Stability of periodic solutions for a model of genetic repression with delays

A technique is discussed for locating the Hopf bifurcation of an n-dimensional system of delay differential equations which arises from a model for control of protein biosynthesis. Certain parameter values are shown to allow a Hopf bifurcation to periodic orbits. At the Hopf bifurcation the periodic orbits are shown to be stable either analytically or numerically depending on the parameter values.On leave from North Carolina State University.Supported in part by N.S.F. Grant # MCS 81-02828     

Positive solutions of an elliptic system arising from a model in evolutionary ecology

We give sufficient and almost necessary conditions for the existence of positive solutions to an elliptic system satisfying various Dirichlet boundary conditions. The elliptic system consists of the steady-state equations of a parabolic system used to model the growth and spread of a particular gene and population living in a bounded region. The model takes into account the fact that the fitness of the individuals in the population may depend on the population size. Some non-existence results are also included.Research partially supported by NSF grant no. DMS-8801968     

Existence and nonexistence of steady-state solutions for a selection-migration model in population genetics

We discuss a selection-migration model in population genetics, involving two alleles A ₁ and A ₂ such that A ₁ is at an advantage over A ₂ in certain subregions and at a disadvantage in others. It is shown that if A ₁ is at an overall disadvantage to A ₂ and the rate of gene flow is sufficiently large than A ₁ must die out; on the other hand, if the two alleles are in some sense equally advantaged overall, then A ₁ and A ₂ can coexist no matter how great the rate of gene flow.     

Stable long-period cycling and complex dynamics in a single-locus fertility model with genomic imprinting

Although long-period population size cycles and chaotic fluctuations in abundance are common in ecological models, such dynamics are uncommon in simple population-genetic models where convergence to a fixed equilibrium is most typical. When genotype-frequency cycling does occur, it is most often due to frequency-dependent selection that results from individual or species interactions. In this paper, we demonstrate that fertility selection and genomic imprinting are sufficient to generate a Hopf bifurcation and complex genotype-frequency cycling in a single-locus population-genetic model. Previous studies have shown that on its own, fertility selection can yield stable two-cycles but not long-period cycling characteristic of a Hopf bifurcation. Genomic imprinting, a molecular mechanism by which the expression of an allele depends on the sex of the donating parent, allows fitness matrices to be nonsymmetric, and this additional flexibility is crucial to the complex dynamics we observe in this fertility selection model. Additionally, we find under certain conditions that stable oscillations and a stable equilibrium point can coexist. These dynamics are characteristic of a Chenciner (generalized Hopf) bifurcation. We believe this model to be the simplest population-genetic model with such dynamics.     

Dynamic models of infectious diseases as regulators of population sizes

Five SIRS epidemiological models for populations of varying size are considered. The incidences of infection are given by mass action terms involving the number of infectives and either the number of susceptibles or the fraction of the population which is susceptible. When the population dynamics are immigration and deaths, thresholds are found which determine whether the disease dies out or approaches an endemic equilibrium. When the population dynamics are unbalanced births and deaths proportional to the population size, thresholds are found which determine whether the disease dies out or remains endemic and whether the population declines to zero, remains finite or grows exponentially. In these models the persistence of the disease and disease-related deaths can reduce the asymptotic population size or change the asymptotic behavior from exponential growth to exponential decay or approach to an equilibrium population size.Research supported by Centers for Disease Control contract 200-87-0515. Support services provided at the University of Iowa Center for Advanced Studies     

Limit cycles in a chemostat model for a single species with age structure

We model an age-structured population feeding on an abiotic resource by combining the Gurtin-MacCamy [Math. Biosci. 43 (1979) 199] approach with a standard chemostat model. Limit cycles arise by Hopf bifurcations at low values of the chemostat dilution rate, even for simple maternity functions for which the original Gurtin-MacCamy model has no oscillatory solutions. We find the exact location in parameter space of the Hopf bifurcations for special cases of our model. The onset of cycling is largely independent of both the form of the resource uptake function and the shape of the maternity function.     

Uncertainty and predictability in population dynamics of a bitrophic ecological model: Mixed-mode oscillations,bistability and sensitivity to parameters

We consider a two-trophic ecological model comprising of two predators competing for their common prey. We cast the model into the framework of a singular perturbed system of equations in one fast variable (prey population density) and two slow variables (predator population densities), mimicking the common observation that the per-capita productivity rate decreases from bottom to top along the trophic levels in Nature. We assume that both predators exhibit Holling II functional response with one of the predators (territorial) having a density dependent mortality rate. Depending on the system parameters, the model exhibits small, intermediate and/or large fluctuations in the population densities. The large fluctuations correspond to periodic population outbreaks followed by collapses (commonly known as cycles of “boom and bust”). The small fluctuations arise due to a singular Hopf bifurcation in the system, and are ecologically more desirable. However, more interestingly, the system exhibits mixed-mode oscillations (which are concatenations of the large amplitude oscillations and the small amplitude oscillations) that indicate the adaptability of the species to prolong the time gap between successive cycles of boom and bust. Numerical simulations are carried out to demonstrate the extreme sensitivity of the system to initial conditions (chaos and bistability of limit cycles are observed) as well as to the system parameters (here we only show the sensitivity to the density dependent mortality rate of the territorial predator). This model throws light at the uncertainties in long term behaviors that are associated with a real ecological system. We show that even very small changes in the system parameters due to natural or human-induced causes can lead to a complete different ecological phenomenon, thus affecting the predictability of the density of the prey population. In this paper, we explain the mechanisms behind the irregular fluctuations in the population sizes in an attempt to understand the dynamics occurring in a natural population and also comment on the inherent uncertainties associated with the system.     

Strong intraspecific variation in genetic diversity and genetic differentiation in Daphnia magna: the effects of population turnover and population size

Theory predicts that genetic diversity and genetic differentiation may strongly vary among populations of the same species depending on population turnover and local population sizes. Yet, despite the importance of these predictions for evolutionary and conservation issues, empirical studies comparing high‐turnover and low‐turnover populations of the same species are scarce. In this study, we used Daphnia magna, a freshwater crustacean, as a model organism for such a comparison. In the southern/central part of its range, D. magna inhabits medium‐sized, stable ponds, whereas in the north, it occurs in small rock pools with strong population turnover. We found that these northern populations have a significantly lower genetic diversity and higher genetic differentiation compared to the southern/central populations. Total genetic diversity across populations was only about half and average within‐population diversity only about a third of that in southern/central populations. Moreover, an average southern population contains more genetic diversity than the whole metapopulation system in the north. We based our analyses both on silent sites and microsatellites. The similarity of our results despite the contrasting mutation rates of these markers suggests that the differences are caused by contemporary rather than by historical processes. Our findings show that variation in population turnover and population size may have a major impact on the genetic diversity and differentiation of populations, and hence may lead to differences in evolutionary processes like local adaptation, hybrid vigour and breeding system evolution in different parts of a species range.     

Density-dependent reproduction causes winter crashes in a common vole population

Common voles in western France exhibit three-year population cycles with winter crashes after large outbreaks. During the winter of 2011–2012, we monitored survival, reproduction, recruitment and population growth rate of common voles at different densities (from low to outbreak densities) in natura to better understand density dependence of demographic parameters. Between October and April, the number of animals decreased irrespective of initial density. However, the decline was more pronounced when October density was higher (loss of ≈54 % of individuals at low density and 95 % at high density). Using capture-mark-recapture models with Pradel's temporal symmetry approach, we found a negative effect of density on recruitment and reproduction. In contrast, density had a slightly positive effect on survival indicating that mortality did not drive the steeper declines in animal numbers at high density. We discuss these results in a population cycle framework, and suggest that crashes after outbreaks could reflect negative effects of density dependence on reproduction rather than changes in mortality rates.     

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     

A disease transmission model in a nonconstant population

A general SIRS disease transmission model is formulated under assumptions that the size of the population varies, the incidence rate is nonlinear, and the recovered (removed) class may also be directly reinfected. For a class of incidence functions it is shown that the model has no periodic solutions. By contrast, for a particular incidence function, a combination of analytical and numerical techniques are used to show that (for some parameters) periodic solutions can arise through homoclinic loops or saddle connections and disappear through Hopf bifurcations.Supported in part by NSERC grant A-8965, the University of Victoria Committee on Faculty Research & Travel, and the Institute for Mathematics and its Applications, Minneapolis, MN, with funds provided by NSF     

Do nomadic avian predators synchronize population fluctuations of small mammals? a field experiment

Three-to-five-year population oscillations of northern small rodents are usually synchronous over hundreds of square kilometers. This regional synchrony could be due to similarity in climatic factors, or due to nomadic predators reducing the patches of high prey density close to the average density of a larger area. We estimated avian predator and small rodent densities in 4–5 predator reduction and 4–5 control areas (c. 3 km² each) during 1989–1992 in western Finland. We studied whether nomadic avian predators concentrate at high prey density areas, and whether this decreases spatial variation in prey density. The yearly mean number of avian predator breeding territories was 0.2–1.0 in reduction areas and 3.0–8.2 in control areas. Hunting birds of prey concentrated in high prey density areas after their breeding season (August), but not necessarily during the breeding season (April to June), when they were constrained to hunt in vicinity of the nest. The experimental reduction of breeding avian predators increased variation in prey density among areas but not within areas. The difference in variation between raptor reduction and control areas was largest in the late breeding season of birds of prey, and decreased rapidly after the breeding season. These results appeared to support the hypothesis that the geographic synchrony of population cycles in small mammals may be driven by nomadic predators concentrating in high prey density areas. Predation and climatic factors apparently are complementary, rather than exclusive, factors in contributing to the synchrony.     

An SIS epidemic model with variable population size and a delay

The SIS epidemiological model has births, natural deaths, disease-related deaths and a delay corresponding to the infectious period. The thresholds for persistence, equilibria and stability are determined. The persistence of the disease combined with the disease-related deaths can cause the population size to decrease to zero, to remain finite, or to grow exponentially with a smaller growth rate constant. For some parameter values, the endemic infective-fraction equilibrium is asymptotically stable, but for other parameter values, it is unstable and a surrounding periodic solution appears by Hopf bifurcation.Research Supported in part by NSERC grant A-8965 and the University of Victoria Committee on Faculty Research & Travel     

具有抑制剂的恒化器竞争模型的分析（英文）

考虑了具有抑制剂的两种群竞争一种微生物的恒化器模型,其中吸收函数和功能反应函数都是营养的一般单调递增函数,并且一种微生物能够分泌一种对另一种微生物起致命影响的抑制剂.利用常微分方程定性理论,首先得到了平衡点的存在条件和局部渐进稳定性;然后讨论了全局渐进稳定性,以及极限环和Hopf分支的存在性.     

Natural enemy specialization and the period of population cycles

The dynamical consequences of multiple‐species interactions remain an elusive and fiercely debated topic. Recently, Murdoch and colleagues proposed a general rule for the dynamics of generalist natural enemies: when periodic, they exhibit single generation cycles (SGCs), similar to single species systems. This contrasts markedly with specialists, which tend to show classic (longer period) consumer–resource cycles. Using a well‐studied laboratory system, we show that this general rule is contradicted when we consider resource age‐structure.     

A mathematical model of a crocodilian population using delay-differential equations

The crocodilia have multiple interesting characteristics that affect their population dynamics. They are among several reptile species which exhibit temperature-dependent sex determination (TSD) in which the temperature of egg incubation determines the sex of the hatchlings. Their life parameters, specifically birth and death rates, exhibit strong age-dependence. We develop delay-differential equation (DDE) models describing the evolution of a crocodilian population. In using the delay formulation, we are able to account for both the TSD and the age-dependence of the life parameters while maintaining some analytical tractability. In our single-delay model we also find an equilibrium point and prove its local asymptotic stability. We numerically solve the different models and investigate the effects of multiple delays on the age structure of the population as well as the sex ratio of the population. For all models we obtain very strong agreement with the age structure of crocodilian population data as reported in Smith and Webb (Aust. Wild. Res. 12, 541-554, 1985). We also obtain reasonable values for the sex ratio of the simulated population.     

Model of microtine cycles caused by lethal toxins in non-preferred food plants

A recent model of microtine cycles has hypothesized that plant chemical defences can drive the precipitous decline phase, through periodic lethal toxin production (PLTP) by non-preferred plant foods. Here we enumerate possible mechanisms using a previously published model of optimal foraging by one consumer (microtine rodent) of two types of food plant (1 preferred and 1 non-preferred). Rate constants for each of the model parameters were sought from the extensive literature on vole cycles. For a range of likely values of input parameters, we evaluated model fit by applying five empirically derived criteria for cyclic behaviour. These were: cycles with a period length of 2-5 yr, peak densities of 100-350 voles per ha and trough densities of 0-25 ha(-1), ratio of peak to trough densities of 10-100, and the occurrence of a catastrophic collapse in the vole population followed by a prolonged low phase. In contrast to previous models of food-induced microtine cycles, the optimal foraging model successfully reproduced the first four criteria and the prolonged low phase. The criterion of population collapse was met if the non-preferred food began producing lethal toxins at a threshold grazing intensity, as predicted by PLTP. Fewer criteria could be met in variations on the model, in which the non-preferred food was equally as nutritious as the preferred food or was continuously toxic.     

Origin of the Koreans: a population genetic study.

A population genetic study was undertaken to investigate the origin of Koreans. Thirteen polymorphic and 7 monomorphic blood genetic markers (serum proteins and red cell enzymes) were studied in a group of 437 Koreans. Genetic distance analyses by both cluster and principal components models were performed between Koreans and eight other populations (Koreans in China, Japanese, Han Chinese, Mongolians, Zhuangs, Malays, Javanese, and Soviet Asians) on the basis of 47 alleles controlled by 15 polymorphic loci. A more detailed analysis using 65 alleles at 19 polymorphic loci was performed on six populations. Both analyses demonstrated genetic evidence of the origin of Koreans from the central Asian Mongolians. Further, the Koreans are more closely related to the Japanese and quite distant from the Chinese. The above evidence of the origin of Koreans fits well with the ethnohistoric account of the origin of Koreans and the Korean language. The minority Koreans in China also maintained their genetic identity.     

Anti-control of periodic firing in HR model in the aspects of position,amplitude and frequency

This paper proposes a novel controller to control position, amplitude and frequency of periodic firing activity in Hindmarsh–Rose model based on Hopf bifurcation theory which is composed of linear control gain and nonlinear control gain. First, we select the activation of the fast ion channel as control parameter. Based on explicit criterion of Hopf bifurcation, a series of conditions are obtained to derive the linear gains of controller responsible for control of the location where the periodic firing activity occurs. Then, based on the control parameter, a series of conditions are obtained to derive the nonlinear gains of controller responsible for controlling the amplitude and frequency of periodic firing activity by using center manifold and normal form. Finally, the numerical experiments show that our controller can make the periodic firing activity occur at designed value and control the amplitude and frequency of periodic firing activity by adjusting nonlinear control gain of controller.