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     

On stochastic logistic population growth models with immigration and multiple births

This paper develops a stochastic logistic population growth model with immigration and multiple births. The differential equations for the low-order cumulant functions (i.e., mean, variance, and skewness) of the single birth model are reviewed, and the corresponding equations for the multiple birth model are derived. Accurate approximate solutions for the cumulant functions are obtained using moment closure methods for two families of model parameterizations, one for badger and the other for fox population growth. For both model families, the equilibrium size distribution may be approximated well using the Normal approximation, and even more accurately using the saddlepoint approximation. It is shown that in comparison with the corresponding single birth model, the multiple birth mechanism increases the skewness and the variance of the equilibrium distribution, but slightly reduces its mean. Moreover, the type of density-dependent population control is shown to influence the sign of the skewness and the size of the variance.     

The birth process with immigration,and the genealogical structure of large populations

This paper studies a version of the birth and immigration process in which families are followed in the order of their appearance. This age structure is related to a number of results from population genetics, in particular the genealogical structure of the infinitely-many neutral alleles model. The asymptotic behavior of this genealogy is an easy consequence of the structure of the age-ordered family size process.     

Dispersal of adult grasshoppers,Mecostethus magister,under the field condition

Estimation of the number of adult grasshoppers, Mecostethus magister, was made by means of the mark-and-recapture method. The birth and death rates are possible to be estimated at the same time, but the immigration and the emigration rate are inevitably involved in these respectively. The immigration and emigration rates must be made clear to know the true birth and death rates. For this purpose the movement of the marked males in 1963 was analyzed. The grasshoppers dominantly moved in the directions of N, NW and W, and the difference in frequency among the movement directions was not so large. The distribution of the dispersal-distance relationship of each quadrate on each released day was fitted approximately to normal distribution. It could be concluded that almost all of the grasshoppers moved within the range of about 31–35m. The emigration rate from the quadrate (12×12m²) was about 0.73–0.77 and the difference in the rate among the released days was small. From these values the emigration rate from the station (84×60m²) was estimated as 0.21–0.23. Subtracting the emigration rate from the death-and-emigration rate, the true death rate was calculated. The death rate was very low until the number of males reached to the peak, then increased gradually. Supposing that immigration rate was equal to the emigration rate, the true birth rate was also estimated. But the presumption might not be pertinent, for the value of birth rates became negative.     

A 3D Individual-Based Model to Study Effects of Chemotaxis,Competition and Diffusion on the Motile-Phytoplankton Aggregation

In this paper, we develop a 3D-individual-based model (IBM) to understand effect of various small-scale mechanisms in phytoplankton cells, on the cellular aggregation process. These mechanisms are: spatial interactions between cells due to their chemosensory abilities (chemotaxis), a molecular diffusion and a demographical process. The latter is considered as a branching process with a density-dependent death rate to take into account the local competition on resources. We implement the IBM and simulate various scenarios under real parameter values for phytoplankton cells. To quantify the effects of the different processes quoted above on the spatial and temporal distribution of phytoplankton, we used two spatial statistics: the Clark–Evans index and the group belonging percentage. Our simulation study highlights the role of the branching process with a weak-to-medium competition in reinforcing the aggregating structure that forms from attraction mechanisms (under suitable conditions for diffusion and attraction forces), and shows by contrast that aggregations cannot form when competition is high.     

Estimation of rates of births, deaths, and immigration from mark-recapture data

Summary .  The analysis of mark–recapture data is undergoing a period of development and expansion. Here we contribute to that by presenting a model which includes both births and immigration, as well as the usual deaths. Data come from a long-term study of the willow tit ( Parus montanus ), where we can assume that all births are recorded, and hence immigrants can also be identified as birds captured as adults for the first time. We model the rates of immigration, birth rate per parent, and death rates of juveniles and adults. Using a hierarchical model allows us to incorporate annual variation in these parameters. The model is fitted to the data using Markov chain Monte Carlo, as a Bayesian analysis. In addition to the model fitting, we also check several aspects of the model fit, in particular whether survival varies with age or immigrant status, and whether capture probability is affected by previous capture history. The latter check is important, as independence of capture histories is a key assumption that simplifies the model considerably. Here we find that the capture probability depends strongly on whether the individual was captured in the previous year.     

A branching model for the spread of infectious animal diseases in varying environments

This paper is concerned with a stochastic model, describing outbreaks of infectious diseases that have potentially great animal or human health consequences, and which can result in such severe economic losses that immediate sets of measures need to be taken to curb the spread. During an outbreak of such a disease, the environment that the infectious agent experiences is therefore changing due to the subsequent control measures taken. In our model, we introduce a general branching process in a changing (but not random) environment. With this branching process, we estimate the probability of extinction and the expected number of infected individuals for different control measures. We also use this branching process to calculate the generating function of the number of infected individuals at any given moment. The model and methods are designed using important infections of farmed animals, such as classical swine fever, foot-and-mouth disease and avian influenza as motivating examples, but have a wider application, for example to emerging human infections that lead to strict quarantine of cases and suspected cases (e.g. SARS) and contact and movement restrictions.     

The effect of immigration on genetic control.

The effect of immigration of already inseminated females into the target area of a genetic control programme is analyzed. The sterile male technique and the use of cytoplasmically incompatible males are handled as special cases of the integrated genetical system proposed by Laven and Aslamakhan. For the sterile male technique explicit formulae are given for the critical immigration rate below which genetic control may be successful. If a strain is available which is advantageous to man, the integrated genetical system may be used to replace rather than to control the target population. An explicit formula is derived for the immigration level which can be sustained by a strain of a given fertility. In comparing the sterile male technique with the use of cytoplasmically incompatible males for control it is concluded that the former is to be chosen since the release rates of males necessary for population suppression would cause population replacement due to the unavoidable escape of even a very few females.     

ESTIMATING THE DURATION OF SPECIATION FROM PHYLOGENIES

Speciation is not instantaneous but takes time. The protracted birth–death diversification model incorporates this fact and predicts the often observed slowdown of lineage accumulation toward the present. The mathematical complexity of the protracted speciation model has barred estimation of its parameters until recently a method to compute the likelihood of phylogenetic branching times under this model was outlined (Lambert et al. 2014 ). Here, we implement this method and study using simulated phylogenies of extant species how well we can estimate the model parameters (rate of initiation of speciation, rate of extinction of incipient and good species, and rate of completion of speciation) as well as the duration of speciation, which is a combination of the aforementioned parameters. We illustrate our approach by applying it to a primate phylogeny. The simulations show that phylogenies often do not contain enough information to provide unbiased estimates of the speciation‐initiation rate and the extinction rate, but the duration of speciation can be estimated without much bias. The estimate of the duration of speciation for the primate clade is consistent with literature estimates. We conclude that phylogenies combined with the protracted speciation model provide a promising way to estimate the duration of speciation.     

A role for mesenchyme dynamics in mouse lung branching morphogenesis

Mammalian airways are highly ramified tree-like structures that develop by the repetitive branching of the lung epithelium into the surrounding mesenchyme through reciprocal interactions. Based on a morphometric analysis of the epithelial tree, it has been recently proposed that the complete branching scheme is specified early in each lineage by a programme using elementary patterning routines at specific sites and times in the developing lung. However, the coupled dynamics of both the epithelium and mesenchyme have been overlooked in this process. Using a qualitative and quantitative in vivo morphometric analysis of the E11.25 to E13.5 mouse whole right cranial lobe structure, we show that beyond the first generations, the branching stereotypy relaxes and both spatial and temporal variations are common. The branching pattern and branching rate are sensitive to the dynamic changes of the mesoderm shape that is in turn mainly dependent upon the volume and shape of the surrounding intrathoracic organs. Spatial and temporal variations of the tree architecture are related to local and subtle modifications of the mesoderm growth. Remarkably, buds never meet after suffering branching variations and continue to homogenously fill the opening spaces in the mesenchyme. Moreover despite inter-specimen variations, the growth of the epithelial tree and the mesenchyme remains highly correlated over time at the whole lobe level, implying a long-range regulation of the lung lobe morphogenesis. Together, these findings indicate that the lung epithelial tree is likely to adapt in real time to fill the available space in the mesenchyme, rather than being rigidly specified and predefined by a global programme. Our results strongly support the idea that a comprehensive understanding of lung branching mechanisms cannot be inferred from the branching pattern or behavior alone. Rather it needs to be elaborated upon with the reconsideration of mesenchyme-epithelium coupled growth and lung tissues mechanics.     

A cost-benefit analysis of Chagas disease control.

Chagas disease transmission can be effectively interrupted by insecticidal control of its triatomine bug vectors. We present here a simple model comparing the costs and benefits of such a programme, designed to eliminate domestic populations of Triatoma infestans throughout its known area of distribution over the seven southernmost countries of Latin America. The model has been simplified to require only four financial estimates relating to the unit cost of housing spraying and benefits due to avoidance of premature death in the acute phase of the disease, avoidance of supportive treatment and care in the chronic phase of the disease, and avoidance of corrective digestive and cardiac surgery. Except for these direct medical costs, all other potential benefits have been ignored. Nevertheless, the model shows that the direct financial benefits of such a programme would far outweigh the costs, and the project would support a remarkably high internal rate of return under the least optimistic estimates.     

Dynamics of mutualist populations that are demographically open

1. Few theoretical studies have examined the impact of immigration and emigration on mutualist population dynamics, but a recent empirical study (A.R. Thompson Oecologia, 143, 61-69) on mutualistic fish and shrimp showed that immigration can prevent population collapse, and that intraspecific competition for a mutualistic partner can curb population expansion. To understand in a theoretical context the implications of these results, and to assess their generality, we present a two-species model that accounts explicitly for immigration and emigration, as well as distinguishing the impacts of mutualism on birth rates, death rates and habitat acquisition. 2. The model confirms that immigration can stabilize mutualistic populations, and predicts that high immigration, along with enhanced reproduction and/or reduced mortality through mutualism, can cause population sizes to increase until habitat availability curbs further expansion. 3. We explore in detail the effects of different forms of habitat limitation on mutualistic populations. Habitat availability commonly limits the density of both populations if mutualists acquire shelter independently. If a mutualist depends on a partner for habitat, densities of that mutualist are capped by the amount of space provided by that partner. The density of the shelter-provider is limited by the environment. 4. If a mutualism solely augments reproduction, and most locally produced individuals leave the focal patch, then the mutualism will have a minimal effect on local dynamics. If the mutualism operates by reducing rates of death or enhancing habitat availability, and there is at least some immigration, then mutualism will affect local dynamics. This finding may be particularly relevant in marine systems, where there is high variability (among species and locations) in the extent to which progeny disperse from natal locations. 5. Overall, our results demonstrate that the consequences of immigration and emigration for the dynamics of mutualists depend strongly on which demographic rate is influenced by mutualism. 6. By relating our model to a variety of terrestrial and aquatic systems, we provide a general framework to guide future empirical studies of the dynamics of mutualistic populations.     

Parameter estimation for discretely observed linear birth‐and‐death processes

Birth‐and‐death processes are widely used to model the development of biological populations. Although they are relatively simple models, their parameters can be challenging to estimate, as the likelihood can become numerically unstable when data arise from the most common sampling schemes, such as annual population censuses. A further difficulty arises when the discrete observations are not equi‐spaced, for example, when census data are unavailable for some years. We present two approaches to estimating the birth, death, and growth rates of a discretely observed linear birth‐and‐death process: via an embedded Galton‐Watson process and by maximizing a saddlepoint approximation to the likelihood. We study asymptotic properties of the estimators, compare them on numerical examples, and apply the methodology to data on monitored populations.     

一类总人口在变化的HIV/AIDS传播的动力学模型

建立了HIV/AIDS传播的具有常数移民和指数出生的SI型模型,其中易感人群按照有无不良行为被分为两组.分别对具双线性传染率和具标准传染率的模型讨论了其无病平衡点和地方病平衡点的存在性,并就某些重要的特殊情况进行了平衡点和稳定性的分析.     

带有移民的两性分支过程极限性质

考虑带有移民的两性分支过程,这里移民数量依赖于当前人口总数,在下临界状态情形,本文证明了此过程收敛于一平衡分布．     

The Probability of Extinction of Infectious Salmon Anemia Virus in One and Two Patches

Single-type and multitype branching processes have been used to study the dynamics of a variety of stochastic birth–death type phenomena in biology and physics. Their use in epidemiology goes back to Whittle’s study of a susceptible–infected–recovered (SIR) model in the 1950s. In the case of an SIR model, the presence of only one infectious class allows for the use of single-type branching processes. Multitype branching processes allow for multiple infectious classes and have latterly been used to study metapopulation models of disease. In this article, we develop a continuous time Markov chain (CTMC) model of infectious salmon anemia virus in two patches, two CTMC models in one patch and companion multitype branching process (MTBP) models. The CTMC models are related to deterministic models which inform the choice of parameters. The probability of extinction is computed for the CTMC via numerical methods and approximated by the MTBP in the supercritical regime. The stochastic models are treated as toy models, and the parameter choices are made to highlight regions of the parameter space where CTMC and MTBP agree or disagree, without regard to biological significance. Partial extinction events are defined and their relevance discussed. A case is made for calculating the probability of such events, noting that MTBPs are not suitable for making these calculations.     

A logistic branching process for population genetics

A logistic (regulated population size) branching process population genetic model is presented. It is a modification of both the Wright-Fisher and (unconstrained) branching process models, and shares several properties including the coalescent time and shape, and structure of the coalescent process with those models. An important feature of the model is that population size fluctuation and regulation are intrinsic to the model rather than externally imposed. A consequence of this model is that the fluctuation in population size enhances the prospects for fixation of a beneficial mutation with constant relative viability, which is contrary to a result for the Wright-Fisher model with fluctuating population size. Explanation of this result follows from distinguishing between expected and realized viabilities, in addition to the contrast between absolute and relative viabilities.     

Modeling ecological traps for the control of feral pigs

Ecological traps are habitat sinks that are preferred by dispersing animals but have higher mortality or reduced fecundity compared to source habitats. Theory suggests that if mortality rates are sufficiently high, then ecological traps can result in extinction. An ecological trap may be created when pest animals are controlled in one area, but not in another area of equal habitat quality, and when there is density‐dependent immigration from the high‐density uncontrolled area to the low‐density controlled area. We used a logistic population model to explore how varying the proportion of habitat controlled, control mortality rate, and strength of density‐dependent immigration for feral pigs could affect the long‐term population abundance and time to extinction. Increasing control mortality, the proportion of habitat controlled and the strength of density‐dependent immigration decreased abundance both within and outside the area controlled. At higher levels of these parameters, extinction was achieved for feral pigs. We extended the analysis with a more complex stochastic, interactive model of feral pig dynamics in the Australian rangelands to examine how the same variables as the logistic model affected long‐term abundance in the controlled and uncontrolled area and time to extinction. Compared to the logistic model of feral pig dynamics, the stochastic interactive model predicted lower abundances and extinction at lower control mortalities and proportions of habitat controlled. To improve the realism of the stochastic interactive model, we substituted fixed mortality rates with a density‐dependent control mortality function, empirically derived from helicopter shooting exercises in Australia. Compared to the stochastic interactive model with fixed mortality rates, the model with the density‐dependent control mortality function did not predict as substantial decline in abundance in controlled or uncontrolled areas or extinction for any combination of variables. These models demonstrate that pest eradication is theoretically possible without the pest being controlled throughout its range because of density‐dependent immigration into the area controlled. The stronger the density‐dependent immigration, the better the overall control in controlled and uncontrolled habitat combined. However, the stronger the density‐dependent immigration, the poorer the control in the area controlled. For feral pigs, incorporating environmental stochasticity improves the prospects for eradication, but adding a realistic density‐dependent control function eliminates these prospects.