Could somebody help me to prove that the equilibrium is stable?
\begin{equation}\begin{cases} u'=u(a_1-b_1u+c_1v+r_1w),\\ v'=v[(1-k)a_2+b_2u-c_2v],\\ w'=kb_3v-(r_2u+q)w.\end{cases}\end{equation}
I have constructed two species mutualistic cooperation models, where species $v$ produces a facilitator $w$ (promoting the growth of species $u$). Considering the biological significance, the model includes the conditions $u(0)\geq 0$, $v(0)\geq 0$, $w(0)\geq 0$, and the correlation coefficients $a_1, b_1, c_1, r_1, a_2, b_2, c_2, b_3, r_2$, and $q>0$, $0\leq k\leq 1$. I need to analyze the equilibrium stability of the differential equations formulated in the model.