  This work is concerned with competitive Lotka-Volterra model with Markov
switching. A novelty of the contribution is that the Markov chain has a
countable state space. Our main objective of the paper is to reduce the
computational complexity by using the two-time-scale systems. Because existence
and uniqueness as well as continuity of solutions for
  Lotka-Volterra ecosystems with Markovian switching in which the switching
takes place in a countable set are not available, such properties are studied
first. The two-time scale feature is highlighted by introducing a small
parameter into the generator of the Markov chain. When the small parameter goes
to 0, there is a limit system or reduced system. It is established in this
paper that if the reduced system possesses certain properties such as
permanence and extinction, etc., then the complex system also has the same
properties when the parameter is sufficiently small. These results are obtained
by using the perturbed Lyapunov function methods.
