MOTILE ORGANISMS DISPERSING AND TRACKING CHEMICAL SIGNALS

Abstract

In this dissertation we draw inspiration from natural systems, and discuss some mathematical models inspired by them. With the goal of describing the motility of copepods, a model typically used to describe bacterial dispersion is extended. From a static diffusion constant and some statistics on velocity, the model provides an effective diffusion constant in the n-dimensional Cartesian space Rn. We then explore interactions between organisms. Chemical signals are described mathematically for stationary and moving sources. The results describe how detecting and signaling organisms are likely to pair up based on motility. Both sections apply mathematical ideas to biological systems, taking a few simple assumptions and discussing their consequences. We introduce a model for dispersion of independent swimmers jumping randomly between multiple translational velocities in arbitrary dimensions. Sample trajectories of the individual swimmers are simulated using the governing stochastic differential equations. The associated Fokker-Planck equations are derived and an analytic prediction is obtained for the effective diffusion constant, which is shown to be consistent with simulations. We show adaptability of the model by fitting to three previous models of swimmers having two or three preferred velocities. We explore how stochastic vs. deterministic velocity changes and restricting certain velocity jumps result in different rates of dispersion. Chemical signals are present over a wide range of scales in nature, from as small as being inhabited by bacteria, all the way up to salmon tracking their home stream in the open ocean. These signals can alert organisms in the presence of predators, help locate food resources, or aid in the effort to find mates. Evolution has had the chance to optimize the motility and sensitivity of organisms to best exploit chemical signals. However, detection is still only possible down to a certain concentration threshold. Motivated by bacteria searching for food patches in a heterogeneous environment we provide calculations to describe the detectable region of a diffusing nutrient patch. From this, a critical chemotactic velocity necessary to reach the origin of the patch can be obtained. Motility is seen to provide additional information only in a donut-shaped region around the source. We then change focus to a moving source. Our calculations show that in ideal conditions access to concentration gradient and sufficient mobility guarantee the ability to find the source once it is detected. Descriptive formulae are provided for the dimensions and shape of the trailing plume from a given release rate and minimum detectable concentration. We discuss optimum types of chaser/source match-ups and provide relevant descriptive calculations. We then apply our results to real world scenarios, demonstrating the usefulness and coherence of our work.