Kinetical systems -- local analysis (Q1978993)

The author deals with systems of the form \(\dot {y}= A G(y),\) where \(A=C-C',\) \(C=(c_{i j})\) and \(C'=(c'_{i j})\) are \(n\times m\)-matrices whose elements are nonnegative integers such that \(0<L=\operatorname {rank}(A)<n\) and \(\det (A(1,\dots ,L))\neq 0\) holds for the corresponding principal submatrix \(A(1,\dots ,L)\) of \(A,\) \(G\) is an \(m\)-vector-valued function defined on \(\mathbb R\times \mathbb R^n\) by \[ G_j(t,y) := -r_j(t) \prod _{i=1}^n y_i^{c_{i j}} +d_j(t) \prod _{i=1}^n y_i^{c'_{i j}}, \quad j=1,\dots ,m, \] and \(r_j,d_j:\mathbb R\mapsto [0,\infty),\) \(j=1,\dots ,m,\) are continuous functions such that \[ \prod _{i=1}^{L}\left [\frac {d_i(t)}{r_i(t)}\right ]{z_{i j}} =\frac {d_j(t)}{r_j(t)}, \quad j=1,\dots ,m, \] and \(\operatorname {col}_j(A)=\sum _{i=1}^L \operatorname {col}_i(A) z_{i j}\) holds for \(j=1,\dots ,m.\) Such systems are called detailed balanced kinetic systems and are often used in chemistry and biology. It is assumed that there exists at least one \(d\times n\)-matrix \(U\) of the rank \(d=n-L\) with nonnegative elements \(u_{i j}\) and such that \(\sum _{i=1}^d u_{i j}>0\) for \(j=1,\dots ,n\) and there exists at least one vector \(b\in \mathbb R^d\) with positive elements such that the linear equation \(Uc=b\) possesses a solution \(c\in \mathbb R^d\) with nonnegative elements and one of such matrices \(U\) and one of such vectors \(b\) are chosen fixed. The author then gives sufficient conditions assuring that any stationary point \(y\in H=\{y: Uy=b\}\) of the given detailed balanced system is asymptotically stable with respect to \(H\).