Periodic Matrix Population Model Research Articles

Demographic consequences of changes in environmental periodicity.

The fate of natural populations is mediated by complex interactions among vital rates, which can vary within and among years. Although the effects of random, among-year variation in vital rates have been studied extensively, relatively little is known about how periodic, nonrandom variation in vital rates affects populations. This knowledge gap is potentially alarming as global environmental change is projected to alter common periodic variations, such as seasonality. We investigated the effects of changes in vital-rate periodicity on populations of three species representing different forms of adaptation to periodic environments: the yellow-bellied marmot (Marmota flaviventer), adapted to strong seasonality in snowfall; the meerkat (Suricata suricatta), adapted to inter-annual stochasticity as well as seasonal patterns in rainfall; and the dewy pine (Drosophyllum lusitanicum), adapted to fire regimes and periodic post-fire habitat succession. To assess how changes in periodicity affect population growth, we parameterized periodic matrix population models and projected population dynamics under different scenarios of perturbations in the strength of vital-rate periodicity. We assessed the effects of such perturbations on various metrics describing population dynamics, including the stochastic growth rate, log λS . Overall, perturbing the strength of periodicity had strong effects on population dynamics in all three study species. For the marmots, log λS decreased with increased seasonal differences in adult survival. For the meerkats, density dependence buffered the effects of perturbations of periodicity on log λS . Finally, dewy pines were negatively affected by changes in natural post-fire succession under stochastic or periodic fire regimes with fires occurring every 30years, but were buffered by density dependence from such changes under presumed more frequent fires or large-scale disturbances. We show that changes in the strength of vital-rate periodicity can have diverse but strong effects on population dynamics across different life histories. Populations buffered from inter-annual vital-rate variation can be affected substantially by changes in environmentally driven vital-rate periodic patterns; however, the effects of such changes can be masked in analyses focusing on inter-annual variation. As most ecosystems are affected by periodic variations in the environment such as seasonality, assessing their contributions to population viability for future global-change research is crucial.

A matrix population model to aid agency response to grass carp (Ctenopharyngodon idella) in the Great Lakes Basin - Lake Erie

Managers and researchers have identified a reproducing population of grass carp (Ctenopharyngodon idella) in the western basin of Lake Erie, generating concern over the potential threat to ecosystem function in the Great Lakes Basin. Capture histories indicate that grass carp may be present at low levels in other areas of Lake Erie, necessitating a large scale, multi-jurisdictional response. As a result, a group of experts and decision makers began a structured decision making exercise to collaboratively address the threat and identify potential response actions. To aid this process, we developed a spatially-explicit periodic matrix population model to project grass carp abundance, and probabilistically evaluate specific management actions. We evaluated four potential management response actions ranging from no action, diffuse removal efforts, and concentrated removal efforts with and without a barrier on the Sandusky River to reduce spawning success. Based on our current knowledge, concentrated removal including a barrier on the Sandusky River provides the most likely path to achieving and maintaining a management target of no more than 10 fish/ha. Our understanding of grass carp ecology in Lake Erie is growing. This model and parameter development methods were designed to flexibly accommodate new information as our understanding of grass carp ecology evolves, or management objectives change. Ultimately, this modeling framework and use of Bayesian methods could facilitate management response efforts for other invasive species occurring over large scales and multiple jurisdictions.

Ecological modeling for the extrapolation of ecotoxicological effects measured during in situ assays in Gammarus.

Evaluating the effects of chemical contamination on populations and ecological communities still constitutes a challenging necessity in environmental management. However, the toxic effects of contaminants are commonly measured by means of organism-level responses. Linking such effects measures with ecological models is a promising way to determine population-level impacts. In this way, population models are currently increasingly used in predictive risk assessment procedures, but their use in environmental diagnostic framework remains limited due to their lack of ecological realism. The present study with the crustacean Gammarus fossarum, a sentinel species in freshwater monitoring, combines a dual field and laboratory experimental approach with a population modeling framework. In this way, we developed an ecologically relevant periodic matrix population model for Gammarus. This model allowed us to capture the population dynamics in the field, and to understand the particular pattern of demographic sensitivities induced by Gammarus life-history phenology. The model we developed provided a robust population-level assessment of in situ-based effects measures recorded during a biomonitoring program on a French watershed impacted by past mining activities. Thus, our study illustrates the potential of population modeling when seeking to decipher the role of environmental toxic contamination in ecological perturbations.

Crop–weed competition rather than temperature limits the population establishment of two annual C4 weeds at the edge of their northern range

SummaryClimate change is predicted to affect range expansion of harmful C4 weeds into the boreal region, given that they are able to successfully colonise both C3 and C4 crops. We studied the impact of a 3°C elevation in temperature on the establishment and maintenance of populations of two annual C4 weeds (Amaranthus retroflexus and Echinochloa crus‐galli) with and without a competing C3 (barley) or C4 (maize) crop. Data obtained from field and glasshouse experiments were modelled using a periodic matrix population model. Competition of a weed with a crop appeared to be a more important factor for limiting the maintenance of weed populations than elevation in temperature, as neither of the weed species was able to maintain populations in competition with crops. Even an increase in the frequency of warm years did not result in viable weed populations establishing. However, A. retroflexus was able to form persistent populations in competition with maize when released from competition every fifth year. Simulations parameterised from glasshouse data predicted that both weed species would persist without competition in the current climate, whereas simulations parameterised from field data suggested only A. retroflexus to be able to persist. These results demonstrate that competition affects the range expansion of arable weed species more than elevation in temperature, necessitating the inclusion of crop–weed interactions in models of range shifts as a consequence of climate change.

Demographic responses of the invasive annual grass Microstegium vimineum to prescribed fires and herbicide

Management of invasive species in forests often includes combinations of prescribed fire and herbicides. However for most efficient management, evaluations of these tools should include whole-population responses of targeted plants. In this study, we evaluated how the timing and frequency of prescribed fire and herbicide application affected population growth of the invasive annual grass Microstegium vimineum (stiltgrass) using periodic matrix population models. We conducted an experiment in M. vimineum-invaded deciduous forests in Indiana, USA to compare effects of spring and fall prescribed fires combined with pre- or post-emergent herbicide on M. vimineum populations and to build matrix population models predicting long-term population responses to these management treatments across multiple life-history stages. We found that spring fires were effective at reducing population growth rates during the year of treatment but there was no effect of burning on M. vimineum populations the following year. Similarly, fall prescribed fires were effective at reducing seed production, as well as numbers of seedlings and adults following fires, but had no long-term effect on population growth rates. Post-emergent herbicide alone was the only treatment that reduced M. vimineum population growth beyond 1year. Seedbank survival had the highest life-stage elasticity across all treatments, indicating that novel management methods specifically designed to exhaust seedbanks for three or more years may be needed to prevent M. vimineum population resurgence after cessation of treatments.

Life‐history phenology strongly influences population vulnerability to toxicants: A case study with the mudsnail Potamopyrgus antipodarum

One of the main objectives of ecological risk assessment is to evaluate the effects of toxicants on ecologically relevant biological systems such as populations or communities. However, the effects of toxicants are commonly measured on selected subindividual or individual endpoints due to their specificity against chemical stressors. Introducing these effects into population models is a promising way to predict impacts on populations. The models currently employed are very simplistic, and their environmental relevance needs to be improved to establish the ecological relevance of hazard assessment. The present study with the gastropod Potamopyrgus antipodarum combines a field experimental approach with a modeling framework. It clarifies the role played by seasonal variability of life-history traits in the population's vulnerability to the alteration of individual performance, potentially due to toxic stress. The present study comprised 3 steps: 1) characterization of the seasonal variability in life-history traits of a local population over 1 yr by using in situ experiments on caged snails, coupled with a demographic follow-up; 2) development of a periodic matrix population model that visualizes the monthly variability of population dynamics; and 3) simulation of the demographic consequences of an alteration in life-history traits (i.e., fertility, juvenile, and adult survival). The results revealed that demographic impacts strongly depend on the season when alterations of individual performance occur. Model analysis showed that this seasonal variability in population vulnerability is strongly related to the phenology of the population. The authors emphasize that improving the realism of population models is a major objective for ecological risk assessment, and that taking into account species phenology in modeling approaches should be a priority.

Sensitivity analysis of periodic matrix population models

Periodic matrix models are frequently used to describe cyclic temporal variation (seasonal or interannual) and to account for the operation of multiple processes (e.g., demography and dispersal) within a single projection interval. In either case, the models take the form of periodic matrix products. The perturbation analysis of periodic models must trace the effects of parameter changes, at each phase of the cycle, on output variables that are calculated over the entire cycle. Here, we apply matrix calculus to obtain the sensitivity and elasticity of scalar-, vector-, or matrix-valued output variables. We apply the method to linear models for periodic environments (including seasonal harvest models), to vec-permutation models in which individuals are classified by multiple criteria, and to nonlinear models including both immediate and delayed density dependence. The results can be used to evaluate management strategies and to study selection gradients in periodic environments.

Climate change, phenological shifts, eco-evolutionary responses and population viability: toward a unifying predictive approach

The debate on emission targets of greenhouse gasses designed to limit global climate change has to take into account the ecological consequences. One of the clearest ecological consequences is shifts in phenology. Linking these shifts to changes in population viability under various greenhouse gasses emission scenarios requires a unifying framework. We propose a box-in-a-box modeling approach that couples population models to phenological change. This approach unifies population modeling with both ecological responses to climate change as well as evolutionary processes. We advocate a mechanistic embedded correlative approach, where the link from genes to population is established using a periodic matrix population model. This periodic model has several major advantages: (1) it can include complex seasonal behaviors allowing an easy link with phenological shifts; (2) it provides the structure of the population at each phase, including the distribution of genotypes and phenotypes, allowing a link with evolutionary processes; and (3) it can incorporate the effect of climate at different time periods. We believe that the way climatologists have approached the problem, using atmosphere–ocean coupled circulation models in which components are gradually included and linked to each other, can provide a valuable example to ecologists. We hope that ecologists will take up this challenge and that our preliminary modeling framework will stimulate research toward a unifying predictive model of the ecological consequences of climate change.

Multiscale topoedaphic heterogeneity increases resilience and resistance of a dominant grassland species to extreme drought and climate change

AbstractIt is argued that the inclusion of spatially heterogeneous environments in biodiversity reserves will be an effective means of encouraging ecosystem resilience and plant community conservation under climate change. However, the resilience and resistance of plant populations to global change, the specific life‐history traits involved and the spatial scale at which environmentally driven demographic variation is expressed remains largely unknown for most plant groups. Here we address these questions by reporting an empirical investigation into the impacts of an unprecedented 3‐year drought on the demography, population growth rates (λ) and biogeographical distribution of core populations of the perennial grassland species Austrostipa aristiglumis in semiarid Australia. We use life‐history analysis and periodic matrix population models to specifically test the hypothesis that patch‐ and habitat‐scale variation in vital life‐history parameters result in spatial differences in the resilience and resistance of A. aristiglumis populations to extreme drought. We show that the development of critical soil water deficits during drought resulted in collapse of adult A. aristiglumis populations (λ≪1), rapid interhabitat phytosociological change and overall contraction towards mesic refugia where populations were both more resistant and resilient to perturbation. Population models, combined with climatic niche analysis, suggest that, even in core areas, a significant reduction in size and habitat range of A. aristiglumis populations is likely under climate change expected this century. Remarkably, however, we show that even minor topographic variation (0.2–3 m) can generate significant variation in demographic parameters that confer population‐level resilience and resistance to drought. Our findings support the hypothesis that extreme climatic events have the capacity to induce rapid, landscape‐level shifts in core plant populations, but that the protection of topographically heterogeneous environments, even at small spatial scales, may play a key role in conserving biodiversity under climate change in the coming century.

Management recommendations for short‐lived weeds depend on model structure and explicit characterization of density dependence

Summary 1. Multiple modelling techniques are currently used to describe population dynamics of established invasions, where intraspecific competition is likely to reduce survival, growth and/or fecundity, suppressing population growth rate. To date, it remains unanswered whether these modelling techniques produce similar management recommendations for density‐dependent weed populations and how to model density dependence to better inform management. 2. We constructed demographic models for a short‐lived weed based on data on multiple manipulated densities in a glasshouse and data from the literature using three germination strategies. We compared management recommendations produced by two main modelling techniques for density‐dependent weed populations and examined whether periodic matrix population models constructed from different densities without characterization of density‐dependent processes (implicit models) produce management recommendations similar to that of the same models with density dependence explicitly characterized and simulated (explicit models). The use of a periodic matrix population model enabled us to target simulated management on either vital rates or entire life stages, and to examine the role of a weed’s germination strategy on model outcomes. 3. Management recommendations differed depending on how density dependence was included in demographic models. Explicit models showed that management conducted after the density‐dependent process driving population dynamics best curbed density‐regulated weed populations, with reductions in seed production having a negligible effect regardless of a germination strategy. By contrast, implicit models constructed from multiple densities produced similar management recommendations for sparse and dense populations, with reductions in survival to a flowering stage, juvenile establishment or seed production leading to the greatest predicted declines in weed density. 4. Our results emphasize the importance of model structure when modelling dynamics of density‐dependent weed populations, suggesting that explicit characterization and inclusion of density dependence in population models is often necessary to inform management. As a weed’s germination strategy had a minor effect on model outcomes, our findings about explicit and implicit modelling techniques can be generalized across annual plants with non‐overlapping generations.

Periodic Matrix Population Models: Growth Rate, Basic Reproduction Number, and Entropy

This article considers three different aspects of periodic matrix population models. First, a formula for the sensitivity analysis of the growth rate lambda is obtained that is simpler than the one obtained by Caswell and Trevisan. Secondly, the formula for the basic reproduction number R0 in a constant environment is generalized to the case of a periodic environment. Some inequalities between lambda and R0 proved by Cushing and Zhou are also generalized to the periodic case. Finally, we add some remarks on Demetrius' notion of evolutionary entropy H and its relationship to the growth rate lambda in the periodic case.

USING MATRIX MODELS TO DETERMINE CROPPING SYSTEM EFFECTS ON ANNUAL WEED DEMOGRAPHY

The design of most agricultural systems ignores the possibility that cropping system characteristics may affect weed population dynamics by altering key demographic rates of weeds. We examined legume green manure and tillage timing effects upon giant foxtail (Setaria faberi) demography with prospective and retrospective perturbation analyses of a periodic matrix population model. Demographic data were collected for S. faberi grown in a wheat–corn–soybean crop sequence in the central USA in 2000 and 2001, with either a wheat sole‐crop (W) or wheat/red clover crop mixture (R) in the wheat phase. Wheat phase residues were incorporated either in fall (FT) or spring (ST) to form four cropping system treatments: FT/W, FT/R, ST/W, and ST/R. Demographic rates estimated from the field data included seed survival from October to March and March to October, seedling recruitment, plant survival, fecundity, and seed predation. The variable efficacy of postemergence weed control was modeled by varying the proportion of seedlings surviving to reproductive maturity from 0.025 to 0.20. Deterministic simulations indicated that there was both interannual and management‐induced variation in S. faberi population growth rate. Stochastic population growth rate depended upon an interaction between legume green manure and tillage‐timing effects, such that stochastic population growth rate was lowest for fall‐tilled wheat and red clover, greater for fall‐tilled wheat, and greatest for both spring‐tilled management regimes. Within the fall‐tilled regime, the cropping system (wheat vs. wheat plus clover) had a greater influence on relative rates of stochastic population growth as plant survival increased. However, plant survival did not affect the stochastic population growth rates in the spring‐tilled treatments. Elasticity analysis suggested that seed survival through the winter, fecundity, and the proportion of seeds escaping predation were important driving variables for this system. Retrospective perturbation analysis supported these results but also indicated that fecundity and predation levels varied more in response to changing management treatments than winter seed survival, leading to their having a greater influence on differences in population growth among the four management treatments. Perturbation analyses of matrix population models will aid the development of integrated weed‐management systems by elucidating cropping‐system effects upon weed demography.

Sensitivity analysis in periodic matrix models: A postscript to Caswell and Trevisan

Periodic matrix population models are a useful approach t@ modelling cyclic variations in demographic rates. Caswell and Trevisan [1] introduced the perturbation analysis (sensitivities and elasticities) @f the per-cycle population growth rate for such models. Although powerful, their method can be time-consuming when the dimension of the matrices is large @r when cycles are composed of many phases. We present a more efficient method, based @n a very simple matrix product. We compared the two methods for matrices of different sizes. We observed a reduction in calculation time on the order of 24% with the new method for a set @f 26 within-year Leslie matrices of size 287 × 287. The time saving may become particularly significant when sensitivities are used in Monte Carlo or bootstrap simulations.