  We study the dynamics of a predator-prey system in a random environment. The
dynamics evolves according to a deterministic Lotka-Volterra system for an
exponential random time after which it switches to a different deterministic
Lotka-Volterra system. This switching procedure is then repeated. The resulting
process is a Piecewise Deterministic Markov Process (PDMP). In the case when
the equilibrium points of the two deterministic Lotka--Volterra systems
coincide we show that almost surely the trajectory does not converge to the
common deterministic equilibrium. Instead, with probability one, the densities
of the prey and the predator oscillate between $0$ and $\infty$. This proves a
conjecture of Takeuchi et al (J. Math. Anal. Appl 2006).
  The proof of the conjecture is a corollary of a result we prove about linear
switched systems. Assume $(Y_t, I_t)$ is a PDMP that evolves according to
$\frac{dY_t}{dt}=A_{I_t} Y_t$ where $A_0,A_1$ are $2\times2$ matrices and $I_t$
is a Markov chain on $\{0,1\}$ with transition rates $k_0,k_1>0$. If the
matrices $A_0$ and $A_1$ are not proportional and have purely imaginary
eigenvalues, then there exists $\lambda >0$ such that $\lim_{t \to \infty}
\frac{\log \| Y_t \|}{t} = \lambda.$
