Invasion speeds for structured populations in fluctuating environments

We live in a time where climate models predict future increases in environmental variability and biological invasions are becoming increasingly frequent. A key to developing effective responses to biological invasions in increasingly variable environments will be estimates of their rates of spatial spread and the associated uncertainty of these estimates. Using stochastic, stage-structured, integrodifference equation models, we show analytically that invasion speeds are asymptotically normally distributed with a variance that decreases in time. We apply our methods to a simple juvenile–adult model with stochastic variation in reproduction and an illustrative example with published data for the perennial herb, Calathea ovandensis. These examples buttressed by additional analysis reveal that increased variability in vital rates simultaneously slow down invasions yet generate greater uncertainty about rates of spatial spread. Moreover, while temporal autocorrelations in vital rates inflate variability in invasion speeds, the effect of these autocorrelations on the average invasion speed can be positive or negative depending on life history traits and how well vital rates “remember” the past.     

Demographic sensitivity and persistence of the threatened white- and orange-bellied frogs of Western Australia

Anurans have been the subject of numerous experimental and monitoring studies, yet their autecology under natural conditions remains poorly understood. Previous work has focussed on the pre-metamorphic life stages, with data suitable for estimating demographic parameters for all life stages collected in few species. This deficiency has almost certainly confounded current views of amphibian autecology and understanding of which mechanisms of decline represent the greatest threats to the persistence of amphibian species. We used a stage-structured metapopulation model, parameterised with comprehensive field data, to explore the sensitivity of two species of myobatrachid frogs from the southwest of Australia, Geocrinia alba and G. vitellina, to changes in demographic parameters at all life stages. The simulation modelling indicated that the population dynamics of both species were consistently most sensitive to changes in juvenile survival, then to fecundity, and third to both adult survival and standard deviation of survival to about equal extent. In practical terms, this indicates that management interventions which attempt to mitigate reductions in juvenile survival are likely to be most successful in arresting anuran metapopulation declines.     

Relating dispersal and range expansion of California sea otters

Linking dispersal and range expansion of invasive species has long challenged theoretical and quantitative ecologists. Subtle differences in dispersal can yield large differences in geographic spread, with speeds ranging from constant to rapidly increasing. We developed a stage-structured integrodifference equation (IDE) model of the California sea otter range expansion that occurred between 1914 and 1986. The non-spatial model, a linear matrix population model, was coupled to a suite of candidate dispersal kernels to form stage-structured IDEs. Demographic and dispersal parameters were estimated independent of range expansion data. Using a single dispersal parameter, alpha, we examined how well these stage-structured IDEs related small scale demographic and dispersal processes with geographic population expansion. The parameter alpha was estimated by fitting the kernels to dispersal data and by fitting the IDE model to range expansion data. For all kernels, the alpha estimate from range expansion data fell within the 95% confidence intervals of the alpha estimate from dispersal data. The IDE models with exponentially bounded kernels predicted invasion velocities that were captured within the 95% confidence bounds on the observed northbound invasion velocity. However, the exponentially bounded kernels yielded range expansions that were in poor qualitative agreement with range expansion data. An IDE model with fat (exponentially unbounded) tails and accelerating spatial spread yielded the best qualitative match. This model explained 94% and 97% of the variation in northbound and southbound range expansions when fit to range expansion data. These otters may have been fat-tailed accelerating invaders or they may have followed a piece-wise linear spread first over kelp forests and then over sandy habitats. Further, habitat-specific dispersal data could resolve these explanations.     

Demographics of Black Vultures in North Carolina

Abstract: Understanding the contributions of vital rates to species population growth is critical to developing new management protocols. We constructed a model population for black vultures (Coragyps atratus) in North Carolina, USA, based on demographic data from a 14-year study. The model population was similar in stage structure to the reference population, and adult survival was the primary contributor to the annual rate of increase (10.6%). We suggest that the North Carolina black vulture population is experiencing high rates of survival and fertility, potentially breeding at an age younger than previously assumed, and growing rapidly.     

Survival and population growth of a long-lived threatened snake species, Drymarchon couperi (Eastern Indigo Snake)

Demographic data provide a basis for understanding the life history and ecology of species, factors which are vital for informing conservation efforts; however, little is known regarding the population ecology of most snake species, including the threatened Eastern Indigo Snake (Drymarchon couperi). We used 11 years (1999–2009) of capture-mark-recapture (CMR) and 2.5 years (2003–2005) of radiotelemetry data from southeastern Georgia, USA, in a CMR modeling framework to estimate apparent survival, capture and transition probabilities, and evaluate factors influencing these parameters. The model-averaged estimate of overall apparent annual survival probability was 0.700 (±0.030 SE) and is comparable to that obtained from known fate analysis (radiotelemetry) at the same site. Body size positively influenced survival, regardless of sex. Capture probability differed seasonally by sex, suggesting lower capture probability for females in fall and males in winter. There was no evidence for effect of precipitation or site-specific differences in survival. Model averaged estimate of annual adult survival estimated using multistate CMR models was 0.738 ± 0.030 and 0.515 ± 0.189 for subadults. We estimated population growth rate (λ) and elasticity (proportional sensitivity) of λ to vital rates using a stage-structured matrix population model. Population growth rate ranged from 0.96 to 1.03 depending on the value of the probability of transitioning from subadult to adult stage. The λ was proportionally most sensitive to changes in adult survival rate, followed by subadult survival. Our results suggest that protecting adult snakes and their habitats would result in the highest likelihood of long-term population stability and growth.     

Disentangling the effects of land use, shrub cover and climate on the invasion speed of native and introduced pines in grasslands

Aims To determine how changes in land use, climate and shrub cover affect the invasion dynamics of native (Pinus sylvestris L.) and introduced (Pinus nigra Arn. subsp. nigra) pines in grasslands. To analyse how these factors interact and affect seedling recruitment, a bottleneck in the lifecycle of many trees. Such information is required to manage the dynamics of these species. Location Grands Causses, calcareous plateaus (Southern France). Methods We used both published and unpublished demographic and dispersal data to assess population growth and invasion speed of invading pines. A demographic and spatially explicit model, which included density dependence and stochasticity in dispersal, demography and environment, was run for different scenarios of sheep grazing pressure (nil, extensive or intensive), shrub cover (0, 10 or 20%) and drought frequency (past‐to‐present or future). For each scenario, population growth rate, invasion speed and elasticity of invasion speed to each demographic and dispersal parameter were computed. Results Grazing was the main factor for limiting invasion speed. Shrub cover reduced tree spread under nil or extensive grazing pressure, but increased it under intensive grazing pressure. Although dry years led to nil seedling establishment rates, an increase in their frequency had surprisingly few effects on pine invasion speed. This last result remained unchanged when very dry years, inducing seedling, but also sapling mortality were introduced. In most environmental conditions, population growth rate and invasion speed were higher for the introduced than for the native pine. Elasticity analysis highlighted the importance of demographic parameters on invasion speed, notably adult and sapling survival. Main conclusion Tree invasion speed may rely at least as much on human activities, like sheep grazing, tree cutting and non‐native trees introduction, as on changes in climate factors. Therefore, human activities need to be explicitly taken into account in the prediction and management of tree dynamics.     

Permanence of single-species stage-structured models

In this paper, we consider population survival by using single-species stage-structured models. As a criterion of population survival, we employ the mathematical notation of permanence. Permanence of stage-structured models has already been studied by Cushing (1998). We generalize his result of permanence, and obtain a condition which guarantees that population survives. The condition is applicable to a wide class of stage-structured models. In particular, we apply our results to the Neubert-Caswell model, which is a typical stage-structured model, and obtain a condition for population survival of the model.The research is partially supported by the Ministry of Education, Science and Culture, Japan, under Grant in Aid for Scientific Research (A) 13304006.     

Species Invasiveness in Biological Invasions: A Modelling Approach

The study of invasiveness, the traits that enable a species to invade a habitat, and invasibility, the habitat characteristics that determine its susceptibility to the establishment and spread of an invasive species, provide a useful conceptual framework to formulate the biological invasion problem in a modelling context. Another important aspect is the complex interaction emerging among the invader species, the noninvader species already present in the habitat, and the habitat itself. Following a modelling approach to the biological invasion problem, we present a spatially explicit cellular automaton model (Interacting Multiple Cellular Automata (IMCA)). We use field parameters from the invader Gleditsia triacanthos and the native Lithraea ternifolia in montane forests of central Argentina as a case study to compare outputs and performance of different models. We use field parameters from another invader, Ligustrum lucidum, and the native Fagara coco from the same system to run the cellular automaton model. We compare model predictions with invasion values from aerial photographs. We discuss in detail the importance of factors affecting species invasiveness, and give some insights into habitat invasibility and the role of interactions between them. Finally, we discuss the relevance of mathematical modelling for studying and predicting biological invasions. The IMCA model provided a suitable context for integrating invasiveness, invasibility, and the interactions. In the invasion system studied, the presence of an invader's juvenile bank not only accelerated the rate of invasion but was essential to ensure invasion. Using the IMCA model, we were able to determine that not only adult survival but particularly longevity of the native species influenced the spread velocity of the invader, at least when a juvenile bank is present. Other factors determining velocity of invasion detected by the IMCA model were seed dispersal distance and age of reproductive maturity. We derived relationships between species' adult survival, fecundity and longevity of both theoretical and applied relevance for biological invasions. Invasion velocities calculated from the aerial photographs agreed well with predictions of the IMCA model.     

Estimation of survival rate and extinction probability for stage-structured populations with overlapping life stages

The development of methods providing reliable estimates of demographic parameters (e.g., survival rates, fecundity) for wild populations is essential to better understand the ecology and conservation requirements of individual species. A number of methods exist for estimating the demographics of stage-structured populations, but inherent mathematical complexity often limits their uptake by conservation practitioners. Estimating survival rates for pond-breeding amphibians is further complicated by their complex migratory and reproductive behaviours, often resulting in nonobservable states and successive cohorts of eggs and tadpoles. Here we used comprehensive data on 11 distinct breeding toad populations (Bufo calamita) to clarify and assess the suitability of a relatively simple method [the Kiritani–Nakasuji–Manly (KNM) method] to estimate the survival rates of stage-structured populations with overlapping life stages. The study shows that the KNM method is robust and provides realistic estimates of amphibian egg and larval survival rates for species in which breeding can occur as a single pulse or over a period of several weeks. The study also provides estimates of fecundity for seven distinct toad populations and indicates that it is essential to use reliable estimates of fecundity to limit the risk of under- or overestimating the survival rates when using the KNM method. Survival and fecundity rates for B. calamita populations were then used to define population matrices and make a limited exploration of their growth and viability. The findings of the study recently led to the implementation of practical conservation measures at the sites where populations were most vulnerable to extinction.     

Reconstruction of a windborne insect invasion using a particle dispersal model,historical wind data,and Bayesian analysis of genetic data

Understanding how invasive species establish and spread is vital for developing effective management strategies for invaded areas and identifying new areas where the risk of invasion is highest. We investigated the explanatory power of dispersal histories reconstructed based on local‐scale wind data and a regional‐scale wind‐dispersed particle trajectory model for the invasive seed chalcid wasp Megastigmus schimitscheki (Hymenoptera: Torymidae) in France. The explanatory power was tested by: (1) survival analysis of empirical data on M. schimitscheki presence, absence and year of arrival at 52 stands of the wasp's obligate hosts, Cedrus (true cedar trees); and (2) Approximate Bayesian analysis of M. schimitscheki genetic data using a coalescence model. The Bayesian demographic modeling and traditional population genetic analysis suggested that initial invasion across the range was the result of long‐distance dispersal from the longest established sites. The survival analyses of the windborne expansion patterns derived from a particle dispersal model indicated that there was an informative correlation between the M. schimitscheki presence/absence data from the annual surveys and the scenarios based on regional‐scale wind data. These three very different analyses produced highly congruent results supporting our proposal that wind is the most probable vector for passive long‐distance dispersal of this invasive seed wasp. This result confirms that long‐distance dispersal from introduction areas is a likely driver of secondary expansion of alien invasive species. Based on our results, management programs for this and other windborne invasive species may consider (1) focusing effort at the longest established sites and (2) monitoring outlying populations remains critically important due to their influence on rates of spread. We also suggest that there is a distinct need for new analysis methods that have the capacity to combine empirical spatiotemporal field data, genetic data, and environmental data to investigate dispersal and invasion.     

Age or Stage Structure?

Discrete stage-structured density-dependent and discrete age-structured density-dependent population models are considered. Regarding the former, we prove that the model at hand is permanent (i.e., that the population will neither go extinct nor exhibit explosive oscillations) and given density dependent fecundity terms we also show that species with delayed semelparous life histories tend to be more stable than species which possess precocious semelparous life histories. Moreover, our findings together with results obtained from other stage-structured models seem to illustrate a fairly general ecological principle, namely that iteroparous species are more stable than semelparous species. Our analysis of various age-structured models does not necessarily support the conclusions above. In fact, species with precocious life histories now appear to possess better stability properties than species with delayed life histories, especially in the iteroparous case. We also show that there are dynamical outcomes from semelparous age-structured models which we are not able to capture in corresponding stage-structured cases. Finally, both age- and stage-structured population models may generate periodic dynamics of low period (either exact or approximate). The important prerequisite is to assume density-dependent survival probabilities.     

Harvest-based Bayesian estimation of sika deer populations using state-space models

We have estimated the number of sika deer, Cervus nippon, in Hokkaido, Japan, with the aim of developing a management program that will reduce the level of agricultural damage caused by these deer. A population index that is defined by the population divided by the population of 1993 is first estimated from the data obtained during a spotlight survey. A generalized linear mixed model (GLMM) with corner point constraints is used in this estimation. We then estimate the population from the index by evaluating the response of index to the known amount of harvest, including hunting. A stage-structured model is used in this harvest-based estimation. It is well-known that estimates of indices suffer from large observation errors when the probability of the observation fluctuates widely; therefore, we apply state-space modeling to the harvest-based estimation to remove the observation errors. We propose the use of Bayesian estimation with uniform prior-distributions as an approximation of the maximum likelihood estimation, without permitting an arbitrary assumption that the parameters fluctuate following prior-distributions. We are able to demonstrate that the harvest-based Bayesian estimation is effective in reducing the observation errors in sika deer populations, but the stage-structured model requires many demographic parameters to be known prior to running the analyses. These parameters cannot be estimated from the observed time-series of the index if there is insufficient data. We then construct a univariate model by simplifying the stage-structured model and show that the simplified model yields estimates that are nearly identical to those obtained from the stage-structured model. This simplification of the model simultaneously clarifies which parameter is important in estimating the population. Electronic supplementary material The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

Dynamic heterogeneity in life histories

Longitudinal data on natural populations have been analysed using multistage models in which survival depends on reproductive stage, and individuals change stages according to a Markov chain. These models are special cases of stage-structured population models. We show that stage-structured models generate dynamic heterogeneity: life-history differences produced by stochastic stratum dynamics. We characterize dynamic heterogeneity in a range of species across taxa by properties of the Markov chain: the entropy, which describes the extent of heterogeneity, and the subdominant eigenvalue, which describes the persistence of reproductive success during the life of an individual. Trajectories of reproductive stage determine survivorship, and we analyse the variance in lifespan within and between trajectories of reproductive stage. We show how stage-structured models can be used to predict realized distributions of lifetime reproductive success. Dynamic heterogeneity contrasts with fixed heterogeneity: unobserved differences that generate variation between life histories. We show by an example that observed distributions of lifetime reproductive success are often consistent with the claim that little or no fixed heterogeneity influences this trait. We propose that dynamic heterogeneity provides a 'neutral' model for assessing the possible role of unobserved 'quality' differences between individuals. We discuss fitness for dynamic life histories, and the implications of dynamic heterogeneity for the evolution of life histories and senescence.     

The stage-structured epidemic: linking disease and demography with a multi-state matrix approach model

Stage-structured epidemic models provide a way to connect the interacting processes of infection and demography. Reproduction and development can replenish the pool of susceptible hosts, and demographic structure leads to heterogeneous transmission and disease risk. Epidemics, in turn, can increase mortality or reduce fertility of the host population. Here we present a framework that integrates both demography and epidemiology in models for stage-structured epidemics. We use the vec-permutation matrix approach to classify individuals jointly by their demographic stage and infection status. We describe demographic and epidemic processes as alternating in time with a periodic matrix model. The application of matrix calculus to this framework allows for the calculation of R₀{\mathcal{R}_0} and sensitivity analysis.     

Optimal barrier zones for stopping the invasion of Aedes aegypti mosquitoes via transgenic or sterile insect techniques

Biological invasions have dramatically altered the natural world by threatening native species and their communities. Moreover, when the invading species is a vector for human disease, there are further substantive public health and economic impacts. The development of transgenic technologies is being explored in relation to new approaches for the biological control of insect pests. We investigate the use of two control strategies, classical sterile insect techniques and transgenic late-acting bisex lethality (Release of Insects carrying a Dominant Lethal), for controlling invasion of the mosquito Aedes aegypti using a spatial stage-structured mathematical model. In particular, we explore the use of a barrier zone of sterile/transgenic insects to prevent or impede the invasion of mosquitoes. We show that the level of control required is not only highly sensitive to the rate at which the sterile/transgenic males are released in the barrier zone but also to the spatial range of release. Our models characterise how the distribution of sterile/transgenic mosquitoes in the barrier zone can be controlled so as to minimise the number of mass-produced insects required for the arrest of species invasion. We predict that, given unknown rates of mosquito dispersal, management strategies should concentrate on larger release areas rather than more intense release rates for optimal control.     

Identifying non-invasible habitats for marine copepods using temperature-dependent <Emphasis Type="Italic">R</Emphasis><Subscript>0</Subscript>

If a non-indigenous species is to thrive and become invasive it must first persist under its new set of environmental conditions. Net reproductive rate (R ₀) represents the average number of female offspring produced by a female over its lifetime, and has been used as a metric of population persistence. We modeled R ₀ as a function of ambient water temperature (T) for the invasive marine calanoid copepod Pseudodiaptomus marinus, which is introduced to west coast of North America from East Asia by ship ballast water. The model was based on temperature-dependent stage-structured population dynamics given by a system of ordinary differential equations. We proposed a methodology to identify habitats that are non-invasible for P. marinus using the threshold of R ₀(T) < 1 in order to identify potentially invasible habitats. We parameterized the model using published data on P. marinus and applied R ₀(T) to identify the range of non-invasible habitats in a global scale based on sea surface temperature data. The model predictions matched the field evidence of species occurrences well.     

A method for the estimation of parameters for natural stage-structured populations

A relatively simple method is proposed for the estimation of parameters of stage-structured populations from sample data for situation where (a) unit time survival rates may vary with time, and (b) the distribution of entry times to stage 1 is too complicated to be fitted with a simple parametric model such as a normal or gamma distribution. The key aspects of this model are that the entry time distribution is approximated by an exponential function withp parameters, the unit time survival rates in stages are approximated by anr parameter exponential polynomial in the stage number, and the durations of stages are assumed to be the same for all individuals. The new method is applied to four Zooplankton data sets, with parametric bootstrapping used to assess the bias and variation in estimates. It is concluded that good estimates of demographic parameters from stagefrequency data from natural populations will usually only be possible if extra information such as the durations of stages is known.     

Integrating demographic data and a mechanistic dispersal model to predict invasion spread of Rhododendron ponticum in different habitats

Understanding the invasion potential of a species in different habitat types within the non-native range is crucial in prioritising management and control efforts, and in the protection of vulnerable habitats through monitoring. Here, using the invasive shrub Rhododendron ponticum as a case study, we integrate information on both the demographics and spatial dynamics within an individual-based, spatially-explicit model to investigate invasion potential in different habitats. Firstly, empirical demographic data were used to establish relationships between demographic traits, such as height and fecundity, and habitat variables. The outputs from models fitted using a Generalised Linear Model approach were then incorporated into an individual-based simulation model of plant spread to investigate the invasion potential in different habitats using a factorial design of treatments. Plant height, and thus seed release height, was the main driver of invasion speed through an increase in dispersal potential, which resulted in the highest invasion speeds predicted for evergreen woodlands, and relatively low speeds for open habitats. Conversely, invasion density was driven by plant fecundity and seedling survival and not dispersal potential; the highest invasion densities were predicted for open habitats, with relatively low densities for evergreen habitats. Deciduous woodland had features resulting in intermediate invasion potential, both in terms of speed and density and may, therefore be the habitat that is most vulnerable to relatively rapid and dense invasion.     

All or nothing: Survival,reproduction and oxidative balance in Spotted Wing Drosophila (Drosophila suzukii) in response to cold

Winter severity and overwintering capacity are key ecological factors in successful invasions, especially in ectotherms. The integration of physiological approaches into the study of invasion processes is emerging and promising. Physiological information describes the mechanisms underlying observed survival and reproductive capacities, and it can be used to predict an organism’s response to environmental perturbations such as cold temperatures. We investigated the effects of various cold treatments on life history and physiological traits of an invasive pest species, Drosophila suzukii, such as survival, fertility and oxidative balance. This species, a native of temperate Asian areas, is known to survive where cold temperatures are particularly harsh and has been recently introduced into Europe and North America. We found that cold treatments had a strong impact on adult survival but no effect on female’s fertility. Although only minor changes were observed after cold treatment on studied physiological traits, a strong sex-based difference was observed in both survival and physiological markers (antioxidant defences and oxidative markers). Females exhibited higher survival, reduced oxidative defences, less damage to nucleic acids, and more damage to lipids. These results suggest that D. suzukii relies on a pathway other than oxidative balance to resist cold injury. Altogether, our results provide information concerning the mechanisms of successful invasion by D. suzukii. These findings may assist in the development of population models that predict the current and future geographic ranges of this species.