The spread of a parasitic infection in a spatially distributed host population

Starting from a stochastic model for the spread of a parasitic infection in a spatially distributed host population we describe the way to a continuum formulation by a deterministic model in terms of a nonlinear partial differential equation and an integro-differential equation. The hosts are assumed to occupy fixed spatial positions, whereas the parasites are mobile, however can propagate only within the hosts. To perform the continuum limit we suppose that the size N_hof the host population, the size N_pof the parasite population, and the ratio N_p/N_htend to infinity. Accordingly, the parameters determining the time evolution of the host and parasite populations are rescaled suitably.Parts of this work have been elaborated during a stay at the Institute of Applied Mathematics at the University of Zürich     

Density-dependent regulation of natural and laboratory rotifer populations

Density-dependent regulation of abundance is fundamentally important in the dynamics of most animal populations. Density effects, however, have rarely been quantified in natural populations, so population models typically have a large uncertainty in their predictions. We used models generated from time series analysis to explore the form and strength of density-dependence in several natural rotifer populations. Population growth rate (r) decreased linearly or non-linearly with increased population density, depending on the rotifer species. Density effects in natural populations reduced r to 0 at densities of 1–10 l^–1 for 8 of the 9 rotifer species investigated. The sensitivities of these species to density effects appeared normally distributed, with a mean r=0 density of 2.3 l^–1 and a standard deviation of 1.9. Brachionus rotundiformis was the outlier with 10–100× higher density tolerance. Density effects in laboratory rotifer populations reduced r to 0 at population densities of 10–100 ml^–1, which is 10⁴ higher than densities in natural populations. Density effects in laboratory populations are due to food limitation, autotoxicity or to their combined effects. Experiments with B. rotundiformis demonstrated the absence of autotoxicity at densities as high as 865 ml^–1, a much higher density than observed in natural populations. It is, therefore, likely that food limitation rather than autotoxicity plays a major role in regulating natural rotifer populations.     

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     

Dynamical modeling of viral spread in spatially distributed populations: stochastic origins of oscillations and density dependence

In order to understand the spatio-temporal structure of epidemics beyond that permitted with classical SIR (susceptible-infective-recovered)-type models, a new mathematical model for the spread of a viral disease in a population of spatially distributed hosts is described. The positions of the hosts are randomly generated in a rectangular habitat. Encounters between any pair of individuals are according to a Poisson process with a mean rate that declines exponentially as the distance between them increases. The contact rate allows the mean rates to be set at a certain number of encounters per day on average. The relevant state variables for each individual at any time are given by the solution of a pair of coupled differential equations for the viral load and the quantity of general immune system effectors which reduce the viral load. The parameters describing within-host viral-immune system dynamics are generated randomly to reflect variability across a population. Transmission is assumed to depend on the viral loads in donors and occurs with a probability ptrans. The initial conditions are such that one randomly chosen individual carries a randomly chosen amount of the virus, whereas the rest of the population is uninfected. Simulations reveal local or whole-population responses. Whole-population disease spread may be in the form of isolated or multiple occurrences, the latter often being approximately periodic. The mechanisms of this oscillatory behaviour are analyzed in terms of several parameters and the distribution of critical points in the host dynamical systems. Increased contact rate, increased probability of transmission and decreased threshold for viral transmission, decreased immune strength and increased viral growth rate all increase the probability of multiple outbreaks and the distribution of the critical points also plays a role.     

Dynamics of maturing populations and their asymptotic behaviour

Summary It is well known that the partial differential equation of the traditional model describing the dynamics of an age-dependent population is of the first order hyperbolic type. An equation of that type cannot simultaneously accommodate a renewal type birth boundary condition and a death boundary condition by old age (accumulation of aging injury) and thus lacks biological realism (mortality by old age). In this paper a governing equation of a parabolic type is derived to represent the expected size of a stochastically maturing population. Using techniques well known for the solution of parabolic partial differential and Volterra integral equations, the asymptotic behaviour of such a maturing population is discussed. Due to a non-local boundary condition, the boundary value problem encountered appears to be new.     

The spread of infectious diseases in spatially structured populations: an invasory pair approximation

The invasion of new species and the spread of emergent infectious diseases in spatially structured populations has stimulated the study of explicit spatial models such as cellular automata, network models and lattice models. However, the analytic intractability of these models calls for the development of tractable mathematical approximations that can capture the dynamics of discrete, spatially-structured populations. Here we explore moment closure approximations for the invasion of an SIS epidemic on a regular lattice. We use moment closure methods to derive an expression for the basic reproductive number, R(0), in a lattice population. On lattices, R(0) should be bounded above by the number of neighbors per individual. However, we show that conventional pair approximations actually predict unbounded growth in R(0) with increasing transmission rates. To correct this problem, we propose an 'invasory' pair approximation which yields a relatively simple expression for R(0) that remains bounded above, and also predicts R(0) values from lattice model simulations more accurately than conventional pair and triple approximations. The invasory pair approximation is applicable to any spatial model, since it takes into account characteristics of invasions that are common to all spatially structured populations.     

Genetic relationships of European populations reflect their ethnohistorical affinities

From 420 records of ethnic locations and movements since 2000 B. C., we computed vectors describing the proportions which peoples of the various European language families contributed to the gene pools within 85 land-based 5 × 5-degree quadrats in Europe. Using these language family vectors, we computed ethnohistorical affinities as arc distances between all pairs of the 85 quadrats. These affinities are significantly correlated with genetic distances based on 26 genetic systems, even when geographic distances, a common causative factor, are held constant. Thus, the ethnohistorical distances explain a significant amount of the genetic variation observed in modern populations. Randomizations of the records by chronology result in loss of significance for the observed partial correlation between genetics and ethnohistory, when geography is held constant. However, a randomization of records by location only results in reduced significance. Thus, while the historical sequence of the movements does not seem to matter in Europe, their geographic locations do. We discuss the implications of these findings. © 1993 Wiley-Liss, Inc.     

Invasion and adaptive evolution for individual-based spatially structured populations

The interplay between space and evolution is an important issue in population dynamics, that is particularly crucial in the emergence of polymorphism and spatial patterns. Recently, biological studies suggest that invasion and evolution are closely related. Here, we model the interplay between space and evolution starting with an individual-based approach and show the important role of parameter scalings on clustering and invasion. We consider a stochastic discrete model with birth, death, competition, mutation and spatial diffusion, where all the parameters may depend both on the position and on the phenotypic trait of individuals. The spatial motion is driven by a reflected diffusion in a bounded domain. The interaction is modelled as a trait competition between individuals within a given spatial interaction range. First, we give an algorithmic construction of the process. Next, we obtain large population approximations, as weak solutions of nonlinear reaction–diffusion equations. As the spatial interaction range is fixed, the nonlinearity is nonlocal. Then, we make the interaction range decrease to zero and prove the convergence to spatially localized nonlinear reaction–diffusion equations. Finally, a discussion of three concrete examples is proposed, based on simulations of the microscopic individual-based model. These examples illustrate the strong effects of the spatial interaction range on the emergence of spatial and phenotypic diversity (clustering and polymorphism) and on the interplay between invasion and evolution. The simulations focus on the qualitative differences between local and nonlocal interactions.      

Some remarks about the wave speed and travelling wave solutions of a nonlinear integral operator

In this paper, we extend the existence of travelling wave solutions of a nonlinear operator in the inferior case to higher dimensions and establish the uniqueness of that solution. In addition we determine the sign of the wave speed in a special case.     

Globally asymptotic properties of proliferating cell populations

This paper presents a general model for the cell division cycle in a population of cells. Three hypotheses are used: (1) There is a substance (mitogen) produced by cells which is necessary for mitosis; (2) The probability of mitosis is a function of mitogen levels; and (3) At mitosis each daughter cell receives exactly one-half of the mitogen present in the mother cell. With these hypotheses we derive expressions for the and curves, the distributions of mitogen and cell cycle times, and the correlation coefficients between mother-daughter (_md) and sister-sister (_ss) cell cycle times.The distribution of mitogen levels is shown to be given by the solution to an integral equation, and under very mild assumptions we prove that this distribution is globally asymptotically stable. We further show that the limiting logarithmic slopes of (t) and (t) are equal and constant, and that _md0 while _ss0. These results are in accord with the experimental results in many different cell lines. Further, the transition probability model of the cell cycle is shown to be a simple special case of the model presented here.     

Monitoring of microbial populations and their cellulolytic activities during the composting of municipal solid wastes

Seasonal changes in microbial populations and the activities of cellulolytic enzymes were investigated during the composting of municipal solid wastes at Damietta compost plant, Egypt. The changes in temperature, pH and carbon/nitrogen (C/N) ratio were also monitored. The results obtained showed that the temperatures of the windrows in all seasons reached the maximum after 3 weeks of composting and then decreased by the end of the composting period (35 days), but did not reach ambient temperature. Marked changes in pH values of the composts in all seasons were found, but generally, the pH was near neutrality. Significant increases in the size of the microbial populations were obtained in autumn and spring seasons compared to summer and winter seasons. The activities of cellulases were also higher in the autumn and spring seasons than in the summer and winter seasons. The decrease in C/N ratio in autumn and spring was higher than in summer and winter. It was evident that the degradation of organic matter increased by an increase in the microflora and its cellulolytic activities.     

Optimization of harvesting age in an integral age-dependent model of population dynamics

The paper deals with optimal control in a linear integral age-dependent model of population dynamics. A problem for maximizing the harvesting return on a finite time horizon is formulated and analyzed. The optimal controls are the harvesting age and the rate of population removal by harvesting. The gradient and necessary condition for an extremum are derived. A qualitative analysis of the problem is provided. The model shows the presence of a zero-investment period. A preliminary asymptotic analysis indicates possible turnpike properties of the optimal harvesting age. Biological interpretation of all results is provided.     

On the importance of dimensionality of space in models of space-mediated population persistence

Spatially explicit models have become widely used in today's mathematical ecology to study persistence of populations. For the sake of simplicity, population dynamics is often analyzed with 1-D models. An important question is: how adequate is such 1-D simplification of 2-D (or 3-D) dynamics for predicting species persistence. Here we show that dimensionality of the environment can play a critical role in the persistence of predator-prey interactions. We consider 1-D and 2-D dynamics of a predator-prey model with the prey growth damped by the Allee effect. We show that adding a second space coordinate into the 1-D model results in a pronounced increase of size of the domain in the parametric space where predator-prey coexistence becomes possible. This result is due to the possibility of formation of a number of 2-D patterns, which is impossible in the 1-D model. The 1-D and the 2-D models exhibit different qualitative responses to variations of system parameters. We show that in ecosystems having a narrow width (e.g. mountain valleys, vegetation patterns along canals in dry areas, etc.), extinction of species is more probable compared to ecosystems having a pronounced second dimension. In particular, the width of a long narrow natural reserve should be large enough to guarantee nonextinction of species via interaction of 2-D population patches.     

Generational coexistence and ancestor's inhibition in bacterial populations

Generational coexistence in structured environments raises the possibility of a competition between ancestors and descendents. This type of kin competition, and in particular, the possibility that descendents might actively repress the ancestor's dominance, has been rarely considered in microbial evolutionary ecology. The recent discovery of the phenomenon of stationary-phase contact-dependent inhibition of bacterial ancestor cells by late descendents provides a new theoretical perspective to analyze intrapopulational evolutionary changes. The ancestor's inhibition effect might accelerate such changes, particularly when the descendents have acquired small adaptive advantages that are insufficient to rapidly displace the well-settled ancestors in a complex niche. Besides this effect of triggering selection of small genetic differences, the opportunities for intergenerational coexistence in bacteria, where ancestor's inhibition might occur, are reviewed in this work. A theoretical analysis is provided about the explanatory possibilities of the ancestor's inhibition effect in the controversies about intraspecific (in a large sense, including intrapopulational) genetic diversification, and the discontinuities observed in such processes, giving rise to the emergence of individualities and therefore differential units of selection.     

On the 'rate of aging' in heterogeneous populations

It is well known that the rate of aging is constant for populations described by the Gompertz law of mortality. However, this is true only when a population is homogeneous. In this note, we consider the multiplicative frailty model with the baseline distribution that follows the Gompertz law and study the impact of heterogeneity on the rate of aging in this population. We show that the rate of aging in this case is a function of age and that it increases in (calendar) time when the baseline mortality rate decreases.     

Establishment and spread of founding populations of an invasive thistle: the role of competition and seed limitation

Successful plant invasions require both the founding and local spread of new populations. High plant densities occur only when founding plants are able to disperse their seeds well locally to quickly colonize and fill the new patch. We test this ability in a 7-year field experiment with Carduus acanthoides, an invasive weed in several North American ecosystems. Founder plants were planted in the center of 64 m² plots and we monitored the recruitment, distribution pattern, mortality, and seed production of the seedlings that originated from these founding plants. Competing vegetation was clipped not at all, once, or twice each year to evaluate the importance of interspecific competition. More seedlings recruited in the intermediate once-clipped plots, and these seedlings also survived better. The control plots had fewer microsites for seedling recruitment; clipping a second time in September stimulated grasses to fill up the gaps. The number of C. acanthoides recruits and their median distances from the founder plants were also explained by the initial seed production of the founding plants. Overall, the experiment shows that the success of founder plants can fluctuate strongly, as 55% of the plots were empty by the sixth year. Our study suggests that the local invasion speed following initial establishment depends strongly on both the propagule pressure and availability of suitable microsites for seedling recruitment and growth.     

The asymptotic speed of propagation of the deterministic non-reducible n-type epidemic

A model has been formulated in [7] to describe the spatial spread of an epidemic involving n types of individuals, when triggered by the introduction of infectives from outside. Wave solutions for such a model have been investigated in [5] and [8] and have been shown only to exist at certain speeds. This paper establishes that the asymptotic speed of propagation, as denned in Aronson and Weinberger [1, 2], of such an epidemic is in fact c₀, the minimum speed at which wave solutions exist. This extends the known result for the one-type and host-vector epidemics.