Mathematical models of biological interactions

Sammanfattning: Mathematical models are used to describe and analyse different types of biological interactions.  From self-propelled particle models capturing the collective motion of fish schools to models in mathematical neuroscience describing the interactions between neurons to individual-based models of ecological interactions. A question that arises for all such models is how we scale from one level to another. How do we scale from fish interactions to the movement of the school of fish? How do we scale from neuronal interactions to the functioning of the brain?  How do we scale from animal competition to population dynamics? It is approaches to this question that we study in this thesis for two different systems. In paper I,  we study a class of spatially explicit individual-based models with contest competition. Based on measures of the spatial statistics, we develop two new approximate descriptions of the spatial population dynamics. The first is based on local interactions of the individuals and approximates the individual-based model well for small dispersal distances. The second approximates the long-range interactions of the individual-based model. Both approximations incorporate the demographic stochasticity from the individual-based model and show that dispersal stabilizes the population dynamics. We calculate extinction probability for the individual-based model and show convergence between the local approximation and the classical mean field approximation of the individual-based model as dispersal distance and population size simultaneously tend to infinity. Taken together, our results deepen the understanding of spatial population dynamics and introduces new approximate analytical descriptions.In paper II,  we propose a model of social burst and glide motion in pairs of fish by combining a well-studied model of neuronal dynamics, the FitzHugh-Nagumo model, with a model of fish motion.  Our model, in which visual stimuli of the position of the other fish affect the internal burst or glide state of the fish, captures a rich set of swimming dynamics found in many species of fish. These include: leader-follower behaviour; periodic changes in leadership; apparently random (i.e. chaotic) leadership change; and pendulum-like tit-for-tat turn taking. Unlike self-propelled particle models, which assume that fish move at a constant speed, the model produces realistic motion of individual fish. Moreover, unlike previous studies where a random component is used for leadership switching to occur, we show that leadership switching, both periodic and chaotic, can be the result of from a deterministic interaction.  We give several empirically testable predictions on how fish interact and discuss our results in light of recently established correlations between fish locomotion and brain activity.