In this paper, we consider a closed Riemannian manifold $$M^{n+1}$$ with dimension $$3\le n+1\le 7$$ , and compact Lie group G acting as isometries on M cohomogeneity at least 3. Suppose the union of non-principal orbits $$M{\setminus } M^{reg}$$ is smooth embedded submanifold most $$n-2$$ . Then for any $$c\in \mathbb {R}$$ show existence nontrivial, smooth, closed, almost embedded, G-invarian...

Whereas the positive equilibrium of a mass-action system with deficiency zero is always globally stable, for deficiency-one networks there are many different scenarios, mainly involving oscillatory behaviour. We present several examples, centers or multiple limit cycles.