Lotka-Volterra models for the market penetration of renewable energy.

We studied trend models for the energy and electricity markets and compared their complexity. Was it associated to the ease of policy interventions to control these markets? Using open data, we applied Lotka-Volterra type models to describe the evolution of the market shares of renewable energy and of electricity from renewable sources and observed that for the latter data more complex models were needed. We then searched for characteristics and anomalies in these models that might indicate immunity to and susceptibility for policy interventions towards reaching carbon neutrality, respectively. However, the evidence linking complexity and receptiveness for policy interventions remained weak.

Parameter estimation of the Solow-Swan fundamental differential equation.

Background: The Solow-Swan model describes the long-term growth of the capital to labor ratio by the fundamental differential equation of Solow-Swan theory. In conventional approaches, this equation was fitted to data using additional information, such as the rates of population growth, capital depreciation, or saving. However, this was not the best possible fit. Objectives: Using the method of least squares, what is the best possible fit of the fundamental equation to the time-series of the capital to labor ratios? Are the best-fit parameters economically sound? Method: For the data, we used the Penn-World Table in its 2021 version and compared six countries and three definitions of the capital to labor ratio. For optimization, we used a custom-made variant of the method of simulated annealing. We also compared different optimization methods and calibrations. Results: When comparing different methods of optimization, our custom-made tool provided reliable parameter estimates. In terms of R-squared they improved upon the parameter estimates of the conventional approach. Except for the USA, the best-fit values of the exponent were unplausible, as they suggested a too large elasticity of output. However, using a different calibration resulted in more plausible values of the best-fit exponent also for France and Pakistan, but not for Argentina and Japan. Conclusion: Our results have shown a discrepancy between the best-fit parameters obtained from optimization and the parameter values that are deemed plausible in economy. We propose a research program to resolve this issue by investigating if suitable calibrations may generate economically plausible best-fit parameter values.

Bertalanffy-Pütter models for avian growth.

This paper explores the ratio of the mass in the inflection point over asymptotic mass for 81 nestlings of blue tits and great tits from an urban parkland in Warsaw, Poland (growth data from literature). We computed the ratios using the Bertalanffy-Pütter model, because this model was more flexible with respect to the ratios than the traditional models. For them, there were a-priori restrictions on the possible range of the ratios. (Further, as the Bertalanffy-Pütter model generalizes the traditional models, its fit to the data was necessarily better.) For six birds there was no inflection point (we set the ratio to 0), for 19 birds the ratio was between 0 and 0.368 (lowest ratio attainable for the Richards model), for 48 birds it was above 0.5 (fixed ratio of logistic growth), and for the remaining eight birds it was in between; the maximal observed ratio was 0.835. With these ratios we were able to detect small variations in avian growth due to slight differences in the environment: Our results indicate that blue tits grew more slowly (had a lower ratio) in the presence of light pollution and modified impervious substrate, a finding that would not have been possible had we used traditional growth curve analysis.

Bertalanffy-Pütter models for the first wave of the COVID-19 outbreak.

The COVID-19 pandemics challenges governments across the world. To develop adequate responses, they need accurate models for the spread of the disease. Using least squares, we fitted Bertalanffy-Pütter (BP) trend curves to data about the first wave of the COVID-19 pandemic of 2020 from 49 countries and provinces where the peak of the first wave had been passed. BP-models achieved excellent fits (R-squared above 99%) to all data. Using them to smoothen the data, in the median one could forecast that the final count (asymptotic limit) of infections and fatalities would be 2.48 times (95% confidence limits 2.42-2.6) and 2.67 times (2.39-2.765) the total count at the respective peak (inflection point). By comparison, using logistic growth would evaluate this ratio as 2.00 for all data. The case fatality rate, defined as the quotient of the asymptotic limits of fatalities and confirmed infections, was in the median 4.85% (confidence limits 4.4%-6.5%). Our result supports the strategies of governments that kept the epidemic peak low, as then in the median fewer infections and fewer fatalities could be expected.

Forecasting the final disease size: comparing calibrations of Bertalanffy-Pütter models.

Using monthly data from the Ebola-outbreak 2013-2016 in West Africa, we compared two calibrations for data fitting, least-squares (SSE) and weighted least-squares (SWSE) with weights reciprocal to the number of new infections. To compare (in hindsight) forecasts for the final disease size (the actual value was observed at month 28 of the outbreak) we fitted Bertalanffy-Pütter growth models to truncated initial data (first 11, 12, …, 28 months). The growth curves identified the epidemic peak at month 10 and the relative errors of the forecasts (asymptotic limits) were below 10%, if 16 or more month were used; for SWSE the relative errors were smaller than for SSE. However, the calibrations differed insofar as for SWSE there were good fitting models that forecasted reasonable upper and lower bounds, while SSE was biased, as the forecasts of good fitting models systematically underestimated the final disease size. Furthermore, for SSE the normal distribution hypothesis of the fit residuals was refuted, while the similar hypothesis for SWSE was not refuted. We therefore recommend considering SWSE for epidemic forecasts.

The growth of domestic goats and sheep: A meta study with Bertalanffy-Pütter models.

Growth literature often uses the Brody, Gompertz, Verhulst, and von Bertalanffy models. Is there a rationale for the preference of these classical named models? The versatile five-parameter Bertalanffy-Pütter (BP) model generalizes these models. We revisited peer-reviewed publications from the years 1970-2019 that fitted growth models to together 122 mass-at-age data of sheep and goats from 19 countries and studied the best-fit BP-models using the least-squares method. None of the named models was ever best-fitting. However, for 70% of the data a single non-sigmoidal model had an acceptable fit (normalized root mean squared error 〈 5% and F-ratio test 〉 5% in comparison to the best-fit): the Brody model. The inherently non-sigmoidal character was further underlined, as there were only 39% of the data, where the best-fitting BP-model had a discernible inflection point. For these data, conclusions of biological interest could be drawn from the sigmoidal best-fit BP-models: the maximal weight gain per day was about 55% higher than the natal weight gain per day.

Comparing growth patterns of three species: Similarities and differences.

Quantitative studies of the growth of dinosaurs have made comparisons with modern animals possible. Therefore, it is meaningful to ask, if extinct dinosaurs grew faster than modern animals, e.g. birds (modern dinosaurs) and reptiles. However, past studies relied on only a few growth models. If these models were false, what about the conclusions? This paper fits growth data to a more comprehensive class of models, defined by the von Bertalanffy-Pütter (BP) differential equation. Applied to data about Tenontosaurus tilletti, Alligator mississippiensis and the Athens Canadian Random Bred strain of Gallus gallus domesticus the best fitting growth curves did barely differ, if they were rescaled for size and lifespan. A difference could be discerned, if time was rescaled for the age at the inception point (maximal growth) or if the percentual growth was compared.

Best fitting tumor growth models of the von Bertalanffy-PütterType.

BACKGROUND: Longitudinal studies of tumor volume have used certain named mathematical growth models. The Bertalanffy-Pütter differential equation unifies them: It uses five parameters, amongst them two exponents related to tumor metabolism and morphology. Each exponent-pair defines a unique three-parameter model of the Bertalanffy-Pütter type, and the above-mentioned named models correspond to specific exponent-pairs. Amongst these models we seek the best fitting one. METHOD: The best fitting model curve within the Bertalanffy-Pütter class minimizes the sum of squared errors (SSE). We investigate also near-optimal model curves; their SSE is at most a certain percentage (e.g. 1%) larger than the minimal SSE. Models with near-optimal curves are visualized by the region of their near-optimal exponent pairs. While there is barely a visible difference concerning the goodness of fit between the best fitting and the near-optimal model curves, there are differences in the prognosis, whence the near-optimal models are used to assess the uncertainty of extrapolation. RESULTS: For data about the growth of an untreated tumor we found the best fitting growth model which reduced SSE by about 30% compared to the hitherto best fit. In order to analyze the uncertainty of prognosis, we repeated the search for the optimal and near-optimal exponent-pairs for the initial segments of the data (meaning the subset of the data for the first n days) and compared the prognosis based on these models with the actual data (i.e. the data for the remaining days). The optimal exponent-pairs and the regions of near-optimal exponent-pairs depended on how many data-points were used. Further, the regions of near-optimal exponent-pairs were larger for the first initial segments, where fewer data were used. CONCLUSION: While for each near optimal exponent-pair its best fitting model curve remained close to the fitted data points, the prognosis using these model curves differed widely for the remaining data, whence e.g. the best fitting model for the first 65 days of growth was not capable to inform about tumor size for the remaining 49 days. For the present data, prognosis appeared to be feasible for a time span of ten days, at most.

Best-fitting growth curves of the von Bertalanffy-Pütter type.

INTRODUCTION: A large body of literature aims at identifying growth models that fit best to given mass-at-age data. The von Bertalanffy-Pütter differential equation is a unifying framework for the study of growth models. PROBLEM: The most common growth models used in poultry science literature fit into this framework, as these models correspond to different exponent-pairs (e.g., Brody, Gompertz, logistic, Richards, and von Bertalanffy models). Here, we search for the optimal exponent-pairs (a and b) amongst all possible exponent-pairs and expect a significantly better fit of the growth curve to concrete mass-at-age data. METHOD: Data fitting becomes more difficult, as there is a large region of nearly optimal exponent-pairs. We therefore develop a fully automated optimization method, with computation time of about 1 to 2 wk per data-set. For the proof of principle, we applied it to literature data about 217 male meat-type chickens, Athens Canadian Random Bred, that were reared under controlled conditions and weighed 28 times during a time span of 170 D. RESULTS: We compared 2 methods of data fitting, least squares using the sum of squared errors (SSE), which is common in literature, and a variant using the sum of squared log-errors SSElog. For these data, the optimal exponent-pairs were (0.43, 4.06) for SSE = 2,208.6 (31% improvement over literature values for the residual standard deviation) and (0.89, 0.93) for SSElog = 0.04599. Both optimal exponents were clearly distinct from the exponent-pairs of the common models in literature. This finding was reinforced by considering the region of nearly optimal exponents. DISCUSSION: We explain, why we recommend using SSElog for data fitting and we discuss prognosis, where data from the first 8 wk of growth would not be enough.

Optimal and near-optimal exponent-pairs for the Bertalanffy-Pütter growth model.

The Bertalanffy-Pütter growth model describes mass m at age t by means of the differential equation dm/dt = p * m a  - q * mb . The special case using the von Bertalanffy exponent-pair a = 2/3 and b = 1 is most common (it corresponds to the von Bertalanffy growth function VBGF for length in fishery literature). Fitting VBGF to size-at-age data requires the optimization of three model parameters (the constants p, q, and an initial value for the differential equation). For the general Bertalanffy-Pütter model, two more model parameters are optimized (the pair a < b of non-negative exponents). While this reduces bias in growth estimates, it increases model complexity and more advanced optimization methods are needed, such as the Nelder-Mead amoeba method, interior point methods, or simulated annealing. Is the improved performance worth these efforts? For the case, where the exponent b = 1 remains fixed, it is known that for most fish data any exponent a < 1 could be used to model growth without affecting the fit to the data significantly (when the other parameters were optimized). We hypothesized that the optimization of both exponents would result in a significantly better fit of the optimal growth function to the data and we tested this conjecture for a data set (20,166 fish) about the mass-growth of Walleye (Sander vitreus), a fish from Lake Erie, USA. To this end, we assessed the fit on a grid of 14,281 exponent-pairs (a, b) and identified the best fitting model curve on the boundary a = b of the grid (a = b = 0.686); it corresponds to the generalized Gompertz equation dm/dt = p * ma  - q * ln(m) * ma . Using the Akaike information criterion for model selection, the answer to the conjecture was no: The von Bertalanffy exponent-pair model (but not the logistic model) remained parsimonious. However, the bias reduction attained by the optimal exponent-pair may be worth the tradeoff with complexity in some situations where predictive power is solely preferred. Therefore, we recommend the use of the Bertalanffy-Pütter model (and of its limit case, the generalized Gompertz model) in natural resources management (such as in fishery stock assessments), as it relies on careful quantitative assessments to recommend policies for sustainable resource usage.

On the exponent in the Von Bertalanffy growth model.

Von Bertalanffy proposed the differential equation m'(t) = p × m(t) a  - q × m(t) for the description of the mass growth of animals as a function m(t) of time t. He suggested that the solution using the metabolic scaling exponent a = 2/3 (Von Bertalanffy growth function VBGF) would be universal for vertebrates. Several authors questioned universality, as for certain species other models would provide a better fit. This paper reconsiders this question. Based on 60 data sets from literature (37 about fish and 23 about non-fish species) it optimizes the model parameters, in particular the exponent 0 ≤ a < 1, so that the model curve achieves the best fit to the data. The main observation of the paper is the large variability in the exponent, which can vary over a very large range without affecting the fit to the data significantly, when the other parameters are also optimized. The paper explains this by differences in the data quality: variability is low for data from highly controlled experiments and high for natural data. Other deficiencies were biologically meaningless optimal parameter values or optimal parameter values attained on the boundary of the parameter region (indicating the possible need for a different model). Only 11 of the 60 data sets were free of such deficiencies and for them no universal exponent could be discerned.

Ensuring sustainability of non-networked sanitation technologies: an approach to standardization.

Non-networked sanitation technologies use no sewer, water or electricity lines. Based on a review of 45 commercially distributed technologies, 12 (representing three concepts) were selected for a detailed audit. They were located in six countries of Africa and Asia. The safety of users was generally assured and the costs per use were not excessive, whereas costs were fully transparent for only one technology surveyed. A main drawback was insufficient quality of the byproducts from on-site treatment, making recycling in agriculture a hygienic and environmental risk. Further, no technology was sufficiently mature (requiring e.g. to shift wastes by hand). In order to promote further development and give producers of mature products a competitive advantage, the paper proposes a certification of technologies to confirm the fulfillment of basic requirements to make them attractive for future users.

A planning-oriented sustainability assessment framework for peri-urban water management in developing countries.

DPSIR and the three-pillar model are well-established frameworks for sustainability assessment. This paper proposes a planning-oriented sustainability assessment framework (POSAF). It is informed by those frameworks but differs insofar as it puts more emphasis on a constructivist conception which recognises that sustainability needs to be defined anew for each planning problem. In finding such a consensus definition, POSAF uses participatory scenario analysis and participatory planning, technical feasibility study, participatory assessment, analysis of trade-offs and social networks in an unusual combination and for goals that differ from the original conceptions of these methods. POSAF was applied in a peri-urban area of Mexico City for the design of improved water service provision, integrating solid waste management. It supported consensus amongst users about the importance of environmental issues, informed planners about the values of stakeholders and users, detected local differences, and identified possible conflicts at an early stage of decision-making.

Why do water and sanitation systems for the poor still fail? Policy analysis in economically advanced developing countries.

The results of an independent evaluation of 60 case studies of water and sanitation infrastructure projects in India, Mexico, and South Africa, most of them implemented since 2000, demonstrate an ongoing problem of failing infrastructure even in economically advanced developing countries. This paper presents a meta-analysis of those project case study results and analyses whether the design of existing policies or other factors contribute to failures. It concludes that the observed failures are due to well-known reasons and recommends how the implementation of the Dublin-Rio Principles can be improved. (They were introduced twenty years ago to avoid such failures by means of more sustainable planning.).

Modeling the final phase of landfill gas generation from long-term observations.

For waste management, methane emissions from landfills and their effect on climate change are of serious concern. Current models for biogas generation that focus on the economic use of the landfill gas are usually based on first order chemical reactions (exponential decay), underestimating the long-term emissions of landfills. The presented study concentrated on the curve fitting and the quantification of the gas generation during the final degradation phase under optimal anaerobic conditions. For this purpose the long-term gas generation (240-1,830 days) of different mechanically biologically treated (MBT) waste materials was measured. In this study the late gas generation was modeled by a log-normal distribution curve to gather the maximum gas generation potential. According to the log-normal model the observed gas sum curve leads to higher values than commonly used exponential decay models. The prediction of the final phase of landfill gas generation by a fitting model provides a basis for CO(2) balances in waste management and some information to which extent landfills serve as carbon sink.

Design of an institutional decision-making process: the case of urban water management.

Urban water infrastructure expenditures cause a major financial burden to municipalities. In the opinion of many policy-makers, public funds may alleviate this burden and facilitate environmental policies. However, practice has shown that despite ambitious policies, funding often follows traditional cost-dominated thinking. In Austria, national policy-makers were interested in new guidelines for funding that increase the transparency of the planning, ensure the adequate treatment of ecological problems, and foster stakeholder involvement, but keep the process as simple as possible, and require minimal changes of the current guidelines. An interdisciplinary project team conducted such a study. Its outcome was tested in two pilot projects. Based on these experiences, policy-makers finally implemented the recommended guidelines for the funding of communal urban water infrastructure projects. A general observation about the policy-making process was a conservative attitude of policy-makers. They prefer simple constraints (precautionary principle) and flexible negotiations (delegation) to complex assessment and decision-aid methodologies.

Feasibility versus sustainability in urban water management.

Decision making in urban water management is exemplified by the case of Austria: Although researchers define a comprehensible concept of sustainability, practitioners emphasize feasibility and accept limitations in sustainability. Could the specification of particular methods, chosen from some decision support methodology, remedy this situation? While an integrative assessment of sustainability should not be based on prescribed or standardized criteria, or even a certain assessment method, it should force the decision makers to make their chosen premise more visible. To this end a change of the decision making process is proposed, which will allow the decision makers to adapt the decision making process to the circumstances of a specific project in a way that is accepted by the stakeholders.