On the blowing up solutions of a system of fractional differential equations

In this paper, we are concerned with the study of the system of fractional differential equations $ {align*} ^cD₀₊α X(t) & = Γ(α)(t+1)k_1Xᵃ(t)Yq(t), 0<α<1, t>0, \\ ^cD₀₊β Y(t) & = Γ(β)(t+1)k_2Yᵇ(t)Xᵖ(t), 0<β<1, t>0, \\ {eq:4-frac sys_sigma} {align*} $ subject to$ {equation*} X(0) = X₀>0, Y(0) = Y₀>0, {eq3:initial_value} {equation*} $where $ Γ(σ) $ stands for the Gamma function, $ ^cD₀₊α $ stands for the Caputo fractional derivative, and $ a,b,p,q, k_1, $ and $ k_2 $ are real numbers that will be specified later. We present sufficient conditions for the non-existence of global solutions. Furthermore, we present the asymptotic growth of blowing-up solutions near the blow up time, and the graphical profile of the blowing-up solutions.