Parameterization of individual-based models: comparisons with deterministic mean-field models

The relating of deterministic, mean-field models into network models, where epidemic spread occurs between interconnected susceptible and infectious individuals or populations, requires careful consideration. Here, we discuss models that consider differently the manner in which contact rate and infectiousness change over time, with different algorithms suitable for different underlying processes. Though these models give coincidental results to the mean-field in the case of large, highly connected networks, the results when sparsely connected networks are considered may differ. Different subsets of the parameters from the mean-field epidemic (R(0), generation time, infectiousness, etc.) are preserved in each case. Despite these differences, simulated epidemics generated under some model architectures are insensitive to the average degree of contact amongst nodes, k. Model-based estimates of k may be model dependent, and must therefore be viewed with caution.     

Resolving discrepancies between deterministic population models and individual-based simulations

ABSTRACT This work ties together two distinct modeling frameworks for population dynamics: an individual-based simulation and a set of coupled integrodifferential equations involving population densities. The simulation model represents an idealized predator-prey system formulated at the scale of discrete individuals, explicitly incorporating their mutual interactions, whereas the population-level framework is a generalized version of reaction-diffusion models that incorporate population densities coupled to one another by interaction rates. Here I use various combinations of long-range dispersal for both the offspring and adult stages of both prey and predator species, providing a broad range of spatial and temporal dynamics, to compare and contrast the two model frameworks. Taking the individual-based modeling results as given, two examinations of the reaction-dispersal model are made: linear stability analysis of the deterministic equations and direct numerical solution of the model equations. I also modify the numerical solution in two ways to account for the stochastic nature of individual-based processes, which include independent, local perturbations in population density and a minimum population density within integration cells, below which the population is set to zero. These modifications introduce new parameters into the population-level model, which I adjust to reproduce the individual-based model results. The individual-based model is then modified to minimize the effects of stochasticity, producing a match of the predictions from the numerical integration of the population-level model without stochasticity.     

Reconciling classical and individual-based approaches in theoretical population ecology: a protocol for extracting population parameters from individual-based models

The two main approaches in theoretical population ecology-the classical approach using differential equations and the approach using individual-based modeling-seem to be incompatible. Linked to these two approaches are two different timescales: population dynamics and behavior or physiology. Thus, the question of the relationship between classical and individual-based approaches is related to the question of the mutual relationship between processes on the population and the behavioral timescales. We present a simple protocol that allows the two different approaches to be reconciled by making explicit use of the fact that processes operating on two different timescales can be treated separately. Using an individual-based model of nomadic birds as an example, we extract the population growth rate by deactivating all demographic processes-in other words, the individuals behave but do not age, die, or reproduce. The growth rate closely matches the logistic growth rate for a wide range of parameters. The implications of this result and the conditions for applying the protocol to other individual-based models are discussed. Since in physics the technique of separating timescales is linked to some concepts of self-organization, we believe that the protocol will also help to develop concepts of self-organization in ecology.     

Coupled map lattice approximations for spatially explicit individual-based models of ecology

Spatially explicit individual-based models are widely used in ecology but they are often difficult to treat analytically. Despite their intractability they often exhibit clear temporal and spatial patterning. We demonstrate how a spatially explicit individual-based model of scramble competition with local dispersal can be approximated by a stochastic coupled map lattice. The approximation disentangles the deterministic and stochastic element of local interaction and dispersal. We are thus able to understand the individual-based model through a simplified set of equations. In particular, we demonstrate that demographic noise leads to increased stability in the dynamics of locally dispersing single-species populations. The coupled map lattice approximation has general application to a range of spatially explicit individual-based models. It provides a new alternative to current approximation techniques, such as the method of moments and reaction-diffusion approximation, that captures both stochastic effects and large-scale patterning arising in individual-based models.     

From local interactions to population dynamics in site-based models of ecology

A central problem in ecology is relating the interactions of individuals-described in terms of competition, predation, interference, etc.-to the dynamics of the populations of these individuals-in terms of change in numbers of individuals over time. Here, we address this problem for a class of site-based ecological models, where local interactions between individuals take place at a finite number of discrete resource sites over non-overlapping generations and, between generations, individuals move randomly between sites over the entire system. Such site-based models have previously been applied to a wide range of ecological systems: from those involving contest or scramble competition for resources to host-parasite interactions and meta-populations. We show how the population dynamics of site-based models can be accurately approximated by and understood through deterministic and stochastic difference equations. Conversely, we use the inverse of this approximation to show what implicit assumptions are made about individual interactions by modelling of population dynamics in terms of difference equations. To this end, we prove a useful and general theorem: that any model in our class of site-based models has a corresponding stochastic difference equation population model, by which it can be approximated. This theorem allows us to calculate long-term population dynamics, evolutionary stable strategies and, by extending our theory to account for large deviations, extinction probabilities for a wide range of site-based systems. Our methodology is then illustrated to various examples of between species competition, predator-prey interactions and co-operation.     

Context-oriented model validation of individual-based models in ecology: A hierarchically structured approach to validate qualitative,compositional and quantitative characteristics

Validation constitutes a vital process in model development and application, as it ensures the applicability of a model for the intended purposes and trustworthy results within the range of model assumptions. Commonly, independent empirical data sets are statistically compared with the generated model results, which is an adequate approach for models which operate on a single hierarchical level, such as most equation-based models. Individual-based models (IBM) can operate on different organisational levels synchronously and have an inherent complex and variable interaction structure for many applications. Thus a plain comparison of data congruity on the result levels might leave too many questions unanswered. However, a more comprehensive assessment of model validity can require additional investigations which encompass also qualitative and structural relationships.Here we describe a hierarchically structured validation which is oriented towards the investigated context of the model and allows organising the validation process in close relation to the different hierarchical levels which are covered in the model. The context oriented organisation protocol for validation includes the following steps: (1) assessing the different model levels separately, then, (2) applying a set of different techniques such as visual inspection, statistical comparison, involvement of experts, aggregation of data on higher integration levels and experimental validation.The context oriented approach accounts for the specificity of individual-based models – i.e., the dynamic self-organisation of model outcomes from biologically underpinned individual interactions without an inherent determination of properties on higher hierarchical levels – and extends the potential of the validation process qualitatively, as it allows to assess complex structural and causal relations and multi-level feedback processes of the developed models.     

The rise of the individual-based model in ecology

Recent advances of three different kinds are driving a change in the way that modelling Is being done in ecology. First, the theory of chaos tells us that short-term predictions of nonlinear systems will be difficult, and long-term predictions will be impossible. The grave Implications this has for ecology are only just beginning to be understood. Second, ecologists have started to recognize the importance of local interactions between individuals in ecological systems. And third, improvements in computer power and software are making computers more inviting as a primary tool for modelling. The combination of these factors may have far-reaching consequences for ecological theory.     

Some theoretical considerations: From landscape ecology to waterscape ecology

Being inspired in the process of teaching and studying on tourism geography and heritage landscapes, the author attempts to broaden the scope of traditional studies in landscape ecology. Upon reviewing the progress, limitations and boundaries of landscape ecology, he criticizes the negligence of waterscapes by academics. The author further examines some waterscape conservation policy programs practiced by different nations. These include 10 National Seashores, 4 National Lakeshores, 4 National Rivers, 13 National Marine Sanctuaries and 1 Marine National Monument in USA, 13 Marine National Parks, 11 Marine Sanctuaries and 4 multiple use Marine Reserves in the State of Victoria, Australia, 3 Marine Nature Reserves and 43 Heritage Coasts in UK, and 272 National Water Parks in China. Based on these explorations, the author proposes a few hypotheses on waterscape ecology. Finally, he concludes that as one among many sub-disciplines of applied ecology, waterscape ecology can only achieve its full-fledged growth through concerted efforts among academics, and the supports from both governments and non-government organizations. Meanwhile, the site-specific policy practices with the rapid increasing social needs will likely facilitate the development of this infant school. Hopefully, the birth and development of waterscape ecology will contribute to the prosperity of global academics and the maintenance of earth level environmental health.     

Some theoretical considerations: From landscape ecology to waterscape ecology

Being inspired in the process of teaching and studying on tourism geography and heritage landscapes, the author attempts to broaden the scope of traditional studies in landscape ecology. Upon reviewing the progress, limitations and boundaries of landscape ecology, he criticizes the negligence of waterscapes by academics. The author further examines some waterscape conservation policy programs practiced by different nations. These include 10 National Seashores, 4 National Lakeshores, 4 National Rivers, 13 National Marine Sanctuaries and 1 Marine National Monument in USA, 13 Marine National Parks, 11 Marine Sanctuaries and 4 multiple use Marine Reserves in the State of Victoria, Australia, 3 Marine Nature Reserves and 43 Heritage Coasts in UK, and 272 National Water Parks in China. Based on these explorations, the author proposes a few hypotheses on waterscape ecology. Finally, he concludes that as one among many sub-disciplines of applied ecology, waterscape ecology can only achieve its full-fledged growth through concerted efforts among academics, and the supports from both governments and non-government organizations. Meanwhile, the site-specific policy practices with the rapid increasing social needs will likely facilitate the development of this infant school. Hopefully, the birth and development of waterscape ecology will contribute to the prosperity of global academics and the maintenance of earth level environmental health.     

Bifurcations and chaos in ecology: lynx returns revisited

One of the most popular data sets in ecology, that of lynx fur returns, is analysed in order to look for evidence for a bifurcation process. This bifurcation seems to be present from the observation of sudden shifts in the amplitude of oscillations of the lynx time series. Schaffer first proposed the possibility for such a bifurcation in 1985, and suggested that a possible source for the qualitative change of lynx's fluctuations was an increased trapping effort, which eventually lead to high-amplitude, chaotic dynamics. By studying the available information from the Hudson Bay Company records, we have found evidence for such an increased trapping pressure that (i) rapidly rose close to the shift from low-amplitude to large-amplitude fluctuations around 1820, and (ii) decreased around 1910, when there is another shift again to damped small oscillations. Although an increase in the top-predator mortality in a three-species food web typically leads to simpler dynamics and eventual predator extinction, here we show that a recent model involving a minimum bound in the lynx population, due to the presence of alternative prey in the lynx diet, consistently supports the presence of a bifurcation phenomenon.     

Return map approach to description of the deterministic chaos in cytosolic calcium oscillations

A modified method of the return map reconstruction is proposed. The method is applied to the analysis of intracellular calcium oscillations. On the basis of the approach developed, these oscillations are recognized as low-dimensional deterministic chaotic process.     

Using deterministic models to assess risk in sediment-impacted estuaries

Terrestrial sediments transported off disturbed watersheds can threaten downstream estuarine and coastal environments. Practitioners in the field of environmental management need to be able to assess how activities that disturb the land are likely to impact on downstream estuarine and coastal receiving waters. Such assessments usually take the form of estimating risk, which has two components: the extent of undesirable consequences and the probability of occurrence of those consequences. The consequence in this case is primarily the ecological damage. In order to estimate risk, a methodology was developed around the use of a deterministic computational model of watershed/estuarine sediment transport. Model simulations are run which cover the physically realistic range of the controlling variables (e.g., wind speed, wind direction, sediment runoff). The model results are related to estuary damage and combined with the probability of occurrence for each scenario, providing a means of quantifying risk. In addition, answers to `long-term average', `most likely' and `worse case' type questions can be made. The procedure has the advantages of being easily implemented, transparent and repeatable. It is also more efficient than a standard Monte-Carlo procedure. The risk associated with changes in landuse can be examined without repeating model simulations. The example used to illustrate the method is that of a developing watershed in which proposed building activities are expected to alter the sediment yield to the downstream estuary during floods, which in turn is expected to smother shellfish beds. By elucidating risk associated with each development scenario, the management debate can be focused on choice of an acceptable risk level.     

Molecular signatures from omics data: From chaos to consensus

In the past 15 years, new "omics" technologies have made it possible to obtain high-resolution molecular snapshots of organisms, tissues, and even individual cells at various disease states and experimental conditions. It is hoped that these developments will usher in a new era of personalized medicine in which an individual's molecular measurements are used to diagnose disease, guide therapy, and perform other tasks more accurately and effectively than is possible using standard approaches. There now exists a vast literature of reported "molecular signatures". However, despite some notable exceptions, many of these signatures have suffered from limited reproducibility in independent datasets, insufficient sensitivity or specificity to meet clinical needs, or other challenges. In this paper, we discuss the process of molecular signature discovery on the basis of omics data. In particular, we highlight potential pitfalls in the discovery process, as well as strategies that can be used to increase the odds of successful discovery. Despite the difficulties that have plagued the field of molecular signature discovery, we remain optimistic about the potential to harness the vast amounts of available omics data in order to substantially impact clinical practice.     

Embedding a grammatical description in deterministic chaos: an experiment in recurrent neural learning

We consider the modeling process of a biological agent by combining the concepts of neuroinformatics and deterministic chaos. We assume that an agent observes a target process as a stochastic symbolic process, which is restricted by grammatical constraints. Our main hypothesis is that an agent would learn the target model by reconstructing an equivalent quasi-stochastic process on its deterministic neural dynamics. We employed a recurrent neural network (RNN), which is regarded as an adjustable deterministic dynamical system. Then, we conducted an experiment to observe how the RNN learns to reconstruct the target process, represented by a stochastic finite state machine in the simulation. The result revealed the capability of the RNN to evolve, by means of learning, toward chaos, which is able to mimic a target's stochastic process. We precisely analyzed the evolutionary process as well as the internal representation of the neural dynamics obtained. This analysis enabled us to clarify an interesting mechanism of the self-organization of chaos by means of neural learning, and also showed how grammar can be embedded in the evolved deterministic chaos.     

Superiority of single covalent modification in specificity: From deterministic to stochastic viewpoint

Covalent modification(s) are required in many signaling pathways. It has been discussed from a deterministic viewpoint that dual covalent modification is more favorable than single covalent modification for signaling specificity. However, whether this conclusion is feasible in stochastic situation has not yet been studied. To study the role of covalent modification in the specificity of a stochastic signaling pathway, we here simulate the dynamics of a transiently stimulated signaling pathway, considering the influence of the stochasticity arising from the low molecule number of reactants. It turns out that the specificity of dual covalent modification would be worse than that of single covalent modification when the number of molecules is in some biologically plausible range. We further discuss some factors that have potential influence on specificity, such as the rates of the upstream reaction cycle of the covalent modification(s), the duration and the magnitude of the transient stimulus. Our numerical results indicate that whether dual or single covalent modification(s) is better in specificity also depends on these factors. Superiority of single covalent modification in specificity would arise if the stimulus is weak and transient, or if it is embedded downstream of a reaction whose activation rate is slow while deactivation rate is fast. The relevance of these conclusions to signal transduction is briefly discussed.