White noise

Quantum mechanics

Ergodicity

Ecology

Biology

Statistics

Markov process

Stochastic process

Applied mathematics

Stationary distribution

Markov chain

Distribution (mathematics)

Economics

Stationary state

Statistical physics

Competition (biology)

Physics

Stationary sequence

Mathematics

Mathematical analysis

Uniqueness

Ergodic theory

Nonlinear Analysis Hybrid Systems

Taking white noises, Markovian switching and Lévy jumps into account, an n−species Lotka–Volterra competitive model with random perturbations is proposed and studied. Sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the model are established. These sufficient conditions are sharp in some cases. Some interesting results are revealed: the white noises and Lévy jumps could make the stationary distribution disappear as well as appear; the Markovian switching could make the stationary distribution appear. Some numerical simulations are also introduced to illustrate the main results.

Stationary distribution and ergodicity of a stochastic hybrid competition model with Lévy jumps