Fitness Taxis in Reaction-Diffusion Systems | Astrid Bro Christensen
| | Abstract | The purpose of this study is to consider a system of partial differential equations describing two spatially distributed populations in a predator-prey interaction with each other. A fitness taxis model is proposed for solving ecological problems in order to analyze the spatial structure of the population densities. This model has the shape of a reaction-advection-diffusion model, where the reaction term expresses the population dynamics. The advection term represents the taxis term, expressed as predator or prey density movement according to the gradient of the net growth rate for the specific species. Undirected movement of the species is represented by diffusion.
It is shown for which conditions Turing-like patterns, known from standard reaction-diffusion systems, are formed by taxis in the model. These conditions are divided i to three different cases for which the equilibrium solution will be unstable to spatial perturbations.
The fitness taxis model only gives rise to results as long as conditions for the third case, CASE 3, are met. For this case only little movement due to taxis is present. For the remaining two cases, i.e. CASE 2 and CASE 1, instabilities in the solution will grow unbounded and no results can be computed from the model. A reformulation of the population dynamics is thus proposed for which the predator is able to eat prey within an area around it. This behavior is implemented with a Gaussian integration kernel.
With the kernel formulation, solutions can be obtained for values where the model meets the conditions for the remaining two cases. The spatial patterns that occur from results computed with the integration kernel turned out to change form from hexagonal arrangement of spots, as seen for simulations with very little movement due to taxis, and into stripes spanning the entire domain. At some point, movement due to taxis increases in a way that a very broad integration kernel is needed in order to compute solutions to the model. As a consequence all instabilities in the solutions gets damped, driving the system towards the stable homogeneous equilibrium solution. | | Type | Master's thesis [Academic thesis] | | Year | 2017 | | Publisher | Technical University of Denmark, Department of Applied Mathematics and Computer Science | | Address | Richard Petersens Plads, Building 324, DK-2800 Kgs. Lyngby, Denmark, compute@compute.dtu.dk | | Series | DTU Compute M.Sc.-2017 | | Note | | | Electronic version(s) | [pdf] | | Publication link | http://www.compute.dtu.dk/English.aspx | | BibTeX data | [bibtex] | | IMM Group(s) | Mathematical Physics |
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