Random Switching between Vector Fields Having a Common Zero

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 $p\in M\mathrm{.}$ 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${I_t = i }.$Weshowthatthebehaviorof$(X,I)$ 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 $(a_{ij}(p))\mathrm{.}$ We introduce two quantities ${\mathrm{\Lambda }}^{-}$ and ${\mathrm{\Lambda }}^{+}$ respectively %called the {\em minimal} and {\em maximal average growth rate.} ${\mathrm{\Lambda }}^{-}$ (respectively ${\mathrm{\Lambda }}^{+}$) is defined as the {\em minimal} (respectively {\em maximal}) {\em growth rate} of $\mathrm{\parallel }{Y}_t\mathrm{\parallel },$ where the minimum (respectively maximum) is taken over all the ergodic measures of the angular process $(\mathrm{\Theta },J)$ with ${\mathrm{\Theta }}_t=\frac{Y_t}{\mathrm{\parallel }{Y}_t\mathrm{\parallel }}\mathrm{.}$ It is shown that ${\mathrm{\Lambda }}^{+}$ coincides with the top Lyapunov exponent (in the sense of ergodic theory) of $(Y,J)$ and that under general assumptions ${\mathrm{\Lambda }}^{-}={\mathrm{\Lambda }}^{+}\mathrm{.}$ We then prove that, under certain irreducibility conditions, $X_t\to p$ exponentially fast when ${\mathrm{\Lambda }}^{+}<0$ 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.