Effects of Dispersal for a Logistic Growth Population in Random Environments

and Applied Analysis 3 = [− 0.5k 1 x 1.5 1 + 0.5k 1 x 1 + 0.5 (r 1 − ε 12 − 0.25σ 2 1 ) x 0.5 1 − 0.5 (r 1 − ε 12 − 0.5σ 2 1 )] dt + [− 0.5k 2 x 1.5 2 + 0.5k 2 x 2 + 0.5 (r 2 − ε 21 − 0.25σ 2 2 ) x 0.5 2 − 0.5 (r 2 − ε 21 − 0.5σ 2 2 )] dt + [0.5 (x −0.5 1 − x −1 1 ) ε 12 x 2 + 0.5 (x −0.5 2 − x −1 2 ) ε 21 x 1 ] dt + 0.5 (x 0.5 1 − 1) σ 1 dB 1 (t) + 0.5 (x 0.5 2 − 1) σ 2 dB 2 (t) . (11) There exists a constant N such that f(x) = x − x < N on t > 0; so we can obtain dV ≤ [− 0.5k 1 x 1.5 1 + 0.5 (k 1 + Nε 12 ) x 1 + 0.5 (r 1 − ε 12 − 0.25σ 2 1 ) x 0.5 1 − 0.5 (r 1 − ε 12 − 0.5σ 2 1 ) ] dt + [− 0.5k 2 x 1.5 2 + 0.5 (k 2 + Nε 21 ) x 2 + 0.5 (r 2 − ε 21 − 0.25σ 2 2 ) x 0.5 2 − 0.5 (r 2 − ε 21 − 0.5σ 2 2 ) ] dt + 0.5 (x 0.5 1 − 1) σ 1 dB 1 (t) + 0.5 (x 0.5 2 − 1) σ 2 dB 2 (t) ≤ Mdt + 0.5 (x0.5 1 − 1) σ 1 dB 1 (t) + 0.5 (x 0.5 2 − 1) σ 2 dB 2 (t) (12) as long as (x 1 , x 2 ) ∈ R 2 + . Integrating both sides from 0 to τ k ∧ T and then taking expectations yield EV (x 1 (τ k ∧ T) , x 2 (τ k ∧ T)) ≤ V (x 1 (0) , x 2 (0)) + ME (τ k ∧ T) ≤ V (x 1 (0) , x 2 (0)) + MT. (13) Denote Ω k = {τ k ≤ T} for k ≥ k 1 , by (8), P(Ω k ) ≥ ε. Note that, for every ω ∈ Ω k , there is some i such that x i (τ k , ω) equals either k or 1/k, and V(x 1 (τ k , ω), x 2 (τ k , ω)) is no less than either√k − 1 − 0.5 ln(k) or 1/√k − 1 − 0.5 ln(1/k). Consequently, V (x 1 (τ k , ω) , x 2 (τ k , ω)) ≥ [√k − 1 − 0.5 ln (k)] ∧ [ 1 √k − 1 − 0.5 ln(1 k )] . (14) It is follows from (13) that V (x 1 (0) , x 2 (0)) + MT ≥ E [I Ωk V (x 1 (τ k , ω) , x 2 (τ k , ω))] ≥ ε ([√k − 1 − 0.5 ln (k)] ∧ [ 1 √k − 1 − 0.5 ln(1 k )]) . (15) Letting k → ∞ leads to the contradiction ∞ > V(x 1 (0) , x 2 (0)) + MT = ∞. (16) So we must have τ ∞ = ∞ a.s. Theorem 1 shows that the solution of system (4) will remain in the positive cone R + . This nice positive invariant property provides us with a great opportunity to construct different types of the Lyapunov functions to discuss the stationary distribution for system (4) in R + in more detail. 4. Stationary Distribution for System (4) In order to prove our main results, we require some results in [25], and the technique we used here is motivated by [26–28]. System (4) can be rewritten as d(x1 (t) x 2 (t) ) = ( x 1 (r 1 − k 1 x 1 ) + ε 12 (x 2 − x 1 ) x 2 (r 2 − k 2 x 2 ) + ε 21 (x 1 − x 2 ) ) dt