An Inhomogeneous Markov Model to Fit the Migration Process

In this paper we study the migration process considering an inhomogeneous Markov model. This is a certain condition to investigate age-dependent population distributions, where the transition probabilities are not constant. We consider also a death process for a population alive in a region at age t and, as a result of this, combined transition probabilities between the states of the concerning Markov chain. The model has non-stationary distribution for t →∞, because the condition of ergodicity does not hold.     

Cell cycle progression

In this paper we consider cell cycle models for which the transition operator for the evolution of birth mass density is a simple, linear dynamical system with a stochastic perturbation. The convolution model for a birth mass distribution is presented. Density functions of birth mass and tail probabilities in n-th generation are calculated by a saddle-point approximation method. With these probabilities, representing the probability of exceeding an acceptable mass value, we have more control over pathological growth. A computer simulation is presented for cell proliferation in the age-dependent cell cycle model. The simulation takes into account the fact that the age-dependent model with a linear growth is a simple linear dynamical system with an additive stochastic perturbation. The simulated data as well as the experimental data (generation times for mouse L) are fitted by the proposed convolution model.     

The two-locus ancestral graph in a subdivided population: convergence as the number of demes grows in the island model

We study the ancestral recombination graph for a pair of sites in a geographically structured population. In particular, we consider the limiting behavior of the graph, under Wrights island model, as the number of subpopulations, or demes, goes to infinity. After an instantaneous sample-size adjustment, the graph becomes identical to the two-locus graph in an unstructured population, but with a time scale that depends on the migration rate and the deme size. Interestingly, when migration is gametic, this rescaling of time increases the population mutation rate but does not affect the population recombination rate. We compare this to the case of a partially-selfing population, in which both mutation and recombination depend on the selfing rate. Our result for gametic migration holds both for finite-sized demes, and in the limit as the deme size goes to infinity. However, when migration occurs during the diploid phase of the life cycle and demes are finite in size, the population recombination rate does depend on the migration rate, in a way that is reminiscent of partial selfing. Simulations imply that convergence to a rescaled panmictic ancestral recombination graph occurs for any number of sites as the number of demes approaches infinity.Send offprint request to: Sabin LessardS. Lessard was supported by grants from the Natural Sciences and Research Council of Canada, the Fonds Québécois de la Recherche sur la Nature et les Technologies, and the Université de Montréal.J. Wakeley was supported by a Career Award (DEB-0133760) and by a grant (DEB-9815367) from the National Science Foundation.     

Hopf bifurcation in a structured population model for the sexual phase of monogonont rotifers

 We are studying a population of monogonont rotifers in the context of non-linear age-dependent models. In the sexual phase of their reproductive cycle we consider the population structured by age, and composed of three subclasses: virgin mictic females, mated mictic females, and haploid males. The model system has a unique stationary population density which is stable as long as a parameter, related to male-female encounter rate, remains below a critical value. When the parameter increases beyond this critical value, the stationary solution becomes unstable and a stable limit cycle (isolated periodic orbit) appears. The occurrence of this supercritical Hopf bifurcation is shown analytically. Received: 2 August 2001 / Revised version: 3 January 2002 / Published online: 26 June 2002     

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.     

Effects of density-dependent migrations on stability of a two-patch predator-prey model

We consider a predator-prey model in a two-patch environment and assume that migration between patches is faster than prey growth, predator mortality and predator-prey interactions. Prey (resp. predator) migration rates are considered to be predator (resp. prey) density-dependent. Prey leave a patch at a migration rate proportional to the local predator density. Predators leave a patch at a migration rate inversely proportional to local prey population density. Taking advantage of the two different time scales, we use aggregation methods to obtain a reduced (aggregated) model governing the total prey and predator densities. First, we show that for a large class of density-dependent migration rules for predators and prey there exists a unique and stable equilibrium for migration. Second, a numerical bifurcation analysis is presented. We show that bifurcation diagrams obtained from the complete and aggregated models are consistent with each other for reasonable values of the ratio between the two time scales, fast for migration and slow for local demography. Our results show that, under some particular conditions, the density dependence of migrations can generate a limit cycle. Also a co-dim two Bautin bifurcation point is observed in some range of migration parameters and this implies that bistability of an equilibrium and limit cycle is possible.     

Screening assays for heart function mutants in Drosophila

The rapid life cycle and genetic tractability of Drosophila make it an ideal organism for large-scale genetic screens. Here we describe a novel assay for pupal heart rate and rhythmicity as well as techniques to measure adult cardiac stress response. These assays can be powerfully combined to concurrently screen for both mutations affecting cardiac function and mutations affecting the age-dependent decline in adult cardiac stress response. Mutations identified in such screens have the potential to contribute greatly to the understanding of both congenital heart disease and the regulation of age-dependent decline in cardiac function in the human population.     

Theoretical insight into three disease-related benefits of migration

Migration (seasonal round-trip movement across relatively large distances) is common within the animal kingdom. This behaviour often incurs extreme costs in terms of time, energy, and/or survival. Climate, food, predation, and breeding are typically suggested as factors favouring the evolution of migration. Although disease regulation has also been considered, few studies consider it as the primary selective pressure for migration. Our aim was to determine, theoretically, under what conditions migration could reduce the long-term disease prevalence within a population, assuming the only benefits of migration are infection-related. We created two mathematical models, one where the population migrates annually and one where the entire population remains on the breeding ground year-round. In each we simulated disease transmission (frequency-dependent and density-dependent) and quantified eventual disease prevalence. In the migration model we varied the time spent migrating, disease-related migration mortality, and the overall migration mortality. When we compared results from the two models, we found that migration generally lowered disease prevalence. We found a population was healthier if it: (1) spent more time migrating (assuming no disease transmission during migration), (2) had higher disease-induced migration mortality, and (3) had an overall higher mortality when migrating (compared to not migrating). These results provide support for two previously proposed mechanisms by which migration can reduce disease prevalence (migratory escape and migratory cull), and also demonstrate that non-selective mortality during migration is a third mechanism. Our findings indicate that migration may be evolutionarily advantageous even if the only migratory benefit is disease control.     

Stable controls in age-dependent population dynamics

The classical age-dependent population model is considered, in which mortality depends on total population and fertility is age-dependent. It is shown that in general such systems are not completely controllable with respect to a control variable in the mortality function, but that in certain circumstances a suitable control can be found to hold the population at a specified level.     

EXTINCTION AND RECOLONIZATION: THEIR EFFECTS ON THE GENETIC DIFFERENTIATION OF LOCAL POPULATIONS

In this paper, we use a model by Slatkin (1977) to investigate the genetic effects of extinction and recolonization for a species whose population structure consists of an array of local demes with some migration among them. In particular, we consider the conditions under which extinction and recolonization might enhance or diminish gene flow and increase or decrease the rate of genetic differentiation relative to the static case with no extinctions. We explicitly take into account the age-structure that is established within the array of populations by the extinction and colonization process. We also consider two different models of the colonization process, the so-called “migrant pool” and “propagule pool” models. Our theoretical studies indicate that the genetic effects of extinction and colonization depend upon the relative magnitudes of K, the number of individuals founding new colonies, and 2Nm, twice the number of migrants moving into extant populations. We find that these genetic effects are surprisingly insensitive to the extinction rate. We conclude that, in order to assess the genetic effects of the population dynamics, we must first answer an important empirical question that is essentially ecological: is colonization a behavior distinct from migration?     

How do cross-migration models arise?

In this paper we present a general method to derive spatio-temporal population models mechanistically. We consider a system of multiple species living in a patchy habitat in which the local population of each species consists of some behavioural groups. We then formulate a continuous-time model where a small positive parameter is present, measuring the time scale of behavioural transitions relative to that of giving birth, death and migration among patches. By the singular perturbation method the model is reduced to a lower dimensional one in which the migration terms are, in general, nonlinear and related to the reaction terms describing the local dynamics. Two examples demonstrating the emergence of cross-migration models, i.e., the models in which the per-capita migration rate of one species depends on the density of some other species, are given.     

General population growth models with Allee effects in a random environment

Allee effects on population growth are quite common in nature, usually studied through deterministic models with a specific growth rate function.In order to seek the qualitative behaviour of populations induced by such effects, one should avoid model-specific behaviours. So, we use as a basis a general deterministic model, i.e. a model with a general growth rate function, to which we add the effect on the growth rate of the random fluctuations in environmental conditions. The resulting model is the general stochastic differential equation (SDE) model that we propose here.We consider two possible cases, weak Allee effects and strong Allee effects, which lead to different qualitative behaviours of the model.We will study the model properties for both cases in terms of existence and uniqueness of the solution, extinction and stationary behaviour of the population. The two cases will be compared with each other and with the general density-dependent SDE model without Allee effects.We then consider as an example the particular case of the classic logistic model and an Allee effect version of it.     

Evolutionarily stable dispersal strategies

Using the idea that life-history parameters are subject to natural selection and should approach values that are stable optima, with the population immune to invasion by mutant individuals, we derive an analytic expression for the evolutionarily stable dispersal rate in a stochastic island model with random site extinction. The results provide interesting contrasts between three different optimization criteria: species survival, individual fitness and gene fitness. We also consider the effects of sexual reproduction, and of localized migration (stepping-stone structure).     

Can a Species Keep Pace with a Shifting Climate?

Consider a patch of favorable habitat surrounded by unfavorable habitat and assume that due to a shifting climate, the patch moves with a fixed speed in a one-dimensional universe. Let the patch be inhabited by a population of individuals that reproduce, disperse, and die. Will the population persist? How does the answer depend on the length of the patch, the speed of movement of the patch, the net population growth rate under constant conditions, and the mobility of the individuals? We will answer these questions in the context of a simple dynamic profile model that incorporates climate shift, population dynamics, and migration. The model takes the form of a growth-diffusion equation. We first consider a special case and derive an explicit condition by glueing phase portraits. Then we establish a strict qualitative dichotomy for a large class of models by way of rigorous PDE methods, in particular the maximum principle. The results show that mobility can both reduce and enhance the ability to track climate change that a narrow range can severely reduce this ability and that population range and total population size can both increase and decrease under a moving climate. It is also shown that range shift may be easier to detect at the expanding front, simply because it is considerably steeper than the retreating back.     

Population and evolutionary dynamics in spatially structured seasonally varying environments

Increasingly imperative objectives in ecology are to understand and forecast population dynamic and evolutionary responses to seasonal environmental variation and change. Such population and evolutionary dynamics result from immediate and lagged responses of all key life‐history traits, and resulting demographic rates that affect population growth rate, to seasonal environmental conditions and population density. However, existing population dynamic and eco‐evolutionary theory and models have not yet fully encompassed within‐individual and among‐individual variation, covariation, structure and heterogeneity, and ongoing evolution, in a critical life‐history trait that allows individuals to respond to seasonal environmental conditions: seasonal migration. Meanwhile, empirical studies aided by new animal‐tracking technologies are increasingly demonstrating substantial within‐population variation in the occurrence and form of migration versus year‐round residence, generating diverse forms of ‘partial migration’ spanning diverse species, habitats and spatial scales. Such partially migratory systems form a continuum between the extreme scenarios of full migration and full year‐round residence, and are commonplace in nature. Here, we first review basic scenarios of partial migration and associated models designed to identify conditions that facilitate the maintenance of migratory polymorphism. We highlight that such models have been fundamental to the development of partial migration theory, but are spatially and demographically simplistic compared to the rich bodies of population dynamic theory and models that consider spatially structured populations with dispersal but no migration, or consider populations experiencing strong seasonality and full obligate migration. Second, to provide an overarching conceptual framework for spatio‐temporal population dynamics, we define a ‘partially migratory meta‐population’ system as a spatially structured set of locations that can be occupied by different sets of resident and migrant individuals in different seasons, and where locations that can support reproduction can also be linked by dispersal. We outline key forms of within‐individual and among‐individual variation and structure in migration that could arise within such systems and interact with variation in individual survival, reproduction and dispersal to create complex population dynamics and evolutionary responses across locations, seasons, years and generations. Third, we review approaches by which population dynamic and eco‐evolutionary models could be developed to test hypotheses regarding the dynamics and persistence of partially migratory meta‐populations given diverse forms of seasonal environmental variation and change, and to forecast system‐specific dynamics. To demonstrate one such approach, we use an evolutionary individual‐based model to illustrate that multiple forms of partial migration can readily co‐exist in a simple spatially structured landscape. Finally, we summarise recent empirical studies that demonstrate key components of demographic structure in partial migration, and demonstrate diverse associations with reproduction and survival. We thereby identify key theoretical and empirical knowledge gaps that remain, and consider multiple complementary approaches by which these gaps can be filled in order to elucidate population dynamic and eco‐evolutionary responses to spatio‐temporal seasonal environmental variation and change.     

Efficient management for the Hokkaido population of sika deer Cervus nippon in Japan: accounting for migration and management cost

We evaluated the fourth stage of the “Conservation and Management Plan for Sika Deer (Cervus nippon) in Hokkaido, Japan (CMPS4)”, focusing on its cost-effectiveness and sika deer migration between two management areas of eastern and western Hokkaido. To clarify these factors, we constructed a stochastic matrix population model that accounts for deer migration and several uncertainties. We assumed four different budget scenarios and simple rules regarding nuisance control, and simulated four alternative management strategies. In the stochastic simulation, we calculated the probability of successfully satisfying the population target given by the CMPS4, an average total actual management cost, and a cost-effectiveness index given four budget conditions of migration rate and budget allocation ratio. The simulation results suggest the following. First, the current management budget is so small that the probability of successfully satisfying the population targets in both areas is only 26–30 %. If the total budget remains small, it should be almost entirely invested in one area, regardless of migration situation, to maximize the probability of successfully meeting the target density in at least that area. However, these probabilities of success decrease with greater migration rate. Second, when the government invests more of its budget in the early management stage, the expected total actual cost decreases and the probability of management success increases. These findings represent cost-effective management strategies for satisfying the CMPS4 targets.     

Evolution of social versus individual learning in an infinite island model

We model the evolution of learning in a population composed of infinitely many, finite-sized islands connected by migration. We assume that there are two discrete strategies, social and individual learning, and that the environment is spatially homogeneous but varies temporally in a periodic or stochastic manner. Using a population-genetic approximation technique, we derive a mathematical condition for the two strategies to coexist stably and the equilibrium frequency of social learners under stable coexistence. Analytical and numerical results both reveal that social learners are favored when island size is large or migration rate between islands is high, suggesting that spatial subdivision disfavors social learners. We also show that the average fecundity of the population under stable coexistence of the two strategies is in general lower than that in the absence of social learners and is minimized at an intermediate migration rate.     

Environmental heterogeneity enhances clonal interference

Clonal interference (CI) is a phenomenon that may be important in several asexual microbes. It occurs when population sizes are large and mutation rates to new beneficial alleles are of significant magnitude. Here we explore the role of gene flow and spatial heterogeneity in selection strength in the adaptation of asexuals. We consider a subdivided population of individuals that are adapting, through new beneficial mutations, and that migrate between different patches. The fitness effect of each mutation depends on the patch and all mutations considered are assumed to be unconditionally beneficial. We find that spatial variation in selection pressure affects the rate of adaptive evolution and its qualitative effects depend on the level of gene flow. In particular, we find that both low migration and high levels of heterogeneity lead to enhanced CI. In contrast, for high levels of migration the rate of fixation of adaptive mutations is higher when environmental heterogeneity is present. In addition, we observe that the level of fitness variation is higher and simultaneous fixation of multiple mutations tends to occur in the regime of low migration rates and high heterogeneity.     

Effective size in management and conservation of subdivided populations

A numerical method for computing the eigenvalue variance effective size of a subdivided population connected by any fixed pattern of migration is described. Using specific examples it is shown that total effective size of a subdivided population can become less than the sum of the subpopulation sizes as a result of directionalities in the pattern of migration. For an extension of the model with threshold harvesting and local deterministic logistic population dynamic we consider the problem of maximizing the total harvesting yield with constraints on the total effective size. For some simple source-sink systems and more complicated population structures where subpopulations differ in their degree of isolation, it is shown to be optimal, for a given total effective size, to raise the harvesting thresholds relatively more in small and in isolated populations. Finally, we show how the method applies to populations which are supplemented, either intentionally or unintentionally. It is shown that the total effective size can be reduced by several orders of magnitude if the captive component of a population is much smaller than the wild component, even with symmetric backward migration.     

Predictions of Taylor's power law, density dependence and pink noise from a neutrally modeled time series

There has recently been increasing interest in neutral models of biodiversity and their ability to reproduce the patterns observed in nature, such as species abundance distributions. Here we investigate the ability of a neutral model to predict phenomena observed in single-population time series, a study complementary to most existing work that concentrates on snapshots in time of the whole community. We consider tests for density dependence, the dominant frequencies of population fluctuation (spectral density) and a relationship between the mean and variance of a fluctuating population (Taylor's power law). We simulated an archipelago model of a set of interconnected local communities with variable mortality rate, migration rate, speciation rate, size of local community and number of local communities. Our spectral analysis showed ‘pink noise’: a departure from a standard random walk dynamics in favor of the higher frequency fluctuations which is partly consistent with empirical data. We detected density dependence in local community time series but not in metacommunity time series. The slope of the Taylor's power law in the model was similar to the slopes observed in natural populations, but the fit to the power law was worse. Our observations of pink noise and density dependence can be attributed to the presence of an upper limit to community sizes and to the effect of migration which distorts temporal autocorrelation in local time series. We conclude that some of the phenomena observed in natural time series can emerge from neutral processes, as a result of random zero-sum birth, death and migration. This suggests the neutral model would be a parsimonious null model for future studies of time series data.