Numerical analysis of a diffuse interface model for plant cell evasion caused by the rice blast fungus

Lead Research Organisation: University of Sussex
Department Name: Sch of Mathematical & Physical Sciences

Abstract

The project will consider a mathematical model for studying plant cell evasion by the pathogen magnaporthe oryzae, commonly known as the rice blast fungus. This fungus results in annual losses of 11 - 18% of the global rice yield whereby a 10% yield loss accounts for crops that would feed 60 million.
We intend to derive, analyse and numerically solve a diffuse interface formulation of the problem. This will involve the coupling of two diffuse interface equations; a scalar phase field one for the evolution of the fungus and a vector valued one for the system of re-action diffusion equations (RDEs) that model the densities of proteins on the tumour's surface. We will discretise these equations using adaptive finite element approximations.
The resulting approximations will be solved using the adaptive nite element tool box ALBERTA.

Publications

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Studentship Projects

Project Reference Relationship Related To Start End Student Name
EP/N509784/1 01/10/2016 30/09/2021
1805391 Studentship EP/N509784/1 01/08/2016 31/01/2020 James Van Yperen
 
Description 1. We derive a new method for the numerical approximation of a diffuse interface approximation to reaction-driven equations on evolving surfaces. We demonstrate this new method gives a much better approximation to the equations compared to the approximations used in the literature. Furthermore, when the velocity of the evolving surface is a phase field approximation to mean curvature flow, we show that our new method still retains a better approximation than the approximations used in the literature.
2. We prove error analysis for the numerical approximation of a mathematical description of a curve moving under mean curvature flow attached orthogonally to a fixed boundary. Furthermore, we prove error bounds for the same setup but also consider an equation satisfied on the moving curve.
3. We analyse and approximate the mathematical model of the rice blast fungus using a diffuse interface method and demonstrate similarities between our approximation and the one derived in the literature. We consider multiple different variations of the model, including a curve approximation focusing on the dynamics in the pore, a full curve approximation, and a phase field approximation to both the curve and surface formulations.
Exploitation Route Further research we have listed as ideas are:
1. Investigating edge smoothing terms for the new finite volume approximation of the diffuse interface approach;
2. Prove uniqueness, stability, and error bounds for the finite volume method;
3. Extend the finite volume approximation from curve evolution to consider surfaces evolution;
4. Extend the finite element error analysis to prove fully discrete error bounds for the system comprising of curve shortening flow attached orthogonally to a fixed boundary coupled to a reaction-diffusion equation on the evolving curve;
5. Generalise the fixed orthogonal boundary condition to consider a moving boundary with the orthogonal condition;
6. Generalise the fixed orthogonal boundary condition to consider surfaces;
7. Apply the edge smoothed finite volume method to the diffuse interface approach of the rice blast model to retract the assumption of larger diffusion constants as well as a fixed ring structure.
Sectors Other