Allee threshold and extinction threshold for spatially explicit metapopulation dynamics with Allee effects

In this paper, we investigate a spatially explicit metapopulation model with Allee effects. We refer to the patch occupancy model introduced by Levins (Bull Entomol Soc Am 15:237–240, 1969) as a spatially implicit metapopulation model, i.e., each local patch is either occupied or vacant and a vacant patch can be recolonized by a randomly chosen occupied patch from anywhere in the metapopulation. When we transform the model into a spatially explicit one by using a lattice model, the obtained model becomes theoretically equivalent to a “lattice logistic model” or a “basic contact process”. One of the most popular or standard metapopulation models with Allee effects, developed by Amarasekare (Am Nat 152:298–302, 1998), supposes that those effects are introduced formally by means of a logistic equation. However, it is easier to understand the ecological meaning of associating Allee effects with this model if we suppose that only the logistic colonization term directly suffers from Allee effects. The resulting model is also well defined, and therefore we can naturally examine it by Monte Carlo simulation and by doublet and triplet decoupling approximation. We then obtain the following specific features of one-dimensional lattice space: (1) the metapopulation as a whole does not have an Allee threshold for initial population size even when each local population follows the Allee effects; and (2) a metapopulation goes extinct when the extinction rate of a local population is lower than that in the spatially implicit model. The real ecological metapopulation lies between two extremes: completely mixing interactions between patches on the one hand and, on the other, nearest neighboring interactions with only two nearest neighbors. Thus, it is important to identify the metapopulation structure when we consider the problems of invasion species such as establishment or the speed of expansion.     

Variation in the outcome of population interactions: bifurcations and catastrophes

 The nature of the association between two species may vary depending on population abundances, age or size of individuals, or environmental conditions. Interactions may switch between beneficial and detrimental depending on the net balance of costs and benefits involved for each species. We study the repercussion of the ecological setting on the outcomes of conditional or variable interactions by means of a model that incorporates density-dependent interaction coefficients; that is, interaction α-functions. These characterize the responsiveness and sensitivity of the association to changes in partner's abundance, and can take positive and negative values. Variable outcomes – and transitions between them – are categorized as homeo- or allo-environmental, that is, occurring under the same ecological setting, or not, respectively. Bifurcation analyses show that these dynamics are moulded by ecological factors that are: intrinsic to the nature of the association (concerning the sensitivity of the interaction), and extrinsic to the association itself (the quality of the environment referred to each species alone). The influence of these factors may be conflicting; consequently, the dynamics involve catastrophic events. In a facultative variable association, stable coexistence is expected when environmental conditions are adverse; otherwise, the exclusion of one species is the likely outcome. Remarkable situations as the switching of victim-exploiter roles illustrate the theoretical perspective. Received: 15 December 2001 / Revised version: 18 November 2002 / Published online: 28 February 2003 Key words or phrases: Variable population interactions – Conditional interactions – Costs and benefits – Density dependent interaction coefficient – Hysteresis – Symbiosis – Mutualism – Parasitism     

Limits to genetic bottlenecks and founder events imposed by the Allee effect

The Allee effect can result in a negative population growth rate at low population density. Consequently, populations below a minimum (critical) density are unlikely to persist. A lower limit on population size should constrain the loss of genetic variability due to genetic drift during population bottlenecks or founder events. We explored this phenomenon by modeling changes in genetic variability and differentiation during simulated bottlenecks of the alpine copepod, Hesperodiaptomus shoshone. Lake surveys, whole-lake re-introduction experiments and model calculations all indicate that H. shoshone should be unlikely to establish or persist at densities less than 0.5–5 individuals m⁻³. We estimated the corresponding range in minimum effective population size using the distribution of habitat (lake) sizes in nature and used these values to model the expected heterozygosity, allelic richness and genetic differentiation resulting from population bottlenecks. We found that during realistic bottlenecks or founder events, >90% of H. shoshone populations in the Sierra Nevada may be resistant to significant changes in heterozygosity or genetic distance, and 70–75% of populations may lose <10% of allelic richness. We suggest that ecological constraints on minimum population size be considered when using genetic markers to estimate historical population dynamics.     

Extinction risk assessments at the population and species level: implications for amphibian conservation

Amphibian populations are declining worldwide and this is causing growing concern. High levels of population declines followed by the expansion of red lists are creating demands for effective strategies to maximize conservation efforts for amphibians. Ideally, integrated and comprehensive strategies should be based on complementary information of population and species extinction risk. Here we evaluate the congruence between amphibian extinction risk assessments at the population level (Declining Amphibian Database––DAPTF) and at species level (GAA––IUCN Red List). We used the Declining Amphibian Database––DAPTF that covers 967 time-series records of amphibian population declines assigned into four levels of declines. We assigned each of its corresponding species into GAA––IUCN red list status, discriminated each species developmental mode, and obtained their geographic range size as well. Extinction risk assessments at the population and species level do not fully coincide across geographic realms or countries. In Paleartic, Neartic and Indo-Malayan realms less than 25% of species with reported population declines are formally classified as threatened. In contrast, more than 60% of all species with reported population declines that occur in Australasia and the Neotropics are indeed threatened according to the GAA––IUCN Red List. Species with aquatic development presented proportionally higher extinction risks at both population and species level than those with terrestrial development, being this pattern more prominent at Australasia, Paleartic, and Neartic realms. Central American countries, Venezuela, Mexico and Australia presented the highest congruence between both population and species risk. We address that amphibian conservation strategies could be improved by using complementary information on time-series population trends and species threat. Whenever feasible, conservation assessments should also include life-history traits in order to improve its effectiveness.     

Daphnia revisited: local stability and bifurcation theory for physiologically structured population models explained by way of an example

We consider the interaction between a general size-structured consumer population and an unstructured resource. We show that stability properties and bifurcation phenomena can be understood in terms of solutions of a system of two delay equations (a renewal equation for the consumer population birth rate coupled to a delay differential equation for the resource concentration). As many results for such systems are available (Diekmann et al. in SIAM J Math Anal 39:1023–1069, 2007), we can draw rigorous conclusions concerning dynamical behaviour from an analysis of a characteristic equation. We derive the characteristic equation for a fairly general class of population models, including those based on the Kooijman–Metz Daphnia model (Kooijman and Metz in Ecotox Env Saf 8:254–274, 1984; de Roos et al. in J Math Biol 28:609–643, 1990) and a model introduced by Gurney–Nisbet (Theor Popul Biol 28:150–180, 1985) and Jones et al. (J Math Anal Appl 135:354–368, 1988), and next obtain various ecological insights by analytical or numerical studies of special cases.     

Evolutionary and population dynamics of host–parasitoid interactions

The role of evolutionary dynamics in understanding host–parasitoid interactions is interlinked with the population dynamics of these interactions. Here, we address the problems in coupling evolutionary and population dynamics of host–parasitoid interactions. We review previous theoretical and empirical studies and show that evolution can alter the ecological dynamics of a host–parasitoid interaction. Whether evolution stabilizes or destabilizes the interaction depends on the direction of evolutionary changes, which are affected by ecological, physiological, and genetic details of the insect biology. We examine the effect of life history correlations on population persistence and stability, embedding two types, one of which is competitively inferior but superior in encapsulation (for parasitoid, virulence), in a Nicholson–Bailey model with intraspecific resource competition for host. If a trade-off exists between intraspecific competitive ability and encapsulation (or virulence, as a countermeasure) in both the host and parasitoid, the trade-off or even positive correlation in the parasitoid is less influential to ecological stability than the trade-off in the host. We comment on the bearing this work has on the broader issues of understanding host–parasitoid interactions, including long-term biological control. Received: November 10, 1998 / Accepted: January 18, 1999     

Modeling population dynamics and conservation of arapaima in the Amazon

To promote understanding of fish population dynamics in tropical river-floodplains, we have synthesized existing information by developing a largely empirical population model for arapaima (Arapaima sp.). Arapaima are characterized by very large bodies, relatively late sexual maturity, small clutches, and large parental investment per offspring, and their populations are overexploited and even declining due to overfishing. We used unparalleled time series data on growth, reproduction, catch-at-age, and size-class abundance estimates for a population that has increased several-fold and undergone drastic changes in fishing practices in the Amazon, Brazil. Model population numbers were close to observed numbers, with generally low mean absolute percentage errors for juveniles (16%), adults (30%), and catch (18%). In using the model to test ecological hypotheses and to investigate management strategies, we found the following: (1) Annual recruitment is directly and positively related to spawner abundance, and it appears to be density-compensatory following a Beverton–Holt relation (R ² = 0.85). (2) Fishing-selectivity of arapaima caused by use of harpoons and gillnets can lower yield potentials dramatically through removal of the faster-growing individuals of the population. That is in part because fewer individuals live long enough to reproduce and survivors take longer to reach reproductive age. (3) Arapaima populations can sustain annual catches of up to 25% of the number of adults in the population the previous year if minimum size (1.5 m) and closed season (December–May) limits are met. (4) When 25% of the number of adults in the population the previous year is harvested under a 1.6 m minimum size limit of catch, catches are slightly smaller but abundance of adults in the population is considerably greater than under a 1.5 m limit. These findings can be used in ongoing management initiatives, but caution is needed because of present biological and ecological uncertainty about these fishes.     

Does the early frog catch the worm? Disentangling potential drivers of a parasite age–intensity relationship in tadpoles

The manner in which parasite intensity and aggregation varies with host age can provide insights into parasite dynamics and help identify potential means of controlling infections in humans and wildlife. A significant challenge is to distinguish among competing mechanistic hypotheses for the relationship between age and parasite intensity or aggregation. Because different mechanisms can generate similar relationships, testing among competing hypotheses can be difficult, particularly in wildlife hosts, and often requires a combination of experimental and model fitting approaches. We used field data, experiments, and model fitting to distinguish among ten plausible drivers of a curvilinear age–intensity relationship and increasing aggregation with host age for echinostome trematode infections of green frogs. We found little support for most of these proposed drivers but did find that the parsimonious explanation for the observed age–intensity relationship was seasonal exposure to echinostomes. The parsimonious explanation for the aggregated distribution of parasites in this host population was heterogeneity in exposure. A predictive model incorporating seasonal exposure indicated that tadpoles hatching early or late in the breeding season should have lower trematode burdens at metamorphosis, particularly with simulated warmer climates. Application of this multi-pronged approach (field surveys, lab experiments, and modeling) to additional parasite–host systems could lead to discovery of general patterns in the drivers of parasite age–intensity and age–distribution relationships.     

Reproductive styles and life history variables relative to exploitation and management ofSebastes stocks

Synopsis The characteristics of lightly and heavily exploited Pacific ocean perch,Sebastes alutus, stocks are evaluated relative to the predictions of life history theory. These long-lived species (50–100 year lifespan) show limited phenotypic plasticity and have little buffering against the effects of reduced lifespan. Reduced stock abundance has generated some compensatory increase in growth rate. Length at first maturity varies only slightly with increased growth rate, although age at maturity may decrease by 1–4 years. Grooth increases yield larger (15–20%) size at age and increased reproductive effort at younger ages, but lower size-specific fecundity for these faster-growing fish. This suggests an energy allocation protocol favouring growth over reproduction in these long-lived animals. Rockfishes have late recruitment to fisheries (ages 10–15), and the detection time for results of management actions is equally long. Their vulnerability to overfishing means that indices of population changes, more representative of fishing effects than the catch rate index presently used, are required. Reproductive value indices are shown to be extremely sensitive and continuous with population abundance changes. Their incorporation into monitoring programs would permit more timely evaluation of management actions. Management policies developed for shorter-lived species are shown to be inappropriate for rockfishes. The need for an improved match in the time frame of the species' life history, and that of management strategies, is stressed.     

Age and Sex Structured Model for Assessing the Demographic Impact of Mother-to-Child Transmission of HIV/AIDS

Age and sex structured HIV/AIDS model with explicit incubation period is proposed as a system of delay differential equations. The model consists of two age groups that are children (0–14 years) and adults (15–49 years). Thus, the model considers both mother-to-child transmission (MTCT) and heterosexual transmission of HIV in a community. MTCT can occur prenatally, at labour and delivery or postnatally through breastfeeding. In the model, we consider the children age group as a one-sex formulation and divide the adult age group into a two-sex structure consisting of females and males. The important mathematical features of the model are analysed. The disease-free and endemic equilibria are found and their stabilities investigated. We use the Lyapunov functional approach to show the local stability of the endemic equilibrium. Qualitative analysis of the model including positivity and boundedness of solutions, and persistence are also presented. The basic reproductive number (ℛ₀) for the model shows that the adult population is responsible for the spread HIV/AIDS epidemic, thus up-to-date developed HIV/AIDS models to assess intervention strategies have focused much on heterosexual transmission by the adult population and the children population has received little attention. We numerically analyse the HIV/AIDS model to assess the community benefits of using antiretroviral drugs in reducing MTCT and the effects of breastfeeding in settings with high HIV/AIDS prevalence ratio using demographic and epidemiological parameters for Zimbabwe.     

A discrete,size-structured model of phytoplankton growth in the chemostat: introduction of inhomogeneous cell division size

 We introduce inhomogeneous, substrate dependent cell division in a time discrete, nonlinear matrix model of size-structured population growth in the chemostat, first introduced by Gage et al. [8] and later analysed by Smith [13]. We show that mass conservation is verified, and conclude that our system admits one non zero globally stable equilibrium, which we express explicitly. Then we run numerical simulations of the system, and compare the predictions of the model to data related to phytoplankton growth, whose obtention we discuss. We end with the identification of several parameters of the system. Received: 9 February 2000 / Revised version: 10 October 2001 / Published online: 23 August 2002 RID="*" ID="*" Present address: Department of Mathematics and Statistics, University of Victoria, B.C., Canada. e-mail: jarino@math.uvic.ca Key words or phrases: Chemostat – Structured population models – Discrete model – Inhomogeneous division size     

Process and measurement errors of population size: their mutual effects on precision and bias of estimates for demographic parameters

Knowing the parameters of population growth and regulation is fundamental for answering many ecological questions and the successful implementation of conservation strategies. Moreover, detecting a population trend is often a legal obligation. Yet, inherent process and measurement errors aggravate the ability to estimate these parameters from population time-series. We use numerical simulations to explore how the lengths of the time-series, process and measurement error influence estimates of demographic parameters. We first generate time-series of population sizes with given demographic parameters for density-dependent stochastic population growth, but assume that these population sizes are estimated with measurement errors. We then fit parameters for population growth, habitat capacity, total error and long-term trends to the ‘measured’ time-series data using non-linear regression. The length of the time-series and measurement error introduce a substantial bias in the estimates for population growth rate and to a lesser degree on estimates for habitat capacity, while process error has little effect on parameter bias. The total error term of the statistical model is dominated by process error as long as the latter is larger than the measurement error. A decline in population size is difficult to document as soon as either error becomes moderate, trends are not very pronounced, and time-series are short (<10–15 seasons). Detecting an annual decline of 1% within 6-year reporting periods, as required for the European Union for the species of Community Interest, appears unachievable.     

Estimating wild boar (Sus scrofa) abundance and density using capture–resights in Canton of Geneva, Switzerland

We estimated wild boar abundance and density using capture–resight methods in the western part of the Canton of Geneva (Switzerland) in the early summer from 2004 to 2006. Ear-tag numbers and transmitter frequencies enabled us to identify individuals during each of the counting sessions. We used resights generated by self-triggered camera traps as recaptures. Program Noremark provided Minta–Mangel and Bowden’s estimators to assess the size of the marked population. The minimum numbers of wild boars belonging to the unmarked population (juveniles and/or piglets) were added to the respective estimates to assess total population size. Over the 3 years, both estimators showed a stable population with a slight diminishing tendency. We used mean home range size determined by telemetry to assess the sampled areas and densities. Mean wild boar population densities calculated were 10.6 individuals/km² ± 0.8 standard deviation (SD) and 10.0 ind/km² ± 0.6 SD with both estimators, respectively, and are among the highest reported from Western Europe. Because of the low proportion of marked animals and, to a lesser extent, of technical failures, our estimates showed poor precision, although they displayed similar population trends compared to the culling bag statistics. Reported densities were consistent with the ecological conditions of the study area.     

Age and growth of invasive round goby <Emphasis Type="Italic">Neogobius melanostomus</Emphasis> from middle Danube

Age and growth of the invasive population of round goby Neogobius melanostomus from the Slovak stretch of River Danube was examined. The samples (n=1130) were collected soon after the invasion was recorded (2004–2005), and later, when the population was already established (2008–2010). Invasive round goby in newly-occupied areas were found to reach smaller body size (15–153 mm standard length) compared to native populations. Age from 0+ to 4+, determined from scales, was recorded in both sexes. Annulus formed in April–May, which varied with age. Growth of freshly established gobies was negative allometric, suggesting increased allocation of their sources to reproduction, which corresponds to less specialized life-history. However, positive allometric growth found in longer established individuals suggests a shift in allocation towards somatic growth, which corresponds to more specialized life-history typical for native populations. None of the three parameters predicted by the theory of alternative ontogenies and invasive potential met the expectations, though two parameters, i.e. growth rate and age at maturation remain equivocal. This can be explained by too short of a time span that has elapsed from the beginning of invasion, or by ecological disturbances that have broken up otherwise stable environment in the habitat studied.     

Self-optimization, community stability, and fluctuations in two individual-based models of biological coevolution

We compare and contrast the long-time dynamical properties of two individual-based models of biological coevolution. Selection occurs via multispecies, stochastic population dynamics with reproduction probabilities that depend nonlinearly on the population densities of all species resident in the community. New species are introduced through mutation. Both models are amenable to exact linear stability analysis, and we compare the analytic results with large-scale kinetic Monte Carlo simulations, obtaining the population size as a function of an average interspecies interaction strength. Over time, the models self-optimize through mutation and selection to approximately maximize a community potential function, subject only to constraints internal to the particular model. If the interspecies interactions are randomly distributed on an interval including positive values, the system evolves toward self-sustaining, mutualistic communities. In contrast, for the predator–prey case the matrix of interactions is antisymmetric, and a nonzero population size must be sustained by an external resource. Time series of the diversity and population size for both models show approximate 1/f noise and power-law distributions for the lifetimes of communities and species. For the mutualistic model, these two lifetime distributions have the same exponent, while their exponents are different for the predator–prey model. The difference is probably due to greater resilience toward mass extinctions in the food-web like communities produced by the predator–prey model.      

An anthropometric comparative study of school children 1990–2000: A retrospective report

The general ecological situation of the Silesian population was described as a global catastrophe. In the first stage (1990–1993) of our study, a population of 1810 Silesian children, 7–9 years of age, was compared to a group of 780 children of the same age living in the Beskidy Mts. The second stage (1998–2000) involved similar populations from the same regions. Appearance of incorrect body posture in mountain children is most often encountered when we compare subgroups of mountain children on the basis of their schools' level above the sea: i.e. the higher they live the less frequent the lesions are. We conclude that the results of the socio-economic transformation undergone by the country and region has changed the distribution of potential ecologically catastrophic areas adding new ones to already existing ones and allowing only slow recovery.     

Breeding migration and population stability

We have modeled habitat shift for reproduction to examine the relationship between the timing of migration and population stability, by modifying Takimoto’s (Am Nat 162:93–109, 2003) consumer–resource model with a consumer’s ontogenetic niche shift. We found that equilibrium was always locally unstable if migration occurs at a fixed time or level of energy storage, whereas it could be stable if the timing of migration was adaptively flexible to maximize reproductive output. The general conditions for stability were safer breeding rather than feeding habitat and abundant resources at the feeding habitat. These results imply that both adopting an adaptive plastic strategy in the timing of migration and choosing to migrate from a rich feeding habitat to a safe breeding habitat can contribute to population stability. We also found that reduced reproductive success with delays in migration, and the survival rate after reproduction, had complicated effects on stability, depending on resource availability at the feeding habitat. The equilibrium was more likely to be stable when reproduction success was only slightly (or greatly) reduced or survival rate was high (or low) if the feeding habitat was rich (or poor). These are significant predictions for ecological study of migrating animals.     

An adaptive management approach to an octocoral fishery based on the Beverton-Holt model

Coral reef species are frequently the focus of bio-prospecting, and when promising bioactive compounds are identified there is often a need for the development of responsible harvesting based on relatively limited data. The Caribbean gorgonian Pseudopterogorgia elisabethae has been harvested in the Bahamas for over a decade. Data on population age structure and growth rates in conjunction with harvest data provide an opportunity to compare fishery practices and outcomes to those suggested by a Beverton-Holt fishery model. The model suggests a minimum colony size limit of 7–9 years of age (21–28 cm height), which would allow each colony 2–4 years of reproduction prior to harvesting. The Beverton-Holt model assumes that colonies at or above the minimum size limit are completely removed. In the P. elisabethae fishery, colonies are partially clipped and can be repeatedly harvested. Linear growth of surviving colonies was up to 3 times that predicted for colonies that were not harvested and biomass increase was up to 9 times greater than that predicted for undisturbed colonies. The survival of harvested colonies and compensatory growth increases yield, and yields at sites that had previously been harvested were generally greater than predicted by the Beverton-Holt model. The model also assumes recruitment is independent of fishing intensity, but lower numbers of young colonies in the fished populations, compared to unfished populations, suggest possible negative effects of the harvest on reproduction. This suggests the need for longer intervals between harvests. Because it can be developed from data that can be collected at a single time, the Beverton-Holt model provides a rational starting point for regulating new fisheries where long-term characterizations of population dynamics are rarely available. However, an adaptive approach to the fishery requires the incorporation of reproductive data.     

Genetic demographic study of shors in Tashtagolskii raion of Kemerovo oblast: Population dynamics and changes in the sex and age composition

The population dynamics and changes in the sex and age structure of the Shor populations of four rural district municipalities of Tashtagolskii raion of Kemerovo oblast (Kyzyl-Shorskii, Ust-Anzasskii, Ust-Kolzasskii, and Ust-Kabyrzinsskii) with time have been analyzed. The Shor populations have been found to have contained a high proportion of people under 18 years of age during two periods, 1940–1955 and 1970–1975 (38.12–46.38 and 40.98–54.97%, respectively). However, the population reproduction pattern changed into the “reduced” one in all the municipalities studied by the early 2000s. Although there are some regional variations, a common trend towards rural population aging has formed: the man age in the Tashtagolskii raion population has increased by 7.52 and 6.94 years for men and women, respectively, during two generations; the natural sex ratio has been disturbed in both the prereproductive and reproductive populations. The total population size and effective reproductive size have decreased in three out of the four rural subpopulations studied.     

Scaling of offspring number and mass to plant and animal size: model and meta-analysis

The scaling of reproductive parameters to body size is important for understanding ecological and evolutionary patterns. Here, we derived allometric relationships for the number and mass of seeds, eggs and neonates from an existing model on population production. In a separate meta-analysis, we collected 79 empirical regressions on offspring mass and number covering different taxa and various habitats. The literature review served as a validation of the model, whereas, vice versa, consistency of isolated regressions with each other and related ecological quantities was checked with the model. The total offspring mass delivered in a reproductive event scaled to adult size with slopes in the range of about 3/4 to 1. Exponents for individual seed, egg and neonate mass varied around 1/2 for most heterotherms and between 3/4 and 1 for most homeotherms. The scaling of the progeny number released in a sowing, clutch or litter was opposite to that of their size. The linear regressions fitted into a triangular envelope where maximum offspring mass is limited by the size of the adult. Minimum seed and egg size scaled with weight exponents of approximately 0 up to 1/4. These patterns can be explained by the influence of parents on the fate of their offspring, covering the continuum of r-strategists (pelagic–aquatic, arial, most invertebrates, heterotherms) and K-strategists (littoral–terrestrial, some invertebrates, homeotherms).