  We consider a simple discrete-time Markov chain with values in
$[0,\infty)^{Z^d}$. The Markov chain describes various interesting examples
such as oriented percolation, directed polymers in random environment, time
discretizations of binary contact path process and the voter model. We study
the phase transition for the growth rate of the "total number of particles" in
this framework. The main results are roughly as follows: If $d \ge 3$ and the
Markov chain is "not too random", then, with positive probability, the growth
rate of the total number of particles is of the same order as its expectation.
If on the other hand, $d=1,2$, or the Markov chain is "random enough", then the
growth rate is slower than its expectation. We also discuss the above phase
transition for the dual processes and its connection to the structure of
invariant measures for the Markov chain with proper normalization.
