Dr. Marcus R Garvie



    Figure 4: Triangulation of a lake with an island

    Unstructured grid generation on a complex geometry using DISTMESH, a mesh generator in Matlab (see SIAM Review Vol. 46, No. 2, pp.329-345, 2004). 281 nodes were used and 312 triangles were created.



    Figure 5: Snapshot at T = 200

    Numerical solution of an expanding wavefront for phytoplankton in a phytoplankton-zooplankton reaction-diffusion model with complex lake geometry (bounding box is [0,400]2), after the localized introduction of predators at (200,380) (marked with an 'X') into a homogeneous distribution of prey. Associated unstructured mesh has 2645 nodes and 4928 triangles. The standard Galerkin FEM using continuous piecewise linear basis functions was employed with zero flux boundary conditions.



    Figure 6: Snapshot at T = 200

    Numerical solution of spatiotemporal 'chaos' for phytoplankton in a phytoplankton-zooplankton reaction-diffusion model with complex lake geometry (bounding box is [0,400]2), after the localized introduction of predators at (250,200) (marked with an 'X'), into a homogeneous distribution of prey. Associated unstructured mesh has 2645 nodes and 4928 triangles. The standard Galerkin FEM using continuous piecewise linear basis functions was employed with zero flux boundary conditions.


Predator-prey models on arbitrary shaped domains

Understanding the relationship between spatial patterns in population densities and environmental heterogeneity is crucial to the understanding of population dynamics and for the management of species in communities. My previous simulations of predator-prey interactions were done on a square domain (habitat). This project extends previous work as it involves generating unstructured grids on arbitrary shaped habitat geometries, for example, in the simulation of plankton dynamics in a lake (see Figure 4). This work is in collaboration with a Research Associate, John Burkardt.
There are four main issues in this project. Firstly, there is the question of what form the input data should take for the lake geometry, i.e., whether we use explicit boundary data or an implicit description (e.g., level set). One possible solution we have used is to create a polygonal domain using the Linux drawing package Inkscape. Secondly, there is the task of generating a 'good' mesh on the computational domain. An example of a good mesh in 2D is one where all triangles are approximately equilateral, and there are many element quality indicators for measuring this. There are a large number of software packages available for unstructured grid generation in 2D. One particularly simple set of routines in Matlab that we have used is called Mesh2d v2.3. Once we have generated our mesh for the computational domain, we need to apply the Finite Element Method discussed above, assemble the resulting linear system, and solve the linear system in an efficient manner. See below for some web-pages that I have written for generating computational regions, and implementing some MATLAB code to simulate predator-prey interactions in arbitary shaped habitats with various boundary conditions: Finally, there are ecological questions concerning how the geometry of the habitat, including possible landscape features (e.g., islands), influences the spatiotemporal dynamics of the solutions. Some preliminary results of our work are shown in Figures 5 and 6 for a ficticious lake with an island. Due to quota constraints I cannot show the animation that I have for these dynamics which are very interesting!

   
   
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Last updated: May 08, 2006 by Marcus R Garvie