Nonlinearities in mathematical ecology: Phenomena and models: Would we live in Volterra's world?

Presented is a critical survey of canonical nonlinear models in theoretical population ecology, namely single-species population, prey–predator, competition, migration within a metapopulation, and trophic chains. Various nonlinear effects, like hysteresis, structural instability, dissipative structures, dynamic chaos, etc., do exist in these models, but the problem how to detect these phenomena in real ecosystems is not yet solved. In the mathematics of nonlinear models, the central question is whether the simplest, i.e., Volterra-type, nonlinearity is sufficient to reproduce a variety of nonlinear phenomena in a given model or we need a more sophisticated formalism. Examples are considered where the Volterra models fail. Although fundamental physical principles, like, e.g., the mass conservation law, should work in ecology too, the ecological origin of the models often causes mathematical effects which are distinct from those in theoretical physics. For example, the trophic-chain model does reveal a kind of chaotic behaviour, but the “ecological strange attractor” occupies an intermediate position between Lorenz's and Feigenbaum's attractors; moreover, the phase volume of our system contracts, hence the system is dissipative (like a Lorenz's one) in spite of its matter conservation property. Nevertheless, when applied properly, physical concepts, like, e.g., the thermodynamic notion of exergy, give better insight both to the patterns of nonlinear ecosystem behaviour and to comparison of the patterns.     

Use of thermodynamic indices as ecological indicators of the development state of lake ecosystems: 2. Exergy and specific exergy indices

The effectiveness of exergy and specific exergy indices as ecological indicators of the trophic state of lake ecosystems is here tested on a small homogeneous set of shallow lakes which, in spite of their similar nutrient concentrations, morphology and hydrology, show a different trophic state and structure, species composition and abundance. The findings reveal that exergy and specific exergy indices have good negative correlation with phytoplankton biomass and Carlson's trophic state index (TSI) and strong positive correlation to water transparency (the relationship between exergy and eutrophication is clearer if the exergy refers to surface units, rather than volume units) and, hence, that they may be used as ecological indicators of the trophic state of lake ecosystems. The relationship between the responses of the thermodynamic approach and other conventional trophic classification methods (Vollenweider's eutrophication model based on phosphorus loading, the Hillbrich-Ilkowska method and the Vollenweider–OECD classification criterion) previously applied to Lake Trasimeno, was also investigated. The decreasing trend of exergy and specific exergy indices with eutrophication increase appears to be essentially due to the change in species composition and trophic structure, rather than to a different trophic potentiality of the ecosystems investigated. Concerning the identification of the environmental factors responsible for exergy and specific exergy trends, the coherence of the correlation structure between water depth, TSI, exergy and specific exergy indices, suggests that the lake's mean water depth plays a significant role in determining the changes in trophic structure and state (and consequently in exergetic indices) within the set of lakes examined and emphasises the importance of lake morphology in the development and ageing of lake ecosystems.     

A review of recent developments in lake modelling

This paper reviews the lake models published the last five years, mainly in Ecological Modelling. The review shows that structurally dynamic modelling and coupling between hydrodynamic and ecological models are applied increasingly. A number of processes that have not been included in lake models before have been proposed. It has been shown that these additional processes in specific case studies are significant, for instance the competition between phytoplankton and macrophytes or cyanobacteria growth and growth of mussels. It is recommended to study these models for the development of models for case studies where these processes are relevant.     

Identifying key species in ecosystems with stochastic sensitivity analysis

The development of approaches to estimate the vulnerability of biological communities and ecosystems to extirpations and reductions of species is a central challenge of conservation biology. One key aim of this challenge is to develop quantitative approaches to estimate and rank interaction strengths and keystoneness of species and functional groups, i.e. to quantify the relative importance of species. Network analysis can be a powerful tool for this because certain structural aspects of ecological networks are good indicators of the mechanisms that maintain co-evolved, biotic interactions. A static view of ecological networks would lead us to focus research on highly-central species in food webs (topological key players in ecosystems). There are a variety of centrality indices, developed for several types of ecological networks (e.g. for weighted and un-weighted webs). However, truly understanding extinction and its community-wide effects requires the use of dynamic models. Deterministic dynamic models are feasible when population sizes are sufficiently large to minimize noise in the overall system. In models with small population sizes, stochasticity can be modelled explicitly. We present a stochastic simulation-based ecosystem model for identification of “dynamic key species” in situations where stochastic models are appropriate. To demonstrate this approach, we simulated ecosystem dynamics and performed sensitivity analysis using data from the Prince William Sound, Alaska ecosystem model. We then compare these results to those of purely topological analyses and deterministic dynamic (Ecosim) studies. We present the relationships between various topological and dynamic indices and discuss their biological relevance. The trophic group with the largest effect on others is nearshore demersals, the species mostly sensitive to others is halibut, and the group of both considerable effect on and sensitivity to others is juvenile herring. The most important trophic groups in our dynamical simulations appear to have intermediate trophic levels.     

Exergetic assessment for ecological economic system: Chinese agriculture

Based on the thermodynamic concept of exergy as a unified measure for environmental resources and economic products, a framework for systems assessment is presented for ecological economies. With a typical systems diagram devised for a general ecological economy with four arm fluxes for free local natural resources, purchased economic investment, environmental impact and economic yield, system indices of the renewability index, exergy yield ratio, exergy investment ratio, environmental resource to yield ratio, system transformity and environmental stress index are defined for a congregated systems ecological assessment with essential implications to sustainability. As a detailed case study to the Chinese agriculture from 1980 to 2000 with cropping, forestry, stockbreeding and fishery sectors, extensive exergy account and systems assessment are carried out with emphasis on annual and structural variations against social political transitions. For the overall agriculture as a congregated ecological stage, the value of the system transformity is found around 10, the typical value for the general ecological hierarchy as well devised by Odum associated with Lindeman's Tenth Law.     

Application of eco-exergy for assessment of ecosystem health and development of structurally dynamic models

Eco-exergy has been widely used in the assessment of ecosystem health, parameter estimations, calibrations, validations and prognoses. It offers insights into the understanding of ecosystem dynamics and disturbance-driven changes. Particularly, structurally dynamic models (SDMs), which are developed using eco-exergy as the goal function, have been applied in explaining and exploring ecosystem properties and changes in community structure driven by biotic and abiotic factors. In this paper, we review the application of eco-exergy for the assessment of ecosystem health and development of structurally dynamic models (SDMs). The limitations and possible future applications of the approach are also addressed.     

The concepts of emergent and collective properties in individual-based models—Summary and outlook of the Bornhöved case studies

Ecology requires the conceptual and technical ability to analyse complex and dynamic systems consisting of a high and variable number of components and relations. These components are part of a variable interaction structure in a spatially heterogeneous context. The components of ecological interaction networks can give rise to self-organised, and scale-dependent interaction patterns and processes, which are the underlying causes of the overall ecological systems states.The individual-based modelling approach provides a widely applicable simulation framework based on a ‘hierarchy theory’ view of ecological systems.Here, we summarise and generalise the theoretical implications of the modelling studies presented in this volume in the field of terrestrial and aquatic, animal and plant ecology. The case studies cover a representative profile of processes related to ecological applications, such as food web interactions, population dynamics, dispersal, energy physiology, nutrient allocation and mutual impact of morphological and physiological development. The generic approach applied in this context allows a hierarchical representation of ecological systems and their components. Model results are obtained as self-organised structural relation networks and as aggregated quantitative states. In order to address different model characteristics we distinguish collective and emergent properties. Collective properties are those that are attributed equally to different organisation levels of the system. Emergent properties result from the activities of lower level entities on a higher organisation level, while not being present on the lower level. They can be subdivided into aggregational and connective properties. Emergent properties that are aggregational are those which emerge as a result of an aggregation procedure by an observer on the higher level which does not make sense or is not applicable on lower levels. Emergent properties that are connective, however, are based on an interaction network of lower level entities, which brings about the specific system characteristic.This classification of model results will allow to generalise the achievements and potential of the individual-based modelling approach in ecology.     

Development of Lake Ladoga ecosystem models: modeling of the phytoplankton succession in the eutrophication process. I

New models of Lake Ladoga ecosystem and the results of modeling are presented. In the first part the model of phytoplankton succession in the process of anthropogenic eutrophication of the lake is considered under the evolution of the phosphorus loading. The still continued anthropogenic eutrophication of the lake started in 1962 when the phosphorus load began to increase. Since 1962 during the evolution of the lake’s state from oligotrophic to developed mezotrophic one, the structure of phytoplankton community dominating species was significantly changed as well as its total productivity. The system state in the model is described by 14 parameters: nine phytoplankton complexes, zooplankton, dissolved organic matter, detritus, dissolved mineral phosphorus and dissolved oxygen. The number of parameters of this model is noticeably larger than that of previous models created by the authors. The relative dynamics of phytoplankton complexes in the lake’s ecosystem evolution was simulated by the new model. It is shown that the modeling results are adequately corresponding to the observation data. The results of phytoplankton structure modeling allow to estimate the impact of phytoplankton on the water quality as well as give the prediction of the lake’s ecosystem evolution with the changes of the phosphorus loading.     

Informing ecological engineering through ecological network analysis,ecological modelling,and concepts of systems and engineering ecology

A shift in the basic philosophy of nature developed by Francis Bacon, Renè Descartes, and Isaac Newton, has been suggested but for the most part rejected within mainstream science. It suggests the need for viewing nature as a deeply organic and connected system of relationships that is not necessarily or readily submissive to reductive thinking and analysis. Ecosystem design within the construct of a field called ecological engineering poses fundamental questions with respect to the philosophy of nature upon which our current scientific paradigm is predominantly based. In an effort to foster development of rigorous, quantitative methods for developing insight into complex ecosystem phenomena we propose systems and engineering ecology—an integrated science comprised of principles from environ theory, ascendency theory, exergy theory, emergy theory, ecological network analysis and ecological modelling, synthesized through the formal agency of systems science. We contend that ecological engineering will be limited in its robustness apart from development of rigorous systems-based sciences that are quantitative and incorporate the complex, emergent properties of ecosystem. We justify our proposed framework on the philosophical paradox of transferring aspects of traditional engineering design into ecological engineering and on the four causes of Aristotle.     

依据(代)生物体对化学品的反应预测对生态系统服务功能的影响

保护生态系统服务功能越来越多地被作为风险评估的目标，但是目前生态风险评估的终点和评估生态系统服务功能受到的潜在影响之间有很大的差距。作者提出了一个框架，将常用的生态毒理学终点与对种群和群落的影响以及生态系统的服务功能联系起来。这个框架建立在机制效应模型的长足进步上，这些模型旨在跨越多种生物组织，并解释各种生物相互作用和反馈。为了说明这一点，作者引入了2个研究案例，它们采用了已完善和已验证的机制效应模型：鱼种群的inSTREAM个体模型和AQUATOX生态系统模型。他们还展示了动态能量平衡理论可以为解释组织级毒性提供一种通用货币。他们认为，一个基于机制模型的框架，可以预测化学品暴露对生态系统服务的影响，再结合经济估值，可以为环境管理提供一种有用的方法。作者强调了使用这个框架的潜在好处以及未来工作中需要解决的挑战。
精选自Forbes, V. E., Salice, C. J., Birnir, B., Bruins, R. J.F., Calow, P., Ducrot, V., Galic, N., Garber, K., Harvey, B. C., Jager, H., Kanarek, A., Pastorok, R., Railsback, S. F., Rebarber, R. and Thorbek, P. (2017), A framework for predicting impacts on ecosystem services from (sub)organismal responses to chemicals. Environmental Toxicology and Chemistry, 36: 845–859. doi: 10.1002/etc.3720
详情请见http://onlinelibrary.wiley.com/doi/10.1002/etc.3720/full
     

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).     

The free energy and information embodied in the amino acid chains of organisms

It is generally accepted as a useful and workable hypothesis that when an ecosystem receives an inflow of exergy (energy that can do work) it will utilize this flow of exergy to move as far away from thermodynamic equilibrium as possible after the exergy (energy) for maintenance has been covered. If more combinations of system components including organisms are offered, the combination of components and processes that will bring the system most away from thermodynamic equilibrium will win.The amino acid sequences of the proteins e.g. enzymes determine and control the life processes of the organisms and may be viewed as information sensu lato. The free energy of oxidation of the amino acids and the peptide bonds of the cell enzymes expresses therefore the exergy content, eco-exergy or work capacity that the information contributes to “moving further away from thermodynamic equilibrium”. In this paper eco-exergy is calculated and plotted versus the β-values (a measure of the information contained in the genome) for different organisms. The eco-exergy density was previously (see [J?rgensen et al., 1995] and [J?rgensen et al., 2005]) proposed to be calculated as the summation of the product of the β-values representing the information of the genome multiplied by the concentrations of the respective ecosystem components. This analysis shows a strong correlation between the β-values and free energy released when oxidizing the enzymes. The β-values can therefore be assumed to represent the free energy that the organisms have invested in genetic information.     

Exergy vs information in ecological successions: Interpreting community changes by a classical thermodynamic approach

This work proposes a methodology based on classical thermodynamics, which allows the variation in ecosystem composition to be interpreted within the framework of the exergy concept. The basic equation of exergy [Mejer, H., Jorgensen, S.E., 1979. Exergy and ecological buffer capacity. State-of-the-art in Ecological Modelling 7, 829–846] was decomposed into three terms – size (C), structural information (I) and concentration (X) – and their significance as indicators of ecosystem state was evaluated by simulating different scenarios of development in a simplified freshwater ecosystem. In order to calculate the exergy terms, the most critical issue in using exergy in an ecological context, i.e. the estimate of reference equilibrium values for organic matter and organisms, had to be faced. With this aim, the equations of classical thermodynamics in solution were applied, and “virtual” values of concentration at equilibrium were calculated for a number of organic compounds (VEC) and freshwater organisms (VECE). The results of the simulation showed that, whereas exergy and the exergy terms inherently connected with the a-biotic component varied consistently with the incorporation of biomass into the ecosystem, the structural information of the biotic component followed different, even opposite, pathways of variation, which were dependent only on the change in the size spectrum of the community. Due to the strict dependence of the VECE values on organism size, the increase of structural information with increasing abundance of large and complex species is also consistent with the general pattern of succession delineated by the classical r–K model. Structural information is therefore proposed as an indicator of the development state, as well as an ecological orientor, whose maximisation is expected during ecosystem development. However, since an increase in structural information is not necessarily accompanied by an increase in exergy, a sort of “antagonism” between these two related orientors emerges, whose resolution may contribute to shed light on the fundamental forces which drive ecosystem development.     

Radial basis function networks with partially classified data

The problem of estimating a classification rule with partially classified observations, which often occurs in biological and ecological modelling, and which is of major interest in pattern recognition, is discussed. Radial basis function networks for classification problems are presented and compared with the discriminant analysis with partially classified data, in situations where some observations in the training set are unclassified. An application on a set of morphometric data obtained from the skulls of 288 specimens of Microtus subterraneus and Microtus multiplex is performed. This example illustrates how the use of both classified and unclassified observations in the estimate of the hidden layer parameters has the potential to greatly improve the network performances.     

Indicating ecosystem integrity — theoretical concepts and environmental requirements

This paper discusses some conceptual fundamentals for the derivation of environmental indicator sets. On the one hand, it defines requirements from environmental politics, environmental management and legislation, reaching from political target hierarchies and sustainable management strategies to holistic protection concepts such as process protection, resource preservation, ecosystem health and ecological integrity. On the other hand, demands from ecosystem theory are described which include the consideration of features such as self-organization, emergence, thermodynamics, gradients and ecological orientors in environmental indicator sets. From that concept, collective and emergent properties are selected and eight holistic ecosystem features are presented that indicate the ecosystemic state as an ensemble. These general indicators of ecosystem integrity are supplemented by variables on structural changes and substance dynamics.     

Die ökologische Bewertung von Seeufern in Deutschland

Goal and Scope

Details about the ecological function of lake shores as ecotones between land and lakes are not well-known. These ecotones are also heavily exploited and, in part, considerably changed. Whereas anthropogenic nutrient loading is decreasing, structural changes are increasing. Unfortunately, there is a deficit in methods of evaluation and decision processes.

Main Focus

Even the EU-water framework directive was no remedy for this deficit, as lake shores were included only implicitly. In this article several evaluation methods and their conceptual groundwork are presented. However, these methods were not developed for lake shore research. Therefore, criteria are proposed which could fulfill the specific demands of lake shore assessments. The management of lakes shores should consider structural and biological parameters, and be agreeable to local residents.

Results and Conclusions

In addition to conventional biodiversity methods, the ecology of lake shores could also be represented by a functional food net, for example in benthic invertebrates. But even quantification of biodiversity alone creates many problems. A simple biodiversity index cannot meet all the demands placed on a method of evaluation in complex situations, especially when coupled with additional information on structure, practicability, costs, etc. For these reasons, assessments for future management cannot be based on such an index.

Outlook

A possible approach to include this complexity in assessments is to apply mathematical models and theoretical order concepts.     

Integrated urban ecosystem evaluation and modeling based on embodied cosmic exergy

This paper presented a thermodynamic synthesis that involved resource accounting, evaluation and modeling of urban ecosystems based on embodied cosmic exergy (EcE), which redefined embodied exergy with the cosmic microwave background radiation (CMBR) as the reference for solar exergy. In a case study of the Beijing urban ecosystem, the major resources supporting the urban ecosystem, both from free natural resources and from the economy, were accounted for, analyzed and evaluated in the same units, Cosmic Joules (Jc). These indicators revealed the current performance of the Beijing urban ecosystem by considering five aspects of the system: EcE sources, EcE intensity, EcE welfare, environmental impacts and economic efficiency. Moreover, through the combination of the EcE synthesis with a systems dynamics, this research constructed an embodied cosmic exergy-based urban system model (EESM) using Beijing as an example of urban development. The results show that the 10 years from 2010 to 2020 will be very critical for the sustainable development of Beijing because many key factors, such as water resources, wastes and urban assets, might be confronted with great changes during this period. These changes will inevitably transform the urban system not only in its external circumstances but also in its inner structure and may lead to serious consequences. Of all the necessary resources, the most sensitive factor is water supply.     

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.