A discrete-time host–parasitoid model with an Allee effect

We introduce a discrete-time host–parasitoid model with a strong Allee effect on the host. We adapt the Nicholson–Bailey model to have a positive density dependent factor due to the presence of an Allee effect, and a negative density dependence factor due to intraspecific competition. It is shown that there are two scenarios, the first with no interior fixed points and the second with one interior fixed point. In the first scenario, we show that either both host and parasitoid will go to extinction or there are two regions, an extinction region where both species go to extinction and an exclusion region in which the host survives and tends to its carrying capacity. In the second scenario, we show that either both host and parasitoid will go to extinction or there are two regions, an extinction region where both species go to extinction and a coexistence region where both species survive.     

The Role of Density Regulation in Extinction Processes and Population Viability Analysis

We review the role of density dependence in the stochastic extinction of populations and the role density dependence has played in population viability analysis (PVA) case studies. In total, 32 approaches have been used to model density regulation in theoretical or applied extinction models, 29 of them are mathematical functions of density dependence, and one approach uses empirical relationships between density and survival, reproduction, or growth rates. In addition, quasi-extinction levels are sometimes applied as a substitute for density dependence at low population size. Density dependence further has been modelled via explicit individual spacing behaviour and/or dispersal. We briefly summarise the features of density dependence available in standard PVA software, provide summary statistics about the use of density dependence in PVA case studies, and discuss the effects of density dependence on extinction probability. The introduction of an upper limit for population size has the effect that the probability of ultimate extinction becomes 1. Mean time to extinction increases with carrying capacity if populations start at high density, but carrying capacity often does not have any effect if populations start at low numbers. In contrast, the Allee effect is usually strong when populations start at low densities but has only a limited influence on persistence when populations start at high numbers. Contrary to previous opinions, other forms of density dependence may lead to increased or decreased persistence, depending on the type and strength of density dependence, the degree of environmental variability, and the growth rate. Furthermore, effects may be reversed for different quasi-extinction levels, making the use of arbitrary quasi-extinction levels problematic. Few systematic comparisons of the effects on persistence between different models of density dependence are available. These effects can be strikingly different among models. Our understanding of the effects of density dependence on extinction of metapopulations is rudimentary, but even opposite effects of density dependence can occur when metapopulations and single populations are contrasted. We argue that spatially explicit models hold particular promise for analysing the effects of density dependence on population viability provided a good knowledge of the biology of the species under consideration exists. Since the results of PVAs may critically depend on the way density dependence is modelled, combined efforts to advance statistical methods, field sampling, and modelling are urgently needed to elucidate the relationships between density, vital rates, and extinction probability.     

Equilibria in systems of interacting structured populations

The existence of a stable positive equilibrium density for a community of k interacting structured species is studied as a bifurcation problem. Under the assumption that a subcommunity of k–1 species has a positive equilibrium and under only very mild restrictions on the density dependent vital growth rates, it is shown that a global continuum of equilibria for the full community bifurcates from the subcommunity equilibrium at a unique critical value of a certain inherent birth modulus for the kth species. Local stability is shown to depend upon the direction of bifurcation. The direction of bifurcation is studied in more detail for the case when vital per unity birth and death rates depend on population density through positive linear functionals of density and for the important case of two interacting species. Some examples involving competition, predation and epidemics are given.     

Demography in relation to population density in two herbivorous marsupials: testing for source–sink dynamics versus independent regulation of population size

We compared demography of populations along gradients of population density in two medium-sized herbivorous marsupials, the common brushtail possum Trichosurus vulpecula and the rufous bettong Aepyprymnus rufescens, to test for net dispersal from high density populations (acting as sources) to low density populations (sinks). In both species, population density was positively related to soil fertility, and variation in soil fertility produced large differences in population density of contiguous populations. We predicted that if source–sink dynamics were operating over this density gradient, we should find higher immigration rates in low-density populations, and positive relationships of measures of individual fitness—body condition, reproductive output, juvenile growth rates and survivorship—to population density. This was predicted because under source–sink dynamics, immigration from high-density sites would hold population density above carrying capacity in low-density sites. The study included 13 populations of these two species, representing a more than 50-fold range of density for each species, but we found that individual fitness, immigration rates and population turnover were similar in all populations. We conclude that net dispersal from high to low density populations had little influence on population dynamics in these species; rather, all populations appeared to be independently regulated at carrying capacity, with a balanced exchange of dispersers among populations. These two species have suffered recent reductions in range, and they are ecologically similar to other species that have declined to extinction in inland Australia. It has been argued that part of the cause of the vulnerability of species like these is that they exhibit source–sink dynamics, and disturbance to source habitats can therefore cause large-scale population collapses. The results of our study argue against this interpretation.     

Evolutionary tracking is determined by differential selection on demographic rates and density dependence

Recent ecological forecasts predict that ~25% of species worldwide will go extinct by 2050. However, these estimates are primarily based on environmental changes alone and fail to incorporate important biological mechanisms such as genetic adaptation via evolution. Thus, environmental change can affect population dynamics in ways that classical frameworks can neither describe nor predict. Furthermore, often due to a lack of data, forecasting models commonly describe changes in population demography by summarizing changes in fecundity and survival concurrently with the intrinsic growth rate (r). This has been shown to be an oversimplification as the environment may impose selective pressure on specific demographic rates (birth and death) rather than directly on r (the difference between the birth and death rates). This differential pressure may alter population response to density, in each demographic rate, further diluting the information combined to produce r. Thus, when we consider the potential for persistence via adaptive evolution, populations with the same r can have different abilities to persist amidst environmental change. Therefore, we cannot adequately forecast population response to climate change without accounting for demography and selection on density dependence. Using a continuous‐time Markov chain model to describe the stochastic dynamics of the logistic model of population growth and allow for trait evolution via mutations arising during birth events, we find persistence via evolutionary tracking more likely when environmental change alters birth rather than the death rate. Furthermore, species that evolve responses to changes in the strength of density dependence due to environmental change are less vulnerable to extinction than species that undergo selection independent of population density. By incorporating these key demographic considerations into our predictive models, we can better understand how species will respond to climate change.     

Effects of habitat quality and size on extinction in experimental populations

Stochastic population theory makes clear predictions about the effects of reproductive potential and carrying capacity on characteristic time-scales of extinction. At the same time, the effects of habitat size and quality on reproduction and regulation have been hotly debated. To trace the causal relationships among these factors, we looked at the effects of habitat size and quality on extinction time in experimental populations of Daphnia magna. Replicate model systems representative of a broad-spectrum consumer foraging on a continuously supplied resource were established under crossed treatments of habitat size (two levels) and habitat quality (three levels) and monitored until eventual extinction of all populations. Using statistically derived estimates of key parameters, we related experimental treatments to persistence time through their effect on carrying capacity and the population growth rate. We found that carrying capacity and the intrinsic rate of increase were each influenced similarly by habitat size and quality, and that carrying capacity and the intrinsic rate of increase were in turn both correlated with time to population extinction. We expected habitat quality to have a greater influence on extinction. However, owing to an unexpected effect of habitat size on reproductive potential, habitat size and quality were similarly important for population persistence. These results support the idea that improving the population growth rate or carrying capacity will reduce extinction risk and demonstrate that both are possible by improving habitat quality or increasing habitat size.     

Scramble competitions can rescue endangered species subject to strong Allee effects

In this article, we study population dynamics of a general two-species discrete-time competition model where each species suffers from both strong Allee effects and scramble intra-specific competitions. We focus on how the combinations of the scramble intra-specific and inter-specific competition affect the extinction and coexistence of these two competing species where each species is subject to strong Allee effects. We derive sufficient conditions on the extinction, essential-like extinction and coexistence for such models. One of the most interesting findings is that scramble competitions can promote coexistence of these two species at their high densities. This is supported by the outcome of single species models with strong Allee effects. In addition, we apply theoretical results to a symmetric competition model with strong Allee effects induced by predator saturations where we give a completed study of its possible equilibria and attractors. Numerical simulations are performed to support our results.     

Deterministic single-locus density-dependent selection

Density-regulated selection is considered for a single, multiallelic gene locus and separated generations. Characteristics resulting from the basic assumption that the average population fitness decreases with increasing density are derived. Under this assumption, it proves to be necessary to distinguish between regions of allelic frequencies which imply limited population growth, unlimited growth, or ultimate extinction when the population stays in the respective region. Particular attention is given to the investigation of the region of limited growth and the carrying capacity function defined on it. Relationships between and the average fitness (adaptive surface) in the non-density dependent model are explained.Besides stability properties of equilibrium points, more general characteristics concerning the asymptotic behavior of population trajectories are treated. In this context, the problems of sudden loss of alleles and of population extinction as a result of large fluctuations in density are discussed.     

Predicting extinctions: interspecific competition,predation and population variability in experimental Daphnia populations

We examine how interspecific competition and two types of size-selective predation affect population density, variability and persistence in laboratory cultures of two species of Daphnia, D. magna and D. longispina. When both species were analysed together, and for D. longispina alone, there were weak negative relationships between mean population density and population variability. Interspecific competition resulted in lower population densities and higher population variability. Extinct populations had lower densities and were also more variable than persisting ones. There was still an effect of population variability on extinction probability after the effect of density on population variability had been accounted for. Hence, the effects of population density and variability on population persistence were partly independent of each other. The effects of size-selective predation on population persistence were more species-specific and not directly related to density or variability. Since the effects of species interactions on persistence were large, we suggest that it is likely that population vulnerability analyses not incorporating effects of interspecific interactions are often misleading.     

Effects of extinction on food web structures on an evolutionary time scale

Extinction affected food web structure in paleoecosystems. Recent theoretical studies that examined the effects of extinction intensity on food web structure on ecological time scales have considered extinction to involve episodic events, with pre-extinction food webs becoming established without dynamics. However, in terms of the paleontological time scale, food web structures are generated from feedback with repeated extinctions, because extinction frequency is affected by food web structure, and food web structure itself is a product of previous extinctions. We constructed a simulation model of changes in tri-trophic-level food webs to examine how continual extinction events affect food webs on an evolutionary time scale. We showed that under high extinction intensity (1) species diversity, especially that of consumer species, decreased; (2) the total population density at each trophic level decreased, while the densities of individual species increased; and (3) the trophic link density of the food web increased. In contrast to previous models, our results were based on an assumption of long-term food web development and are able to explain overall trends posited by empirical investigations based on fossil records.     

Host-Parasitoid Dynamics and the Success of Biological Control When Parasitoids Are Prone to Allee Effects

In sexual organisms, low population density can result in mating failures and subsequently yields a low population growth rate and high chance of extinction. For species that are in tight interaction, as in host-parasitoid systems, population dynamics are primarily constrained by demographic interdependences, so that mating failures may have much more intricate consequences. Our main objective is to study the demographic consequences of parasitoid mating failures at low density and its consequences on the success of biological control. For this, we developed a deterministic host-parasitoid model with a mate-finding Allee effect, allowing to tackle interactions between the Allee effect and key determinants of host-parasitoid demography such as the distribution of parasitoid attacks and host competition. Our study shows that parasitoid mating failures at low density result in an extinction threshold and increase the domain of parasitoid deterministic extinction. When proned to mate finding difficulties, parasitoids with cyclic dynamics or low searching efficiency go extinct; parasitoids with high searching efficiency may either persist or go extinct, depending on host intraspecific competition. We show that parasitoids suitable as biocontrol agents for their ability to reduce host populations are particularly likely to suffer from mate-finding Allee effects. This study highlights novel perspectives for understanding of the dynamics observed in natural host-parasitoid systems and improving the success of parasitoid introductions.     

Survival of small populations under demographic stochasticity.

We estimate the mean time to extinction of small populations in an environment with constant carrying capacity but under stochastic demography. In particular, we investigate the interaction of stochastic variation in fecundity and sex ratio under several different schemes of density dependent population growth regimes. The methods used include Markov chain theory, Monte Carlo simulations, and numerical simulations based on Markov chain theory. We find a strongly enhanced extinction risk if stochasticity in sex ratio and fluctuating population size act simultaneously as compared to the case where each mechanism acts alone. The distribution of extinction times deviates slightly from a geometric one, in particular for short extinction times. We also find that whether maximization of intrinsic growth rate decreases the risk of extinction or not depends strongly on the population regulation mechanism. If the population growth regime reduces populations above the carrying capacity to a size below the carrying capacity for large r (overshooting) then the extinction risk increases if the growth rate deviates from an optimal r-value.     

Responses of Metapopulation Dynamics to Two Different Kinds of Habitat Destruction Caused by Human Activities

Habitat destruction can be classified into instantaneous destruction and continuous destruction by the different ways of human destroying habitat. Previous studies, however, always focused on instantaneous destruction. In this study, we develop a universal model, Multi-time scale N-species model, to study and compare the responses of metapopulation dynamics to both kinds of habitat destruction. The model explores that: (1) under instantaneous habitat destruction, species extinction is determined by the proportion of habitat destruction (D) and the structure of metapopulation (q). When D>q, species will go extinct ranked from the best competitor to the worst. When D≤ q, no species will go extinct, but the equilibrium abundances of odd-ranked competitors will decrease, and the equilibrium abundances of even-ranked competitors will increase; (2) under continuous destruction, species extinction is dependent on the speed of habitat destruction and the metapopulation structure. The higher the speed of habitat destruction and the bigger q are, the earlier species go extinct. Usually, there are two possible mechanisms of species extinction: one is that all species go extinct collectively following complete destruction, and the other is that species go extinct in ranked competitive order from best to worst, and the survivals, if they exist, will go extinct collectively following complete destruction. The oscillation amplitudes of inferior competitors are so large as to increase the probability of stochastic extinction under instantaneous destruction. Therefore, it is relatively propitious for the persistence of rare species under slow and continuous destruction, especially when continuous destruction stops.     

Evolutionary dynamics and strong Allee effects

We describe the dynamics of an evolutionary model for a population subject to a strong Allee effect. The model assumes that the carrying capacity k(u), inherent growth rate r(u), and Allee threshold a(u) are functions of a mean phenotypic trait u subject to evolution. The model is a plane autonomous system that describes the coupled population and mean trait dynamics. We show bounded orbits equilibrate and that the Allee basin shrinks (and can even disappear) as a result of evolution. We also show that stable non-extinction equilibria occur at the local maxima of k(u) and that stable extinction equilibria occur at local minima of r(u). We give examples that illustrate these results and demonstrate other consequences of an Allee threshold in an evolutionary setting. These include the existence of multiple evolutionarily stable, non-extinction equilibria, and the possibility of evolving to a non-evolutionary stable strategy (ESS) trait from an initial trait near an ESS.     

Considering threats to population viability of the endangered Korean long-tailed goral (Naemorhedus caudatus) using VORTEX

Population viability analysis (PVA) has frequently been used in conservation biology to predict extinction rates for threatened or endangered species. In this study, we used VORTEX to model Korean long-tailed goral (Naemorhedus caudatus) using previously collected ecological data. We focused on modelling population extinction, mean population size and heterozygosity. The minimum viable population size was found to be at least 50 gorals for 100 years, regardless of carrying capacity. However, populations with fewer than 50 gorals could not remain successful in the model. Inbreeding depression, catastrophes and supplementation also affected patterns of population extinction, mean population size and heterozygosity. Supplementation with new individuals had the strongest effect on extinction, mean population size and heterozygosity, followed by initial population size, inbreeding, catastrophes and carrying capacity. These results suggest that a supplementation by extra goral individuals from goral proliferation facilities would be the most helpful means for the restoration programme. More Korean goral-specific information regarding demographic and habitat parameters is needed for further PVA of the species.     

Population size dependence, competitive coexistence and habitat destruction

1. Spatial dynamics can lead to coexistence of competing species even with strong asymmetric competition under the assumption that the inferior competitor is a better colonizer given equal rates of extinction. Patterns of habitat fragmentation may alter competitive coexistence under this assumption.
2. Numerical models were developed to test for the previously ignored effect of population size on competitive exclusion and on extinction rates for coexistence of competing species. These models neglect spatial arrangement.
3. Cellular automata were developed to test the effect of population size on competitive coexistence of two species, given that the inferior competitor is a better colonizer. The cellular automata in the present study were stochastic in that they were based upon colonization and extinction probabilities rather than deterministic rules.
4. The effect of population size on competitive exclusion at the local scale was found to have little consequence for the coexistence of competitors at the metapopulation (or landscape) scale. In contrast, population size effects on extinction at the local scale led to much reduced landscape scale coexistence compared to simulations not including localized population size effects on extinction, especially in the cellular automata models. Spatially explicit dynamics of the cellular automata vs. deterministic rates of the numerical model resulted in decreased survival of both species. One important finding is that superior competitors that are widespread can become extinct before less common inferior competitors because of limited colonization.
5. These results suggest that population size–extinction relationships may play a large role in competitive coexistence. These results and differences are used in a model structure to help reconcile previous spatially explicit studies which provided apparently different results concerning coexistence of competing species.     

demoniche – an R‐package for simulating spatially‐explicit population dynamics

demoniche is a freely available R‐package which simulates stochastic population dynamics in multiple populations of a species. A demographic model projects population sizes utilizing several transition matrices that can represent impacts on species growth. The demoniche model offers options for setting demographic stochasticity, carrying capacity, and dispersal. The demographic projection in each population is linked to spatially‐explicit niche values, which affect the species growth. With the demoniche package it is possible to compare the influence of scenarios of environmental changes on future population sizes, extinction probabilities, and range shifts of species.     

Density‐dependent and density‐independent drivers of population change in Barton Springs salamanders

Understanding population change is essential for conservation of imperiled species, such as amphibians. Worldwide amphibian declines have provided an impetus for investigating their population dynamics, which can involve both extrinsic (density‐independent) and intrinsic (density‐dependent) drivers acting differentially across multiple life stages or age classes. In this study, we examined the population dynamics of the endangered Barton Springs Salamander (Eurycea sosorum) using data from a long‐term monitoring program. We were interested in understanding both the potential environmental drivers (density‐independent factors) and demographic factors (interactions among size classes, negative density dependence) to better inform conservation and management activities. We used data from three different monitoring regimes and multivariate autoregressive state‐space models to quantify environmental effects (seasonality, discharge, algae, and sediment cover), intraspecific interactions among three size classes, and intra‐class density dependence. Results from our primary data set revealed similar patterns among sites and size classes and were corroborated by our out‐of‐sample data. Cross‐correlation analysis showed juvenile abundance was most strongly correlated with a 9‐month lag in aquifer discharge, which we suspect is related to inputs of organic carbon into the aquifer. However, sedimentation limited juvenile abundance at the surface, emphasizing the importance of continued sediment management. Recruitment from juveniles to the sub‐adult size class was evident, but negative density‐dependent feedback ultimately regulated each size class. Negative density dependence may be an encouraging sign for the conservation of E. sosorum because populations that can reach carrying capacity are less likely to go extinct compared to unregulated populations far below their carrying capacity. However, periodic population declines coupled with apparent migration into the aquifer complicate assessments of species status. Although both density‐dependent and density‐independent drivers of population change are not always apparent in time series of animal populations, both have important implications for conservation and management of E. sosorum.     

Population extinction risks of three Neotropical small mammal species

The population persistence and extinction probabilities of three small mammal species were analyzed by estimating growth and extinction properties obtained from 10 years of live-trapping data at two different habitat types in semiarid Chile. We used a stochastic formulation with an exponential growth model known as a Wiener-drift process, out of which growth and extinction quantities were estimated. The rodent Phyllotis darwini showed the lowest rates of growth, and the lowest infinitesimal variances, whereas the opposite trend was found for the rodent Akodonolivaceus. The marsupial Thylamys elegans showed intermediate values for growth rates and infinitesimal variances. The rodent P. darwini showed the lowest extinction risk in the study site. We also detected spatial differences between mesic and xeric habitats in the growth rates of P. darwini and T. elegans, and in the extinction risks of the three species studied. Although the population growth of these three species can be approximated by purely stochastic processes, the introduction of density dependence through autoregressive log-linear models reduced the extinction times of all species analyzed. Received: 16 May 1997 / Accepted: 22 January 1998     

Effects of predation pressure and prey density on short-term indirect interactions between two prey species that share a common predator

1. Generalist predators are important contributors to reliable conservation biological control. Indirect interactions between prey species that share a common generalist predator can influence both community dynamics and the efficacy of biological control. 2. Laboratory cage experiments investigated the impact of the combined consumptive and non-consumptive effects of predation by adult Hippodamia convergens as a shared predator on the population growth and relative abundance of Acyrthosiphon pisum and Aphis gossypii as prey species. Predation pressure and prey density were varied. 3. At low predation pressure the indirect interaction between aphid species was asymmetrical with a proportionally greater negative impact of predation on A. gossypii than on A. pisum. At intermediate predation pressure, the indirect interaction became symmetrical. At high predation pressure and higher levels of prey density, it was asymmetrical with greater negative impact on A. pisum, often driven to local extinction while A. gossypii populations persisted. 4. A linear mixed-effects model including early population growth of both aphid species and predation pressure explained 96% and 92% of the variation in the population growth of A. pisum and A. gossypii, respectively, over an 8-day period. The overall effect of shared predation on the indirect interaction between the two aphid species is best described as apparent commensalism, where A. pisum benefited from early population growth of A. gossypii, while A. gossypii was unaffected by early population growth of A. pisum. Considering these indirect interactions is important for conservation biological control efforts to be successful.