Nicholas F. Britton

Aggregation and the competitive exclusion principle

A mathematical model for aggregation in a single animal population is set up. It relies on two premises. First, there is an advantage to individuals in the population in grouping together, for example for social purposes or to reduce the risk of predation. Second, the intra-specific competition at a point depends not simply on the population density at that point but on the average population density near the point, since the animals may move to find resources. The model is then extended to competing populations, and inter-specific competition is also assumed to depend on an average population density. It is shown that the resulting aggregation may lead to the co-existence of populations one of which would otherwise be excluded by the other. This finding is discussed with regard to the Competitive Exclusion Principle.

A predator-prey reaction-diffusion system with nonlocal effects

We consider a predator-prey system in the form of a coupled system of reaction-diffusion equations with an integral term representing a weighted average of the values of the prey density function, both in past time and space. In a limiting case the system reduces to the Lotka Volterra diffusion system with logistic growth of the prey. We investigate the linear stability of the coexistence steady state and bifurcations occurring from it, and expressions for some of the bifurcating solutions are constructed. None of these bifurcations can occur in the degenerate case when the nonlocal term is in fact local.

Stochastic simulation of benign avascular tumour growth using the Potts model

We simulate the growth of a benign avascular tumour embedded in normal tissue, including cell sorting that occurs between tumour and normal cells, due to the variation of adhesion between different cell types. The simulation uses the Potts model, an energy minimisation method. Trial random movements of cell walls are checked to see if they reduce the adhesion energy of the tissue. These trials are then accepted with Boltzmann weighted probability. The simulated tumour initially grows exponentially, then forms three concentric shells as the nutrient level supplied to the core by diffusion decreases: the outer shell consists of live proliferating cells, the middle of quiescent cells and the centre is a necrotic core, where the nutrient concentration is below the critical level that sustains life. The growth rate of the tumour decreases at the onset of shell formation in agreement with experimental observation. The tumour eventually approaches a steady state, where the increase in volume due to the growth of the proliferating cells equals the loss of volume due to the disintegration of cells in the necrotic core. The final thickness of the shells also agrees with experiments.

Deciding on a new home: how do honeybees agree?

A swarm of honeybees (Apis mellifera) is capable of selecting one nest–site when faced with a choice of several. We adapt classical mathematical models of disease, information and competing beliefs to such decision–making processes. We show that the collective decision may be arrived at without the necessity for any bee to make any comparison between sites.

Chimpanzees and the mathematics of battle

Recent experiments have demonstrated the importance of numerical assessment in animal contests. Nevertheless, few attempts have been made to model explicitly the relationship between the relative number of combatants on each side and the costs and benefits of entering a contest. One framework that may be especially suitable for making such explicit predictions is Lanchesters theory of combat, which has proved useful for understanding combat strategies in humans and several species of ants. We show, with data from a recent series of playback experiments, that a model derived from Lanchesters ‘square law’ predicts willingness to enter intergroup contests in wild chimpanzees (Pan troglodytes). Furthermore, the model predicts that, in contests with multiple individuals on each side, chimpanzees in this population should be willing to enter a contest only if they outnumber the opposing side by a factor of 1.5. We evaluate these results for intergroup encounters in chimpanzees and also discuss potential applications of Lanchesters square and linear laws for understanding combat strategies in other species.

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.

The persistence of parasitic plasmids

The conditions under which plasmids are predicted to persist remain controversial. Here, we reevaluate the ordinary differential equations used previously to model plasmid persistence and conclude that the parameter space required for maintenance is far less stringent than has been supposed. Strikingly, our model demonstrates that purely parasitic plasmids may persist, even in the absence of heterogeneity in the host population, and that this persistence is expressed by oscillations or damped oscillations between the plasmid-bearing and the plasmid-free class.

Coupled oscillators and activity waves in ant colonies

We investigated the phenomenon of activity cycles in ants, taking into account the spatial structure of colonies. In our study species,Leptothorax acervorum, there are two spatially segregated groups in the nest. We developed a model that considers the two groups as coupled oscillators which can produce synchronized activity. By investigating the effects of noise on the model system we predicted how the return of foragers affects activity cycles in ant colonies. We tested these predictions empirically by comparing the activity of colonies under two conditions: when foragers are and are not allowed to return to the nest. The activity of the whole colony and of each group within the colony was studied using image analysis. This allowed us to reveal the spatial pattern of activity wave propagation in ant colonies for the first time.

A mathematical model of the gate control theory of pain

The first test which any theory of pain must pass is that it must be able to explain the phenomena observed in acute pain in humans. This criterion is used to test the major theory of pain at present, the gate control theory of Melzack & Wall (1965, 1982). The theory is explicit enough to be cast in mathematical terms, and the mathematical model is shown to explain the observations considered. It also points up a common misconception on the consequences of the theory, and thus demolishes an argument which has been used against it. A hypothesis of the origin of rhythmic pain is then made, and consequent testable predictions given. This is the first time that the gate control theory has been used to explain any quality of pain. It has important consequences for the treatment of such pain. Finally, the applicability of the gate control theory as an explanation for chronic pain is discussed.

Analysis of a vector-bias model on malaria transmission.

We incorporate a vector-bias term into a malaria-transmission model to account for the greater attractiveness of infectious humans to mosquitoes in terms of differing probabilities that a mosquito arriving at a human at random picks that human depending on whether he is infectious or susceptible. We prove that transcritical bifurcation occurs at the basic reproductive ratio equalling 1 by projecting the flow onto the extended centre manifold. We next study the dynamics of the system when incubation time of malaria parasites in mosquitoes is included, and find that the longer incubation time reduces the prevalence of malaria. Also, we incorporate a random movement of mosquitoes as a diffusion term and a chemically directed movement of mosquitoes to humans expressed in terms of sweat and body odour as a chemotaxis term to study the propagation of infected population to uninfected population. We find that a travelling wave occurs; its speed is calculated numerically and estimated for the lower bound analytically.