  Let $E$ be a finite set, $\{F^i\}_{i \in E}$ a family of vector fields on
$\mathbb{R}^d$ leaving positively invariant a compact set $M$ and having a
common zero $p \in M.$ We consider a piecewise deterministic Markov process
$(X,I)$ on $M \times E$ defined by $\dot{X}_t = F^{I_t}(X_t)$ where $I$ is a
jump process controlled by $X:$ $\Pr(I_{t+s} = j | (X_u, I_u)_{u \leq t}) =
a_{i j}(X_t) s + o(s)$ for $i \neq j$ on $\{I_t = i \}.$
  We show that the behavior of $(X,I)$ is mainly determined by the behavior of
the linearized process $(Y,J)$ where $\dot{Y}_t = A^{J_t} Y_t,$ $A^i$ is the
Jacobian matrix of $F^i$ at $p$ and $J$ is the jump process with rates
$(a_{ij}(p)).$
  We introduce two quantities $\Lambda^-$ and $\Lambda^+$ respectively %called
the {\em minimal} and {\em maximal average growth rate.} $\Lambda^-$
(respectively $\Lambda^+$) is defined as the {\em minimal} (respectively {\em
maximal}) {\em growth rate} of $\|Y_t\|,$ where the minimum (respectively
maximum) is taken over all the ergodic measures of the angular process
$(\Theta, J)$ with $\Theta_t = \frac{Y_t}{\|Y_t\|}.$ It is shown that
$\Lambda^+$ coincides with the top Lyapunov exponent (in the sense of ergodic
theory) of $(Y,J)$ and that under general assumptions $\Lambda^- = \Lambda^+.$
We then prove that, under certain irreducibility conditions, $X_t \to p$
exponentially fast when $\Lambda^+ < 0$ and $(X,I)$ converges in distribution
at an exponential rate toward a (unique) invariant measure supported by $M
\setminus \{p\} \times E$ when $\Lambda^- > 0.$ Some applications to certain
epidemic models in a fluctuating environment are discussed and illustrate our
results.
