Mathematics applied to the climate system: outstanding challenges and recent progress

The societal need for reliable climate predictions and a proper assessment of their uncertainties is pressing. Uncertainties arise not only from initial conditions and forcing scenarios, but also from model formulation. Here, we identify and document three broad classes of problems, each representing what we regard to be an outstanding challenge in the area of mathematics applied to the climate system. First, there is the problem of the development and evaluation of simple physically based models of the global climate. Second, there is the problem of the development and evaluation of the components of complex models such as general circulation models. Third, there is the problem of the development and evaluation of appropriate statistical frameworks. We discuss these problems in turn, emphasizing the recent progress made by the papers presented in this Theme Issue. Many pressing challenges in climate science require closer collaboration between climate scientists, mathematicians and statisticians. We hope the papers contained in this Theme Issue will act as inspiration for such collaborations and for setting future research directions.

The AVOID programme’s new simulations of the global benefits of stringent climate change mitigation

Quantitative simulations of the global-scale benefits of climate change mitigation are presented, using a harmonised, self-consistent approach based on a single set of climate change scenarios. The approach draws on a synthesis of output from both physically-based and economics-based models, and incorporates uncertainty analyses. Previous studies have projected global and regional climate change and its impacts over the 21st century but have generally focused on analysis of business-as-usual scenarios, with no explicit mitigation policy included. This study finds that both the economics-based and physically-based models indicate that early, stringent mitigation would avoid a large proportion of the impacts of climate change projected for the 2080s. However, it also shows that not all the impacts can now be avoided, so that adaptation would also therefore be needed to avoid some of the potential damage. Delay in mitigation substantially reduces the percentage of impacts that can be avoided, providing strong new quantitative evidence for the need for stringent and prompt global mitigation action on greenhouse gas emissions, combined with effective adaptation, if large, widespread climate change impacts are to be avoided. Energy technology models suggest that such stringent and prompt mitigation action is technologically feasible, although the estimated costs vary depending on the specific modelling approach and assumptions.

Modelling human visual navigation using multi-view scene reconstruction

It is often assumed that humans generate a 3D reconstruction of the environment, either in egocentric or world-based coordinates, but the steps involved are unknown. Here, we propose two reconstruction-based models, evaluated using data from two tasks in immersive virtual reality. We model the observer’s prediction of landmark location based on standard photogrammetric methods and then combine location predictions to compute likelihood maps of navigation behaviour. In one model, each scene point is treated independently in the reconstruction; in the other, the pertinent variable is the spatial relationship between pairs of points. Participants viewed a simple environment from one location, were transported (virtually) to another part of the scene and were asked to navigate back. Error distributions varied substantially with changes in scene layout; we compared these directly with the likelihood maps to quantify the success of the models. We also measured error distributions when participants manipulated the location of a landmark to match the preceding interval, providing a direct test of the landmark-location stage of the navigation models. Models such as this, which start with scenes and end with a probabilistic prediction of behaviour, are likely to be increasingly useful for understanding 3D vision.

The time-course of morphological constraints: a study of plurals inside derived words

The avoidance of regular but not irregular plurals inside compounds (e.g. *rats eater vs. mice eater) has been one of the most widely studied morphological phenomena in the psycholinguistics literature. To examine whether the constraints that are responsible for this contrast have any general significance beyond compounding, we investigated derived word forms containing regular and irregular plurals in two experiments. Experiment 1 was an offline acceptability judgment task, and experiment 2 measured eye movements during reading derived words containing regular and irregular plurals and uninflected base nouns. The results from both experiments show that the constraint against regular plurals inside compounds generalizes to derived words. We argue that this constraint cannot be reduced to phonological properties, but is instead morphological in nature. The eye-movement data provide detailed information on the time-course of processing derived word forms indicating that early stages of processing are affected by a general constraint that disallows inflected words from feeding derivational processes, and that the more specific constraint against regular plurals comes in at a subsequent later stage of processing. We argue that these results are consistent with stage-based models of language processing.

Ecosystem effects of CO2 concentration: evidence from past climates

Atmospheric CO2 concentration has varied from minima of 170-200 ppm in glacials to maxima of 280-300 ppm in the recent interglacials. Photosynthesis by C-3 plants is highly sensitive to CO2 concentration variations in this range. Physiological consequences of the CO2 changes should therefore be discernible in palaeodata. Several lines of evidence support this expectation. Reduced terrestrial carbon storage during glacials, indicated by the shift in stable isotope composition of dissolved inorganic carbon in the ocean, cannot be explained by climate or sea-level changes. It is however consistent with predictions of current process-based models that propagate known physiological CO2 effects into net primary production at the ecosystem scale. Restricted forest cover during glacial periods, indicated by pollen assemblages dominated by non-arboreal taxa, cannot be reproduced accurately by palaeoclimate models unless CO2 effects on C-3-C-4 plant competition are also modelled. It follows that methods to reconstruct climate from palaeodata should account for CO2 concentration changes. When they do so, they yield results more consistent with palaeoclimate models. In conclusion, the palaeorecord of the Late Quaternary, interpreted with the help of climate and ecosystem models, provides evidence that CO2 effects at the ecosystem scale are neither trivial nor transient.

Effects of noise and by-catch on a Danish harbour porpoise population

Ships and wind turbines generate noise, which can have a negative impact on marine mammal populations by scaring animals away. Effective modelling of how this affects the populations has to take account of the location and timing of disturbances. Here we construct an individual-based model of harbour porpoises in the Inner Danish Waters. Individuals have their own energy budgets constructed using established principles of physiological ecology. Data are lacking on the spatial distribution of food which is instead inferred from knowledge of time-varying porpoise distributions. The model produces plausible patterns of population dynamics and matches well the age distribution of porpoises caught in by-catch. It estimates the effect of existing wind farms as a 10% reduction in population size when food recovers fast (after two days). Proposed new wind farms and ships do not result in further population declines. The population is however sensitive to variations in mortality resulting from by-catch and to the speed at which food recovers after being depleted. If food recovers slowly the effect of wind turbines becomes negligible, whereas ships are estimated to have a significant negative impact on the population. Annual by-catch rates ≥10% lead to monotonously decreasing populations and to extinction, and even the estimated by-catch rate from the adjacent area (approximately 4.1%) has a strong impact on the population. This suggests that conservation efforts should be more focused on reducing by-catch in commercial gillnet fisheries than on limiting the amount of anthropogenic noise. Individual-based models are unique in their ability to take account of the location and timing of disturbances and to show their likely effects on populations. The models also identify deficiencies in the existing database and can be used to set priorities for future field research.

Is behavioral ecology important for understanding and predicting population dynamics?

Population ecology is a discipline that studies changes in the number and composition (age, sex) of the individuals that form a population. Many of the mechanisms that generate these changes are associated with individual behavior, for example how individuals defend their territories, find mates or disperse. Therefore, it is important to model population dynamics considering the potential influence of behavior on the modeled dynamics. This study illustrates the diversity of behaviors that influence population dynamics describing several methods that allow integrating behavior into population models and range from simpler models that only consider the number of individuals to complex individual-based models that capture great levels of detail. A series of examples shows the importance of explicitly considering behavior in population modeling to avoid reaching erroneous conclusions. This integration is particularly relevant for conservation, as incorrect predictions regarding the dynamics of populations of conservation interest can lead to inadequate assessment and management. Improved predictions can favor effective protection of species and better use of the limited financial and human conservation resources.

The status and challenge of global fire modelling

Biomass burning impacts vegetation dynamics, biogeochemical cycling, atmospheric chemistry, and climate, with sometimes deleterious socio-economic impacts. Under future climate projections it is often expected that the risk of wildfires will increase. Our ability to predict the magnitude and geographic pattern of future fire impacts rests on our ability to model fire regimes, either using well-founded empirical relationships or process-based models with good predictive skill. A large variety of models exist today and it is still unclear which type of model or degree of complexity is required to model fire adequately at regional to global scales. This is the central question underpinning the creation of the Fire Model Intercomparison Project - FireMIP, an international project to compare and evaluate existing global fire models against benchmark data sets for present-day and historical conditions. In this paper we summarise the current state-of-the-art in fire regime modelling and model evaluation, and outline what essons may be learned from FireMIP.

Spatial housing economics: a survey

This introduction to the Virtual Special Issue surveys the development of spatial housing economics from its roots in neo-classical theory, through more recent developments in social interactions modelling, and touching on the role of institutions, path dependence and economic history. The survey also points to some of the more promising future directions for the subject that are beginning to appear in the literature. The survey covers elements hedonic models, spatial econometrics, neighbourhood models, housing market areas, housing supply, models of segregation, migration, housing tenure, sub-national house price modelling including the so-called ripple effect, and agent-based models. Possible future directions are set in the context of a selection of recent papers that have appeared in Urban Studies. Nevertheless, there are still important gaps in the literature that merit further attention, arising at least partly from emerging policy problems. These include more research on housing and biodiversity, the relationship between housing and civil unrest, the effects of changing age distributions - notably housing for the elderly - and the impact of different international institutional structures. Methodologically, developments in Big Data provide an exciting framework for future work.

Truth in complex adaptive systems models should be based on truth by constructive verification

It is argued that the truth status of emergent properties of complex adaptive systems models should be based on an epistemology of proof by constructive verification and therefore on the ontological axioms of a non-realist logical system such as constructivism or intuitionism. ‘Emergent’ properties of complex adaptive systems (CAS) models create particular epistemological and ontological challenges. These challenges bear directly on current debates in the philosophy of mathematics and in theoretical computer science. CAS research, with its emphasis on computer simulation, is heavily reliant on models which explore the entailments of Formal Axiomatic Systems (FAS). The incompleteness results of Gödel, the incomputability results of Turing, and the Algorithmic Information Theory results of Chaitin, undermine a realist (platonic) truth model of emergent properties. These same findings support the hegemony of epistemology over ontology and point to alternative truth models such as intuitionism, constructivism and quasi-empiricism.

Predictive control using feedback linearization based on dynamic neural models

This paper presents a hybrid control strategy integrating dynamic neural networks and feedback linearization into a predictive control scheme. Feedback linearization is an important nonlinear control technique which transforms a nonlinear system into a linear system using nonlinear transformations and a model of the plant. In this work, empirical models based on dynamic neural networks have been employed. Dynamic neural networks are mathematical structures described by differential equations, which can be trained to approximate general nonlinear systems. A case study based on a mixing process is presented.

e-learning in project management using simulation models: a case study based on the replication of an experiment

Current e-learning systems are increasing their importance in higher education. However, the state of the art of e-learning applications, besides the state of the practice, does not achieve the level of interactivity that current learning theories advocate. In this paper, the possibility of enhancing e-learning systems to achieve deep learning has been studied by replicating an experiment in which students had to learn basic software engineering principles. One group learned these principles using a static approach, while the other group learned the same principles using a system-dynamics-based approach, which provided interactivity and feedback. The results show that, quantitatively, the latter group achieved a better understanding of the principles; furthermore, qualitatively, they enjoyed the learning experience