Investigations into a plankton population model: Mortality and its importance in climate change scenarios

The potential for marine plankton ecosystems to influence climate by the production of dimethylsulphide (DMS) has been an important topic of recent research into climate change. Several General Circulation Models, used to predict climate change, have or are being modified to include interactions of ecosystems with climate. Climate change necessitates that parameters within ecosystem models must change during long-term simulations, especially mortality parameters that increase as organisms are pushed toward the boundaries of their thermal tolerance. Changing mortality parameters can have profound influences on ecosystem model dynamics. There is therefore a pressing need to understand the influence of varying mortality parameters on the long-term behaviour of ecosystem models. This work examines the sensitivity of a model of DMS production by marine ecosystems to variations in three linear mortality coefficients. Significant differences in behaviour are observed, and we note the importance of these results in formulating ecosystem models for application in simulations of climate change.     

Linking species- and ecosystem-level impacts of climate change in lakes with a complex and a minimal model

To study the interaction between species- and ecosystem-level impacts of climate change, we focus on the question of how climate-induced shifts in key species affect the positive feedback loops that lock shallow lakes either in a transparent, macrophyte-dominated state or, alternatively, in a turbid, phytoplankton-dominated state. We hypothesize that climate warming will weaken the resilience of the macrophyte-dominated clear state. For the turbid state, we hypothesize that climate warming and climate-induced eutrophication will increase the dominance of cyanobacteria. Climate change will also affect shallow lakes through a changing hydrology and through climate change-induced eutrophication. We study these phenomena using two models, the full ecosystem model PCLake and a minimal dynamic model of lake phosphorus dynamics. Quantitative predictions with the complex model show that changes in nutrient loading, hydraulic loading and climate warming can all lead to shifts in ecosystem state. The minimal model helped in interpreting the non-linear behaviour of the complex model. The main output parameters of interest for water quality managers are the critical nutrient loading at which the system will switch from clear to turbid and the much lower critical nutrient loading – due to hysteresis – at which the system switches back. Another important output parameter is the chlorophyll-a level in the turbid state. For each of these three output parameters we performed a sensitivity analysis to further understand the dynamics of the complex model PCLake. This analysis showed that our model results are most sensitive to changes in temperature-dependence of cyanobacteria, planktivorous fish and zooplankton. We argue that by combining models at various levels of complexity and looking at multiple aspects of climate changes simultaneously we can develop an integrated view of the potential impact of climate change on freshwater ecosystems.     

Feasibility and stability in randomly assembled Lotka-Volterra models

For a Lotka-Volterra model to represent a viable ecosystem it's nontrivial equilibrium must be feasible. If m is the number of species, it is shown that in a set of randomly assembled Lotka-Volterra models, the fraction of models with a feasible equilibrium is some function of m which behaves like 2^?m. Moreover a subset of Lotka-Volterra models, each of which has a feasible equilibrium, has the same stability property as a set of linear models which is assembled randomly in the same manner. This contradicts a recent claim that a Lotka-Volterra model with a feasible equilibrium tends to be stable. Thus for two reasons the probability that a Lotka-Volterra model represents a viable and stable ecosystem decreases rapidly with the number of species. This supports the theme developed by May that stability in model ecosystems decreases with diversity.     

Nonlinear relationships between vital rates and state variables in demographic models

To accurately estimate population dynamics and viability, structured population models account for among-individual differences in demographic parameters that are related to individual state. In the widely used matrix models, such differences are incorporated in terms of discrete state categories, whereas integral projection models (IPMs) use continuous state variables to avoid artificial classes. In IPMs, and sometimes also in matrix models, parameterization is based on regressions that do not always model nonlinear relationships between demographic parameters and state variables. We stress the importance of testing for nonlinearity and propose using restricted cubic splines in order to allow for a wide variety of relationships in regressions and demographic models. For the plant Borderea pyrenaica, we found that vital rate relationships with size and age were nonlinear and that the parameterization method had large effects on predicted population growth rates, X (linear IPM, 0.95; nonlinear IPMs, 1.00; matrix model, 0.96). Our results suggest that restricted cubic spline models are more reliable than linear or polynomial models. Because even weak nonlinearity in relationships between vital rates and state variables can have large effects on model predictions, we suggest that restricted cubic regression splines should be considered for parameterizing models of population dynamics whenever linearity cannot be assumed.     

Representing mediating effects and species reintroductions in Ecopath with Ecosim

Ecosystem models play an important role in supporting ecosystem approaches to management. To improve the representation of how ecosystems work, ecosystem models should be able to represent mediating effects (e.g., habitat provision) that species provide to each other as well as species (re)introductions, both common situations that can strongly influence ecosystem dynamics. We examine how such processes can be incorporated into Ecopath with Ecosim (EwE), a widely used tool for represent aquatic ecosystems with the potential to support ecosystem-based management. We used the reintroduction of sea otters (Enhydralutris) to the west coast of Vancouver Island, British Columbia, Canada as a case study. The model demonstrates how to account for benefits provided by kelp forests by contributing to primary production, increased feeding areas and food availability through prey retention. It also demonstrates how the reintroduction and range expansion of sea otters can be represented in Ecospace, and the implications of these options.     

Eigengroup analyses of linear ecosystem models

Eigenvalue analysis has been widely used for characterization of model ecosystems, and utilization of the total eigensystem information has been demonstrated for ecosystem models. Solution of the unique eigenvectors, termed eigengroups, associated with specified perturbation initial conditions allows proper assignment of the ecosystem model content to each exponential decay function. Such eigengroup solutions of Monte Carlo assigned perturbations are reported here for a nine-compartment tropical moist forest phosphorus model and for a set of seven six-compartment general nutrient cycling models. Confirmed by CSMP III STIFF numerical integration, results indicate that the dynamics of these multi-compartment models, each having many internal closed cycles, are adequately described by only one or two exponential decay functions, even at massive perturbation levels. For these models, conjugate complex noncritical eigenvalues were found to be common, but the decay rates of the envelopes of oscillation are consistently so rapid that system-intrinsic oscillations are undetectable. The techniques used here are presented as efficient means of identifying the transient response space of linear ecosystem models.     

Increased spatial variance accompanies reorganization of two continental shelf ecosystems

Phase transitions between alternate stable states in marine ecosystems lead to disruptive changes in ecosystem services, especially fisheries productivity. We used trawl survey data spanning phase transitions in the North Pacific (Gulf of Alaska) and the North Atlantic (Scotian Shelf) to test for increases in ecosystem variability that might provide early warning of such transitions. In both time series, elevated spatial variability in a measure of community composition (ratio of cod [Gadus sp.] abundance to prey abundance) accompanied transitions between ecosystem states, and variability was negatively correlated with distance from the ecosystem transition point. In the Gulf of Alaska, where the phase transition was apparently the result of a sudden perturbation (climate regime shift), variance increased one year before the transition in mean state occurred. On the Scotian Shelf, where ecosystem reorganization was the result of persistent overfishing, a significant increase in variance occurred three years before the transition in mean state was detected. However, we could not reject the alternate explanation that increased variance may also have simply been inherent to the final stable state in that ecosystem. Increased variance has been previously observed around transition points in models, but rarely in real ecosystems, and our results demonstrate the possible management value in tracking the variance of key parameters in exploited ecosystems.     

A bio-inspired computing model as a new tool for modeling ecosystems: The avian scavengers as a case study

The models used for ecosystems modeling are generally based on differential equations. However, in recent years new computational models based on biological processes, or bioinspired models, have arisen, among which are P systems. These are inspired by the functions of cells and present important advantages with respect to traditional models, such as a high computational efficiency, modularity and their ability to work in parallel. They are simple, individual-based models that use biological parameters that can be obtained experimentally. In this work, we present the framework for a model based on P systems applied to the study of an ecosystem in which three avian scavengers (predators) interact with 10 wild and domestic ungulates (preys). The computation time for 100 repetitions, corresponding to 14 simulation years each, with an initial population composed of 385,422 individuals, was 30 min. Our results suggest that the model presented, based on P systems, correctly simulates the population dynamics in the period of time analyzed. We discuss the usefulness of this tool in simulating complex ecosystems dynamics to aid managers, conservationists and policy-makers in making appropriate decisions for the improvement of management and conservation programs.     

Multistate modeling of habitat dynamics: factors affecting Florida scrub transition probabilities

Many ecosystems are influenced by disturbances that create specific successional states and habitat structures that species need to persist. Estimating transition probabilities between habitat states and modeling the factors that influence such transitions have many applications for investigating and managing disturbance-prone ecosystems. We identify the correspondence between multistate capture-recapture models and Markov models of habitat dynamics. We exploit this correspondence by fitting and comparing competing models of different ecological covariates affecting habitat transition probabilities in Florida scrub and flatwoods, a habitat important to many unique plants and animals. We subdivided a large scrub and flatwoods ecosystem along central Florida's Atlantic coast into 10-ha grid cells, which approximated average territory size of the threatened Florida Scrub-Jay (Aphelocoma coerulescens), a management indicator species. We used 1.0-m resolution aerial imagery for 1994, 1999, and 2004 to classify grid cells into four habitat quality states that were directly related to Florida Scrub-Jay source-sink dynamics and management decision making. Results showed that static site features related to fire propagation (vegetation type, edges) and temporally varying disturbances (fires, mechanical cutting) best explained transition probabilities. Results indicated that much of the scrub and flatwoods ecosystem was resistant to moving from a degraded state to a desired state without mechanical cutting, an expensive restoration tool. We used habitat models parameterized with the estimated transition probabilities to investigate the consequences of alternative management scenarios on future habitat dynamics. We recommend this multistate modeling approach as being broadly applicable for studying ecosystem, land cover, or habitat dynamics. The approach provides maximum-likelihood estimates of transition parameters, including precision measures, and can be used to assess evidence among competing ecological models that describe system dynamics.     

Modelling coral reef ecosystems with limited observational data

Ecosystem models represent potentially powerful tools for coral reef ecosystem managers. They can provide insight into ecosystem dynamics not achievable through alternative means allowing coral reef managers to assess the potential outcome of any given management decision. One of the main limitations in the applicability of ecosystem models is that they often require detailed empirical data and this can restrict their applicability to ecosystems that are either currently well studied or have the resources available to collect the required data. This study describes the development of a coral reef ecosystem model that can be calibrated to an ecosystem with limited empirical data. Based on the assumption that coral reef ecological structure is generic across all tropical coral reefs and that the magnitude of the interactions between ecological components is reef specific, the dynamics of the ecosystem can be replicated based on limited empirical data. The model successfully replicated the dynamics of three individual reef systems including an inshore and oceanic reef within the Great Barrier Reef and a Caribbean reef system. It highlighted the importance of understanding the specific dynamics of a given reef and that a positive management intervention in one system may result in a negative outcome for another. The model was also used to assess the importance of various interactions within coral reef ecosystems. It identified the interactions between hard corals and other non-algal benthic components as being an important (but currently understudied) facet of coral reef ecology. The development of this modelling approach provides access to ecosystem modelling tools for coral reef managers previously excluded due to a lack of resources or technical expertise.     

Mortality and predation in ecosystem models: is it important how these are expressed?

The effects of the form of the grazing and mortality terms used in plankton models are well known. The same cannot be said for ecosystem models. As ecosystem models become more popular more needs to be known about the effects of model formulation on model behaviour and performance. The impact of the form of the grazing response function and mortality terms used in a biogeochemical ecosystem model are considered here. We show that in the large and inter-linked webs used in ecosystem models, model behaviour is far more sensitive to the form of the grazing term than to that of the mortality terms that close the modelled food web.When using biogeochemical ecosystem models in shallow marine ecosystems, the most dynamic and sophisticated functional responses describing grazing require more parameters and validation than the simpler Holling disk equation, but usually still lead to the same general conclusions about the system state and the effects of changes in forcing functions. Thus, the use of more complex functional responses is not necessarily warranted in many cases. Similarly, the extra effort and data required to explicitly represent the top predators (sharks, mammals and birds) is not necessary if they are not the focus of the study. A quadratic mortality term applied to intermediate predators (such as piscivores) is sufficient to achieve plausible model behaviour. It should be noted, however, that some degree of sophistication is required in the grazing and mortality terms. Use of simple linear functional responses and mortality terms is unsuitable for models used to consider a range of nutrient loading or harvesting scenarios.     

On the latent state estimation of nonlinear population dynamics using Bayesian and non-Bayesian state-space models

Nonlinear state-space models have been increasingly applied to study population dynamics and data assimilation in environmental sciences. State-space models can account for process error and measurement error simultaneously to correct for the bias in the estimates of system state and model parameters. However, few studies have compared the performance of different nonlinear state-space models for reconstructing the state of population dynamics from noisy time series. This study compared the performance of the extended Kalman filter (EKF), unscented Kalman filter (UKF) and Bayesian nonlinear state-space models (BNSSM) through simulations. Synthetic population time series were generated using the theta logistic model with known parameters, and normally distributed process and measurement errors were introduced using the Monte Carlo simulations. At higher levels of nonlinearity, the UKF and BNSSM had lower root mean square error (RMSE) than the EKF. The BNSSM performed reliably across all levels of nonlinearity, whereas increased levels of nonlinearity resulted in higher RMSE of the EKF. The Metropolis–Hastings algorithm within the Gibbs algorithm was used to fit the theta logistic model to synthetic time series to estimate model parameters. The estimated posterior distribution of the parameter θ indicated that the 95% credible intervals included the true values of θ (=0.5 and 1.5), but did not include 1.0 and 0.0. Future studies need to incorporate the adaptive Metropolis algorithm to estimate unknown model parameters for broad applications of Bayesian nonlinear state-space models in ecological studies.     

Efficient and sustainable management of complex forest ecosystems

A large range of models has been developed for the analysis of optimal forest management strategies, with the well-known Faustmann models dating back to the mid-19th century. To date, however, there has been relatively little attention for the implications of complex ecosystem dynamics for optimal forest management. This paper examines the implications of irreversible ecosystem responses for efficient and sustainable forest management. The paper is built around two forest models that comprise two ecosystem components, forest cover and topsoil, the interactions between these components, and the supply of the ecosystem services ‘wood’ and ‘erosion control’. The first model represents a forest that responds in a reversible way to overharvesting. In the second model, an additional ecological process has been included and the ecosystem irreversibly collapses below certain thresholds in forest cover and topsoil depth. The paper presents a general model, and demonstrates the implications of pursuing efficient as well as sustainable forest management for the two forest ecosystems. Both fixed and variable harvesting cycles are examined. Efficient and sustainable harvesting cycles are compared, and it is shown that irreversible ecosystem behaviour reduces the possibilities to reconcile efficient and sustainable forest management through a variable harvesting cycle.     

Indicators of ecosystem function identify alternate states in the sagebrush steppe

Models of ecosystem change that incorporate nonlinear dynamics and thresholds, such as state-and-transition models (STMs), are increasingly popular tools for land management decision-making. However, few models are based on systematic collection and documentation of ecological data, and of these, most rely solely on structural indicators (species composition) to identify states and transitions. As STMs are adopted as an assessment framework throughout the United States, finding effective and efficient ways to create data-driven models that integrate ecosystem function and structure is vital. This study aims to (1) evaluate the utility of functional indicators (indicators of rangeland health, IRH) as proxies for more difficult ecosystem function measurements and (2) create a data-driven STM for the sagebrush steppe of Colorado, USA, that incorporates both ecosystem structure and function. We sampled soils, plant communities, and IRH at 41 plots with similar clayey soils but different site histories to identify potential states and infer the effects of management practices and disturbances on transitions. We found that many IRH were correlated with quantitative measures of functional indicators, suggesting that the IRH can be used to approximate ecosystem function. In addition to a reference state that functions as expected for this soil type, we identified four biotically and functionally distinct potential states, consistent with the theoretical concept of alternate states. Three potential states were related to management practices (chemical and mechanical shrub treatments and seeding history) while one was related only to ecosystem processes (erosion). IRH and potential states were also related to environmental variation (slope, soil texture), suggesting that there are environmental factors within areas with similar soils that affect ecosystem dynamics and should be noted within STMs. Our approach generated an objective, data-driven model of ecosystem dynamics for rangeland management. Our findings suggest that the IRH approximate ecosystem processes and can distinguish between alternate states and communities and identify transitions when building data-driven STMs. Functional indicators are a simple, efficient way to create data-driven models that are consistent with alternate state theory. Managers can use them to improve current model-building methods and thus apply state-and-transition models more broadly for land management decision-making.     

Age-dependent population dynamics with several interior variables and spatial spread

The problems of the asymptotic behavior of age-dependent population models with interior and spatial structures are considered. It is proved that the existence and uniqueness of the stable state and its exact form is founded for general linear models. Problems on the speed of convergence to stable state and transitional effects are investigated. Methods of solving two special classes of nonlinear models (separate models and models of the Gurtin-MacCami type) are suggested. A model of forest stand dynamics on the basis of conception of layer-mosaic characteristics of the spatial-temporal structure of stands is examined as an example of the application of given results.     

Biological responses to environmental forcing: the linear tracking window hypothesis

Determining the relative contributions of intrinsic and extrinsic processes to the regulation of biological populations has been a recurrent ecological issue. Recent discussions concerning ecosystem "regime shifts" again raise the question of whether population fluctuations are mainly controlled by external forcing. Results of nonlinear time series analyses indicate that pelagic populations typically do not passively track stochastic environmental variables. Rather, population dynamics are better described as nonlinear amplification of physical forcing by biological interactions. However, we illustrate that in some cases populations do show linear tracking of the physical environment. To explain why population dynamics can sometimes be linear, we propose the linear tracking window hypothesis: populations are most likely to track the stochastic environmental forcing when their generation time matches the characteristic time scale of the environmental signal. While our observations follow this hypothesis well, our results indicate that the linear tracking window is a necessary but not a sufficient condition.     

Aggregation of Dynamic Systems and the Existence of a Regeneration Function

In environmental economics, complex ecosystems are often represented by low-dimensional models. The question of whether the results of these models can be applied to a more complex reality leads to the investigation of an aggregation problem in a nonlinear dynamic setting. We show that restrictive assumptions are needed for aggregation and that the low-dimensional model has to be linear. On the basis of these results, we argue that the aggregation of complex ecosystems is often oversimplifying and that substantial gains can be expected from the use of more complex models.     

A note concerning time parameterization of Markovian models of ecosystem flow analysis

A continuous Markovian model of resource flow in a steady state ecosystem model is developed. This model calculates the mean and variance of the frequency of intercompartmental cyclings and duration of compartmental residence times. This model is compared with an analogous discrete Markovian flow model to demonstrate the sensitivity of discrete and continuous ecosystem flow analyses. Appropriate time parameterization of of discrete Markovian flow models is then discussed with special reference to Shannon's theorem of dynamic system sampling.     

Ecosystem services as a counterpart of emergy flows to ecosystems

A generic input-state-output scheme has been used to represent ecosystem dynamics. Systemic approaches to ecosystems use functions that are based either on inputs, state or outputs of the system. Some examples of approaches that use a combination of functions have been recently proposed. For example the use of eco-exergy to emergy flow can be seen as a mixed input-state approach; more recently, to connect the state to the output of the ecosystem, the relation of eco-exergy and ecosystems services has been proposed. This paper studies the link between the useful output of an ecosystems and its input through the relation between ecosystem services and emergy flow, in a kind of grey/black box scheme (i.e., without considering the state and the structure of the ecosystem). No direct connection between the two concepts can be determined, but identifying and quantifying the emergy flows feeding an ecosystem and the services to humans coming from them facilitate the sustainable conservation of Nature and its functions. Furthermore, this input-output relation can be established in general by calculating the ratio of the value of the ecosystem services to the emergy flow that supports the system. In particular, the ratio of the world ecosystem services to the emergy flow supporting the entire biosphere has been calculated showing that, at least at the global level, Nature is more efficacious in producing “money” (in form of ecosystem services) than economic systems (e.g., national economies and their GDP).     

Vegetation competition model for water and light limitation. II: Spatial dynamics of groundwater and vegetation

In temperate climates groundwater can have a profound effect on vegetation, because it strongly influences the spatio-temporal distribution of soil moisture in the rootzone and therefore the occurrence of water and oxygen stress of vegetation. This article focuses on vegetation and groundwater dynamics along a hill slope by developing and evaluating a fully coupled hydrological-vegetation model for a temperate forest ecosystem. The vegetation model is described in part 1 of this series of two papers. To simulate the hydrology an extended version of the saturated-unsaturated hydrological model STARWARS has been used. The coupled model is used to investigate both the short and long-term dynamics for a system of two species. Both compete for light and water where one is adapted to wet conditions and the other to dry conditions. The daily dynamics show that the influence of groundwater is particularly strong in spring when waterlogging occurs due to decreased evapotranspiration in winter. Long simulation runs of 1000 years were performed to study the equilibrium state for the two species. Comparison of simulation results with observations of groundwater depth and vegetation types along a dry-wet gradient in a natural forest shows that a reductionist approach is able to capture these patterns well. Sensitivity analysis shows that the border between wet- and dry-adapted species moves upslope with increased rainfall, decreased slope angle and decreased aquifer thickness. These results are similar to previous findings which were based on global maximization of ecosystem evaporation or minimizing ecosystem stress. Comparison of runs with a fixed and a dynamic groundwater table shows that a dynamic groundwater table facilitates a wider transition zone between vegetation types along the hill slope. In this transition the biomass of vegetation is higher in the case of a dynamic groundwater than in case of a static groundwater table. This underlines the importance of incorporating spatial groundwater dynamics in models of groundwater influenced ecosystems.