The subject of this talk concerns the rigorous numerical analysis of two well-known reaction-diffusion systems modeling nonlinear predator-prey interactions. The particular models involve a Holling type II functional response for the predators and logistic growth of the prey, which is the most frequently studied case in the ecology community. Numerical results are presented for two fully-practical piecewise linear finite element methods. A priori estimates and error bounds for the semi-discrete and fully-discrete finite element approximations are discussed. Numerical results illustrating the theoretical results and spatiotemporal phenomena (spiral waves and chaos) are presented in one and two space dimensions. There are several implementational advantages of the finite element methods, for example, they have equivalent finite difference representations, and under mild restrictions of the time-step the coefficient matrices of the resulting linear systems are strictly diagonally dominant. The simplicity of the schemes means we need only 80 lines of Matlab code to solve a 2D problem with 2 million degrees of freedom many thousands of times.