Graeme P. Boswell

Habitat fragmentation, percolation theory and the conservation of a keystone species

Many species survive in specialized habitats. When these habitats are destroyed or fragmented the threat of extinction looms. In this paper, we use percolation theory to consider how an environment may fragment. We then develop a stochastic, spatially explicit, individual–based model to consider the effect of habitat fragmentation on a keystone species (the army ant Eciton burchelli) in a neo tropical rainforest. The results suggest that species may become extinct even in huge reserves before their habitat is fully fragmented; this has important implications for conservation. We show that sustainable forest–harvesting strategies may not be as successful as is currently thought. We also suggest that habitat corridors, once thought of as the saviour for fragmented environments, may have a detrimental effect on population persistence.

Growth and function of fungal mycelia in heterogeneous environments.

As decomposer organisms, pathogens, plant symbionts and nutrient cyclers, fungi are of fundamental importance in the terrestrial environment. Moreover, in addition to their well-known applications in industry, many species also have great potential in environmental biotechnology. The study of this important class of organisms is difficult through experimental means alone due to the heterogeneity of their natural growth habitat and the microscopic scale of growth. In this work we present a mathematical model for colony expansion that is derived through consideration of the growth characteristics on the microscale. The model equations are of mixed hyperbolic-parabolic type and are treated with a numerical scheme that preserves positivity and conserves mass. The numerical solutions are compared against experimental results in a variety of environments. Thus the effect of different translocation mechanisms on fungal growth and function are identified. The derivation and analysis of an approximation to the full model yields further results concerning basic properties of mycelial growth. Finally, the acidification of the growth habitat is considered and the model thus provides important predictions on the functional consequences of the redistribution of internally-located material.

Solubilization of calcium phosphate as a consequence of carbon translocation by Rhizoctonia solani

A model system based on arrays of three concentric rings of discrete agar droplets is described which allowed study of fungal growth in vitro in nutritionally-heterogeneous conditions. Droplets containing different combinations of glucose and calcium phosphate were used to study the consequences of spatially separating these components in relation to metal phosphate solubilization by Rhizoctonia solani. A pH indicator, bromocresol purple, was added to the agar to visualise the localised production of acidity by the fungus. In the presence of the fungus, solubilization of calcium phosphate on homogeneous agar plates only occurred when glucose was present in the underlying medium. However, solubilization occurred in droplets containing calcium phosphate, but no glucose, when glucose was present in other droplets within the tessellation and where fungal hyphae spanned the droplets. This demonstrates that substrate was transported via mycelia from glucose-containing domains, with the functional consequence of metal phosphate solubilization. In another design, where the inner ring of droplets contained glucose and the outer ring contained only calcium phosphate, acidification of all droplets in the outer ring was observed when the inner droplets contained glucose. However, solubilization of calcium phosphate only occurred when the concentration of glucose in the inner droplets was greater than 2% (w/v). This indicated that a threshold concentration of carbon source may be required before such mechanisms of solubilization are invoked. There was also evidence for reverse translocation of substrate from newly colonised glucose-containing droplets in the outer ring to the central droplets, where fungal growth had originated.

Solubilization of metal phosphates by Rhizoctonia solani

The effects of different temperatures and pH on the growth and solubilization of insoluble calcium phosphate, cobalt phosphate, manganese phosphate, strontium hydrogen phosphate and zinc phosphate by Rhizoctonia solani on solidified media were assessed. Solubilization of the metal phosphates was monitored by the production of a clear zone around or underneath the fungus. R. solani efficiently solubilized all five metal phosphates except cobalt phosphate when grown on medium at pH 7. Solubilization activity by R. solani decreased with increasing pH on medium containing calcium phosphate but increased on strontium hydrogen phosphate-amended medium. The uptake of metals by the mycelia was unaffected by the pH of the medium or the growth temperature. Small quantities of crystals were produced in the agar when R. solani was grown on calcium phosphate- and strontium hydrogen phosphate-amended media and these were identified as calcium or strontium sulphates respectively: there appeared to be little or no production of insoluble oxalates although a role for oxalate in the overall solubilization process cannot be discounted. These results are discussed in relation to their physiological and environmental significance, and the important roles of fungi in effecting transformations of insoluble metal-containing compounds in the environment.

A positive numerical scheme for a mixed-type partial differential equation model for fungal growth

We describe a numerical scheme for the solution of a mixed-type PDE system which arises in the modelling of fungal growth. Given the application, conservation of mass and preservation of positivity are of paramount importance. The scheme employs a method of lines approach in which the system is split into hyperbolic and parabolic parts. Positivity and conservation of mass are ensured by the use of generalised flux functions and, in particular, flux limiters. The spatial discretisation results in stiff and non-stiff components which are solved using implicit and explicit methods respectively. Properties of the scheme are investigated via comparison with experimental data and performance is compared with another method of solution.

Modelling mycelial networks in structured environments

The growth habitat of most filamentous fungi is complex and displays a range of nutritional, structural, and temporal heterogeneities. There are inherent difficulties in obtaining and interpreting experimental data from such systems, and hence in this article a cellular automaton model is described to augment experimental investigation. The model, which explicitly includes nutrient uptake, translocation, and anastomosis, is calibrated for Rhizoctonia solani and is used to simulate growth in a range of three-dimensional domains, including those exhibiting soil-like characteristics. Results are compared with experimental data, and it is shown how the structure of the growth domain significantly influences key properties of the model mycelium. Thus, predictions are made of how environmental structure can influence the growth of fungal mycelia.

Power variation strategies for cycling time trials: A differential equation model

Abstract In cycling time trials, competitors aim to ride a course in the fastest possible time and the implementation of a pacing strategy is therefore essential. In this study, a differential equation model of a cyclist incorporating continuous changes in velocity is formulated and applied to a selection of theoretical courses and athletes. The model is augmented with a constraint corresponding to a mean work rate and various pacing strategies are considered. The inclusion of continuous accelerations experienced by the cyclist forms an essential component in a model for courses comprising many changes of gradient, and a steady-state approximation, which has previously been used to assess pacing strategies, is not suitable. In addition to formulating a result on the mathematically optimal solution of the model equations subject to the mean power constraint, it is also shown that substantial time savings can be realized by cyclists increasing their work rates on uphill sections and suitably reducing their work rates elsewhere. However, the amount of time saved is highly course- and athlete-dependent with the greatest gains arising on courses with the longest continuous ascents by cyclists of greatest mass.

Mathematical modelling of fungal growth and function.

This contribution is based on the six presentations given at the Special Interest Group meeting on Mathematical modelling of fungal growth and function held during IMC9. The topics covered aspects of fungal growth ranging across several orders of magnitude of spatial and temporal scales from the bio-mechanics of spore ejection, vesicle trafficking and hyphal tip growth to the form and function of mycelial networks. Each contribution demonstrated an interdisciplinary approach to questions at specific scales. Collectively, they represented a significant advance in the multi-scale understanding of fungal biology.

Arms races and the evolution of big fierce societies

The causes of biological gigantism have received much attention, but only for individual organisms. What selection pressures might favour the evolution of gigantic societies? Here we consider the largest single–queen insect societies, those of the Old World army ant Dorylus, single colonies of which can have 20 million workers. We propose that colony gigantism in Dorylus arises as a result of an arms race and test this prediction by developing a size–structured mathematical model. We use this model for exploring and potentially explaining differences in colony size, colony aggression and colony propagation strategies in populations of New World army ants Eciton and Old World army ants Dorylus. The model shows that, by determining evolutionarily stable strategies (ESSs), differences in the trophic levels at which these army ants live feed forwards into differences in their densities and collision rates and, hence, into different strategies of growth, aggression and propagation. The model predicts large colony size and the occurrence of battles and a colony–propagation strategy involving highly asymmetrical divisions in Dorylus and that Eciton colonies should be smaller, non–combative and exhibit equitable binary fission. These ESSs are in excellent agreement with field observations and demonstrate that gargantuan societies can arise through arms races.

Modelling combat strategies in fungal mycelia

Fungal mycelia have a well-established role in nutrient cycling and are widely used as agents in biological control and in the remediation of polluted landscapes. Competition and combat between different fungal communities is common in these contexts and its outcome impacts on local biodiversity and the success of such biotechnological applications. In this investigation a mathematical model representing mycelia as a system of partial differential equations is used to simulate combat between two fungal colonies growing into a nutrient-free domain. The resultant equations are integrated numerically and the model simulates well-established outcomes of combat between fungal communities. The outcome of pairwise combat is shown to depend on numerous factors including the suppression of advancing hyphae in rivals, the degradation of a rivals established biomass and the utilization and redistribution of available nutrient resources. It is demonstrated how non-transitive hierarchies in fungal communities can be established through switching mechanisms, mirroring observations reported in experimental studies, and how specialized defensive structures can emerge through changes in the redistribution of internal resources.