DSpace at VNU: Predator-prey System with the Effect of Environmental Fluctuation

VNU Journal of Science: Mathematics – Physics, Vol 30, No (2014) 49-69 Predator-prey System with the Effect of Environmental Fluctuation Le Hong Lan* Faculty of Basic Sciences, Hanoi University of Communications and Transport, Lang Thuong, Dong Da, Hanoi, Vietnam Received 18 July 2014 Revised 27 August 2014; Accepted 15 September 2014 Abstract: In this paper we study the trajectory behavior of Lotka - Volterra predator - prey systems with periodic coefficients under telegraph noises We describe the ω - limit set of the solution, give sufficient conditions for the persistence and prove the existence of a Markov periodic solution Keywords: Key words and phrases, Lotka-Volterra Equation, Predator - Prey, Telegraph noise, ω - limit set, Markov periodic solution Introduction* The Kolmogorov equation  x ( t ) = x f t , x ( t ) , y ( t )   y ( t ) = y g t , x ( t ) , y ( t )  with the functions f t , x ( t ) , y ( t )  ; g t , x ( t ) , y ( t ) periodic in t is a strong tool to describe the evolution of prey-predator communities depending on the changing of seasons There is a lot of work dealing with the asymptotic behavior of such systems as the existence of periodic solutions, the persistence [1-4] In particular, the classical model for a system consisting of two species in preypredator relation  x ( t ) = x ( t )  a ( t ) − b ( t ) x ( t ) − c ( t ) y ( t )    y ( t ) = y ( t ) −  d ( t ) + e ( t ) x ( t ) − f ( t ) y ( t )  (1.1) with the periodic coefficients a; b; c; d; e; f is well investigated in [5-10], where x ( t ) (resp y ( t ) ) is the quantity of the prey (resp of predator) at time t _ * Tel.: 84- 989060885 Email: honglanle229@gmail.com 49 L.H Lan / VNU Journal of Science: Mathematics – Physics, Vol 30, No (2014) 49-69 50 In almost of these works, one supposes that the communities develop under an environment without random perturbation However, it is clear that it is not the case in reality because in general, annual seasonal living conditions of the communities are not the same Therefore, it is important to take into account not only in the changing of seasons but also in the fluctuation of stochastic factors, which may have important consequences on the dynamics of the communities For the stochastic Lotka - Volterra equation, a systematic review has been given in [11-13] In our separate paper [14], we analyze the Lotka - Volterra predator-prey system with constant coefficients under the telegraph noises, i.e., environmental variability causes the parameter switching between two systems Then we have described some parts of ω -set of solutions and show out the existence of a stationary distribution In this paper, we want to consider predator-prey models under the influence of stochastic fluctuation of environment and changing periodically of season as well We describe completely the omega limit set of the positive solutions of Equation (1.1) with the periodic coefficients under the telegraph noises Also, the existence of a Markov periodic solution that attracts the other solutions of Equation (2.4), starting in » + × » + under certain conditions is proved The rest of the paper is divided into three sections Section details the model Some properties of the solution and the set of omega limit are shown in section The last section is some simulations and discussions Preliminary Let (Ω, F , P ) be a complete probability space and {ξ (t ) : t ≥ 0} be a continuous-time Markov chain defined on (Ω, F , P ) , whose state space is a two-element set M = {−, +} and whose generator is given by q Q =  11  q21 q12   −α = q22   β α   −β  with α > and β > It follows that, ϖ = ( p, q ) , the stationary distribution of {ξ ( t ) :t ≥ 0} satisfying the system of equations ϖ Q =   p + q =1 is given by β  P {ξ ( t ) = 1} =  p = tlim → +∞ α+β   q = lim P {ξ ( t ) = 2} = β t → +∞  α+β (2.1) Such a two-state Markov chain is commonly referred to as telegraph noise because of the nature of its sample paths The trajectory of {ξ t } is piecewise-constant, cadlag functions Suppose that L.H Lan / VNU Journal of Science: Mathematics – Physics, Vol 30, No (2014) 49-69 = τ < τ < τ < < τ n < 51 (2.2) are its jump times Put σ := τ − τ , σ := τ − τ , , σ n := τ n − τ n −1 (2.3) It is known that the sequence {σ k }k =1 is an independent random variables in the condition of n { } given sequence ξτ k n k =1 (see [15, 16]) Note that if ξ is given then ξτ n is constant since the process {ξ t } takes only two values Hence, (σ k )∞k =1 is a sequence of conditionally independent random variables, valued in [0, + ∞ ] Moreover, if ξ0 = + then σ n +1 has the exponential density α1[0, + ∞ ) e−α t and σ n +1 has the density β 1[0, + ∞ ) e − β t Conversely, if ξ0 = − then σ 2n has the exponential density α1[0, + ∞ ) e−α t and σ n +1 has the density β 1[0, + ∞ ) e − β t (see [15]) Here 1 , t ≥ 1[0, + ∞ ) =  0 , t < Denote ℑ0n = σ (τ k , k ≤ n ) ; ℑ∞n = σ (τ k −τ n , k > n ) We see that ℑ0n is independent of ℑ∞n for any n ∈ » in the condition that ξ0 given Let ξ0 have the distribution Ρ {ξ0 = +} = p ; Ρ {ξ0 = −} = q then {ξt } is a stationary process Therefore, there exists a group θ t , t ∈ » of P− preserving measure transformations θ t : Ω → Ω such that ξt (ω ) = ξ0 (θ t ω ) , ω ∈ Ω We consider the periodic predator-prey equation under a random environment Suppose that the quantity x of the prey and the quantity y of the predator are described by a Lotka - Volterra equation  x = x  a (ξt , t ) − b (ξt , t ) x − c (ξt , t ) y    y = y  −d (ξt , t ) + e (ξt , t ) x − f (ξt , t ) y  (2.4) where g : Ε → » + for g = a, b, c, d , e, f such that g ( i,.) are continuous and periodic functions with period T > for any i ∈ Ε Moreover, m ≤ g ( i , t ) ≤ M ; in which m and M are two positive constants In case where the noise {ξt } intervenes virtually into Equation (2.4), it makes a switching between the deterministic periodic system  x+ ( t ) = x+ ( t )  a ( +, t ) − b ( +, t ) x+ ( t ) − c ( +, t ) y+ ( t )    y+ ( t ) = y+ ( t )  −d ( +, t ) + e ( +, t ) x+ ( t ) − f ( +, t ) y+ ( t ) (2.5) and another  x− ( t ) = x− ( t )  a ( −, t ) − b ( −, t ) x− ( t ) − c ( −, t ) y− ( t )    y− ( t ) = y− ( t )  −d ( −, t ) + e ( −, t ) x− ( t ) − f ( −, t ) y− ( t )  (2.6) L.H Lan / VNU Journal of Science: Mathematics – Physics, Vol 30, No (2014) 49-69 52 Thus, the relationship of these two systems will determine the trajectory behavior of Equation (2.4) System (2.4) without the noise {ξt } , i.e., g (ξt , t ) = g ( t ) for any g = a, b, , f is studied in [9] They show that Theorem 2.1 Consider the system  x ( t ) = x ( t )  a ( t ) − b ( t ) x ( t ) − c ( t ) y ( t )    y ( t ) = y ( t )  −d ( t ) + e ( t ) x ( t ) − f ( t ) y ( t ) (2.7) where a, b, , f are T-periodic functions a) If a d  inf   > sup   b e (2.8) b c inf   > sup   e d  (2.9) then system (2.7) has a positive T-periodic solution ( x *( t ) , y *( t ) ) satisfying ( x ( t ) − x (t ) , y (t ) − y (t )) → ( 0,0) * t →∞ * (2.10) b) If d  a inf   > sup   e   b (2.11) then the (unique) periodic solution u * ( t ) of the equation u ( t ) = u ( t )  a ( t ) − b ( t ) u ( t ) is stable and ( x ( t ) − u (t ) , y (t )) → ( 0,0 ) * t →+∞ (2.12) for any positive solution ( x ( t ) , y ( t ) ) to (2.7) Figure Coexistence of predator and prey Figure Extinction of predators L.H Lan / VNU Journal of Science: Mathematics – Physics, Vol 30, No (2014) 49-69 53 Lemma 2.2 Consider the system  x ( t ) = f ( x, y , t )   y ( t ) = g ( x, y, t ) where f , g : » × [0, + ∞ ) → » × [0, + ∞ ) are T-periodic functions in t Suppose that this system has a globally asymptotically stable T- periodic solution ( x (t ) , y (t )) := ( x (t ,0, z ) , y (t ,0, z )) , := ( x , y ) is the initial point Then, for every * where z0* * * * * * ε > and a compact set K, we can find a T * = T * ( ε , Κ ) > such that for all t ≥ T * , s ≥ , ( x0 , y0 ) ∈ Κ we have x ( t + s , s , x0 , y0 ) − x* ( t + s ) + y ( t + s , s , x0 , y0 ) − y * ( t + s ) ≤ ε (2.13) Proof Since f , g are T − periodic, we can suppose that ≤ s ≤ T Moreover, it is easy to show that if ( x0 , y0 ) ∈ Κ and ≤ s ≤ T , there is a compact set Κ ′ such that ( x ( T , s, x , y0 ) , y (T , s, x0 , y0 ) ) ∈ Κ ′ Due to the periodicity of parameters, it is therefore sufficient to verify (2.13) for s = Since ( x* ( t ) , y* ( t ) ) is stable, we can find a δ ε > such that if x − x0* + y − y0* ≤ δ ε then x ( t ,0, x , y ) − x* ( t ) + y ( t ,0, x , y ) − y* ( t ) ≤ ε , ∀t ≥ ( 2.14 ) On the one hand, ( x* ( t ) , y * ( t ) ) is globally asymptotic then for every ( x0 , y0 ) ∈Κ , there exist a T( x0 , y0 ) = k( x0 , y0 )T , k( x0 , y0 ) ∈ » such that ( ) ( ) ( ) ( ) x T( x0 , y0 ) ,0, x , y − x* T( x0 , y0 ) + y T( x0 , y0 ) ,0, x , y − y* T( x0 , y0 ) ≤ δ ε By the continuous dependence of solutions on the initial data, there is a neighborhood of ( x0 , y0 ) , denoted by Vx0 , y0 , such that ( ) ( ( ) ) ( ) x T( x0 , y0 ) ,0, x , y − x* T( x , y ) + y T( x0 , y0 ) ,0, x , y − y * T( x0 , y0 ) ≤ δ ε , ∀ ( x , y ) ∈Vx0 , y0 ( 2.15) As a result of (2.14) and (2.15), x ( t ,0, x , y ) − x* ( t ) + y ( t ,0, x , y ) − y * ( t ) ≤ δ ε , ∀ ( x , y ) ∈Vx0 , y0 , t ≥ T( x0 , y0 ) { The family Vx0 , y0 : ( x0 , y0 ) ∈ Κ finite family {V } is an open covering of } such that Κ ⊂ ∪ V n x10 , y10 , , Vxn , y n 0 i =1 x0i , y0i point ( x0 , y0 ) ∈ Κ and for all t > T , we have: * x ( t ,0, x0 , y0 ) − x* ( t ) + y ( t ,0, x0 , y0 ) − y * ( t ) < ε The proof is complete ( 2.16 ) Κ Since Κ is compact then there is a By choosing T * = max T ( x0i , y0i ) , for any 1≤ i ≤ n L.H Lan / VNU Journal of Science: Mathematics – Physics, Vol 30, No (2014) 49-69 54 Dynamic behavior of the solution ( x0 , y0 ) ∈ » 2+ Denote by ( x ( t ,0, x0 , y0 ) , y ( t ,0, x0 , y0 ) ) the solution of initial condition ( x ( 0,0, x0 , y0 ) , y ( 0,0, x0 , y0 ) ) = ( x0 , y0 ) For the sake of write ( x ( t ) , y ( t ) ) for ( x ( t ,0, x0 , y0 ) , y ( t ,0, x0 , y0 ) ) if there is no confusion Let (2.4) satisfying the simplification, we Proposition 3.1 The system (2.4) is dissipative and the rectangle ( 0, M / m ] × ( 0, M / m − 1 is forward invariant Proof By the uniqueness of the solution, it is easy to show that both the nonnegative and positive cones of » 2+ are positively invariant for (2.4) From the first equation of system (2.4) we see that x = x  a (ξt , t ) − b (ξt , t ) x − c (ξt , t ) y  < x  a (ξt , t ) − b (ξt , t ) x  < x ( M − mx ) By the comparison theorem, it follows that if x ( ) ≥ then x ( t ) ≤ M / m , ∀ t > t0 for some t0 > Similarity, y = y −  d (ξ t , t ) + e (ξ t , t ) x − f (ξt , t ) y  < y −  d (ξ t , t ) + e (ξt , t ) M / m − f (ξ t , t ) y  < y ( −m + M / m − my ) , which follows that y ( t ) ≤ M / m − , ∀ t > t1 for some t1 > t0 From these estimates, we also see that the rectangle ( 0, M / m ] × ( 0, M / m − 1 is forward invariant The proof is complete Proposition 3.2 There exists A δ > such that limsup x ( t ,0, x0 , y0 ) ≥ δ for any ( x0 , y0 ) with t → +∞ probability Proof By the system (2.4), there exist δ > 0, ε > such that − d (ξt , t ) + e (ξt , t ) x − f (ξt , t ) y < −ε ; ∀ < x < δ , < y ≤ M / m − and a (ξ t , t ) − b (ξ t , t ) x − c (ξt , t ) y > ε for all ≤ x, y ≤ δ ( 3.1) Assume that limsup x ( t ,0, x0 , y0 ) < δ with a positive probability Then, there is a t3 > such that t → +∞ x ( t ) < δ , y ( t ) ≤ M / m − ∀t ≥ t3 , which implies that y ( t ) < −ε y ( t ) Therefore, for some t4 > t3 , y ( t ) < δ , ∀ t ≥ t4 From (3.1) we see x ( t ) > ε x ( t ) , ∀ t ≥ t4 , which follows that lim x ( t ) = ∞ t →+ ∞ This contradiction implies the assertion of this proposition Proposition 3.3 There exists a positive number xmin satisfying: if t > such that x ( t ,0, x0 , y0 ) ≥ xmin for all t ≥ t ( x0 , y0 ) ∈ » 2+ we can find L.H Lan / VNU Journal of Science: Mathematics – Physics, Vol 30, No (2014) 49-69 55 Proof With δ mentioned in 3.2, there exists t > such that x ( t ) > δ Let < ε1 ≤ δ such that −δ1 := −m + M ε1 < If x ( t ) ≥ ε1 for all t > t then the proposition is proved Otherwise, x ( t ) < ε1 { } for a t > t Let h1 = inf s > t : x ( s ) < ε1 We see that if x ( t ) ≤ ε for t > h1 then y = y −  d (ξ t , t ) + e (ξ t , t ) x − f (ξ t , t ) y  ≤ y ( −m + M ε ) = −δ1 y for all t ∈ ( h1 , h2 ) which implies that y ( t ) ≤ y ( h1 ) e −δ1 (t − h1 ) ≤ ymax e −δ1 (t − h1 ) , ∀ t ∈ ( h1 , h2 ) Hence, ( x = x  a (ξ t , t ) − b (ξ t , t ) x − c (ξt , t ) y  ≥ x m − M x − M ymax e − δ1 ( t − h1 ) ) , ∀t ∈(h , h ) Put t ( n ( t ) = ∫ m − M ymax e −δ1 ( t − h1 ) t ) ds ; N (t ) = ∫ e ( ) ds h1 n s h1 By comparison theorem we get x (t ) ≥ ε1 e n (t ) , ∀t ∈ ( h1 , h2 ) + ε M N (t ) ε1 e n (t ) Let α = > It is clear that α does not depend on ( x ( ) , y ( ) ) and h1 Let t > h + ε M N (t ) xmin = {α , ε1} we see that x ( t ) > xmin , ∀ t > t The proof is complete As is known, the property of solutions of Lotka -Volterra systems near to boundary is dependent of two marginal equations In the case where the prey is absent, the quantity v(t ) of predator at the time t satisfies the equation v = −d (ξ t , t ) v − f (ξ t , t ) v Thus, v(t ) decreases exponentially to Similarly, without the predator, the quantity u (t ) of the prey at the time t satisfies the logistic equation u = u  a (ξt , t ) − b (ξ t , t ) u  , < u ( ) ∈ » + (3.2) If u (t ) is a solution of (3.2) then {ξ t , u ( t )} is Markov processes A random process {φt } , valued in a measurable space (S; S), is said to be periodic with period T if ( for any t1 , t2 , , tn ∈ » , the simultaneous distribution of φ t1 + k T ,φ t2 + k T , ,φ tn + k T ) does not depend on k ∈» We show that Equation (3.2) has a unique solution u * ( t ) such that (ξt , u * ( t ) ) is a periodic process Indeed, put u* ( t ) = e t A( t ) ∫ b (ξ , s ) e s −∞ A( s ) ds L.H Lan / VNU Journal of Science: Mathematics – Physics, Vol 30, No (2014) 49-69 56 t where A ( t ) = ∫ a (ξ s , s ) ds Firstly, we see that t +T ∫ a ξ s (ω ), s  ds e u* (t + T ,ω ) = s t +T ∫ b ξ (ω ) , s  e ∫ a ξτ (ω ), τ  dτ ds s −∞ t +T e = T ∫ a ξs−T (θ ω ), s −T  ds s t +T ∫ b ξ (θ ω ) , s − T  e T s −T T ∫ a ξτ −T (θ ω ), τ −T  dτ ds −∞ t T ∫ a ξ s (θ ω ), s  ds e −T = T ∫ a ξ s (θ ω ), s  ds e −T s t ∫ b ξ (θ ω ) , s  e T T ∫ a ξτ (θ ω ), τ  dτ s ds −∞ t T ∫ a ξs (θ ω ), s  ds e0 = s ∫ a ξτ (θ ω ), τ  dτ e b ξ s (θ T ω ) , s  ds ∫ t T = u* (t , θ T , ω ) −∞ Hence, by virtue of P - preserving measure property of θ , for any continuous function h , for any t1 < t2 < < tn ; k ∈ » we have { } Ε h ξt1 + k T , u * ( t1 + k T ) , ξt2 + k T , u * ( t2 + k T ) , , ξ tn + k T , u * ( tn + k T ) { } = Ε h ξ t1 (θ kT ) , u * ( t1 , θ kT ) , ξ t2 (θ kT ) , u * ( t2 , θ kT ) , , ξtn (θ kT ) , u * ( tn , θ kT )  { } = Ε h ξ t1 (.) , u * ( t1 , ) , ξt2 (.) , u * ( t2 , ) , , ξ tn (.) , u * ( tn , ) This means that (ξt , u * ( t ) ) is a periodic process with period T The uniqueness follows from the following lemma: Lemma 3.4 For any u0 > 0, lim u ( t ) − u * ( t )  = a.s., where u ( t ) t→+ ∞ is the solution of the equation (3.2) satisfying u ( ) = u0 1 − we have z = −a z Thus, by virtue of the bounded below property by u u* positive constant of z we follow the result Proof Put z = Lemma 3.5 [Law of large numbers for periodic processes] For any continuous, bounded function h ( t , i , u ) , periodic in t with period T we have L.H Lan / VNU Journal of Science: Mathematics – Physics, Vol 30, No (2014) 49-69 t  lim h  s , ξ s , u * ( s ) ds = Ε  t →+ ∞ t ∫   T T ∫ h  s , ξ s  , u * ( s ) ds   57 (3.3) Proof Put ( n +1) T ∫ Xn = h  s , ξ s , u * ( s ) ds nT Since {ξt , u * ( t )} is periodic then { X n } is a stationary process By the law of large numbers we have lim n →∞ n n ∑X k = Ε [ X / J ] a s., k =0 where J is the σ − algebra of the invariant sets However, (ξt ) is ergodic and u * ( t ) has no non-trivial invariant set then we follow that J = {Φ, Ω} This implies that 1 [ ] *   h s ξ u s ds = Xk = Ε[ X ] , , lim ( ) ∑ s t →+ ∞ t ∫  t →+∞ T t / T T [ ] k =0 t t /T lim = T   Ε  ∫ h  s , ξ s , u * ( s ) ds  T 0  Where, [ x ] denotes the integer number such that [ x ] ≤ x < [ x ] + Lemma is proved We study conditions that ensure the persistence of y (t ) of the Equation (2.4) with x(0) > and y (0) > Proposition 3.6 Put λ := T   Ε  ∫  − d (ξt , t ) + e (ξt , t ) u * ( t ) dt  T 0  (3.4) a) If λ > then limsup y ( t ) > δ > with probability t→+ ∞ b) In case λ < 0, lim y ( t ) = with probability t→+ ∞ Proof By comparison theorem, if x ( ) = u ( ) we have u ( t ) ≥ x ( t ) , ∀t Therefor, by virtue of Lemma 3.4 we have liminf ln u * ( t ) − ln x ( t ) ≥ t t→+∞ a) From Equations (3.2) and (2.4) we have ln u * ( t ) − ln u * ( ) t ln x ( t ) − ln x ( ) t t t = t 1 a (ξ s , s ) ds − ∫ b (ξ s , s ) u * ( s ) ds ∫ t t t = (3.5) t 1 a (ξ s , s ) ds − ∫ b (ξ s , s )  x ( s ) − u * ( s ) ds − ∫ t t t 1 − ∫ b (ξ s , s ) u * ( s ) ds − ∫ c (ξ s , s ) y ( s ) d t t (3.6) L.H Lan / VNU Journal of Science: Mathematics – Physics, Vol 30, No (2014) 49-69 58 On subtracting (3.6) from (3.5) we obtain t t 1  ≤ liminf  ∫ c (ξ s , s ) y ( s ) ds − ∫ b (ξ s , s ) u * ( s ) − x ( s ) ds  t t→+∞  t  t t 1  ≤ liminf  ∫ M y ( s ) ds − ∫ m u * ( s ) − x ( s )  ds  t t→+∞   t Hence, t t 1 M  liminf  ∫ y ( s ) ds − ∫ u * ( s ) − x ( s )  ds  ≥ t t→+ ∞   t m Otherwise, y (t ) y (t ) = − d (ξt , t ) + e (ξ t , t ) x ( t ) − f (ξt , t ) y ( t ) follows ln y ( t ) − ln y ( ) t t =  − d (ξ s , s ) + e (ξ s , s ) u * ( s )  ds − ∫ t t − (3.7) t 1 e (ξ s , s ) u * ( s ) − x ( s )  ds − ∫ f (ξ s , s ) y ( s ) ds ∫ t t and t t 1 e (ξ s , s ) u * ( s ) − x ( s ) ds + ∫ f (ξ s , s ) y ( s ) ds = ∫ t t = t ln y ( t ) − ln y ( )  − d (ξ s , s ) + e (ξ s , s ) u * ( s )  ds − ∫ t t  ln y ( t ) − ln y ( )  Moreover, y (t ) is bounded above then liminf −  ≥ and we apply the law t t→+∞   t of large numbers (Lemma 3.5), lim ∫  − d (ξ s , s ) + e (ξ s , s ) u * ( s ) ds =λ , consequently, t→+∞ t t 1 t  liminf  ∫ e (ξ s , s ) u * ( s ) − x ( s ) ds + ∫ f (ξ s , s ) y ( s ) ds  = t t→+ ∞  t  t ln y ( t ) − ln y ( )  1 = lim inf  ∫  − d (ξ s , s ) + e (ξ s , s ) u * ( s ) ds −  t t→+ ∞  t  1 t   ln y ( t ) − ln y ( )  ≥ lim inf  ∫  − d (ξ s , s ) + e (ξ s , s ) u * ( s ) ds  + lim inf − ≥λ t t→+ ∞   t → + ∞   t Hence, t t 1  liminf  ∫ u * ( s ) − x ( s )  ds + ∫ y ( s ) ds  ≥ t t→+ ∞   t L.H Lan / VNU Journal of Science: Mathematics – Physics, Vol 30, No (2014) 49-69 59 t t  λ f (ξ s , s ) 1 e (ξ s , s ) *   liminf  ∫ u ( s ) − x ( s )  ds + ∫ y ( s ) ds  ≥  t M t→+ ∞   M t M By (3.7) plus (3.8), we obtain t liminf t→+ ∞ M   + 1 y ( s ) ds ≥ liminf ∫ t 0m t→+ ∞  t t 1 M  y ( s ) ds − ∫ u * ( s ) − x ( s )  ds  +  ∫ t  t m  t 1 t  λ + liminf  ∫ u * ( s ) − x ( s ) ds + ∫ y ( s ) ds  ≥ t t→+ ∞  t  M t then liminf t→+ ∞ m y ( s ) ds ≥ λ ≥ and limsup y ( t ) > δ > ∫ t M ( M + m) t→+∞ b) From the second equality of systems (2.4) and λ > we have limsup ln y ( t ) − ln y ( ) t t→+ ∞ 1 = lim sup  ∫  − d (ξ s , s ) + e (ξ s , s ) u * ( s )  ds − t→+∞  t t − t t  1 *   e , s u s x s ds f (ξ s , s ) y ( s ) ds  − − ξ ( ) ( ) ( ) s ∫ ∫   t t  t ≤ λ − limsup t→+ ∞ t 1 e (ξ s , s ) u * ( s ) − x ( s )  ds − limsup ∫ f (ξ s , s ) y ( s ) ds < ∫ t t t→+ ∞ which implies that lim y ( t ) = The proof is complete t→+∞ Remark 3.7 The conditions (3.4) is easily to be checked by simulation method based on the law of large numbers Moreover, by (ξt , u * ( t ) ) is solution of equation (3.2), we have λ := T   Ε  ∫  − d (ξt , t ) + e (ξt , t ) u * ( t )  dt  T   = T   e (ξ t , t )   Ε  ∫  − d (ξt , t ) + b (ξ t , t ) u * ( t ) dt  T   b (ξ t , t )   ≥ T      e ( ± , t )  Ε  ∫  − d (ξt , t ) + inf  b (ξ t , t ) u * ( t ) dt   t→+∞ T    b (ξ t , t )     = T      e ( ± , t )  Ε  ∫  − d (ξt , t ) + inf  a (ξt , t )  dt  +  t→+∞ T    b (ξ t , t )     T  e ( ± , t )    + inf  Ε  ∫  a (ξt , t ) − b (ξt , t ) u * ( t ) dt   t→+∞   b (ξ t , t )  T  Note that L.H Lan / VNU Journal of Science: Mathematics – Physics, Vol 30, No (2014) 49-69 60 T   Ε  ∫  a (ξ t , t ) − b (ξ t , t ) u * ( t )  dt  = T    e ( ± , t )  and that − d ( ± , t ) + inf   a ( ± , t ) ≥ , ∀t > t→+∞  b ( ± , t )  provided that  a ( ± , t )   d ( ± , t )  inf   > sup    b ( ± , t )  t → + ∞  e ( ± , t )  (3.9) t→+∞ Then, λ > under the condition 3.9, which is similar to (2.8) From now on, we suppose that λ > Lemma 3.8 With probability 1, there are infinitely many sn = sn (ω ) > such that sn > sn −1 , lim sn = ∞ and x ( sn ) ≥ δ , y ( sn ) ≥ δ , ∀n ∈ » n →+ ∞ Proof By Proposition 3.3 we can find t > such that x ( t ) ≥ xmin , for all t > t On the other hand, there exists δ < xmin and a random sequences {sn } ↑ ∞ , sn > t such that y ( sn ) > δ , ∀ n ∈ » The proof is complete For the sake of simplicity, we suppose ξ = + a.s and set xn := x (τ n , x , y ) , yn := y (τ n , x , y ) ℑ0n = σ (τ k , k ≤ n ) ; ℑ∞n = σ (τ k −τ n , k > n ) It is clear that ( xn , yn ) is ℑ0n measurable ℑ0n is independent ℑ∞0 if ξ0 is given Hypotheses 3.9 On the quadrant int » 2+ , the system (2.5) has a stable positive T − periodic solution ( x+* , y+* ) such that ( x (t ) − x (t ) , y (t ) − y (t )) → ( 0,0) + * + + * + t →+∞ Lemma 3.10 Suppose that Hypothesis 3.9 holds and λ > , we can find an ∆ > such that with probability 1, there are infinitely many n ∈ » such that ∆ ≤ xn , yn ≤ M * Moreover, we can find ∆ > { such that the events x2 k +1 > ∆ , y2 k +1 > ∆ } as well as {x 2k > ∆ , y2 k > ∆ } occur infinitely many often Proof Let {ℑt } be the filtration generated by {ξ ( t )} It is obvious that {ξ ( t ) , x ( t ) , y ( t )} is a strong Feller-Markov process with respect to the filtration {ℑt } For a stopping time ς , the σ − algebra at ς is ℑς = { A ∈ ℑ∞ : A ∩ {ς ≤ t} ∈ ℑt , ∀t ∈ » + } Fix a T1 > , by Lemma 3.8, we can define almost surely finite stopping times η1 = inf {t > 0: x ( t ) ≥ δ , y ( t ) ≥ δ } η2 = inf {t > η1 + T1 : x ( t ) ≥ δ , y ( t ) ≥ δ } ηn = inf {t > ηn −1 + T1 : x ( t ) ≥ δ , y ( t ) ≥ δ } L.H Lan / VNU Journal of Science: Mathematics – Physics, Vol 30, No (2014) 49-69 For a stopping time ς , we write τ (ς ) for the first jump of ξ ( t ) { 61 after ς , i.e., } τ (ς ) = inf {t > ς : ξ ( t ) ≠ ξ (ς )} Let σ (ς ) = τ (ς ) − ς and Ak = σ (ηk ) < T1 , k ∈ » Obviously, Ak is in the σ − algebra generated by {ξ (η n + s ): s ≥ 0} and Ak ∈ ℑηk +1 also Therefore, in view of the strong Markov property of (ξ ( t ) , x ( t ) , y ( t ) ) and [see 15, Theorem 5, p 59] we have Ρ  Ak ξ (η k ) = ±  = Ρ σ ( ) > T1 ξ ( ) = ±    Hence, ( ) Ρ Ak = Ρ σ (η k ) > T1 ξ (η k ) = +  Ρ ξ (η k ) = +  + Ρ σ (η k ) > T1 ξ (η k ) = −  Ρ ξ (η k ) = −  = Ρ σ ( ) > T1 ξ ( ) = +  Ρ ξ (η k ) = +  + Ρ σ ( ) > T1 ξ ( ) = −  Ρ ξ (η k ) = −  ≤ p {( )} )} = where p = max Ρ σ ( ) > T1 ξ = ± < Moreover, { Ε 1Ak +11Ak ξ (η k +1 ) , x (η k +1 ) , y (η k +1 { = Ε {1 Ε 1 = Ε {1 Ε 1  } )} = Ε Ε 1Ak +11Ak ℑηk +1  ξ (η k +1 ) , x (η k +1 ) , y (η k +1 )   Ak Ak ℑηk +1  ξ (η k +1 ) , x (η k +1 ) , y (η k +1 Ak +1 Ak +1 (ξ (η ) , x (η ) , y (η ) ) k +1 k +1 k +1 } ξ (η k +1 ) , x (η k +1 ) , y (η k +1 )  { = Ε 1Ak ξ (η k +1 ) , x (η k +1 ) , y (η k +1 )  Ε 1Ak +1 (ξ (η k +1 ) , x (η k +1 ) , y (η k +1 ) )  } which implies that { } { }{ } Ρ Ak +1 ∩ Ak ξ (η k +1 ) = ± = Ρ Ak +1 ξ (η k +1 ) = ± Ρ Ak ξ (η k +1 ) = ± (3.10) Therefore, from (3.10) and the equation ) { ( } { } Ρ Ak +1 ∩ Ak = Ρ Ak +1 ∩ Ak ξ (η k +1 ) = + Ρ {ξ (η k +1 ) = +} + Ρ Ak +1 ∩ Ak ξ (η k +1 ) = − Ρ {ξ (η k +1 ) = −} it follows ( ) { } { } Ρ Ak +1 ∩ Ak ≤ p Ρ Ak ξ (η k +1 ) = + Ρ {ξ (η k +1 ) = +} + p Ρ Ak ξ (η k +1 ) = − Ρ {ξ (η k +1 ) = −} ≤ p Continuing this way, we conclude that  n   n  Ρ  ∪ Ai  = − Ρ  ∩ Ai  ≤ − p  i=k   i =k  ( ) Consequently,  +∞ +∞  Ρ  ∩∪ Ai  =  k =1 i = k  n − k +1 L.H Lan / VNU Journal of Science: Mathematics – Physics, Vol 30, No (2014) 49-69 62 Let ∆ = { x± ( t + s , t , x0 , y0 ) , y± ( t + s , t , x0 , y0 ) : t ∈ [0, T1 ] , s ∈ [ 0, T ] , x0 , y0 ∈ [δ , M ]} > , if Ak occurs, xτ (ηk ) > ∆ , yτ (ηk ) > ∆ , which directly implies the first assertion As a result, we are able to define finite stopping times η = inf {n > 0: ∆ ≤ xn , yn ≤ M * } η = inf {t > η1 : ∆ ≤ xn , yn ≤ M * } η k = inf {t > ηk −1 : ∆ ≤ xn , yn ≤ M * } { } ∆ = x± ( t + s , t , x0 , y0 ) , y± ( t + s , t , x0 , y0 ) : t ∈ [ 0, T1 ] , s ∈ [0, T ] , x0 , y0 ∈  ∆, M *  > Put { Note that if the event Bk = σ η k +1 < T1 } occurs then ∆ ≤ xηk , yηk , xηk +1 , yηk +1 ≤ M * Using arguments similar to the previous part of this proof, we can show that Bk occurs infinitely often Consequently, we obtain the second assertion of this lemma due to the fact that η k is odd then η k + is even and conversely Next, we will describe the ω − limit sets of the system (2.4) Denoted by Ω ( x, y , ω ) the ω − limit set of the solution ( x ( t ,0, x, y ) , y ( t ,0, x, y ) ) (ω ) starting in ( x, y ) To simplify the notations, for t ≥ s ≥ , we denote π t+,s ( x, y ) := ( x+ ( t , s, x, y ) , y+ ( t , s, x, y ) ) ; resp π t−,s ( x, y ) := ( x− ( t , s, x, y ) , y− ( t , s, x, y ) ) ) is the solution to the system (2.5) (resp (2.6)) starting at ( x, y ) ∈ » 2+ at time s Suppose that the solution starting at γ +* ( ) = ( x+* ( ) , y+* ( ) ) at time is a periodic solution to the system (2.5), we now describe the pathwise dynamic behavior of the solutions of system (2.4) Put Γ= {( x, y ) = π ( −1) tn , tn−1 n } .π t−3 , t2 π t+3 , t2 (γ +* ( ) ) : = t1 ≤ t2 ≤ ≤ tn ; n ∈ » (3.11) where γ +* ( ) is mentioned above Let us ( x0 , y0 ) ∈ » 2+ Theorem 3.11 Suppose that on the quadrant int » 2+ , the system (2.5) has unique stable T − periodic solution ( x+* ( t ) , y+* ( t ) ) and with λ mentioned in Proposition 3.6, let λ > Then, a) With probability 1, the closure Γ of Γ is a subset of the ω − limit set Ω ( x0 , y0 , ω ) ( )  h (+ ,t , z ) h (− ,t , z )  ≠0 det    h (+ ,t , z ) h (−,t , z)   b) If there exists a z = x , y such that the point z = π t+, γ +* ( ) satisfies the following condition 2 (3.12) L.H Lan / VNU Journal of Science: Mathematics – Physics, Vol 30, No (2014) 49-69 ( ( 63 ) ( ) ( ) ( ) ) ( ) ( ) ( ) h ξ , t , z = a ξ , t − b ξ , t x − c ξ , t y t t t  t where   h2 ξt , t , z = − d ξt , t + e ξt , t x − f ξt , t y  Then, with probability 1, the closure Γ of Γ is the ω - limit set Ω ( x0 , y0 , ω ) Moreover, Γ absorbs all positive solutions in the sense that for any initial value ( x0 , y0 ) ∈ int » 2+ , the value γ (ω ) = inf {t > 0:( x ( s,0, x0 , y0 , ω ) , y ( s,0, x0 , y0 , ω ) ) ∈ Γ , ∀s > t} is finite outside a P-null set Proof Denote Η := Η ∆ , M * We construct a sequence of stopping times η1 = inf { 2k : ( x2 k , y2 k ) ∈ Η} η2 = inf {2k > η1 : ( x2 k , y2 k ) ∈ Η} ηn = inf {2k > ηn −1 : ( x2 k , y2 k ) ∈ Η} It is easy to see that the events {ηk = n} ∈ ℑ0n for any k; n Thus the event {ηk = n} is independent of ℑ∞0 if ξ is given By the Proposition 3.1 and Lemma 3.10, ηn < ∞ a.s for all n For simplicity, we put mod ( t ) = t − kt T where kt is such a integer that kt T ≤ t ≤ ( kt + 1) T As a convention, the notation mod ( t ) ∈ ( −δ , δ ) means mod ( t ) ∈ [0, δ ) ∪ (T − δ , T ) By U ε ( x, y ) , we denote neighborhood of point ( x, y ) with radius ε > and φ ( t , s, x, y ) = ( x ( t , s, x, y ) , y ( t , s, x, y ) ) Firstly, we prove that for any ε1 > 0, δ1 > 0, there are infinitely many odd stopping times such that ( x2n+1 , y2n+1 ) ∈U ε γ +* ( ) and mod (τ n +1 ) ∈ ( −δ1 , δ1 ) We have π +t +τη k +1 where , τηk +1 (x η k +1 ) ( ( ) , yηk +1 = π +t + mod τ , mod τ x ,y = π +t − T + mod τ , x , y ( ηk +1 ) ( ηk +1 ) ηk +1 ηk +1 ( ηk +1 ) (x, y ) = π + ( T , mod τηk +1 ) (x η k +1 ) ) , yηk +1 Therefore, applying the Lemma 2.2 obtains, for any neighborhood U ε1 γ +* ( )  , there exists T * > and δ so that ) This is equivalent to t ∈ ( − mod (τ ) + K T − δ , − mod (τ ) + K T + δ ) K ≥ K is the smallest natural number satisfying − mod (τ ) + K T − δ > T Note that, − mod (τ ) + K T < T + T := T π +t +τη k , τηk (x ηk ) ( , yηk ∈U ε1 γ +* ( )  ; ∀t > T * , mod t + τ ηk ∈ ( −δ , δ ) ηk ηk * ηk +1 * ηk +1 Now, let δ = {δ1 , δ } ; for any u > 0, δ > 0, k ∈ » , put , in which K ∈ » L.H Lan / VNU Journal of Science: Mathematics – Physics, Vol 30, No (2014) 49-69 64 { ( ( ) ( )} ) Ak = ω : σ ηk +1 ∈ − mod τηk +1 + K T − δ , − mod τ ηk +1 + K T + δ Note that if X has the exponential distribution then Ρ {t < X < t + a} ≥ Ρ {s < X < s + a} whenever t ≤ s Using the strong Markov property of {ξ ( t ) , x ( t ) , y ( t )} and noting that we have already known the value of ξτη , we have the estimation k { } { ( ( ) ( ) Ρ Ak = Ρ σ ηk +1 ∉ − mod τηk +1 + K T − δ , − mod τηk +1 + K T + δ +∞ = ∫ { ( )} τη k = t { ( )} τηk = t , ξτη = + { ( )} ξτη = + { ( Ρ σ ηk +1 ∉ − mod ( t ) + K T − δ , − mod ( t ) + K T + δ +∞ = ∫ Ρ σ ηk +1 ∉ − mod ( t ) + K T − δ , − mod ( t ) + K T + δ +∞ = ∫ Ρ σ ηk +1 ∉ − mod ( t ) + K T − δ , − mod ( t ) + K T + δ +∞ ∫ Ρ σ ηk +1 ∉ T − δ , T + δ )} ξτη = + k +∞ { ( = Ρ σ ∉ T − δ3 , T + δ3 )} ∫ Ρ{τ { × Ρ τ ηk ∈ d t } { { × Ρ τ ηk ∈ d t k k ≤ )} { × Ρ τ ηk ∈ d t { × Ρ τ ηk ∈ d t } } } ( ∈ dt = Ρ σ ∉ T − δ , T + δ ηk } )} := ϕ < { We now estimate Ρ Ak ∩ Ak + (ξ ( t ) , x ( t ) , y ( t ) ) { } Since Ak ∈ ℑηk +2 , applying the strong Markov property of we have } { } } = Ε 1 Ε{1 { } = −} ≤ ϕ Ε (1 ) = ϕ  Ρ Ak ∩ Ak +1 = Ε  Ε 1A 1A ℑηk +1  = Ε 1A Ε 1A ℑηk +1  k k +1 k +1    k  { = Ε 1A Ε 1A ℑηk +1 k +1  k Ak Ak +1 ξτη k +1 Ak Continuing this way, we have, { ( ( ) ( ) } ) Ρ {∩+k =∞1 ∪ +i =∞k Ai } = Ρ ω : σ ηk +1 ∈ − mod τηk +1 + K T − δ , − mod τ ηk +1 + K T + δ i o of n = The even Ak occurs infinitely means that, with probability 1, for any δ1 > , for any U ε1 γ +* ( )  , there are infinitely n = n (ω ) ∈ » many such that mod (τ n +1 ) ∈ ( −δ , δ ) ⊂ ( −δ1 , δ1 ) Thus γ +* ( ) ∈ Ω ( x0 , y , ( x2n , y2n ) ∈U ε (γ ) 2 and {π γ ( ) : t ≥ }∈ Ω ( x , y , ω ) a.s To this, we show that for (γ ) , ∀δ > , there are infinitely many even stopping times such that Secondly, we prove γ := π t−,0 γ +* ( )  , ∀U ε ( x2n+1 , y2n+1 ) ∈U ε γ +* ( ) ω ) − t,0 * + 0 and mod (τ n ) ∈ ( mod ( t1 − δ ) , mod ( t1 + δ ) ) By continuity of solutions with L.H Lan / VNU Journal of Science: Mathematics – Physics, Vol 30, No (2014) 49-69 ε3 > , δ5 > , δ6 > respect to initial conditions, there are ∀ ( x, y ) ∈ U ε γ * + ( ) , ∀t ∈ ( t1 − δ , t1 + δ ) π t−,0 γ +* ( ) − π t−,0 γ +* ( ) < π t−,0 ( x , y ) − π t−, γ +* ( )  < π t−+ s , s ( x , y ) − π t−,0 ( x , y ) < 65 small enough so that if and ∀ mod ( s ) ∈ ( −δ , δ ) then ε2 ε2 ε2 Therefore, π t−+ s , s ( x , y ) − π t−, γ +* ( )  ≤ π t−+ s , s ( x , y ) − π t−,0 ( x , y ) + π t−,0 ( x , y ) − π t−,0 γ +* ( ) + π t−, γ +* ( )  − π t−1 ,0 γ +* ( ) < ε , ∀ ( x , y ) ∈U ε γ +* ( )  , ∀t ∈ ( t1 − δ , t1 + δ ) , ∀ mod ( s ) ∈ ( −δ , δ ) Put { = inf { 2k + > ς1 = inf 2k + 1: ( x2 k +1 , y2 k +1 ) ∈ U ε γ +* ( ) , mod (τ k +1 ) ∈ ( −δ , δ ) ς2 } ς : ( x2 k +1 , y2 k +1 ) ∈ U ε γ +* ( ) , mod (τ k +1 ) ∈ ( −δ , δ ) } { } ς n = inf 2k + > ς n −1 : ( x2 k +1 , y2 k +1 ) ∈ U ε γ +* ( ) , mod (τ k +1 ) ∈ ( −δ , δ ) From the previous part of this proof, it follows that ς k < + ∞ {ς k = n}∈ ℑ , {ς k } { n and lim ς k = + ∞ a.s Since k→+∞ is independent of ℑ Put t = {δ , δ } By the same argument as above we ∞ n ( ) ( ) } ∈ ( t − t , t + t ) which implies ( x , y ) ∈ U (γ ) for many infinite k ∈ » and mod (τ ) ∈ ( mod ( t − t ) , mod ( t + t ) ) ⊂ ( mod ( t − δ ) , mod ( t + δ ) ) obtain Ρ ω : σ ς n +1 ∈ t1 − t , t1 + t i o of n = This relation says that xς k , yς k ∈ U ε γ +* ( )  and σς k +1 ς k +1 ς k +1 ς k +1 ε2 4 This means γ ∈ Ω ( x0 , y0 , ω ) a.s Lastly, by similar way and induction, we conclude that Γ is a subset of Ω ( x0 , y0 , ω ) Because Ω ( x0 , y0 , ω ) is a close set, we have Γ ⊂ Ω ( x0 , y0 , ω ) a.s ( b) We now prove the second assertion of this theorem Let z = x , y ) satisfying the condition (3.12) By the existence and continuous dependence on the initial values of the solutions, there exist two numbers a > and b > () such that the function ϕ ( s, t ) = π t+, s π t−, s z continuously differentiable in ( −a , a ) × ( −b , b ) is defined and L.H Lan / VNU Journal of Science: Mathematics – Physics, Vol 30, No (2014) 49-69 66 We note that ( ( ) ) ( ( ) ( ) ( ) )  x h +, x, y x h1 − , x , y   ∂ϕ ∂ϕ    det  , = det     ∂s ∂t  (t , t )  y h2 + , x , y y h2 − , x , y     h +, z h1 − , z    ≠ = x y det    h2 + , z h2 − , z    ( ( ) ) Therefore, by Theorem of Inverse Function, there exist < a1 < a , < b1 < b such that ϕ ( s, t ) is a diffeomorphism between V = ( 0, a1 ) × ( 0, b1 ) and U = ϕ (V ) As a consequence, U is an open set Moreover, for ( x, y ) = π t+ , s π t−, s * * every * ( x, y ) ∈ U , there exists ( z ) ∈ Γ Hence, U ⊂ Γ ⊂ Ω ( x a ( s* , t * ) ∈ V = ( 0, a1 ) × ( 0, b1 ) such that , y0 , ω ) Thus, there is a stopping time γ < + ∞ a.s such that ( x (γ ) , y ( γ ) ) ∈ U Since Γ is a forward invariant set and U ⊂ Γ , it follows that ( x (t ) , y (t )) ∈ Γ , ∀ t > γ with probability The fact ( x (t ) , y (t )) ∈ Γ for all t > γ implies that Ω ( x0 , y0 , ω ) ⊂ Γ By combining with the part a) we get Ω ( x0 , y0 , ω ) = Γ a.s The proof is complete Simulation and discussion Noting that λ can be estimated by using the law of large number and formula (3.4) for an initial concrete set We will illustrate the above model by following numerical examples in three cases Example I λ > and the coexistence case presents in both states (see figure 3) It corresponds to α = 0.6 ; β = 0.4 ; a ( + ) = 10 + sin t ; b ( + ) = + d (+) = 1− cos t ; c ( + ) = ; 1  π  π cos  t −  ; e ( + ) = 1.8 ; f ( + ) = 3.1 + sin  t +  ; 3   6 a ( − ) = 11.7 − sin ( t + π ) ; b ( − ) = 1.5 + cos t ; c ( − ) = 1.4 − 1  π sin  t +  ; d ( − ) = 2.1 + sin ( t + π ) ;  2 e ( − ) = 1.2 + 1  π cos t ; f ( − ) = 2.7 − cos  t +   5 L.H Lan / VNU Journal of Science: Mathematics – Physics, Vol 30, No (2014) 49-69 67 the initial condition ( x ( ) , y ( ) ) = ( 2.5 ; 2.8 ) and number of switching n = 300 In this example, the periodic T = 2π , the solution of (2.4) switches between two positive periodic orbit of the systems (2.5) and (2.6) Figure Orbit of the system (2.4) in example I Example II λ > and one state is coexistence, the other is extinction of predator The system (2.5) with coefficients a ( + ) = 12 + sin π t ; b ( + ) = 2.8 + cos π t ; 1 π   c ( + ) = 2.4 + sin (π t + π ) ; d ( + ) = 1.2 − cos  π t −  ; 12   e ( + ) = 2.4 − 1 π  sin π t ; f ( + ) = 2.4 + sin  π t +  ; 6  has a stable positive periodic solution and the system (2.6) with coefficients a ( − ) = 6.1 − sin (π t + π ) ; b ( − ) = 1.6 + cos π t ; c ( − ) = 2.4 − π π   sin  π t +  ; d ( − ) = + sin  π t +  ; 2 2   e ( − ) = 0.5 + 1 π  cos π t ; f ( − ) = 1.9 − cos  π t +  5  has predator tending to The number of switching n = 300 , transition intensities α = 0.3 , β = 0.7 and initial condition ( x ( ) , y ( ) ) = (1.2, 3.4 ) Since λ > , the system (2.4) is persistent (see figure 4) 68 L.H Lan / VNU Journal of Science: Mathematics – Physics, Vol 30, No (2014) 49-69 Figure Orbit of the system (2.4) in example II This work provides some results about the asymptotic behavior of a system of two coupled deterministic predator-prey models switching at random The formula for the value λ can not be explicitly computed However, it is easy to approximate it by simulation When λ > the dynamics of the predator-prey system leads to the existence of a periodic Markov process, which plays an important role in the study of the development of communities References [1] K Gopalsamy Global asymptotic stability in a periodic Lotka-Volterra system, J.Austral Math Soc Ser , B 27 (1985), pp 66-72 [2] A Tineo Permanence of a large class periodic predator-prey systems, J.Math.Anal.Appl, 241 (2000), pp 83-91 [3] P.Yang, R Xu Global attractivity of the periodic Lotka-Volterra system, J Math Anal Appl, 233 (1999), no 1, pp 221-232 [4] J Zhao, W Chen Global asymptotic stability of a periodic ecological model, Appl.Math.Comput, 147 (2004), pp 881-892 [5] Z Amine, R Ortega A periodic prey-predator system, J.Math.Anal.Appl, 185 (1994), pp 477-489 [6] M.Bardi Predator-prey models in periodically uctuating environments, J.Math.Biology, 12 (1981), pp 127-140 [7] J M Cushing 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varying environment: extinction or survival Bull Math Biol, 46 (1984), no 1, pp 175-184 [14] P Auger, N H Du, N T Hieu Evolution of Lotka-Volterra predator-prey systems under telegraph noise Math Biosci Eng, (2009), no 4, 683-700 [15] I.I Gihman and A.V Skorohod The Theory of Stochastic Processes Springer -Verlag Berlin Heidelberg New York 1979 [16] R.S Lipshter and Shyriaev Statistics of Stochastic Processes Nauka, Moscow 1974 (in Russian)  ... Journal of Science: Mathematics – Physics, Vol 30, No (2014) 49-69 52 Thus, the relationship of these two systems will determine the trajectory behavior of Equation (2.4) System (2.4) without the. .. influence of stochastic fluctuation of environment and changing periodically of season as well We describe completely the omega limit set of the positive solutions of Equation (1.1) with the periodic... changing of seasons but also in the fluctuation of stochastic factors, which may have important consequences on the dynamics of the communities For the stochastic Lotka - Volterra equation, a systematic