Some stochastic differential equation models of an aquatic ecosystem

A set of stochastic differential equations has been used to model an aquatic ecosystem. The randomness in the system has been introduced through initial conditions of the state variables, parameters, and input variables (light and temperature). These models were analysed using Monte Carlo simulation procedures and the results were similar to those observed in the experimental and field data. They were different, however, from the results of a deterministic simulation. This approach allows us to incorporate the maximum degree of information in the model and to study the behavior of the system without arbitrarily manipulating the values of the parameters. Some possible refinements and generalizations of this approach are also discussed.     

A hierarchical analysis of terrestrial ecosystem model Biome-BGC: Equilibrium analysis and model calibration

The increasing complexity of ecosystem models represents a major difficulty in tuning model parameters and analyzing simulated results. To address this problem, this study develops a hierarchical scheme that simplifies the Biome-BGC model into three functionally cascaded tiers and analyzes them sequentially. The first-tier model focuses on leaf-level ecophysiological processes; it simulates evapotranspiration and photosynthesis with prescribed leaf area index (LAI). The restriction on LAI is then lifted in the following two model tiers, which analyze how carbon and nitrogen is cycled at the whole-plant level (the second tier) and in all litter/soil pools (the third tier) to dynamically support the prescribed canopy. In particular, this study analyzes the steady state of these two model tiers by a set of equilibrium equations that are derived from Biome-BGC algorithms and are based on the principle of mass balance. Instead of spinning-up the model for thousands of climate years, these equations are able to estimate carbon/nitrogen stocks and fluxes of the target (steady-state) ecosystem directly from the results obtained by the first-tier model. The model hierarchy is examined with model experiments at four AmeriFlux sites. The results indicate that the proposed scheme can effectively calibrate Biome-BGC to simulate observed fluxes of evapotranspiration and photosynthesis; and the carbon/nitrogen stocks estimated by the equilibrium analysis approach are highly consistent with the results of model simulations. Therefore, the scheme developed in this study may serve as a practical guide to calibrate/analyze Biome-BGC; it also provides an efficient way to solve the problem of model spin-up, especially for applications over large regions. The same methodology may help analyze other similar ecosystem models as well.     

A note on the sufficiency of low resolution ecosystem models

Analysis of a system of non-linear differential equations illustrates the effects of interactions between biotic and abiotic components of a complex aquatic ecosystem model. A stochastic analysis shows that the variance of the abiotic variables is related in a simple manner to the autocorrelation function of the biotic variables. The results suggest that for oligotrophic and eutrophic conditions, relatively simple ecosystem models may be sufficient for studies of an aquatic environment. Under mesotrophic conditions, the high state variable resolution of a complex model may be necessary.     

Mathematical modelling of a fish pond ecosystem

A mathematical model is constructed for a fish breeding pond for carp, silver carp and bighead. The model is a system of ordinary differential equations describing the material transformations in the ecosystem. It allows a choice of optimal regimes of the aeration, feeding and fertilization of a pond for different climatic conditions in order to maximize the yield.     

Combined effect of fire and water scarcity on vegetation patterns in arid lands

In many arid zones around the word, the vegetation spontaneously forms regular patterns to optimize the use of the scarce water resources. The patterns act as early warning signal that fragile ecosystems may suddenly undergo irreversible shifts, thus, interpreting the structural shape of vegetation patterns is crucial to deciphering the ecosystem history and its expected further development. The sudden and irreversible shift of delicate ecosystems as a consequence of minor variation of the climatic forcing has been studied extensively in the past. The attitude of the ecosystem to recover after a catastrophic event, such as fire, did not receive as much attention so far. Here we modelled fire, as a sudden shift of the ecosystem state variables and functionality and evaluated post-fire scenarios under the hypothesis that two major feedbacks shaped the vegetation patterns: a positive feedback between preferential infiltration and plant growth, and a second feedback between infiltration and vegetation burning. A simple model solving a set of partial differential equations for soil moisture, plant biomass, surface water and dead biomass balance predicted significantly diverse post-fire vegetation patterns depending on the fire severity and on the degree of soil water repellency induced by the vegetation burning.     

A grassland ecosystem model of the Xilingol steppe, Inner Mongolia, China

The natural grassland ecosystem of the Xilingol steppe has traditionally been the source of the most productive and highest quality agriculture in northern China. Unfortunately, the area is now experiencing degradation due to resource overuse. In an attempt to forecast grassland production and to sustain the ecosystem, we built a time-dependent simulation model of the ecosystem based on long-range weather forecasts (several weeks to several months). The model incorporated five state variables including above- and belowground biomass, the amount of standing dead plant material, livestock (sheep) weight, and the amount of excrement per unit ground area. Within the model, solar light energy is fixed by grassland vegetation and flows through the other variables via a variety of organism-environment interactions. The model was written using a set of simultaneous differential equations and was numerically analyzed. The values of the time-dependent parameters controlling energy flow were determined based on data accumulated in experiments and field surveys executed at a grassland experimental station located in Xilingol, as well as by reference to related literature. We used daily meteorological data including air temperature and rainfall recorded at the Xilinhot Meteorological Observatory. Simulated results for several stocking densities coincided well with the data of aboveground plant biomass observed at the experimental station in 1990, 1993, and 1997. We obtained reasonable simulation results for five stocking densities, three air temperature patterns, and five rainfall patterns. When a month-long drought, which sometimes occurs in this area, was forecast by a local weather station, a decrease in grassland production was forecast by the model. Such forecasts will assist in the management of livestock, forage preservation, and grassland conservation.     

Modelling the distribution and effect of heavy metals in an aquatic ecosystem

Models are reviewed describing the distribution and effect of heavy metals in an aquatic ecosystem. Since a model used for an impact statement should give the maximum concentration level rather than the seasonal variation, a model focussing on this situation is suggested. The basic differential equations describe (1) the variation in concentration of the toxicant per biomass dry matter in a given trophic level, and (2) the exchange of toxicant between sediment and water. Furthermore, since a substantial part of the heavy metal in an aquatic ecosystem is bound to suspended matter, an equation describing the equilibrium between dissolved and suspended matter must be included.A literature review has been carried out on the parameters used in the above mentioned equations and a demonstration, showing how it is possible to find approximate values for such parameters as excretion coefficient and uptake coefficient on the basis of a relationship between these two parameters and the size of an organism, is given.     

Simulation of aquatic food web and species interactions by adaptive agents embodied with evolutionary computation: a conceptual framework

Individual-based and state variable-based adaptive agents (AA) are discussed regarding their relevance to different types of ecosystems. Individual-based AA proved applicable to a spatially explicit simulation of highly simplified terrestrial food webs. State variable-based AA with evolutionary computation (EC) embodied are suggested for the simulation of aquatic food webs and plankton species interactions. Embodiment of EC in AA can be achieved by evolving predictive rules (ER), differential equations (EDE) or artificial neural networks (ANN) derived from a diverse lake database. In order to provide ecosystem simulation with resilience to environmental change, agent banks can be created containing alternative agents for same species or functional groups from different lakes. State variable-based AA are currently tested for aquatic ecosytem simulation by means of a diverse lake database. It promises to overcome constraints by the rigidity of traditional lake ecosystem models.     

EcoTroph: Modelling marine ecosystem functioning and impact of fishing

EcoTroph (ET) is a model articulated around the idea that the functioning of aquatic ecosystems may be viewed as a biomass flow moving from lower to higher trophic levels, due to predation and ontogenetic processes. Thus, we show that the ecosystem biomass present at a given trophic level may be estimated from two simple equations, one describing biomass flow, the other their kinetics (which quantifies the velocity of biomass transfers towards top predators). The flow kinetic of prey partly depends on the abundance of their predators, and a top-down equation expressing this is included in the model. Based on these relationships, we simulated the impact on a virtual ecosystem of various exploitation patterns. Specifically, we show that the EcoTroph approach is able to mimic the effects of increased fishing effort on ecosystem biomass expected from theory. Particularly, the model exhibits complex patterns observed in field data, notably cascading effects and ‘fishing down the food web’. EcoTroph also provides diagnostic tools for examining the relationships between catch and fishing effort at the ecosystem scale and the effects of strong top-down controls and fast-flow kinetics on ecosystems resilience. Finally, a dynamic version of the model is derived from the steady-state version, thus allowing simulations of time series of ecosystem biomass and catches. Using this dynamic model, we explore the propagation of environmental variability in the food web, and illustrated how exploitation can induce a decrease of ecosystem stability. The potential for applying EcoTroph to specific ecosystems, based on field data, and similarities between EcoTroph and Ecopath with Ecosim (EwE) are finally discussed.     

Implementation of ecological modeling as an effective management and investigation tool: Lake Kinneret as a case study

The need for scientifically based management of lakes, as key water resources, requires the establishment of quantitative relationships between in-lake processes responsible for water quality (WQ) and the intensity of major management measures (MM, e.g. nutrient loading). In this paper, we estimate the impact of potential changes in nutrient loading on the Lake Kinneret ecosystem. Following validation of the model against a comprehensive dataset, we applied an approach that goes beyond scenario testing by linking the lake ecosystem model DYRESM–CAEDYM with a set of ecosystem variables included in a pre-assessed system of water quality indices. The emergent properties of the ecosystem predicted from the model simulations were also compared with lake data as a form of indirect validation of the model. Model output, in good agreement with lake data, indicated differential effects of nitrogen and phosphorus nutrient loading on concentrations, and major in-lake fluxes, of TN and TP, and dynamics and algal community structure. Both model output and lake data indicated a strong relationship between nitrogen loading and in-lake TN values. This relationship is not apparent for phosphorus and only a weak relationship exists between phosphorus loading and in-lake TP. The modeling results, expressed in terms of water quality, allowed establishment of critical/threshold values for the nutrient loads. Implementation of the ecological modeling supplemented with the quantified set of WQ indices allowed us to take a step towards establishment of the association between permissible ranges for water quality and major management measures, i.e. towards sustainable management.     

On modelling temporal patterns in stressed ecosystems — Recreation in a coniferous forest

A system of ordinary differential equations is presented as being appropriate for modelling the impact of stress on the temporal behaviour of certain components within an ecosystem. The modelling problem is fist discussed in general terms and then in terms specifically relating to the impact of recreational activities (hunting, fishing, sightseeing, etc.) on a coniferous forest ecosystem.Using crude data, the model is used to simulate over a 3-year period the biomass levels of four compartments of the ecosystem (viz, timber, deer, fish and forage) in the absence of recreational activities. These results are then contrasted with simulation results obtained by introducing a “moderate” and then “high” degree of recreational activity, as well as the response of the system under moderate recreation to management strategies involving the construction of dams and the harvesting of timber.     

Persistence for ecosystem microcosm models

A qualitative analysis of a system of autonomous differential equations modelling an ecosystem microcosm is carried out from the point of view of persistence. Necessary and sufficient conditions are given that no trajectory with positive initial conditions has a component that tends to zero asymptotically or reaches zero in finite time. The results are stated in terms of threshold levels for the input nutrient parameter in the system.     

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.     

Water and carbon fluxes above European coniferous forests modelled with artificial neural networks

Artificial neural networks are used to select a minimal set of input variables to model water vapour and carbon exchange of coniferous forest ecosystems, independently of tree species and without detailed physiological information. Neural networks are used because of their power to fit highly non-linear relations between input and output-variables. Radiation, temperature, vapour pressure deficit and time of the day showed to be the dynamic input variables that determine ecosystem water fluxes. The same variables, together with projected leaf area index are needed for modelling CO₂-fluxes. The results for the individual sites show that the neural networks found mean water and carbon flux responses to the driving variables valid for all sites. The sensitivity analysis of the derived neural networks shows that the LAI-effect of the CO₂-flux model is overfitted because of the low variability of LAI. However, the predictions of CO₂-fluxes of sites not included in the calibration set indicate that the LAI-response of the network is reliable and that results can be used as a first estimate of the net ecosystem carbon exchange of the forest sites. Independent predictions of forest ecosystem vapour fluxes were equally satisfying as empirical models specifically calibrated for the individual sites. The results indicate that both short term water and carbon fluxes of European coniferous forests can be modelled without using detailed physiological and site specific information.     

Large scale systems approach to estuarine water quality modelling with multiple constituents

A matrix model for simulating concentration distributions of water quality constituents with coupled reactions in an estuary is developed from a large scale systems approach. The model is an approximation to the set of coupled partial differential equations describing the process. This steady state approximation is formulated as a large algebraic system consisting of coupled subsystems. The large algebraic system is solved by an efficient iterative method. Results utilizing actual field data are presented for the nitrogen cyle with five constituent forms of nitrogen for Corpus Christi Bay, Texas. Simulated and observed concentrations are compared.     

A Markovian model for ecosystem flow analysis

Nutrient flow through ecosystems is modeled as a discrete Markov chain whose transition probabilities are stationary or time inhomogenous according to whether a steady state or dynamic ecosystem is modeled, respectively. Expected residence time and number of intercompartmental transfers for a nutrient within a set of compartments are derived. Variances of these random variables are also considered. A measure for ecosystem resource recycling is given as a weighted sum of probabilities.     

A general two-species competition model with time-varying rates

The classical mathematical model for the behaviour of the populations of two competing biological species has previously been generalized by the author, by assuming that the coefficients of intrinsic growth, self-inhibition and interaction were all functions of time and, for a certain class of the governing differential equations, the exact solutions were obtained: an example was given in which the coefficients were periodic functions. In the differential equations of this model (as well as in the autonomous Lotka-Volterra equations), each of the Malthusian growth-rates was assumed to be diminished by a linear function of the populations of the two species, without there being any rigorous justification for this assumption. We here generalize the differential equations by assuming for these diminution functions general nonlinear forms having time-varying coefficients. The exact solutions are given for four classes of the resulting strongly nonlinear non-autonomous differential equations. Various conclusions about the growth modes of the two populations and their asymptotic behaviour are drawn, both when specific and when arbitrary forms are assumed for the coefficient functions. Cases are examined in which the Competitive Exclusion Principle holds and others in which it does not.     

Steady states and sensitivities of commonly used pelagic ecosystem model components

Pelagic, coupled ocean circulation-ecosystem models, are widely used in climate research. These tools aim to quantify fluxes of nutrients and carbon in the ocean and are, increasingly, the base of future projections. For this purpose it is crucial to quantify and identify the sources of uncertainties. In contrast to physical models, the underlying equations for ecosystem models are derived from empirical relationships rather than based on first principles. This resulted in the development of a multitude of different ecosystem models - different in respect to both, underlying principles and complexity. Clearly, the question arises, to what extent the sensitivities of these models are comparable.This study focuses on the intrinsic dynamics of some widely used, simple (containing 2-3 prognostic variables) ecosystem models in a 0-D framework (i.e., comprising only the well-mixed oceanic surface layer). A suite of differing model approaches is tuned such that their behavior is similar. The setup resembles the well-mixed oceanic surface layer in the Baltic proper. It is illustrated that strong differences between the model approaches appear due to exemplary, anticipated changes in the external nutrient and light conditions. Herewith, we demonstrate the well-known, but rarely demonstrated fact that, apparent consistency between modeled prognostic variables with today's data bases is not necessarily a good measure of forecast skill. The causes which lead to the different sensitivities are illustrated by considering the steady state solutions. It is pointed out, that apparently small changes in the model formulations can result in very different dynamical behavior and an enormous spread between the model approaches, despite the feasibility to tune a common behavior in a limited range of light and nutrient supply. In our examples, the sensitivity is mainly a function of the formulation of the loss rate of phytoplankton. It is thus, in particular, the formulation of highly unknown heteorotrophic processes that determines the model sensitivity.     

On the role of entropy in water quality modelling

The use of the entropy principle in phenomenological water quality models is not only necessary, but also of great advantage. A deterministic ecosystem model must obey the 2nd law of thermodynamics. Gibb's equation is a constraint additional to the balances of mass, energy and momentum. The entropy principle supports the unified treatment of physical, chemical and biological processes in water bodies, offers stability criteria and controls the further development of the aquatic ecosystems. Thermodynamic criteria also allow the determination of the bifurcation points of the model equations. Especially near these points the state and structure of the ecosystem can be strongly changed by fluctuations of the variables and parameters of the ecosystem.Results of the thermodynamic theory of selforganizing systems (Glansdorff and Prigogine, 1971; Nicolis and Prigogine, 1977) are of very great importance for water quality modelling. Furthermore, the entropy principle bridges the phenomenological, stochastic and cybernetic approaches to water quality modelling.While the paper deals with general aspects of the role of entropy in water quality modelling, the basic system of equations, taking the entropy principle into account, can be found in a previous paper (Mauersberger, 1978).     

Potential impact model to assess agricultural pressure to landscape ecological functions

Conservation and protection of soil and water resources and visual aspects of landscape, as well as the promotion of biodiversity, are some of the central tasks of environmental policy development and social politics in the future. One of the main questions is: ‘which agricultural systems are able to guarantee sustained resource-conserving land use?’ Based on the ecological risk concepts of the 1970s and 1980s, a potential impact model was developed using a universal assessment algorithm derived from fuzzy logic. The model estimated the potential impact of agricultural land use on ecosystem function using a few resource indicators. Intervention intensities of agricultural land-use are set in relation to site conditions and aggregated for each of several defined potential impact categories. The interpretation with respect to risk and the calculation of potential impact values are explained.