In this paper, a positivity-preserving difference scheme is studied for two-dimensional Fisher-Kolmogorov-Petrovsky-Piscounov equation (Fisher-KPP). The energy analysis method is used to prove that the difference scheme has a series of mathematical properties, including preserving positivity, preserving boundedness, preserving monotonicity, and has a convergence order of $O(\tau+h_x^2+h_y^2)$ in maximum norm, when the ratio of temporal meshsize to spatial meshsizes satisfy $R_{x}+R_{y}+[b\tau(p-1)]/2\leq \frac{1}{2}$. Then by developing Richardson extrapolation methods (REMs), the extrapolation solutions with convergence orders of $O(\tau^{2}+h_{x}^{4}+h_{y}^{4})$ are obtained. Finally, two numerical examples are given to confirm that the numerical results agree well with the theoretical results. It is worthwhile meaning that the condition of the ratios of temporal meshsize to spatial meshsizes as the current REMs are used are the same as the one of the original numerical method.