This paper studies a system of semi-linear fractional diffusion equations
which arise in competitive predator-prey models by replacing the second-order derivatives
in the spatial variables with fractional derivatives of order less than two. Moving
finite element methods are proposed to solve the system of fractional diffusion equations
and the convergence rates of the methods are proved. Numerical examples are
carried out to confirm the theoretical findings. Some applications in anomalous diffusive
Lotka-Volterra and Michaelis-Menten-Holling predator-prey models are studied.