<p>In evolutionary and ecological sciences, describing the adaptive dynamics of species affected by the phenotype of maturation time is challenging. In this work, considering the maturation time as a quantitative trait, we propose a stage-structured two-species competition model in a discrete environment. When competition leads to long-term bistable dynamics, we obtain the existence of pulsating waves of the population by appealing to monotone dynamical system theories. Furthermore, we numerically explore the relationship between the outcome of competition, characterised by the spreading direction of bistable pulsating waves, and the maturation time of species. When the maturation time has an inverse relationship with the competitiveness of the mature population, there may exist an optimal maturation time suggesting an evolutionarily stable strategy(ESS). Interestingly, the survival pressure from the increase of adult mortality or from spatial heterogeneity can lead to a decrease in the maturation time of ESS.</p>