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1.  Taylor's law and body size in exploited marine ecosystems 
Ecology and Evolution  2012;2(12):3168-3178.
Taylor's law (TL), which states that variance in population density is related to mean density via a power law, and density-mass allometry, which states that mean density is related to body mass via a power law, are two of the most widely observed patterns in ecology. Combining these two laws predicts that the variance in density is related to body mass via a power law (variance-mass allometry). Marine size spectra are known to exhibit density-mass allometry, but variance-mass allometry has not been investigated. We show that variance and body mass in unexploited size spectrum models are related by a power law, and that this leads to TL with an exponent slightly <2. These simulated relationships are disrupted less by balanced harvesting, in which fishing effort is spread across a wide range of body sizes, than by size-at-entry fishing, in which only fish above a certain size may legally be caught.
PMCID: PMC3539009  PMID: 23301181
Balanced harvesting; density-mass allometry; fishing; power law; size spectrum; size-at-entry; variance-mass allometry
2.  Models of collective cell behaviour with crowding effects: comparing lattice-based and lattice-free approaches 
Individual-based models describing the migration and proliferation of a population of cells frequently restrict the cells to a predefined lattice. An implicit assumption of this type of lattice-based model is that a proliferative population will always eventually fill the lattice. Here, we develop a new lattice-free individual-based model that incorporates cell-to-cell crowding effects. We also derive approximate mean-field descriptions for the lattice-free model in two special cases motivated by commonly used experimental set-ups. Lattice-free simulation results are compared with these mean-field descriptions and with a corresponding lattice-based model. Data from a proliferation experiment are used to estimate the parameters for the new model, including the cell proliferation rate, showing that the model fits the data well. An important aspect of the lattice-free model is that the confluent cell density is not predefined, as with lattice-based models, but an emergent model property. As a consequence of the more realistic, irregular configuration of cells in the lattice-free model, the population growth rate is much slower at high cell densities and the population cannot reach the same confluent density as an equivalent lattice-based model.
PMCID: PMC3479911  PMID: 22696488
cell migration; cell proliferation; lattice-based; lattice-free; random walk
3.  A mathematical framework for modelling cambial surface evolution using a level set method 
Annals of Botany  2011;108(6):1001-1011.
Background and Aims
During their lifetime, tree stems take a series of successive nested shapes. Individual tree growth models traditionally focus on apical growth and architecture. However, cambial growth, which is distributed over a surface layer wrapping the whole organism, equally contributes to plant form and function. This study aims at providing a framework to simulate how organism shape evolves as a result of a secondary growth process that occurs at the cellular scale.
The development of the vascular cambium is modelled as an expanding surface using the level set method. The surface consists of multiple compartments following distinct expansion rules. Growth behaviour can be formulated as a mathematical function of surface state variables and independent variables to describe biological processes.
Key Results
The model was coupled to an architectural model and to a forest stand model to simulate cambium dynamics and wood formation at the scale of the organism. The model is able to simulate competition between cambia, surface irregularities and local features. Predicting the shapes associated with arbitrarily complex growth functions does not add complexity to the numerical method itself.
Despite their slenderness, it is sometimes useful to conceive of trees as expanding surfaces. The proposed mathematical framework provides a way to integrate through time and space the biological and physical mechanisms underlying cambium activity. It can be used either to test growth hypotheses or to generate detailed maps of wood internal structure.
PMCID: PMC3189832  PMID: 21470972
Dynamic model; level sets; surface growth; vascular cambium; wood formation
4.  Assessing Lévy walks as models of animal foraging 
The hypothesis that the optimal search strategy is a Lévy walk (LW) or Lévy flight, originally suggested in 1995, has generated an explosion of interest and controversy. Long-standing empirical evidence supporting the LW hypothesis has been overturned, while new models and data are constantly being published. Statistical methods have been criticized and new methods put forward. In parallel with the empirical studies, theoretical search models have been developed. Some theories have been disproved while others remain. Here, we gather together the current state of the art on the role of LWs in optimal foraging theory. We examine the body of theory underpinning the subject. Then we present new results showing that deviations from the idealized one-dimensional search model greatly reduce or remove the advantage of LWs. The search strategy of an LW with exponent μ = 2 is therefore not as robust as is widely thought. We also review the available techniques, and their potential pitfalls, for analysing field data. It is becoming increasingly recognized that there is a wide range of mechanisms that can lead to the apparent observation of power-law patterns. The consequence of this is that the detection of such patterns in field data implies neither that the foragers in question are performing an LW, nor that they have evolved to do so. We conclude that LWs are neither a universal optimal search strategy, nor are they as widespread in nature as was once thought.
PMCID: PMC3140726  PMID: 21632609
animal movement; efficiency; heavy-tailed; power law; random search; random walk
5.  Random walk models in biology 
Mathematical modelling of the movement of animals, micro-organisms and cells is of great relevance in the fields of biology, ecology and medicine. Movement models can take many different forms, but the most widely used are based on the extensions of simple random walk processes. In this review paper, our aim is twofold: to introduce the mathematics behind random walks in a straightforward manner and to explain how such models can be used to aid our understanding of biological processes. We introduce the mathematical theory behind the simple random walk and explain how this relates to Brownian motion and diffusive processes in general. We demonstrate how these simple models can be extended to include drift and waiting times or be used to calculate first passage times. We discuss biased random walks and show how hyperbolic models can be used to generate correlated random walks. We cover two main applications of the random walk model. Firstly, we review models and results relating to the movement, dispersal and population redistribution of animals and micro-organisms. This includes direct calculation of mean squared displacement, mean dispersal distance, tortuosity measures, as well as possible limitations of these model approaches. Secondly, oriented movement and chemotaxis models are reviewed. General hyperbolic models based on the linear transport equation are introduced and we show how a reinforced random walk can be used to model movement where the individual changes its environment. We discuss the applications of these models in the context of cell migration leading to blood vessel growth (angiogenesis). Finally, we discuss how the various random walk models and approaches are related and the connections that underpin many of the key processes involved.
PMCID: PMC2504494  PMID: 18426776
angiogenesis; chemotaxis; diffusion; dispersal; navigation; random walk
6.  An event-based model of superspreading in epidemics 
Many recent disease outbreaks (e.g. SARS, foot-and-mouth disease) exhibit superspreading, where relatively few individuals cause a large number of secondary cases. Epidemic models have previously treated this as a demographic phenomenon where each individual has an infectivity allocated at random from some distribution. Here, it is shown that superspreading can also be regarded as being caused by environmental variability, where superspreading events (SSEs) occur as a stochastic consequence of the complex network of interactions made by individuals. This interpretation based on SSEs is compared with data and its efficacy in evaluating epidemic control strategies is discussed.
PMCID: PMC2197209  PMID: 17255000
superspreading; epidemiology; branching process; environmental stochasticity

Results 1-6 (6)