Consistent micro, macro and state-based population modelling

A population system can be modelled using a micro model focusing on the individual entities, a macro model where the entities are aggregated into compartments, or a state-based model where each possible discrete state in which the system can exist is represented. However, the concepts, building blocks, procedural mechanisms and the time handling for these approaches are very different. For the results and conclusions from studies based on micro, macro and state-based models to be consistent (contradiction-free), a number of modelling issues must be understood and appropriate modelling procedures be applied. This paper presents a uniform approach to micro, macro and state-based population modelling so that these different types of models produce consistent results and conclusions. In particular, we demonstrate the procedures (distribution, attribute and combinatorial expansions) necessary to keep these three types of models consistent. We also show that the different time handling methods usually used in micro, macro and state-based models can be regarded as different integration methods that can be applied to any of these modelling categories. The result is free choice in selecting the modelling approach and the time handling method most appropriate for the study without distorting the results and conclusions.     

From local interactions to population dynamics in site-based models of ecology

A central problem in ecology is relating the interactions of individuals-described in terms of competition, predation, interference, etc.-to the dynamics of the populations of these individuals-in terms of change in numbers of individuals over time. Here, we address this problem for a class of site-based ecological models, where local interactions between individuals take place at a finite number of discrete resource sites over non-overlapping generations and, between generations, individuals move randomly between sites over the entire system. Such site-based models have previously been applied to a wide range of ecological systems: from those involving contest or scramble competition for resources to host-parasite interactions and meta-populations. We show how the population dynamics of site-based models can be accurately approximated by and understood through deterministic and stochastic difference equations. Conversely, we use the inverse of this approximation to show what implicit assumptions are made about individual interactions by modelling of population dynamics in terms of difference equations. To this end, we prove a useful and general theorem: that any model in our class of site-based models has a corresponding stochastic difference equation population model, by which it can be approximated. This theorem allows us to calculate long-term population dynamics, evolutionary stable strategies and, by extending our theory to account for large deviations, extinction probabilities for a wide range of site-based systems. Our methodology is then illustrated to various examples of between species competition, predator-prey interactions and co-operation.     

Numerical simulation of a stochastic model for cancerous cells submitted to chemotherapy

A stochastic model is proposed to study the problem of inherent resistance by cell populations when chemotherapeutic agents are used to control tumor growth. Stochastic differential equations are introduced and numerically integrated to simulate expected response to the chemotherapeutic strategies as a function of different parameters. Satisfactory demonstration runs of the model indicate that it could represent a useful tool in verifying the results of experimental and clinical chemotherapy courses and planning treatment strategies. Some types of behaviour are illustrated graphically.Work supported by the C.N.R. Grants: 85.02652.01; 86.02116.01     

On the time to extinction for a two-type version of Bartlett's epidemic model

We are interested in how the addition of type heterogeneities affects the long time behaviour of models for endemic diseases. We do this by analysing a two-type version of a model introduced by Bartlett under the restriction of proportionate mixing. This model is used to describe diseases for which individuals switch states according to susceptible-->infectious-->recovered and immune, where the immunity is life-long. We describe an approximation of the distribution of the time to extinction given that the process is started in the quasi-stationary distribution, and we analyse how the variance and the coefficient of variation of the number of infectious individuals depends on the degree of heterogeneity between the two types of individuals. These are then used to derive an approximation of the time to extinction. From this approximation we conclude that if we increase the difference in infectivity between the two types the expected time to extinction decreases, and if we instead increase the difference in susceptibility the effect on the expected time to extinction depends on which part of the parameter space we are in, and we can also obtain non-monotonic behaviour. These results are supported by simulations.     

A comparison of three different stochastic population models with regard to persistence time

Results are summarized from the literature on three commonly used stochastic population models with regard to persistence time. In addition, several new results are introduced to clearly illustrate similarities between the models. Specifically, the relations between the mean persistence time and higher-order moments for discrete-time Markov chain models, continuous-time Markov chain models, and stochastic differential equation models are compared for populations experiencing demographic variability. Similarities between the models are demonstrated analytically, and computational results are provided to show that estimated persistence times for the three stochastic models are generally in good agreement when the models are consistently formulated. As an example, the three stochastic models are applied to a population satisfying logistic growth. Logistic growth is interesting as different birth and death rates can yield the same logistic differential equation. However, the persistence behavior of the population is strongly dependent on the explicit forms for the birth and death rates. Computational results demonstrate how dramatically the mean persistence time can vary for different populations that experience the same logistic growth.     

Persistence of a fig wasp population and evolution of dioecy in figs: a simulation study

The relationship between fig trees and their pollinator wasps is a well-known example of species-specific obligate mutualism. In this article we present a stochastic model of this mutualistic system, referring to data on a dioecious fig (Ficus schwarzii) in Borneo, and examine the conditions for the persistence of a wasp population for a given period. (1) When the average duration of the flowering interval of fig trees is short, even a small fig population can sustain a wasp population successfully. A population whose average period of flowering cycle is half that of another population can sustain a wasp population with a number of trees less than half of the other population. (2) The wasp survival rate (WSR) is higher when (a) the variation of the interval periods of fig flowering is smaller, (b) the fig population size is larger, and (c) figs can prolong their receptivity to wait for a wasp if no wasps are available. (3) WSR is predictable from the average proportion of the fig's receptive phases, in which wasps are available, to their total receptive phases. (4) The persistence period of a wasp population increases exponentially with the number of fig trees. Based on these results we propose a new hypothesis, as a possible scenario, on the evolution of dioecy from monoecy in Ficus. Received: November 13, 1998 / Accepted: July 14, 1999     

Meta-models as a straightforward approach to the sensitivity analysis of complex models

Complex simulation models are important tools in applied ecological and conservation research. However sensitivity analysis of this important class of models can be difficult to conduct. High level interactions and non-linear responses are common in complex simulations, and this necessitates a global sensitivity analysis, where each parameter is tested at a range of values, and in combination with changes in many other parameters. We reviewed the literature, searching for population viability analyses that used simulation models. We found only 9 out of the 122 simulation population viability analysis used global sensitivity analysis. This result is typical of other simulation models in applied ecology, where global sensitivity analysis is rare. We then demonstrate how to conduct a meta-modeling sensitivity analysis, where a simpler statistically fit function (the meta-model, also known as the surrogate model or emulator) is used to approximate the behavior of the complicated simulation. This simpler meta-model is interrogated to inform on the behavior of simulation model. We fit two example meta-models, a generalized linear model and a boosted regression tree, to exemplify the approach. Our hope is that by going through these techniques thoroughly they will become more widely adopted.     

Research of population with impulsive perturbations based on dynamics of a neutral delay equation and ecological quality system

This paper studies the global behaviors of a nonlinear autonomous neutral delay differential population model with impulsive perturbation. This model may be suitable for describing the dynamics of population with long larval and short adult phases. It is shown that the system may have global stability of the extinction and positive equilibria, or grow without being bounded under some conditions.     

A comparison of persistence-time estimation for discrete and continuous stochastic population models that include demographic and environmental variability

A discrete-time Markov chain model, a continuous-time Markov chain model, and a stochastic differential equation model are compared for a population experiencing demographic and environmental variability. It is assumed that the environment produces random changes in the per capita birth and death rates, which are independent from the inherent random (demographic) variations in the number of births and deaths for any time interval. An existence and uniqueness result is proved for the stochastic differential equation system. Similarities between the models are demonstrated analytically and computational results are provided to show that estimated persistence times for the three stochastic models are generally in good agreement when the models satisfy certain consistency conditions.     

Orientational dynamics of magnetotactic bacteria in Earth’s magnetic field—a simulation study

We investigate through simulations the phenomena of magnetoreception to enable an understanding of the minimum requirements of a fail-safe mechanism, operational at the cellular level, to sense a weak magnetic field at ambient temperature in a biologically active environment. To do this, we use magnetotactic bacteria (MTB) as our model system. The magnetic field sensing ability of these bacteria is due to the presence of magnetosomes, which are internal membrane-bound organelles that contain an iron-based magnetic mineral crystal. These magnetosomes are usually found arranged in a chain aligned with the long axis of the bacterial body. This arrangement yields an overall magnetic dipole moment to the bacterial cell. To simulate this orientation process, we set up a rotational Langevin stochastic differential equation and solve it repeatedly over appropriate time steps for isolated spherical shaped MTB as well as for a more realistic model of spheroidal MTB with flagella. The orientation process appears to depend on shape parameters with spheroidal MTB showing a slower response time compared to spherical MTB. Further, our simulation also reveals that the alignment to the external magnetic field is more robust for an MTB when compared to single magnetosome. For the simulation involving magnetosomes, we include an extra torque that arises from the twisting of an attachment tether and enhance the viscosity of the surrounding medium to mimic intracellular conditions in the governing Langevin equation. The response time of alignment is found to be substantially reduced when one includes a dipole interaction term with a neighboring magnetosome and the alignment becomes less robust with increase in inter dipole distance. The alignment process can thereby be said to be very sensitively dependent on the distance between magnetosomes. Simulating the process of alignment between two neighboring magnetosomes, both in the absence and presence of an ambient magnetic field, we conclude that alignment between these dipoles at the distances typical in an MTB is highly probable and it would be the locked unit that responds to changes in the external magnetic field.     

The large graph limit of a stochastic epidemic model on a dynamic multilayer network

We consider a Markovian SIR-type (Susceptible → Infected → Recovered) stochastic epidemic process with multiple modes of transmission on a contact network. The network is given by a random graph following a multilayer configuration model where edges in different layers correspond to potentially infectious contacts of different types. We assume that the graph structure evolves in response to the epidemic via activation or deactivation of edges of infectious nodes. We derive a large graph limit theorem that gives a system of ordinary differential equations (ODEs) describing the evolution of quantities of interest, such as the proportions of infected and susceptible vertices, as the number of nodes tends to infinity. Analysis of the limiting system elucidates how the coupling of edge activation and deactivation to infection status affects disease dynamics, as illustrated by a two-layer network example with edge types corresponding to community and healthcare contacts. Our theorem extends some earlier results describing the deterministic limit of stochastic SIR processes on static, single-layer configuration model graphs. We also describe precisely the conditions for equivalence between our limiting ODEs and the systems obtained via pair approximation, which are widely used in the epidemiological and ecological literature to approximate disease dynamics on networks. The flexible modeling framework and asymptotic results have potential application to many disease settings including Ebola dynamics in West Africa, which was the original motivation for this study.     

Timescales of population rarity and commonness in random environments

This is a mathematical study of the interactions between non-linear feedback (density dependence) and uncorrelated random noise in the dynamics of unstructured populations. The stochastic non-linear dynamics are generally complex, even when the deterministic skeleton possesses a stable equilibrium. There are three critical factors of the stochastic non-linear dynamics; whether the intrinsic population growth rate (lambda) is smaller than, equal to, or greater than 1; the pattern of density dependence at very low and very high densities; and whether the noise distribution has exponential moments or not. If lambda < 1, the population process is generally transient with escape towards extinction. When lambda > or = 1, our quantitative analysis of stochastic non-linear dynamics focuses on characterizing the time spent by the population at very low density (rarity), or at high abundance (commonness), or in extreme states (rarity or commonness). When lambda >1 and density dependence is strong at high density, the population process is recurrent: any range of density is reached (almost surely) in finite time. The law of time to escape from extremes has a heavy, polynomial tail that we compute precisely, which contrasts with the thin tail of the laws of rarity and commonness. Thus, even when lambda is close to one, the population will persistently experience wide fluctuations between states of rarity and commonness. When lambda = 1 and density dependence is weak at low density, rarity follows a universal power law with exponent -3/2. We provide some mathematical support for the numerical conjecture [Ferriere, R., Cazelles, B., 1999. Universal power laws govern intermittent rarity in communities of interacting species. Ecology 80, 1505-1521.] that the -3/2 power law generally approximates the law of rarity of 'weakly invading' species with lambda values close to one. Some preliminary results for the dynamics of multispecific systems are presented.     

Rainbow trout: a population simulation based on individual responses to varying environmental and demographic parameters

Synopsis An age-structured simulation model of the growth and population dynamics of a migratory rainbow trout population is presented. The model includes all principal life-history intervals and incorporates the food density-temperature relationships of salmonid growth efficiency proposed by Brett et al. (1969) and Shelbourn et al. (1973). Population size, mean weight, and biomass are adjusted and output monthly over time in age, sex, and location categories; a simulation run may continue for as long as 100 years. A variety of environmental and and biological parameters are utilized in the simulation which can be altered as a user option. Simulation results compare favorably with field data. Sensitivity analyses illustrate the importance of age-sex specific maturation ratios, age-class strength fluctuations, and natural mortality rates in determining population size.     

Simultaneous effects of food limitation and inducible resistance on herbivore population dynamics

Many herbivore populations fluctuate temporally, but the causes of those fluctuations remain unclear. Plant inducible resistance can theoretically cause herbivore population fluctuations, because herbivory may induce plant changes that reduce the survival or reproduction of later-feeding herbivores. Herbivory can also simply reduce the quantity of food available for later feeders and this, too, can cause population fluctuations. Inducible resistance and food limitation often occur simultaneously, yet whether they jointly facilitate or suppress herbivore fluctuations remains largely unexplored. We present models that suggest that food limitation and inducible resistance may have synergistic effects on herbivore population dynamics. The population-level response of the food plant to herbivory and the details of how inducible resistance affects herbivore performance both influence the resulting herbivore dynamics. Our results identify some biological properties of plant-herbivore systems that might determine whether or not cycles occur, and suggest that future empirical and theoretical population dynamics studies should account for the effects of both food limitation and inducible resistance.     

Limit theorems for the population size of a birth and death process allowing catastrophes

The linear birth and death process with catastrophes is formulated as a right continuous random walk on the non-negative integers which evolves in continuous time with an instantaneous jump rate proportional to the current value of the process. It is shown that distributions of the population size can be represented in terms of those of a certain Markov branching process. The ergodic theory of Markov branching process transition probabilities is then used to develop a fairly complete understanding of the behaviour of the population size of the birth-death-catastrophe process.Research done while on leave at Colorado State University from the University of Western Australia and partially supported by N.S.F. grant DMS-8501763     

长白山北坡与西坡岳桦种群差异分析

本研究通过植被调查,从密度、胸径、树高、树龄、世代和种群分布格局等方面分析长白山北坡与西坡岳桦(Betula ermanii Cham.)种群的坡向差异及随海拔变化的坡向差异,探讨植被坡向差异的干扰因素。结果显示,整体上北坡岳桦种群胸径、二代木比重显著大于西坡,一代木比重则相反,说明尽管西坡生境优于北坡,但西坡植被演替落后于北坡,同时受干扰的因素多于北坡。北坡与西坡岳桦种群特征随海拔变化的趋势存在统一性,岳桦种群密度随海拔的升高均呈"单峰"型变化;其中北坡岳桦胸径、树高、树龄、二代木比重随海拔升高逐渐减小,种群空间分布格局呈聚集-随机-聚集变化趋势,说明在较低和较高海拔岳桦只能占据部分生境,符合一般山地植被特征。西坡岳桦树龄、树高随海拔波动减小,种群呈聚集分布;同时不同海拔的岳桦种群特征坡向差异也不同,海拔1600～1700 m,西坡岳桦胸径、树高、树龄小于北坡,而种群密度大于北坡;海拔1800～2100 m,西坡岳桦胸径、树高、树龄大于北坡,而种群密度小于北坡。研究结果说明西坡的干扰随海拔呈随机性,具有风干扰的特征,推测除已知的火山喷发、火山灰侵蚀干扰以及生境条件差异外,风干扰也是产...     

Relationships of nitrogen and phosphorus to a tropical phytoplankton population

Summary The relationships of nitrogen and phosphorus in the plankton of tropical Lake George (Uganda) have been investigated. The data used are from analyses of samples collected from mid-lake areas during one year, and along transects of increasing plankton concentration, at various times when conditions allowed, and also from results of nutrient uptake experiments. These data are discussed in relation to previously reported information about general limnological conditions in Lake George, and in relation to physiological effects of nutrient enrichments. The features are compared with other waters.The peculiarities of the lake make difficult the precise calculations of long term primary production using oxygen exchange techniques, so that the determination of nutrient turnover rates based upon metabolic activity measured in this way could be misleading. Nevertheless, reasonable overall assessment of nitrogen and phosphorus turnover might be obtained from the proportions of these elements in the plankton. This could be done mainly due to the low variability of the limnological conditions, the apparent limitation of both nitrogen and phosphorus nutrients and the extreme predominance of algal metabolism over the remaining biota.It was also considered that the multiplicity of growth limiting factors which the nitrogen and phosphorus conditions impart contribute to an inherantly stable biomass.

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Zusammenfassung Die Beziehungen zwischen Stickstoff und Phosphor im Plankton des tropischen Sees Lake George (Uganda) wurden untersucht. Die benutzten Daten stammen aus Analysen vom Proben, die über ein Jahr hinweg aus einem Gebiet in Seemitte entnommen wurden. Des weiteren wurden Proben entlang von Profilen mit steigender Planktonkonzentration untersucht. Ferner wurden Ergebnisse von Nährstoffaufnahme-experimenten benutzt. Die Daten wurden mit denen verglichen, die bereits über die algemeinen limnologischen Bedingungen des Sees veröffentlicht wurden. Auch wurden sie in Bezug auf die physiologischen Einflüsse einer erhöhten Nährstoffzufuhr diskutiert. Die Merkmale des Sees wurden mit denen anderer Gewässern verglichen.Die charakteristischen Besonderheiten des Sees machen eine genaue Berechnung von lang anhaltender Primärproduktion unter Anwendung der Sauerstoffaustausch-Methode schwierig, so dass die Bestimmung der Raten des Nährstoffwechsels, basiert auf gemessener metabolischer Aktivität, ein falsches Bild geben könnte. Immerhin konnten annehmbare allgemeine Werte von Stickstoff und Phosphorstoffwechsel aus der Verteilung dieser Elemente im Plankton gewonnen werden. Diese Werte konnten vor allem durch die geringen Schwankungen in den limnologischen Bedingungen, durch die auffallende Beschränkung im Auftreten von Stickstoff und Phosphor und durch das extreme Überwicht vom Algenmetabolismus über die übrige Biota, gewonnen werden.Es wird angenommen, dass die Vielfalt der wachstumbegrenzenden Faktoren, die mit dem Stickstoff- und Phosphorhaushalt zusammenhängen, zu einer von Natur aus stabilen Biomasse führen.

     

From snapshot information to long-term population dynamics of Acacias by a simulation model

The African Acacia species A. raddiana is believed to be endangered in the Negev desert of Israel. The ecology of this species is not well understood. The main idea of our study is to learn more about the long-term population dynamics of these trees using snapshot information in the form of size frequency distributions. These distributions are highly condensed indices of population dynamics acting over many years. In this paper, we analyse field data on recruitment, growth, and mortality and use an existing simulation model of the population dynamics of A. raddiana (SAM) to produce contrasting scenarios of these live history processes that are based on the analysed field evidence. The main properties of simulated as well as observed tree size frequency distributions are characterised with Simpson's index of dominance and a new permutation index. Finally, by running the SAM model under the different scenarios, we study the effect of these different processes on simulated size frequency distributions (pattern) and we compare them to size distributions observed in the field, in order to identify the processes acting in the field. Our study confirms rare recruitment events as a major factor shaping tree size frequency distributions and shows that the paucity of recruitment has been a normal feature of A. raddiana in the Negev over many years. Irregular growth, e.g., due to episodic rainfall, showed a moderate influence on size distributions. Finally, the size frequency distributions observed in the Negev reveal the information that, in this harsh environment, mortality of adult A. raddiana is independent of tree size (age).     

Computation of spiking activity for a stochastic spatial neuron model: effects of spatial distribution of input on bimodality and CV of the ISI distribution

We obtain computational results for a new extended spatial neuron model in which the neuronal electrical depolarization from resting level satisfies a cable partial differential equation and the synaptic input current is also a function of space and time, obeying a first order linear partial differential equation driven by a two-parameter random process. The model is first described explicitly with the inclusion of all biophysical parameters. Simplified equations are obtained with dimensionless space and time variables. A standard parameter set is described, based mainly on values appropriate for cortical pyramidal cells. When the noise is small and the mean voltage crosses threshold, a formula is derived for the expected time to spike. A simulation algorithm, involving one-dimensional random processes is given and used to obtain moments and distributions of the interspike interval (ISI). The parameters used are those for a near balanced state and there is great sensitivity of the firing rate around the balance point. This sensitivity may be related to genetically induced pathological brain properties (Rett's syndrome). The simulation procedure is employed to find the ISI distribution for some simple patterns of synaptic input with various relative strengths for excitation and inhibition. With excitation only, the ISI distribution is unimodal of exponential type and with a large coefficient of variation. As inhibition near the soma grows, two striking effects emerge. The ISI distribution shifts first to bimodal and then to unimodal with an approximately Gaussian shape with a concentration at large intervals. At the same time the coefficient of variation of the ISI drops dramatically to less than 1/5 of its value without inhibition.