Simple Finite Element Methods for Approximating Predator–Prey Dynamics in Two Dimensions Using Matlab

Abstract

We describe simple finite element schemes for approximating spatially extended predator–prey dynamics with the Holling type II functional response and logistic growth of the prey. The finite element schemes generalize ‘Scheme 1’ in the paper by Garvie (Bull Math Biol 69(3):931–956, 2007). We present user-friendly, open-source Matlab code for implementing the finite element methods on arbitrary-shaped two-dimensional domains with Dirichlet, Neumann, Robin, mixed Robin–Neumann, mixed Dirichlet–Neumann, and Periodic boundary conditions. Users can download, edit, and run the codes from http://www.uoguelph.ca/~mgarvie/. In addition to discussing the well posedness of the model equations, the results of numerical experiments are presented and demonstrate the crucial role that habitat shape, initial data, and the boundary conditions play in determining the spatiotemporal dynamics of predator–prey interactions. As most previous works on this problem have focussed on square domains with standard boundary conditions, our paper makes a significant contribution to the area.

Appendix: Assembly of the Matrix \(K\) and \(L\)

Consider a generic triangle $\tau$ with nodes labelled $P_i$, $P_j$, and $P_k$, with coordinates $(x_i,y_i)$, $(x_j,y_j)$ and $(x_k,y_k)$, respectively. Then, the linear basis function associated with node $P_k$ can be expressed as⁷

$$\begin{aligned} \varphi _k(x,y): = \frac{h_{ji}(x,y)}{h_{ji}(x_k,y_k)}, \quad \text{ where }\quad h_{ji}(x,y): = (x-x_i)(y_j-y_i) - (x_j-x_i)(y-y_i), \end{aligned}$$

($h_{ji}(x_k,y_k)\ne 0$), and the basis functions for the other nodes are defined analogously. Now as the gradient of the basis functions are constant on each triangle $\tau$, all contributions to the Stiffness matrix $K$ have the form $\int _\tau \nabla \varphi _s\cdot \nabla \varphi _p\,\mathrm{d}\mathbf{x} = \nabla \varphi _s\cdot \nabla \varphi _p |\tau |$, where the area of the triangle $\tau$ is given by $|\tau | = | x_j y_k-x_k y_j-x_i y_k+x_k y_i+x_i y_j-x_j y_i |/2$. Elementary calculations yield

$$\begin{aligned} \nabla \varphi _k\cdot \nabla \varphi _i = \frac{(y_j-y_i)(y_k-y_j)+(x_i-x_j)(x_j-x_k)}{h_{ji}(x_k,y_k) h_{kj}(x_i,y_i)}, \end{aligned}$$

with similar expressions obtained for $\nabla \varphi _k\cdot \nabla \varphi _j$, $\nabla \varphi _i\cdot \nabla \varphi _j$, $|\nabla \varphi _k|^2$, $|\nabla \varphi _i|^2$ and $|\nabla \varphi _j|^2$. The contributions to the matrix $L:=\widehat{M}^{-1}K$ are readily calculated after noting $\int _\tau \varphi _i\,\mathrm{d}\mathbf{x} = \frac{1}{3}|\tau |$, where we used the fact that the volume of a tetrahedron is a third of the base area times the height.