Steady states in population models with monotone,stochastic dynamics

Existence, uniqueness and asymptotic stability of stochastic equilibrium are established in multi-dimensional population models with monotone dynamics.     

Population dynamics with a stable efficient equilibrium

We propose a game-theoretic dynamics of a population of replicating individuals. It consists of two parts: the standard replicator one and a migration between two different habitats. We consider symmetric two-player games with two evolutionarily stable strategies: the efficient one in which the population is in a state with a maximal payoff and the risk-dominant one where players are averse to risk. We show that for a large range of parameters of our dynamics, even if the initial conditions in both habitats are in the basin of attraction of the risk-dominant equilibrium (with respect to the standard replication dynamics without migration), in the long run most individuals play the efficient strategy.     

Population models with environmental stochasticity

Two discrete population models, one with stochasticity in the carrying capacity and one with stochasticity in the per capita growth rate, are investigated. Conditions under which the corresponding Markov processes are null recurrent and positively recurrent are derived.     

Population models for diseases with no recovery

An S I epidemic model with a general shape of density-dependent mortality and incidence rate is studied. The asymptotic behaviour is global convergence to an endemic equilibrium, above a threshold, and to a disease-free equilibrium, below the threshold. The effect of vaccination is then examined.     

On simulating strongly interacting, stochastic population models. II. Multiple compartments

In Wick and Stelf [Math. Biosci. 187 (2004) 1], we showed how to simulate a pair of strongly interacting biological populations evolving stochastically over many orders-of-magnitude. Here we generalize the method to any (finite) number of compartments; transitions including births, deaths, progression through life-stages, and mitoses; and arbitrary rate functions. We illustrate the technique for a seven-compartment model of the cellular immune response to a viral infection.     

Population growth with randomly distributed jumps

 The growth of populations with continuous deterministic and random jump components is treated. Three special models in which random jumps occur at the time of events of a Poisson process and admit formal explicit solutions are considered: A) Logistic growth with random disasters having exponentially distributed amplitudes; B) Logistic growth with random disasters causing the removal of a uniformly distributed fraction of the population size; and C) Exponential decay with sudden increases (bonanzas) in the population and with each increase being an exponentially distributed fraction of the current population. Asymptotic and numerical methods are employed to determine the mean extinction time for the population, qualitatively and quantitatively. For Model A, this time becomes exponentially large as the carrying capacity becomes much larger than the mean disaster size. Implications for colonizing species for Model A are discussed. For Models B and C, the practical notion of a small, but positive, effective extinction level is chosen, and in these cases the expected extinction time rises rapidly with population size, yet at less than an e xponentially large order. Received 21 June 1996; received in revised form 17 February 1997     

Effects of predation on host-pathogen dynamics in SIR models

The integration of infectious disease epidemiology with community ecology is an active area of research. Recent studies using SI models without acquired immunity have demonstrated that predation can suppress infectious disease levels. The authors recently showed that incorporating immunity (SIR models) can produce a “hump”-shaped relationship between disease prevalence and predation pressure; thus, low to moderate levels of predation can boost prevalence in hosts with acquired immunity. Here we examine the robustness of this pattern to realistic extensions of a basic SIR model, including density-dependent host regulation, predator saturation, interference, frequency-dependent transmission, predator numerical responses, and explicit resource dynamics. A non-monotonic relationship between disease prevalence and predation pressure holds across all these scenarios. With saturation, there can also be complex responses of mean host abundance to increasing predation, as well as bifurcations leading to unstable cycles (epidemics) and pathogen extinction at larger predator numbers. Firm predictions about the relationship between prevalence and predation thus require one to consider the complex interplay of acquired immunity, host regulation, and foraging behavior of the predator.     

Population dynamics of shrews on small islands accord with the equilibrium model

Three of the six species of shrew in Finland, Sorex araneus, S. caecutiens, and S. minutus , are common on the mainland and widespread on islands in lakes. The islands range from 0.01 to 500 ha in area, and from 10 to 3000 m in isolation (distance from the mainland). The species-area relationship, the lack of importance of habitat diversity, the increasing frequency of unoccupied small islands with isolation, and direct observations of small populations, all suggest that populations on small islands have a high extinction rate. Demographic stochasticity is the main cause of extinctions in the superior competitor, S. araneus , which occurs consistently on islands greater than 2 ha. The small species, S. caecutiens and S. minutus , are more sensitive to environmental stochasticity than is S. araneus , and are inferior to it in interspecific competition; these factors probably contribute to the absence of the small species from many islands tens of hectares in area. Frequent colonization of islands less than 500 m from the mainland is indicated by large numbers of shrews trapped from tiny islets where breeding is not possible, by increasing epigenetic divergence of island populations with isolation, and by observations of dispersal to and colonization of islands. Dispersal ability decreases with decreasing individual size, which may partly explain the absence of the small shrews from many relatively large islands. The shrew populations persist in a dynamic equilibrium on the islands. Epigenetic morphological variation is a useful tool in ecological studies of island populations.     

Population genetic models of duplicated genes

Various population genetic models of duplicated genes are introduced. The problems covered in this review include the fixation process of a duplicated copy, copy number polymorphism, the fates of duplicated genes and single nucleotide polymorphism in duplicated genes. Because of increasing evidence for concerted evolution by gene conversion, this review introduces recently developed gene conversion models. In the first half, models assuming independent evolution of duplicated genes are introduced, and then the effect of gene conversion is considered in the second half.     

Do “infectious” prey select for high levels of natural antibodies in tropical pythons?

Natural antibodies (NAbs) constitute an important component in vertebrate immune system, but, in spite of this, have often been dismissed as “non-specific background” signals. We observed a significant positive relationship between water python (Liasis fuscus) body length/age and levels of antibodies reactive with two administered antigens (tetanus and diphtheria). However, no humoral immune response to the antigens was observed. The lack of elevated immune response, and the age-associated increase in antibody titres, strongly suggest that the antibodies consisted of polyreactive NAbs, and that absence of an elevated immune response was caused by such high levels of NAbs that they were able to mask the epitopes of the antigens. In our study area pythons feed mainly on rodents that frequently, before being killed, are able to inflict numerous bites to the snakes. The bites most likely transmit pathogens such as bacteria. As NAbs have been shown to act as a first line defence against bacterial infections, the high levels of NAbs in the pythons may be an adaptation to reduce pathogenic effects of bacteria transmitted by the prey when the snakes are feeding. Thus, the results from present study suggest that NAbs may have an important immunological function by reducing deleterious effects of pathogens in wild populations.     

生长季节和非生长季节交替出现的种群动态模型及环境变化的影响

有些生物的生长季节和非生长季节交替出现,本文建立了描述这种生物种群动态的方程,并研究了在环境稳定、随机波动、定向变化的情况下种群的变化方式,还讨论了种群的危害及濒危情况．生长季节延长时种群增大,有害生物的危害加重,濒危生物濒危程度减轻;生长季节缩短时种群减小,有害生物的危害减轻,濒危生物更加濒危或灭绝．     

一类基于比率且具双线性密度制约的捕食-食饵系统的全局稳定性分析

研究一类基于比率和具双线性密度制约的捕食-食饵系统．由Bendixson环域定理及比较定理给出了奇点（0,0）和（1,0）全局渐近稳定的条件．通过对等倾线的研究给出了正平衡点存在的条件,并构造了Dulac函数讨论其全局渐近稳定性．     

关于一类时滞人口模型的全局吸引性

给出了保证时滞人口模型N'(t)=r(t)N(t) t≤0的每一正解N(t)趋于正平衡点 N*=1(t→∞)的一族充分条件．改进了Joseph和Yu的相关结果．     

Lichenicolous fungi show population subdivision by host species but do not share population history with their hosts

Lichenicolous fungi are a species-rich biological group growing on lichen thalli. Here, we analyze the genetic structure of the lichenicolous basidiomycete Tremella lobariacearum and three host species (Lobaria pulmonaria, Lobaria macaronesica, and Lobaria immixta) in Macaronesia. We used ordination and analysis of molecular variance to investigate the structuring of genetic variation, and a simulation test to investigate whether rDNA haplotypes of T. lobariacearum were significantly associated with host species. To investigate the evolutionary and demographic history of the lichenicolous fungus and its hosts, we used coalescent samplers to generate trees, and Bayesian skyline plots. We found that the hosts were most important in structuring populations of the lichenicolous species. Despite their wide geographic distribution, the same haplotypes of T. lobariacearum consistently associated with a given host species. Our results suggest that the Lobaria hosts create a selective environment for the lichenicolous fungus. Both the pathogen and the host populations exhibited substantial genetic structure. However, evolutionary and demographic histories differed between the parasite and its hosts, as evidenced by different divergence times and tree topologies.     

Multilocus models in the infinite island model of population structure

Different methods have been developed to consider the effects of statistical associations among genes that arise in population genetics models: kin selection models deal with associations among genes present in different interacting individuals, while multilocus models deal with associations among genes at different loci. It was pointed out recently that these two types of models are very similar in essence. In this paper, we present a method to construct multilocus models in the infinite island model of population structure (where deme size may be arbitrarily small). This method allows one to compute recursions on allele frequencies, and different types of genetic associations (including associations between different individuals from the same deme), and incorporates selection. Recursions can be simplified using quasi-equilibrium approximations; however, we show that quasi-equilibrium calculations for associations that are different from zero under neutrality must include a term that has not been previously considered. The method is illustrated using simple examples.     

Illustration of some limits of the Markov assumption for transition between groups in models of spread of an infectious pathogen in a structured herd

In epidemic models concerning a structured population, sojourn times in a group are usually described by an exponential distribution. For livestock populations, realistic distributions may be preferred for group changes (e.g. depending on sojourn time). We illustrated the effect on pathogen spread of the use of an exponential distribution, instead of the true distribution of the transition time, between groups for a population separated into two groups (youngstock, adults) when this true distribution is a triangular one. Concerning the epidemic process, two assumptions were defined: one type of excreting animal (SIR model), and two types of excreting animals (transiently or persistently infected animals). The study was conducted with two indirect-transmission levels between groups. Among the adults, the epidemic size and the last infection time were significantly different. For persistence, epidemic sizes (in the entire population and in youngstock) and first infection time, results varied according to models (excretion assumption, indirect-transmission level).     

Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission

A precise definition of the basic reproduction number, , is presented for a general compartmental disease transmission model based on a system of ordinary differential equations. It is shown that, if , then the disease free equilibrium is locally asymptotically stable; whereas if , then it is unstable. Thus, is a threshold parameter for the model. An analysis of the local centre manifold yields a simple criterion for the existence and stability of super- and sub-threshold endemic equilibria for near one. This criterion, together with the definition of , is illustrated by treatment, multigroup, staged progression, multistrain and vector–host models and can be applied to more complex models. The results are significant for disease control.     

Epidemic models with spatial spread due to population migration

A spatial diffusion operator that governs the migration of polymorphic populations is derived and some specific epidemic models are analyzed in the presence of this type of diffusion. Threshold criteria and asymptotic behavior of solutions are derived, and it is shown that spatially heterogeneous steady states can occur in these models.The work of this author was partially supported by the National Science Foundation's Ecosystem Studies Program under Interagency Agreement No. DED80-21024 with the U.S. Department of Energy under contract W-7405-eng-26 with Union Carbide Corporation