Environmental periodicity and ecosystem stability

Simple ecosystem models involving a primary producer, nutrients, and a herbivore are developed with metabolic dependence on periodic functions representing solar radiation and temperature. Simulation results suggests that the models yield generally stable output provided that equilibrium values for the state variables are positive in the nonperiodic analog. Linear dependence of plant growth on nutrient supply leads to a larger stability domain than that produced by hyperbolic dependence.     

Grass feedbacks on fire stabilize savannas

Savannas commonly consist of a discontinuous cover of overstory trees and a groundcover of grasses. Savanna models have previously demonstrated that vegetation feedbacks on fire frequency can limit the density of overstory trees, thereby maintaining savannas. Positive feedbacks of either savanna trees alone or trees and grasses together on fire frequency have been shown to result in a stable savanna equilibrium. Grass feedbacks on fire frequency, in contrast, have resulted in stable equilibria in either a grassland or forest state, but not in a savanna. These results, however, were derived from a system of differential equations that assumes that fire occurrence is strictly deterministic and that vegetation losses due to fire are continuous in time. We develop an alternative formulation of the grass-fire feedback model that assumes that fires are discrete and occur stochastically in time to examine the influence of these assumptions on the predicted state of the system. We show that incorporating fire as a discrete event can produce a recurring temporal refuge in which both grass and trees co-occur in a stable, bounded savanna. In our model, tree abundance is limited without invoking demographic bottlenecks in the transition from fire-sensitive to fire-resistant life history stages. An increasing strength of grass feedback on fire results in regular, predictable fires, which suggests that the system can also be modeled using a set of difference equations. We implement this discrete system using modified Leslie/Gower difference equations and demonstrate the existence of a bounded savanna state in this model framework. Our results confirm the potential for grass feedbacks to result in stable savannas, and indicate the importance of modeling fire as a discrete event rather than as a loss rate that is continuous in time.     

Effects of surrounding land use and water depth on seagrass dynamics relative to a catastrophic algal bloom

Seagrasses are the foundation of many coastal ecosystems and are in global decline because of anthropogenic impacts. For the Indian River Lagoon (Florida, U.S.A.), we developed competing multistate statistical models to quantify how environmental factors (surrounding land use, water depth, and time [year]) influenced the variability of seagrass state dynamics from 2003 to 2014 while accounting for time‐specific detection probabilities that quantified our ability to determine seagrass state at particular locations and times. We classified seagrass states (presence or absence) at 764 points with geographic information system maps for years when seagrass maps were available and with aerial photographs when seagrass maps were not available. We used 4 categories (all conservation, mostly conservation, mostly urban, urban) to describe surrounding land use within sections of lagoonal waters, usually demarcated by land features that constricted these waters. The best models predicted that surrounding land use, depth, and year would affect transition and detection probabilities. Sections of the lagoon bordered by urban areas had the least stable seagrass beds and lowest detection probabilities, especially after a catastrophic seagrass die‐off linked to an algal bloom. Sections of the lagoon bordered by conservation lands had the most stable seagrass beds, which supports watershed conservation efforts. Our results show that a multistate approach can empirically estimate state‐transition probabilities as functions of environmental factors while accounting for state‐dependent differences in seagrass detection probabilities as part of the overall statistical inference procedure.     

Declines in migrant shorebird populations from a winter‐quarter perspective

Many long‐distance migrating shorebird (i.e., sandpipers, plovers, flamingos, oystercatchers) populations are declining. Although regular shorebird monitoring programs exist worldwide, most estimates of shorebird population trends and sizes are poor or nonexistent. We built a state‐space model to estimate shorebird population trends. Compared with more commonly used methods of trend estimation, state‐space models are more mechanistic, allow for the separation of observation and state process, and can easily accommodate multivariate time series and nonlinear trends. We fitted the model to count data collected from 1990 to 2013 on 18 common shorebirds at the 2 largest coastal wetlands in southern Africa, Sandwich Harbour (a relatively pristine bay) and Walvis Bay (an international harbor), Namibia. Four of the 12 long‐distance migrant species declined since 1990: Ruddy Turnstone (Arenaria interpres), Little Stint (Calidris minuta), Common Ringed Plover (Charadrius hiaticula), and Red Knot (Calidris canutus). Populations of resident species and short‐distance migrants increased or were stable. Similar patterns at a key South African wetland suggest that shorebird populations migrating to southern Africa are declining in line with the global decline, but local conditions in southern Africa's largest wetlands are not contributing to these declines. State‐space models provide estimates of population levels and trends and could be used widely to improve the current state of water bird estimates.     

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.     

Renewable Resources in an Overlapping Generations Economy Without Capital

We incorporate a renewable resource into an overlapping generations model without capital and with quasi-linear preferences. Besides being an input for production the resource serves as a store of value. We characterize the dynamics, efficiency, and stability of the steady-state equilibria. The stability properties are sensitive to the type of resource growth. For constant growth there is only one steady-state equilibrium which is stable and efficient. In the general case of the concave growth function, there are usually at least two steady-state equilibria, one of which is stable and the other one unstable. The unstable steady state is efficient, but the stable one may or may not be. We study the robustness of our results by assuming a logarithmic utility function. We show that for the Cobb–Douglas production function the steady state is unique and stable regardless of whether the equilibrium is efficient or inefficient. Our analytical results are illustrated by numerical calculations.     

Age,stages, and the role of generation time in matrix models

The earliest matrix models, proposed in the 1940s, consider age classes, and were later proved to be equivalent to the discrete time version of the stable population theory. In this theory and models, besides the asymptotic growth rate, a very important characteristic is the turnover of individuals, measured in various ways by generation time. Models considering stages, on the contrary, do not take into account the age of individuals and seem largely preferable to age-structured models for many populations in which demographic characteristics are related to biological stages (such a seed, rosette, flowering plant, etc.) rather than to age per itself. These two kinds of models can be embedded as particular cases of stage by age models or multistate models. Theses general models can be used to develop a multistate stable population theory with many advantages. This general theory is reviewed with emphasis on general rules for sensitivity analyses in which generation time plays a central role.     

Model of a chemostat utilising phenol as inhibitory substrate

A pure culture of Acinetobacter lwoffi was grown in a batch and continuous culture using phenol as the limiting substrate. Five kinetic models were applied to the data to determine the kinetic parameters governing growth of the organisms. A non-linear leastsquares technique was used to fit the data to the different functions. No significant differences were found on statistical basis between the models and so a choice was made on the grounds of mathematical simplicity. The so-called ‘Haldane function’ was chosen as the best and utilised in the model describing the steady-state behaviour of the chemostat. This model takes into account the inhibitory character of phenol and also the maintenance energy of the bacteria. Analysis of local stability shows that one can expect to obtain three steady states of which one is the trivial washout state, another is unstable and the third is a high conversion stable steady state.This model was tested with a pure culture of Acinetobacter lwoffi and phenol inlet concentrations of 200, 500 and 1000 mgl^?1. Good agreement was found between the model and the results of the continuous experiments.     

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.     

Combining multistate capture-recapture data with tag recoveries to estimate demographic parameters

Matrix population models that allow an animal to occupy more than one state over time are important tools for population and evolutionary ecologists. Definition of state can vary, including location for metapopulation models and breeding state for life history models. For populations whose members can be marked and subsequently reencountered, multistate mark-recapture models are available to estimate the survival and transition probabilities needed to construct population models. Multistate models have proved extremely useful in this context, but they often require a substantial amount of data and restrict estimation of transition probabilities to those areas or states subjected to formal sampling effort. At the same time, for many species, there are considerable tag recovery data provided by the public that could be modeled in order to increase precision and to extend inference to a greater number of areas or states. Here we present a statistical model for combining multistate capture-recapture data (e.g., from a breeding ground study) with multistate tag recovery data (e.g., from wintering grounds). We use this method to analyze data from a study of Canada Geese (Branta canadensis) in the Atlantic Flyway of North America. Our analysis produced marginal improvement in precision, due to relatively few recoveries, but we demonstrate how precision could be further improved with increases in the probability that a retrieved tag is reported.     

Estimating parameters of hidden Markov models based on marked individuals: use of robust design data

Development and use of multistate mark-recapture models, which provide estimates of parameters of Markov processes in the face of imperfect detection, have become common over the last 20 years. Recently, estimating parameters of hidden Markov models, where the state of an individual can be uncertain even when it is detected, has received attention. Previous work has shown that ignoring state uncertainty biases estimates of survival and state transition probabilities, thereby reducing the power to detect effects. Efforts to adjust for state uncertainty have included special cases and a general framework for a single sample per period of interest. We provide a flexible framework for adjusting for state uncertainty in multistate models, while utilizing multiple sampling occasions per period of interest to increase precision and remove parameter redundancy. These models also produce direct estimates of state structure for each primary period, even for the case where there is just one sampling occasion. We apply our model to expected-value data, and to data from a study of Florida manatees, to provide examples of the improvement in precision due to secondary capture occasions. We have also implemented these models in program MARK. This general framework could also be used by practitioners to consider constrained models of particular interest, or to model the relationship between within-primary-period parameters (e.g., state structure) and between-primary-period parameters (e.g., state transition probabilities).     

An inverse ecosystem model of year-to-year variations with first order approximation to the annual mean fluxes

Ecosystems exhibit nonlinear dynamics that are often difficult to capture in models. Consequently, linearization is commonly applied to remove some of the uncertainties associated with the nonlinear terms. However, since the true model is unknown and the operating point to linearize the model about is uncertain, developing linear ecosystems models is non-trivial. To develop a linear ecosystem model, we assume that the annual mean state of an ecosystem is a minor bias from the long-term mean state. A first order approximation inverse model to govern the year-to-year dynamics of ecosystems whose characteristic time scales are less than 1 year is developed, through theoretically formulation, on the basis of steady state analysis, time scale separation and nondimensionalization. The approach is adept at predicting year-to-year variations and to tracking system response to changes in environmental drivers when compared to data generated with a standard nonlinear NPZD model.     

Alternative stable states and phase shifts in coral reefs under anthropogenic stress

Ecosystems with alternative stable states (ASS) may shift discontinuously from one stable state to another as environmental parameters cross a threshold. Reversal can then be difficult due to hysteresis effects. This contrasts with continuous state changes in response to changing environmental parameters, which are less difficult to reverse. Worldwide degradation of coral reefs, involving "phase shifts" from coral to algal dominance, highlights the pressing need to determine the likelihood of discontinuous phase shifts in coral reefs, in contrast to continuous shifts with no ASS. However, there is little evidence either for or against the existence of ASS for coral reefs. We use dynamic models to investigate the likelihood of continuous and discontinuous phase shifts in coral reefs subject to sustained environmental perturbation by fishing, nutrification, and sedimentation. Our modeling results suggest that coral reefs with or without anthropogenic stress can exhibit ASS, such that discontinuous phase shifts can occur. We also find evidence to support the view that high macroalgal growth rates and low grazing rates on macroalgae favor ASS in coral reefs. Further, our results suggest that the three stressors studied, either alone or in combination, can increase the likelihood of both continuous and discontinuous phase shifts by altering the competitive balance between corals and algae. However, in contrast to continuous phase shifts, we find that discontinuous shifts occur only in model coral reefs with parameter values near the extremes of their empirically determined ranges. This suggests that continuous shifts are more likely than discontinuous shifts in coral reefs. Our results also suggest that, for ecosystems in general, tackling multiple human stressors simultaneously maximizes resilience to phase shifts, ASS, and hysteresis, leading to improvements in ecosystem health and functioning.     

Validation of ecological state space models using the Laplace approximation

Many statistical models in ecology follow the state space paradigm. For such models, the important step of model validation rarely receives as much attention as estimation or hypothesis testing, perhaps due to lack of available algorithms and software. Model validation is often based on a naive adaptation of Pearson residuals, i.e. the difference between observations and posterior means, even if this approach is flawed. Here, we consider validation of state space models through one-step prediction errors, and discuss principles and practicalities arising when the model has been fitted with a tool for estimation in general mixed effects models. Implementing one-step predictions in the R package Template Model Builder, we demonstrate that it is possible to perform model validation with little effort, even if the ecological model is multivariate, has non-linear dynamics, and whether observations are continuous or discrete. With both simulated data, and a real data set related to geolocation of seals, we demonstrate both the potential and the limitations of the techniques. Our results fill a need for convenient methods for validating a state space model, or alternatively, rejecting it while indicating useful directions in which the model could be improved.     

Lotka-Volterra competition models for sessile organisms

Markov models are widely used to describe the dynamics of communities of sessile organisms, because they are easily fitted to field data and provide a rich set of analytical tools. In typical ecological applications, at any point in time, each point in space is in one of a finite set of states (e.g., species, empty space). The models aim to describe the probabilities of transitions between states. In most Markov models for communities, these transition probabilities are assumed to be independent of state abundances. This assumption is often suspected to be false and is rarely justified explicitly. Here, we start with simple assumptions about the interactions among sessile organisms and derive a model in which transition probabilities depend on the abundance of destination states. This model is formulated in continuous time and is equivalent to a Lotka-Volterra competition model. We fit this model and a variety of alternatives in which transition probabilities do not depend on state abundances to a long-term coral reef data set. The Lotka-Volterra model describes the data much better than all models we consider other than a saturated model (a model with a separate parameter for each transition at each time interval, which by definition fits the data perfectly). Our approach provides a basis for further development of stochastic models of sessile communities, and many of the methods we use are relevant to other types of community. We discuss possible extensions to spatially explicit models.     

Regression analysis of forest damage by marginal models for correlated ordinal responses

Studies on forest damage generally cannot be carried out by common regression models, for two main reasons: Firstly, the response variable, damage state of trees, is usually observed in ordered categories. Secondly, responses are often correlated, either serially, as in a longitudinal study, or spatially, as in the application of this paper, where neighbourhood interactions exist between damage states of spruces determined from aerial pictures. Thus so-called marginal regression models for ordinal responses, taking into account dependence among observations, are appropriate for correct inference. To this end we extend the binary models of Liang and Zeger (1986) and develop an ordinal GEEI model, based on parametrizing association by global cross-ratios. The methods are applied to data from a survey conducted in Southern Germany. Due to the survey design, responses must be assumed to be spatially correlated. The results show that the proposed ordinal marginal regression models provide appropriate tools for analysing the influence of covariates, that characterize the stand, on the damage state of spruce.     

Rapid cultural adaptation can facilitate the evolution of large-scale cooperation

Over the past several decades, we have argued that cultural evolution can facilitate the evolution of large-scale cooperation because it often leads to more rapid adaptation than genetic evolution, and, when multiple stable equilibria exist, rapid adaptation leads to variation among groups. Recently, Lehmann, Feldman, and colleagues have published several papers questioning this argument. They analyze models showing that cultural evolution can actually reduce the range of conditions under which cooperation can evolve and interpret these models as indicating that we were wrong to conclude that culture facilitated the evolution of human cooperation. In the main, their models assume that rates of cultural adaption are not strong enough compared to migration to maintain persistent variation among groups when payoffs create multiple stable equilibria. We show that Lehmann et al. reach different conclusions because they have made different assumptions. We argue that the assumptions that underlie our models are more consistent with the empirical data on large-scale cultural variation in humans than those of Lehmann et al., and thus, our models provide a more plausible account of the cultural evolution of human cooperation in large groups.     

A holistic approach to ecological modelling

Models of different complexity were used to examine how the ecological buffer capacity, β (defined as the change in loading relative to the change in a considered state variable) varies when the loading, e.g. the input of phosphorus, is changed. It was found that while β = ΔP(total)/ΔP(soluble) increases with increasing complexity of the model at low P-loading, the β-value will — at medium P-loadings — have a maximum value at a certain degree of complexity, and will be a decreasing function of complexity at high phosphorus loadings. This might explain why very eutrophic lakes, rivers polluted with organic matter or other stressed ecosystems are stable although their complexity is low.The more complex ecosystems seem best able to cope with increasing variations in climatic factors.In the models considered the thermodynamic function exergy correlates well with the sum of relevant buffer capacities. High exergy levels mean that the structure is more able to meet changes in external factors.