The effect of emigration and immigration on the dynamics of a discrete-generation population

The recent paper of Sinha and Parthasarathy investigated the effect of modifying the Ricker and logistic population models to simulate the effects of immigration to, and emigration from, the population. Immigration of a fixed number of individuals was shown to reduce the probability of observing chaos in the Ricker model but not the logistic one. Here, isocline analysis is used to investigate why these effects occur. The stabilization effect for the Ricker equation occurs over a wide range of values of the immigration parameter. There are no values of the parameter, however, which increase the stability of the logistic equation substantially. In contrast density-dependent immigration is found to destabilize both the Ricker and logistic models. Density-dependent emigration serves to reduce the propensity of both models to exhibit chaos.     

On linear perturbations of the Ricker model

A class of linearly perturbed discrete-time single species scramble competition models, like the Ricker map, is considered. Perturbations can be of both recruitment and harvesting types. Stability (bistability) is considered for models, where parameters of the map do not depend on time. For models with recruitment, the result is in accordance with Levin and May conjecture [S.A. Levin, R.M. May, A note on difference delay equations, Theor. Pop. Biol. 9 (1976) 178]: the local stability of the positive equilibrium implies its global stability. For intrinsic growth rate r-->infinity the way to chaos is broken down to get extinction of population for the depletion case and to establish a stable two-cycle period for models with immigration. The latter behaviour is also studied for models with random discrete constant perturbations of recruitment type. Extinction, persistence and existence of periodic solutions are studied for the perturbed Ricker model with time-dependent parameters.     

Behaviour of simple population models under ecological processes

The two most popular and extensively-used discrete models of population growth display the generic bifurcation structure of a hierarchy of period-doubling sequence to chaos with increasing growth rates. In this paper we show that these two models, though they belong to a general class of one-dimensional maps, show very different dynamics when important ecological processes such as immigration and emigration/depletion, are considered. It is important that ecologists recognize the differences between these models before using them to describe their data—or develop optimization strategies—based on these models.     

褐飞虱持续暴发成因——以江西泰和县为例

为了明确泰和县近年来晚稻田褐飞虱Nilaparvata lugens(Stl)连年暴发的规律。通过分析泰和县2008—2013年褐飞虱的田间调查及灯诱数据,并结合气象条件,研究了泰和县晚稻田褐飞虱暴发的主要影响因素。结果表明:(1)危害早稻田的褐飞虱以2、3代为主,其田间种群数量的多少主要受春季迁入虫量的限制,本地早稻收割后的滞留虫源对晚稻种群数量的影响很小。(2)危害晚稻田的褐飞虱以5、6代为主,后期迁入虫量、秋季温度以及晚稻田初期外来迁入虫量是影响晚稻田后期暴发的关键因子。(3)不同年份间晚稻田褐飞虱暴发类型不同。2008和2009年属后期迁入虫量与"暖秋"共同影响类型;2010年是"暖秋"影响类型;2011和2012年是后期迁入虫量影响类型;2013年是晚稻田早期迁入虫量积累影响类型。后期迁入虫量、秋季温度以及晚稻田早期迁入虫量是导致泰和县晚稻田褐飞虱连年暴发的主要原因。这些结果为褐飞虱暴发规律的阐明及可持续治理提供了理论依据和参考价值。     

Density-dependent immigration promotes population stability in a long-distance migratory bird

The spatial structure of populations determines the relative importance of reproduction, survival and movement on population dynamics. However, the mechanisms by which local individuals and immigrants interact and the subsequent effects of immigrants on productivity are poorly known. We developed an integrated population model (IPM) to study the extent and consequences of immigration on the dynamics of a neotropical migrant (American redstart, Setophaga ruticilla) over an 11-year period in Ontario, Canada. New immigrants represented the majority of the study population each year with higher immigration rates for males than females and for first-year breeders than breeders in their second year or older. Immigration was negatively density dependent, with immigrants replacing previously established breeders in a compensatory manner following their death or emigration. Because of the tradeoff between immigration and apparent survival, neither had a strong influence on population growth and reproductive output was most strongly correlated with a change in abundance between years. However, if immigration ceased, the study population would become locally extinct within 7 years and thus immigrants were essential for local population persistence. We found no evidence for reduced breeding success when immigrants represented a higher proportion of the study population. Our research highlights the importance of movement in the stability of open populations and the strong correlation between the fates of local breeders and the number of immigrants entering the population. We recommend the use of IPMs to address the spatial scale over which immigration occurs and how different scales influence its contribution to population dynamics.     

Generalized aphid population growth models with immigration and cumulative-size dependent dynamics

Mechanistic models in which the per-capita death rate of a population is proportional to cumulative past size have been shown to describe adequately the population size curves for a number of aphid species. Such previous cumulative-sized based models have not included immigration. The inclusion of immigration is suggested biologically as local aphid populations are initiated by migration of winged aphids and as reproduction is temperature-dependent. This paper investigates two models with constant immigration, one with continuous immigration and the other with restricted immigration. Cases of the latter are relatively simple to fit to data. The results from these two immigration models are compared for data sets on the mustard aphid in India.     

Population Floors and the Persistence of Chaos in Ecological Models

Chaotic dynamics have been observed in a wide range of population models. Here we describe the effects of perturbing several of these models so as to introduce a non-zero minimum population size. This perturbation generally reduces the likelihood of observing chaos, in both discrete and continuous time models. The extent of this effect depends on whether chaos is generated through period-doubling, quasiperiodicity, or intermittence. Chaos reached via the quasiperiodic route is more robust against the perturbation than period-doubling chaos, whilst the inclusion of a population floor in a model exhibiting intermittent chaos may increase the frequency of population bursts although these become non-chaotic.     

Density-dependent birth rate, birth pulses and their population dynamic consequences

 In most models of population dynamics, increases in population due to birth are assumed to be time-independent, but many species reproduce only during a single period of the year. We propose a single-species model with stage structure for the dynamics in a wild animal population for which births occur in a single pulse once per time period. Using the discrete dynamical system determined by the stroboscopic map, we obtain an exact periodic solution of systems which are with Ricker functions or Beverton-Holt functions, and obtain the threshold conditions for their stability. Above this threshold, there is a characteristic sequence of bifurcations, leading to chaotic dynamics, which implies that the dynamical behaviors of the single species model with birth pulses are very complex, including small-amplitude annual oscillations, large-amplitude multi-annual cycles, and chaos. This suggests that birth pulse, in effect, provides a natural period or cyclicity that allows for a period-doubling route to chaos. Received: 13 June 2001 / Revised version: 7 September 2001 / Published online: 8 February 2002     

Nestboxes and immigration drive the growth of an urban Peregrine Falcon Falco peregrinus population

Drivers of wildlife population dynamics are generally numerous and interacting. Some of these drivers may impact demographic processes that are difficult to estimate, such as immigration into the focal population. Populations may furthermore be small and subject to demographic stochasticity. All of these factors contribute to blur the causal relationship between past management action and current population trends. The urban Peregrine Falcon Falco peregrinus population in Cape Town, South Africa, increased from three pairs in 1997 to 18 pairs in 2010. Nestboxes were installed over this period to manage the interface between new urban pairs of Falcons and the human users of colonized buildings, and incidentally to improve breeding success. We used integrated population models (IPMs) formally to combine information from a capture–mark–recapture study, monitoring of reproductive success and counts of population size. As all local demographic processes were directly observed, the IPM approach also allowed us to estimate immigration by difference. The provision of nestboxes, as a possible stimulant of population growth, improved breeding success and accounted for an estimated 3–26% of the population increase. The most important driver of growth, however, was immigration. Despite low sample sizes, the IPM approach allowed us to obtain relatively precise estimates of the population‐level impact of nestbox deployment. The goal of conservation interventions is often to increase population size, so the effectiveness of such interventions should ideally be assessed at the population level. IPMs are powerful tools in this context for combining demographic information that may be limited due to small population size or practical constraints on monitoring. Our study quantitatively documented both the immigration process that led to growth of a small population and the effect of a management action that helped the process.     

Evolutionary consequences of asymmetric dispersal rates

We study the consequences of asymmetric dispersal rates (e.g., due to wind or current) for adaptive evolution in a system of two habitat patches. Asymmetric dispersal rates can lead to overcrowding of the "downstream" habitat, resulting in a source-sink population structure in the absence of intrinsic quality differences between habitats or can even cause an intrinsically better habitat to function as a sink. Source-sink population structure due to asymmetric dispersal rates has similar consequences for adaptive evolution as a source-sink structure due to habitat quality differences: natural selection tends to be biased toward the source habitat. We demonstrate this for two models of adaptive evolution: invasion of a rare allele that improves fitness in one habitat but reduces it in the other and antagonistic selection on a quantitative trait determined by five additive loci. If a habitat can sustain a population without immigration, the conditions for adaptation to that habitat are most favorable if there is little or no immigration from the other habitat; the influence of emigration depends on the magnitude of the allelic effects involved and other parameters. If, however, the population is initially unable to persist in a given habitat without immigration, our model predicts that the population will be most likely to adapt to that habitat if the dispersal rates in both directions are high. Our results highlight the general message that the effect of gene flow upon local adaptation should depend profoundly on the demographic context of selection.     

Dental perspectives on the population history of Southeast Asia

This article uses metric and nonmetric dental data to test the "two-layer" or immigration hypothesis whereby Southeast Asia was initially occupied by an "Australo-Melanesian" population that later underwent substantial genetic admixture with East Asian immigrants associated with the spread of agriculture from the Neolithic period onwards. We examined teeth from 4,002 individuals comprising 42 prehistoric and historic samples from East Asia, Southeast Asia, Australia, and Melanesia. For the odontometric analysis, dental size proportions were compared using factor analysis and Q-mode correlation coefficients, and overall tooth size was also compared between population samples. Nonmetric population affinities were estimated by Smith's distances, using the frequencies of 16 tooth traits. The results of both the metric and nonmetric analyses demonstrate close affinities between recent Australo-Melanesian samples and samples representing early Southeast Asia, such as the Early to Middle Holocene series from Vietnam, Malaysia, and Flores. In contrast, the dental characteristics of most modern Southeast Asians exhibit a mixture of traits associated with East Asians and Australo-Melanesians, suggesting that these populations were genetically influenced by immigrants from East Asia. East Asian metric and/or nonmetric traits are also found in some prehistoric samples from Southeast Asia such as Ban Kao (Thailand), implying that immigration probably began in the early Neolithic. Much clearer influence of East Asian immigration was found in Early Metal Age Vietnamese and Sulawesi samples. Although the results of this study are consistent with the immigration hypothesis, analysis of additional Neolithic samples is needed to determine the exact timing of population dispersals into Southeast Asia.     

Can noise induce chaos?

An important component of the mathematical definition of chaos is sensitivity to initial conditions. Sensitivity to initial conditions is usually measured in a deterministic model by the dominant Lyapunov exponent (LE), with chaos indicated by a positive LE. The sensitivity measure has been extended to stochastic models; however, it is possible for the stochastic Lyapunov exponent (SLE) to be positive when the LE of the underlying deterministic model is negative, and vice versa. This occurs because the LE is a long-term average over the deterministic attractor while the SLE is the long-term average over the stationary probability distribution. The property of sensitivity to initial conditions, uniquely associated with chaotic dynamics in deterministic systems, is widespread in stochastic systems because of time spent near repelling invariant sets (such as unstable equilibria and unstable cycles). Such sensitivity is due to a mechanism fundamentally different from deterministic chaos. Positive SLE's should therefore not be viewed as a hallmark of chaos. We develop examples of ecological population models in which contradictory LE and SLE values lead to confusion about whether or not the population fluctuations are primarily the result of chaotic dynamics. We suggest that "chaos" should retain its deterministic definition in light of the origins and spirit of the topic in ecology. While a stochastic system cannot then strictly be chaotic, chaotic dynamics can be revealed in stochastic systems through the strong influence of underlying deterministic chaotic invariant sets.     

Seroprevalence of chagas infection in the donor population

We retrospectively calculated the prevalence and epidemiologic characteristics of Chagas infection in the New York blood donor population over three years utilizing the New York Blood Center's database of the New York metropolitan area donor population. Seventy Trypanosoma cruzi positive donors were identified from among 876,614 donors over a 3-year period, giving an adjusted prevalence of 0.0083%, with 0.0080% in 2007, 0.0073% in 2008, and 0.0097% in 2009. When filtered only for self-described "Hispanic/Latino" donors, there were 52 Chagas positive donors in that 3-year period (among 105,122 self-described Hispanic donors) with an adjusted prevalence of 0.052%, with 0.055% in 2007, 0.047% in 2008, and 0.053% in 2009. In conclusion, we found a persistent population of patients with Chagas infection in the New York metropolitan area donor population. There was geographic localization of cases which aligned with Latin American immigration clusters.     

Effect of immigration on a chaotic insect population

Recently, the most convincing evidence of complex dynamics and chaos in biological populations has been presented for Tribolium castaneaum, a classic laboratory model insect. In this note, the robustness of this system is investigated and a constant immigration term is added to the adult population equation. It has been found that such perturbation to the model can either have a complicating effect (when the isolated system is periodic) or a simplifying one (when the system is chaotic in isolation).     

Routes to chaos in a model of a bursting neuron.

Chaotic regimens have been observed experimentally in neurons as well as in deterministic neuronal models. The R15 bursting cell in the abdominal ganglion of Aplysia has been the subject of extensive mathematical modeling. Previously, the model of Plant and Kim has been shown to exhibit both bursting and beating modes of electrical activity. In this report, we demonstrate (a) that a chaotic regime exists between the bursting and beating modes of the model, and (b) that the model approaches chaos from both modes by a period doubling cascade. The bifurcation parameter employed is the external stimulus current. In addition to the period doubling observed in the model-generated trajectories, a period three "window" was observed, power spectra that demonstrate the approaches to chaos were generated, and the Lyaponov exponents and the fractal dimension of the chaotic attractors were calculated. Chaotic regimes have been observed in several similar models, which suggests that they are a general characteristic of cells that exhibit both bursting and beating modes.     

Population dynamic models of the spread of Wolbachia

Wolbachia are endosymbionts that are found in many insect species and can spread rapidly when introduced into a naive host population. Most Wolbachia spread when their infection frequency exceeds a threshold normally calculated using purely population genetic models. However, spread may also depend on the population dynamics of the insect host. We develop models to explore interactions between host population dynamics and Wolbachia infection frequency for an age-structured insect population regulated by larval density dependence. We first derive a new expression for the threshold frequency that extends existing theory to incorporate important details of the insect's life history. In the presence of immigration and emigration, the threshold also depends on the form of density-dependent regulation. We show how the type of immigration (constant or pulsed) and the temporal dynamics of the host population can strongly affect the spread of Wolbachia. The results help understand the natural dynamics of Wolbachia infections and aid the design of programs to introduce Wolbachia to control insects that are disease vectors or pests.     

Structured models of metapopulation dynamics

I develop models of metapopulation dynamics that describe changes in the numbers of individuals within patches. These models are analogous to structured population models, with patches playing the role of individuals. Single species models which do not include the effect of immigration on local population dynamics of occupied patches typically lead to a unique equilibrium. The models can be used to study the distributions of numbers of individuals among patches, showing that both metapopulations with local outbreaks and metapopulations without outbreaks can occur in systems with no underlying environmental variability. Distributions of local population sizes (in occupied patches) can vary independently of the total population size, so both patterns of distributions of local population sizes are compatible with either rare or common species. Models which include the effect of immigration on local population dynamics can lead to two positive equilibria, one stable and one unstable, the latter representing a threshold between regional extinction and persistence.     

Immigration can destabilize tri-trophic interactions: implications for conservation of top predators

Top predators often have large home ranges and thus are especially vulnerable to habitat loss and fragmentation. Increasing connectance among habitat patches is therefore a common conservation strategy, based in part on models showing that increased migration between subpopulations can reduce vulnerability arising from population isolation. Although three-dimensional models are appropriate for exploring consequences to top predators, the effects of immigration on tri-trophic interactions have rarely been considered. To explore the effects of immigration on the equilibrium abundances of top predators, we studied the effects of immigration in the three-dimensional Rosenzweig-MacArthur model. To investigate the stability of the top predator equilibrium, we used MATCONT to perform a bifurcation analysis. For some combinations of model parameters with low rates of top predator immigration, population trajectories spiral towards a stable focus. Holding other parameters constant, as immigration rate is increased, a supercritical Hopf bifurcation results in a stable limit cycle and thus top predator populations that cycle between high and low abundances. Furthermore, bistability arises as immigration of the intermediate predator is increased. In this case, top predators may exist at relatively low abundances while prey become extinct, or for other initial conditions, the relatively higher top predator abundance controls intermediate predators allowing for non-zero prey population abundance and increased diversity. Thus, our results reveal one of two outcomes when immigration is added to the model. First, over some range of top predator immigration rates, population abundance cycles between high and low values, making extinction from the trough of such cycles more likely than otherwise. Second, for relatively higher intermediate predator migration rates, top predators may exist at low values in a truncated system with impoverished diversity, again with extinction more likely.     

Immigration rates in fragmented landscapes--empirical evidence for the importance of habitat amount for species persistence

Background

The total amount of native vegetation is an important property of fragmented landscapes and is known to exert a strong influence on population and metapopulation dynamics. As the relationship between habitat loss and local patch and gap characteristics is strongly non-linear, theoretical models predict that immigration rates should decrease dramatically at low levels of remaining native vegetation cover, leading to patch-area effects and the existence of species extinction thresholds across fragmented landscapes with different proportions of remaining native vegetation. Although empirical patterns of species distribution and richness give support to these models, direct measurements of immigration rates across fragmented landscapes are still lacking.

Methodology/Principal Findings

Using the Brazilian Atlantic forest marsupial Gray Slender Mouse Opossum (Marmosops incanus) as a model species and estimating demographic parameters of populations in patches situated in three landscapes differing in the total amount of remaining forest, we tested the hypotheses that patch-area effects on population density are apparent only at intermediate levels of forest cover, and that immigration rates into forest patches are defined primarily by landscape context surrounding patches. As expected, we observed a positive patch-area effect on M. incanus density only within the landscape with intermediate forest cover. Density was independent of patch size in the most forested landscape and the species was absent from the most deforested landscape. Specifically, the mean estimated numbers of immigrants into small patches were lower in the landscape with intermediate forest cover compared to the most forested landscape.

Conclusions/Significance

Our results reveal the crucial importance of the total amount of remaining native vegetation for species persistence in fragmented landscapes, and specifically as to the role of variable immigration rates in providing the underlying mechanism that drives both patch-area effects and species extinction thresholds.     

A combined inverse method and multivariate approach for exploring population trends of Florida manatees

Given the expense and time required to monitor marine mammal populations effectively, approaches that fully exploit the resulting data certainly are warranted. We employed a two‐step modeling approach to estimate key demographic parameters, including immigration, from aerial surveys of manatees (Trichechus manatus latirostris) in the Northwest management unit of Florida. Abundances of adults and calves were predicted by multivariate adaptive regression spline (MARS) models, after accounting for heterogeneous detection rates caused by variable environmental conditions. The resulting predictions were incorporated into a stage‐structured, deterministic model that used an inverse method to estimate parameters with and without immigration. The model without immigration estimated mean survival probabilities of 0.966, 0.923, and 0.794 for adults, subadults and calves, respectively, with a per capita reproductive rate of 0.135. These parameter estimates yielded an overall mean population growth rate of approximately 1.037, which is comparable to rates from mark‐recapture studies. When we added an immigration term that accounted for the greater slope in adult counts since 1999, as identified by the MARS model, the estimated per capita reproductive rate was 0.122, with survival probabilities for adults, subadults and calves of 0.926, 0.920, and 0.833, respectively. These rates were coupled with an estimated mean winter immigration rate corresponding to roughly 5.2% of the adult and subadult population. In this latter scenario, the number of manatees in the core population of the Northwest management unit was predicted to remain constant, with a population growth rate near one, and additional manatees counted during aerial surveys were deemed to be immigrants. While further studies could certainly expound on the potential effects of migrants on population indices, we present this first published immigration estimate for wintering manatees in northwest Florida.