly depending on the cell quota. Received 10 October 1995; received in revised form 26 March 1996     

A generalized transport model for biased cell migration in an anisotropic environment

 A generalized transport model is derived for cell migration in an anisotropic environment and is applied to the specific cases of biased cell migration in a gradient of a stimulus (taxis; e.g., chemotaxis or haptotaxis) or along an axis of anisotropy (e.g., contact guidance). The model accounts for spatial or directional dependence of cell speed and cell turning behavior to predict a constitutive cell flux equation with drift velocity and diffusivity tensor (termed random motility tensor) that are explicit functions of the parameters of the underlying random walk model. This model provides the connection between cell locomotion and the resulting persistent random walk behavior to the observed cell migration on longer time scales, thus it provides a framework for interpreting cell migration data in terms of underlying motility mechanisms. Received: 8 April 1999     

Extinction of top-predator in a three-level food-chain model

 In this paper we extend the Lyapunov functions, constructed by A. Ardito and P. Ricciardi for predator–prey system [1], to the three level food chain models. We first consider a general three-level food-chain model. A criterion for the extinction of top predator will be given. Then we restrict our attentions to the case in which the prey is of logistic growth and predators have Holling’s type II functional responses. Received: 10 October 1997     

Gravity-induced absorbance changes in Phycomyces: a novel method for detecting primary responses of gravitropism

 The negative gravitropism of the sporangiophores of Phycomyces blakesleeanus Burgeff is elicited by different sensory inputs, which include flexure of the growing zone, buoyance of lipid globules and sedimentation of paracrystalline proteins, so-called octahedral crystals (C. Schimek et al., 1999a, Planta 210: 132–142). Gravity-induced absorbance changes (GIACs), which are associated with primary events of gravity sensing, were detected in the growing zones of sporangiophores. After placing sporangiophores horizontally, GIACs were detected after a latency of about 5 min, i.e. 15–25 min prior to gravitropic bending. The spectroscopic properties of the GIACs indicate that gravitropic stimulation could imply the reduction of cytochromes. The GIACs were spectrally distinct from light-induced absorbance changes (LIACs), showing that the primary responses of the light and gravity transduction chains are different. A dual stimulation with gravity and light generated GIAC-LIACs which were distinct from the absorbance changes occurring after the single stimuli and which indicate that light and gravity interact early in the respective transduction chains. Received: 2 September 1999 / Accepted: 9 November 1999     

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.