Mathematical Biology

The Moran process models the evolutionary dynamics between two competing types in a population, traditionally assuming a fixed population size. We investigate an extension to this process which adds ecological aspects through variable population sizes. For the original Moran process, birth and death events are correlated to maintain a constant population size. Here we decouple the two events and derive the stochastic differential equation that represents the dynamics in a well-mixed population and captures its behaviour as the population size becomes arbitrarily large. Our analysis explores the impact of this decoupling on two key determinants of the evolutionary process: fixation probabilities and fixation times. In evolutionary graph theory, these statistics depend significantly on the population structure, such that structures have been identified that act as ‘amplifiers’ of selection while others are ‘suppressors’ of selection. However, these features are crucially dependent on the sequence of events, such as birth-death vs death-birth – a seemingly small change with significant consequences. In our extension of the Moran process this distinction is no longer necessary or possible. We determine the fixation probabilities and times for the well-mixed population, regular graphs as well as amplifiers and suppressors, and compare them to the original Moran process.

Skeletal muscle is composed of cells collectively referred to as fibers, which themselves contain contractile proteins arranged longtitudinally into sarcomeres. These latter respond to signals from the nervous system, and contract; unlike cardiac muscle, skeletal muscles can respond to voluntary control. Muscles react to mechanical forces - they contain connective tissue and fluid, and are linked via tendons to the skeletal system - but they also are capable of activation via stimulation (and hence, contraction) of the sacromeres. The restorative along-fibre force introduce strong mechanical anisotropy, and depend on departures from a characteristic length of the sarcomeres; diseases such as cerebral palsy cause this characteristic length to change, thereby impacting muscle force. In the 1910s, A.V. Hill [1] posited a mathematical description of skeletal muscles which approximated muscle as a 1-dimensional nonlinear and massless spring. This has been a remarkably successful model, and remains in wide use. Yet skeletal muscle is three dimensional, has mass, and a fairly complicated structure. Are these features important? What insights are gained if we include some of this complexity in our models? Many mathematical questions of interest in skeletal muscle mechanics arise: how to model this system, how to discretize it, and what theoretical properties does it have? In this talk, we survey recent work on the modeling, parameter estimation, simulation and validation of a fully 3-D continuum elasticity approach for skeletal muscle dynamics. This is joint work based on a long-standing collaboration with James Wakeling (Dept. of Biomedical Physiology and Kinesiology, SFU).

The evolution and maintenance of cooperation is a fundamental problem in evolutionary biology. Because cooperative behaviors impose a cost, Cooperators are vulnerable to exploitation by Defectors that do not pay the cost to cooperate but still benefit from the cooperation of others. The bacteriophage $\Phi_6$ exhibits cooperative and defective phenotypes in infection: during replication, phages produce essential proteins in the host cell cytoplasm. Coinfection between multiple phages is possible. A given phage cannot guarantee exclusive access to its own proteins, so Cooperators contribute to the common pool of proteins while Defectors contribute less and instead appropriate proteins from Cooperators. Previous experimental work found that $\Phi_6$ was trapped in a prisoner's dilemma, predicting that the cooperative phenotype should disappear. Here we propose that environmental feedback, or interplay between phage and host densities, can maintain cooperation in $\Phi_6$ populations by modulating the rate of co-infection and shifting the advantages of cooperation vs. defection. We build and analyze an ODE model and find that for a wide range of parameter values, environmental feedback allows Cooperation to survive.

Evolutionary game theory is a discipline devoted to studying populations of individuals that are subject to evolutionary pressures, and whose success generally depends on the composition of the population. In biological contexts, individuals could be molecules, simple organisms or animals, and evolutionary pressures often take the form of natural selection and mutations. In socioeconomic contexts, individuals could be humans, firms or other institutions, and evolutionary pressures often derive from competition for scarce resources and experimentation.

In this talk I will give a very basic introduction to agent-based evolutionary game theory, a bottom-up approach to modelling and analyzing these systems. The defining feature of this modelling approach is that the individual units of the system and their interactions are explicitly and individually represented in the model. The models thus defined can be usefully formalized as stochastic processes, whose dynamics can be explored using computer simulation and approximated using various mathematical theories.

For over a century, scientists have studied striking spatiotemporal patterns during the continual tooth replacement of reptiles. Aside from the compelling aesthetics of this phenomenon, it is thought that understanding the underlying mechanisms may provide the insight required to trigger adult tooth replacement in humans. Theoretical frameworks have long been proposed to understand the rules behind the observed spatiotemporal order, but have only been analyzed mathematically more recently. Starting from Edmund's observations in crocodiles and proposed theory of replacement waves, we show how a simple model consisting of a row of non-interacting phase oscillators predicts several experimental observations. Next, inspired by the hypothesis put forth by Osborn, we consider a variation of the phase model with ODEs that account for mutual inhibition between tooth sites, and use continuation methods to thoroughly search parameter space for experimentally validated solutions. We then extend the model to a PDE that explicitly accounts for the diffusion of inhibitory signals between teeth, yielding some novel solution types. Using continuation methods once again, we delineate parameter regimes with solutions that closely resemble experimental observations in leopard geckos.