This talk focuses on the study of a system of fractional differential equations with power-type nonlinearity that arises in nature. Using the comparison method, we establish sufficient conditions for the non-existence of global solutions and analyze the growth rate of the blowing-up solutions near the critical time. A key challenge in analyzing the growth rate of solutions near the blow-up time arises from the presence of the (weakly) singular kernel involved in the fractional differential operator. This difficulty is addressed by extending the analysis to the complex plane, where we uncover detailed asymptotic behavior near the critical blow-up time.