  The recently proposed {\em cybersecurity dynamics} approach aims to
understand cybersecurity from a holistic perspective by modeling the evolution
of the global cybersecurity state. These models describe the interactions
between the various kinds of cyber defenses and the various kinds of cyber
attacks. We study a particular kind of cybersecurity dynamics caused by the
interactions between preventive and reactive defenses (e.g., filtering and
malware detection) against push- and pull-based cyber attacks (e.g., malware
spreading and "drive-by download" attacks). The dynamics was previously shown
to be globally stable in a {\em special} regime of the parameter universe, but
little is known beyond this special regime. In this paper, we resolve an open
problem in this domain by proving that the dynamics is globally stable in the
{\em entire} parameter universe (i.e., the dynamics always converges to a
unique equilibrium). We discuss the cybersecurity meanings and implications of
this theoretic result. We also prove that the dynamics converges {\em
exponentially} to the equilibrium except for a special parameter regime, in
which case the dynamics converges {\em polynomially}. Since it is often
difficult to compute the equilibrium, we propose new bounds of the equilibrium
and numerically show that these bounds are tighter than those proposed in the
literature.
