We consider the competitive population dynamics of two species described by the Lotka-Volterra model in the presence of spatial diffusion. The model is described by the diffusion coefficient ($d_{\alpha }$) and proliferation rate ($r_{\alpha }$) of the species $\alpha$ ($\alpha =1,2$ is the species label). Propagating wave front solutions in one dimension are investigated analytically and by numerical solutions. It is found that the wave profiles and wave speeds are determined by the speed parameters, $v_{\alpha }\equiv 2\sqrt{d_{\alpha }{r}_{\alpha }}$, of the two species, and the phase diagrams for various inter- and intracompetitive scenarios are determined. The steady wave front speeds are obtained analytically via nonlinear dynamics analysis and verified by numerical solutions. The effect of the intermediate stationary state is investigated and propagating wave profiles beyond the simple Fisher wave fronts are revealed. The wave front speed of a species can display abrupt increase as its speed parameter is increased. In particular for the case in which both species are aggressive, our results show that the speed parameter is the deciding factor that determines the ultimate surviving species, in contrast to the case without diffusion in which the final surviving species is decided by its initial population advantage. Possible relations to the biological relevance of modeling cancer development and wound healing are also discussed.

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