Long-time behavior of a non-autonomous stochastic predator-prey model with jumps
aa r X i v : . [ m a t h . P R ] F e b Modern Stochastics: Theory and Applications 0 (0) (2020) 0–0https://doi.org/
Long-time behavior of a non-autonomousstochastic predator-prey model with jumps
Olga Borysenko a,0 , Oleksandr Borysenko b,1a
Department of Mathematical Physics, National Technical University ofUkraine, 37, Prosp.Peremohy, Kyiv, 03056, Ukraine b Department of Probability Theory, Statistics and Actuarial Mathematics,Taras Shevchenko National University of Kyiv , Ukraine, 64 VolodymyrskaStr., Kyiv, 01601 Ukraine [email protected] (Olg. Borysenko), [email protected] (O. Borysenko)
Abstract
It is proved the existence and uniqueness of the global positive solution tothe system of stochastic differential equations describing a non-autonomous stochas-tic predator-prey model with modified version of Leslie-Gower term and Holling-typeII functional response disturbed by white noise, centered and non-centered Poissonnoises. We obtain sufficient conditions of stochastic ultimate boundedness, stochasticpermanence, non-persistence in the mean, weak persistence in the mean and extinc-tion of the solution to the considered system.
Keywords
Stochastic Predator-Prey Model, Leslie-Gower and Holling-type IIfunctional response, Global Solution, Stochastic Ultimate Boundedness, StochasticPermanence, Extinction, Non-Persistence in the Mean, Weak Persistence in theMean
The deterministic predator-prey model with modified version of Leslie-Gowerterm and Holling-type II functional response is studied in [1]. This model has Corresponding author.
Preprint submitted to VTeX / Modern Stochastics: Theory and Applications
Let (Ω , F , P) be a probability space, w i ( t ) , i = 1 , , t ≥ are independent stan-dard one-dimensional Wiener processes on (Ω , F , P) , and ν i ( t, A ) , i = 1 , areindependent Poisson measures defined on (Ω , F , P) independent on w i ( t ) , i =1 , . Here E[ ν i ( t, A )] = t Π i ( A ) , i = 1 , , ˜ ν i ( t, A ) = ν i ( t, A ) − t Π i ( A ) , i = 1 , , Π i ( · ) , i = 1 , are finite measures on the Borel sets in R . On the probabilityspace (Ω , F , P) we consider an increasing, right continuous family of completesub- σ -algebras {F t } t ≥ , where F t = σ { w i ( s ) , ν i ( s, A ) , s ≤ t, i = 1 , } .We need the following assumption. Assumption 1.
It is assumed, that a i ( t ) , b ( t ) , c i ( t ) , σ i ( t ) , γ i ( t, z ) , δ i ( t, z ) , i =1 , , m ( t ) are bounded, continuous on t functions, a i ( t ) > , i = 1 , , b > , c i inf > , i = 1 , , m inf = inf t ≥ m ( t ) > , and ln(1+ γ i ( t, z )) , ln(1+ δ i ( t, z )) , i =1 , are bounded, Π i ( R ) < ∞ , i = 1 , .In what follows we will assume that Assumption 1 holds. Theorem 1.
There exists a unique global solution X ( t ) to the system (3) forany initial value X (0) = X ∈ R , and P { X ( t ) ∈ R } = 1 . Proof.
Let us consider the system of stochastic differential equations dξ i ( t ) = (cid:20) a i ( t ) − b i ( t ) exp { ξ i ( t ) } − c i ( t ) exp { ξ ( t ) } m ( t ) + exp { ξ ( t ) } − β i ( t ) (cid:21) dt + σ i ( t ) dw i ( t ) + Z R ln(1 + γ i ( t, z ))˜ ν ( dt, dz ) + Z R ln(1 + δ i ( t, z ))˜ ν ( dt, dz ) ,v i (0) = ln x i , i = 1 , . (4)The coefficients of the system (4) are local Lipschitz continuous. So, for any ini-tial value ( ξ (0) , ξ (0)) there exists a unique local solution Ξ( t ) = ( ξ ( t ) , ξ ( t )) on [0 , τ e ) , where sup t<τ e | Ξ( t ) | = + ∞ (cf. Theorem 6, p.246, [8]). There-fore, from the Itô formula we derive that the process X ( t ) = (exp { ξ ( t ) } , exp { ξ ( t ) } ) is a unique, positive local solution to the system (3). To show thissolution is global, we need to show that τ e = + ∞ a.s. Let n ∈ N be sufficientlylarge for x i ∈ [1 /n , n ] , i = 1 , . For any n ≥ n we define the stopping time τ n = inf (cid:26) t ∈ [0 , τ e ) : X ( t ) / ∈ (cid:18) n , n (cid:19) × (cid:18) n , n (cid:19)(cid:27) . It is easy to see that τ n is increasing as n → + ∞ . Denote τ ∞ = lim n →∞ τ n ,whence τ ∞ ≤ τ e a.s. If we prove that τ ∞ = ∞ a.s., then τ e = ∞ a.s. and X ( t ) ∈ R a.s. for all t ∈ [0 , + ∞ ) . So we need to show that τ ∞ = ∞ a.s. If itis not true, there are constants T > and ε ∈ (0 , , such that P { τ ∞ < T } > ε .Hence, there is n ≥ n such that P { τ n < T } > ε, ∀ n ≥ n . (5) For the non-negative function V ( X ) = P i =1 k i ( x i − − ln x i ) , X = ( x , x ) , x i > , k i > , i = 1 , by the Itô formula we obtain dV ( X ( t )) = X i =1 k i ( x i ( t ) − (cid:20) a i ( t ) − b i ( t ) x i ( t ) − c i ( t ) x ( t ) m ( t ) + x ( t ) (cid:21) + β i ( t ) + Z R δ i ( t, z ) x i ( t )Π ( dz ) dt + X i =1 k i ( x i ( t ) − σ i ( t ) dw i ( t )+ Z R [ γ i ( t, z ) x i ( t ) − ln(1 + γ i ( t, z ))]˜ ν ( dt, dz )+ Z R [ δ i ( t, z ) x i ( t ) − ln(1 + δ i ( t, z ))]˜ ν ( dt, dz ) . (6)Let us consider the function f ( t, x , x ) = φ ( t, x ) + ψ ( t, x , x ) , x > ,x > where φ ( t, x ) = − k b ( t ) x + k (cid:16) α ( t ) + b ( t ) (cid:17) x + k β ( t ) + k β ( t ) − k a ( t ) − k a ( t ) ,ψ ( t, x , x ) = ( m ( t ) + x ) − h − k c ( t ) x + (cid:16) k α ( t ) − k c ( t ) (cid:17) x x + (cid:16) k α ( t ) m ( t ) + k c ( t ) + k c ( t ) (cid:17) x i . Under Assumption 1 there is a constant L ( k , k ) > , such that φ ( t, x ) ≤ k h − b x + ( α + b ) x i + β max ( k + k ) ≤ L ( k , k ) . If α ≤ , then for the function ψ ( t, x , x ) we have ψ ( t, x , x ) ≤ − k c x + ( k + k ) c max x m ( t ) + x ≤ L ( k , k ) . If α > , then for k = k c α there is a constant L ( k , k ) > , suchthat ψ ( t, x , x ) ≤ n − k c x + ( k α − k c ) x x + h k α m sup + ( k + k ) c max i x o ( m ( t ) + x ) − = k m ( t ) + x (cid:26) − c c α x + (cid:20) c m sup + (cid:18) c α (cid:19) c max (cid:21) x (cid:27) ≤ L ( k , k ) . Therefore there is a constant L ( k , k ) > , such that f ( t, x , x ) ≤ L ( k , k ) .So from (6) we obtain by integrating V ( X ( T ∧ τ n )) ≤ V ( X ) + L ( k , k )( T ∧ τ n )+ X i =1 k i T ∧ τ n Z ( x i ( t ) − σ i ( t ) dw i ( t ) + T ∧ τ n Z Z R [ γ i ( t, z ) x i ( t ) − ln(1+ γ i ( t, z ))] ˜ ν ( dt, dz ) + T ∧ τ n Z Z R [ δ i ( t, z ) x i ( t ) − ln(1 + δ i ( t, z ))]˜ ν ( dt, dz ) . (7)Taking expectations we derive from (7)E [ V ( X ( T ∧ τ n ))] ≤ V ( X ) + L ( k , k ) T. (8)Set Ω n = { τ n ≤ T } for n ≥ n . Then by (5) , P(Ω n ) = P { τ n ≤ T } > ε , ∀ n ≥ n . Note that for every ω ∈ Ω n there is at least one of x ( τ n , ω ) and x ( τ n , ω ) equals either n or /n . So V ( X ( τ n )) ≥ min { k , k } min { n − − ln n, n − n } . From (8) it follows V ( X ) + L ( k , k ) T ≥ E[ Ω n V ( X ( τ n ))] ≥ ε min { k , k } min { n − − ln n, n − n } , where Ω n is the indicator function of Ω n . Letting n → ∞ leads to the con-tradiction ∞ > V ( X ) + L ( k , k ) T = ∞ . This completes the proof of thetheorem. Lemma 1.
The density of prey population x ( t ) obeys lim sup t →∞ ln( m + x ( t )) t ≤ , ∀ m > , a.s. (9) Proof.
By Itô formula for the process e t ln( m + x ( t )) we have e t ln( m + x ( t )) − ln( m + x ) = Z t e s (cid:26) ln( m + x ( s ))+ x ( s ) m + x ( s ) (cid:20) a ( s ) − b ( s ) x ( s ) − c ( s ) x ( s ) m ( s ) + x ( s ) (cid:21) − σ ( s ) x ( s )2( m + x ( s )) + Z R (cid:20) ln (cid:18) γ ( s, z ) x ( s ) m + x ( s ) (cid:19) − γ ( s, z ) x ( s ) m + x ( s ) (cid:21) Π ( dz ) ds + Z t e s σ ( s ) x ( s ) m + x ( s ) dw ( s ) + t Z Z R e s ln (cid:18) γ ( s, z ) x ( s ) m + x ( s ) (cid:19) ˜ ν ( ds, dz ) + t Z Z R e s ln (cid:18) δ ( s, z ) x ( s ) m + x ( s ) (cid:19) ν ( ds, dz ) . (10)Let us denote for < κ ≤ the process ζ κ ( t ) = Z t e s σ ( s ) x ( s ) m + x ( s ) dw ( s )+ t Z Z R e s ln (cid:18) γ ( s, z ) x ( s ) m + x ( s ) (cid:19) ˜ ν ( ds, dz )+ t Z Z R e s ln (cid:18) δ ( s, z ) x ( s ) m + x ( s ) (cid:19) ν ( ds, dz ) − κ Z t e s σ ( s ) (cid:18) x ( s ) m + x ( s ) (cid:19) ds − κ t Z Z R "(cid:18) γ ( s, z ) x ( s ) m + x ( s ) (cid:19) κe s − − κe s ln (cid:18) γ ( s, z ) x ( s ) m + x ( s ) (cid:19) Π ( dz ) ds − κ t Z Z R "(cid:18) δ ( s, z ) x ( s ) m + x ( s ) (cid:19) κe s − Π ( dz ) ds. By virtue of the exponential inequality ([6], Lemma 2.2) for any
T > , < κ ≤ , β > we have P { sup ≤ t ≤ T ζ κ ( t ) > β } ≤ e − κβ . (11)Choose T = kτ, k ∈ N , τ > , κ = εe − kτ , β = θe kτ ε − ln k , < ε < , θ > weget P { sup ≤ t ≤ T ζ κ ( t ) > θe kτ ε − ln k } ≤ k θ . By Borel-Cantelli lemma for almost all ω ∈ Ω , there is a random integer k ( ω ) ,such that ∀ k ≥ k ( ω ) and ≤ t ≤ kτ we have Z t e s σ ( s ) x ( s ) m + x ( s ) dw ( s )+ t Z Z R e s ln (cid:18) γ ( s, z ) x ( s ) m + x ( s ) (cid:19) ˜ ν ( ds, dz )+ t Z Z R e s ln (cid:18) δ ( s, z ) x ( s ) m + x ( s ) (cid:19) ν ( ds, dz ) ≤ ε e kτ Z t e s (cid:18) σ ( s ) x ( s ) m + x ( s ) (cid:19) ds + e kτ ε t Z Z R "(cid:18) γ ( s, z ) x ( s ) m + x ( s ) (cid:19) εe s − kτ − − εe s − kτ ln (cid:18) γ ( s, z ) x ( s ) m + x ( s ) (cid:19)(cid:21) Π ( dz ) ds + e kτ ε t Z Z R "(cid:18) δ ( s, z ) x ( s ) m + x ( s ) (cid:19) εe s − kτ − Π ( dz ) ds + θe kτ ln kε . (12) By using the inequality x r ≤ r ( x − , ∀ x ≥ , ≤ r ≤ for x =1 + γ ( s,z ) x ( s ) m + x ( s ) , r = εe s − kτ , then for x = 1 + δ ( s,z ) x ( s ) m + x ( s ) , r = εe s − kτ , we derivethe estimates e kτ ε t Z Z R "(cid:18) γ ( s, z ) x ( s ) m + x ( s ) (cid:19) εe s − kτ − − εe s − kτ ln (cid:18) γ ( s, z ) x ( s ) m + x ( s ) (cid:19)(cid:21) Π ( dz ) ds ≤ t Z Z R e s (cid:20) γ ( s, z ) x ( s ) m + x ( s ) − ln (cid:18) γ ( s, z ) x ( s ) m + x ( s ) (cid:19)(cid:21) Π ( dz ) ds, (13) e kτ ε t Z Z R "(cid:18) δ ( s, z ) x ( s ) m + x ( s ) (cid:19) εe s − kτ − Π ( dz ) ds ≤ t Z Z R e s δ ( s, z ) x ( s ) m + x ( s ) Π ( dz ) ds. (14)From (10) , by using (12) – (14) we get e t ln( m + x ( t )) ≤ ln( m + x ) + Z t e s ln( m + x ( s ))+ x ( s ) m + x ( s ) (cid:20) a ( s ) − b ( s ) x ( s ) − c ( s ) x ( s ) m ( s ) + x ( s ) (cid:21) − σ ( s ) x ( s )2( m + x ( s )) × (cid:0) − εe s − kτ (cid:1) + Z R δ ( s, z ) x ( s ) m + x ( s ) Π ( dz ) ds + θe kτ ln kε , a.s. (15)It is easy to see that, under Assumption 1, for any x > there exists a constant L > independent on k, s and x , such that ln( m + x ) − x b ( s ) m + x + xα ( s ) m + x ≤ L. So, from (15) for any ( k − τ ≤ t ≤ kτ we have (a.s.) ln( m + x ( t ))ln t ≤ e − t ln( m + x )ln t + L ln t (1 − e − t ) + θe kτ ln kεe ( k − τ ln( k − τ . Therefore lim sup t →∞ ln( m + x ( t ))ln t ≤ θe τ ε , ∀ θ > , τ > , ε ∈ (0 , , a.s. If θ ↓ , τ ↓ , ε ↑ , then we obtain lim sup t →∞ ln( m + x ( t ))ln t ≤ , a.s.So lim sup t →∞ ln( m + x ( t )) t ≤ , a.s. Corollary 1.
The density of prey population x ( t ) obeys lim sup t →∞ ln x ( t ) t ≤ , a.s. Lemma 2.
The density of predator population x ( t ) has the property that lim sup t →∞ ln x ( t ) t ≤ , a.s. Proof.
Making use of Itô formula we get e t ln x ( t ) − ln x = Z t e s ( ln x ( s ) + a ( s ) − c ( s ) x ( s ) m ( s ) + x ( s ) − σ ( s )2+ Z R (cid:20) ln (cid:16) γ ( s, z ) (cid:17) − γ ( s, z ) (cid:21) Π ( dz ) ds + ψ ( t ) , (16)where ψ ( t ) = Z t e s σ ( s ) dw ( s ) + t Z Z R e s ln (cid:16) γ ( s, z ) (cid:17) ˜ ν ( ds, dz )+ t Z Z R e s ln (cid:16) δ ( s, z ) (cid:17) ν ( ds, dz ) . By virtue of the exponential inequality (11) we have P { sup ≤ t ≤ T ζ κ ( t ) > β } ≤ e − κβ , ∀ < κ ≤ , β > , where ζ κ ( t ) = ψ ( t ) − κ t Z e s σ ( s ) ds − κ t Z Z R h (1 + γ ( s, z )) κe s − − κe s ln(1 + γ ( s, z ))] Π ( dz ) ds − κ t Z Z R h (1 + δ ( s, z )) κe s − i Π ( dz ) ds. Choose T = kτ, k ∈ N , τ > , κ = e − kτ , β = θe kτ ln k , θ > we get P { sup ≤ t ≤ T ζ κ ( t ) > θe kτ ln k } ≤ k θ . By the same arguments as in the proof of Lemma 1, using Borel-Cantellilemma, we derive from (16) e t ln x ( t ) ≤ ln x + Z t e s ( ln x ( s ) + a ( s ) − c ( s ) x ( s ) m ( s ) + x ( s ) − σ ( s )2 (cid:0) − e s − kτ (cid:1) + Z R δ ( s, z )Π ( dz ) ds + θe kτ ln k, a.s. (17)for all sufficiently large k ≥ k ( ω ) and ≤ t ≤ kτ .Using inequality ln x − cx ≤ − ln c − , ∀ x ≥ , c > for x = x ( s ) , c = c ( s ) m ( s )+ x ( s ) , we derive from (17) the estimate e t ln x ( t ) ≤ ln x + t Z e s ln (cid:16) m sup + x ( s ) (cid:17) ds + L ( e t −
1) + θe kτ ln k, for some constant L > .So for ( k − τ ≤ t ≤ kτ , k ≥ k ( ω ) we have lim sup t →∞ ln x ( t ) t ≤ lim sup t →∞ t t Z e s − t ln (cid:16) m sup + x ( s ) (cid:17) ds ≤ , by virtue of Lemma 1. Lemma 3.
Let p > . Then for any initial value x > , the p th-moment ofprey population density x ( t ) obeys lim sup t →∞ E [ x p ( t )] ≤ K ( p ) , (18)where K ( p ) > is independent of x .For any initial value x > , the expectation of predator population density x ( t ) obeys lim sup t →∞ E [ x ( t )] ≤ K , (19)where K > is independent of x . Proof.
Let τ n be the stopping time defined in Theorem 1. Applying the Itôformula to the process V ( t, x ( t )) = e t x p ( t ) , p > we obtain V ( t ∧ τ n , x ( t ∧ τ n )) = x p + t ∧ τ n Z e s x p ( s ) ( p " a ( s ) − b ( s ) x ( s ) − c ( s ) x ( s ) m ( s ) + x ( s ) (cid:21) + p ( p − σ ( s )2 + Z R [(1 + γ ( s, z )) p − − pγ ( s, z )] Π ( dz )+ Z R [(1 + δ ( s, z )) p −
1] Π ( dz ) ds + t ∧ τ n Z pe s x p ( s ) σ ( s ) dw ( s )+ t ∧ τ n Z Z R e s x p ( s ) [(1 + γ ( s, z )) p −
1] ˜ ν ( ds, dz )+ t ∧ τ n Z Z R e s x p ( s ) [(1 + δ ( s, z )) p −
1] ˜ ν ( ds, dz ) . (20)Under Assumption 1 there is constant K ( p ) > , such that e s x p ( p (cid:20) a ( s ) − b ( s ) x − c ( s ) x m ( s ) + x (cid:21) + p ( p − σ ( s )2 ++ Z R [(1+ γ ( s, z )) p − − pγ ( s, z )] Π ( dz )+ Z R [(1 + δ ( s, z )) p −
1] Π ( dz ) ≤ K ( p ) e s . (21)From (20) and (21) , taking expectations, we obtain E[ V ( t ∧ τ n , x ( t ∧ τ n ))] ≤ x p + K ( p ) e t . Letting n → ∞ leads to the estimate e t E[ x p ( t )] ≤ x p + e t K ( p ) . (22)So from (22) we derive (18) .Let us prove the estimate (19) . Applying the Itô formula to the process U ( t, X ( t )) = e t [ k x ( t ) + k x ( t )] , k i > , i = 1 , we obtain dU ( t, X ( t )) = e t ( k x ( t ) + k x ( t ) + k " a ( t ) x ( t ) − b ( t ) x ( t ) − c ( t ) x ( t ) x ( t ) m ( t ) + x ( t ) (cid:21) + k " a ( t ) x ( t ) − c ( t ) x ( t ) m ( t ) + x ( t ) + X i =1 k i Z R x i ( t ) δ i ( t, z )Π ( dz ) dt + e t ( X i =1 k i " x i ( t ) σ i ( t ) dw i ( t )+ Z R x i ( t ) γ i ( t, z )˜ ν ( dt, dz ) + Z R x i ( t ) δ i ( t, z )˜ ν ( dt, dz ) . (23) For the function f ( t, x , x ) = 1 m ( t ) + x (cid:26) k h − b ( t ) x + (cid:16) a ( t )+ ¯ δ ( t ) − b ( t ) m ( t ) (cid:17) x + m ( t ) (cid:16) a ( t )+ ¯ δ ( t ) (cid:17) x i + h k (cid:16) a ( t ) + ¯ δ ( t ) (cid:17) − k c ( t ) i x x + k h − c ( t ) x + m ( t ) (cid:16) a ( t ) + ¯ δ ( t ) (cid:17) x i (cid:27) , where ¯ δ i ( t ) = Z R δ i ( t, z )Π ( dz ) , i = 1 , we have f ( t, x , x ) ≤ φ ( x , x ) + φ ( x ) m ( t ) + x , where φ ( x , x ) = k h − b x + (cid:16) d − b m inf (cid:17) x + m sup d x i + h k d − k c i x x ,φ ( x ) = k (cid:2) − c x + m sup d x (cid:3) , d i = 1+ a i sup + | ¯ δ i | sup , i = 1 , . For k = k c /d there is a constant L ′ > , such that φ ( x , x ) ≤ L ′ k and φ ( x ) ≤ L ′ k . So there is a constant L > , such that f ( t, x , x ) ≤ Lk . (24)From (23) and (24) by integrating and taking expectation, we derive E[ U ( t ∧ τ n , X ( t ∧ τ n ))] ≤ k (cid:20) x + c d x + Le t (cid:21) . Letting n → ∞ leads to the estimate e t E (cid:20) x ( t ) + c d x ( t ) (cid:21) ≤ x + c d x + Le t . So E[ x ( t )] ≤ (cid:18) d c x + x (cid:19) e − t + d c L. (25)From (25) we have (19) . Lemma 4. If p > , where p ( t ) = a ( t ) − β ( t ) , then for any initial value x > , the predator population density x ( t ) satisfies lim sup t →∞ E "(cid:18) x ( t ) (cid:19) θ ≤ K ( θ ) , < θ < , (26) Proof.
For the process U ( t ) = 1 /x ( t ) by the Itô formula we derive U ( t ) = U (0) + t Z U ( s ) c ( s ) x ( s ) m ( s ) + x ( s ) − a ( s ) + σ ( s )+ Z R γ ( s, z )1 + γ ( s, z ) Π ( dz ) ds − t Z U ( s ) σ ( s ) dw ( s ) − t Z Z R U ( s ) γ ( s, z )1 + γ ( s, z ) ˜ ν ( ds, dz ) − t Z Z R U ( s ) δ ( s, z )1 + δ ( s, z ) ν ( ds, dz ) . Then, by applying Itô formula, we derive, for < θ < U ( t )) θ = (1 + U (0)) θ + t Z θ (1 + U ( s )) θ − (1 + U ( s )) U ( s ) × c ( s ) x ( s ) m ( s ) + x ( s ) − a ( s ) + σ ( s )+ Z R γ ( s, z )1 + γ ( s, z ) Π ( dz ) + θ − U ( s ) σ ( s )+ 1 θ Z R " (1 + U ( s )) (cid:18) U ( s ) + γ ( s, z )(1 + γ ( s, z ))(1 + U ( s )) (cid:19) θ − ! + θ (1 + U ( s )) U ( s ) γ ( s, z )1 + γ ( s, z ) (cid:21) Π ( dz )+ 1 θ Z R (1 + U ( s )) "(cid:18) U ( s ) + δ ( s, z )(1 + δ ( s, z ))(1 + U ( s )) (cid:19) θ − Π ( dz ) ds − t Z θ (1 + U ( s )) θ − U ( s ) σ ( s ) dw ( s )+ t Z Z R "(cid:18) U ( s )1 + γ ( s, z ) (cid:19) θ − (1 + U ( s )) θ ˜ ν ( ds, dz )+ t Z Z R "(cid:18) U ( s )1 + δ ( s, z ) (cid:19) θ − (1 + U ( s )) θ ˜ ν ( ds, dz ) = (1 + U (0)) θ + t Z θ (1 + U ( s )) θ − J ( s ) ds − I ,stoch ( t ) + I ,stoch ( t ) + I ,stoch ( t ) , (27)where I j,stoch ( t ) , j = 1 , are the corresponding stochastic integrals in (27) .Under the Assumption 1 there exists constants | K ( θ ) | < ∞ , | K ( θ ) | < ∞ such, that for the process J ( t ) we have the estimate J ( t ) ≤ (1 + U ( t )) U ( t ) − a ( t ) + c U − ( t ) m inf + σ ( t )+ Z R γ ( s, z )1 + γ ( s, z ) Π ( dz ) + θ − U ( s ) σ ( s )+ 1 θ Z R " (1 + U ( s )) (cid:18)
11 + γ ( s, z ) + 11 + U ( s ) (cid:19) θ − ! + θ (1 + U ( s )) U ( s ) γ ( s, z )1 + γ ( s, z ) (cid:21) Π ( dz )+ 1 θ Z R (1 + U ( s )) "(cid:18)
11 + δ ( s, z ) + 11 + U ( s ) (cid:19) θ − Π ( dz ) ≤ U ( t ) − a ( t ) + σ ( t )2 + Z R γ ( t, z )Π ( dz ) + θ σ ( t )+ 1 θ Z R [(1 + γ ( t, z )) − θ − ( dz ) + 1 θ Z R [(1 + δ ( t, z )) − θ − ( dz ) + K ( θ ) U ( t ) + K ( θ ) = − K ( t, θ ) U ( t ) + K ( θ ) U ( t ) + K ( θ ) , where we used the inequality ( x + y ) θ ≤ x θ + θx θ − y , < θ < , x, y > . Dueto lim θ → θ σ ( t ) + 1 θ Z R [(1 + γ ( t, z )) − θ − ( dz )+ 1 θ Z R [(1 + δ ( t, z )) − θ − ( dz ) + Z R ln(1 + γ ( t, z ))Π ( dz )+ Z R ln(1 + δ ( t, z ))Π ( dz ) = lim θ → ∆( θ ) = 0 , and condition p > we can choose a sufficiently small < θ < to satisfy K ( θ ) = inf t ≥ K ( t, θ ) = inf t ≥ [ p ( t ) − ∆( θ )] = p − ∆( θ ) > . So from (27) and the estimate for J ( t ) we derive d (cid:2) (1 + U ( t )) θ (cid:3) ≤ θ (1 + U ( t )) θ − [ − K ( θ ) U ( t ) + K ( θ ) U ( t ) + K ( θ )] dt − θ (1 + U ( t )) θ − U ( t ) σ ( t ) dw ( t ) + Z R "(cid:18) U ( t )1 + γ ( t, z ) (cid:19) θ − (1+ U ( t )) θ ˜ ν ( dt, dz )+ Z R "(cid:18) U ( t )1 + δ ( t, z ) (cid:19) θ − (1+ U ( t )) θ ˜ ν ( dt, dz ) . (28)By the Itô formula and (28) we have d (cid:2) e λt (1 + U ( t )) θ (cid:3) = λe λt (1 + U ( t )) θ dt + e λt d (cid:2) (1 + U ( t )) θ (cid:3) ≤ e λt θ (1 + U ( t )) θ − (cid:20) − (cid:18) K ( θ ) − λθ (cid:19) U ( t ) + (cid:18) K ( θ ) + 2 λθ (cid:19) U ( t )+ K ( θ ) + λθ (cid:21) dt − θe λt (1 + U ( t )) θ − U ( t ) σ ( t ) dw ( t )+ e λt Z R "(cid:18) U ( t )1 + γ ( t, z ) (cid:19) θ − (1 + U ( t )) θ ˜ ν ( dt, dz )+ e λt Z R "(cid:18) U ( t )1 + δ ( t, z ) (cid:19) θ − (1 + U ( t )) θ ˜ ν ( dt, dz ) . (29)Let us choose λ = λ ( θ ) > such that K ( θ ) − λ/θ > . Then there is aconstant K > , such that (1 + U ( t )) θ − (cid:20) − (cid:18) K ( θ ) − λθ (cid:19) U ( t )+ (cid:18) K ( θ ) + 2 λθ (cid:19) U ( t ) + K ( θ ) + λθ (cid:21) ≤ K. (30)Let τ n be the stopping time defined in Theorem 1. Then by integrating (29) ,using (30) and taking the expectation we obtain E h e λ ( t ∧ τ n ) (1 + U ( t ∧ τ n )) θ i ≤ (cid:18) x (cid:19) θ + θλ K (cid:0) e λt − (cid:1) . Letting n → ∞ leads to the estimate e t E (cid:2) (1 + U ( t )) θ (cid:3) ≤ (cid:18) x (cid:19) θ + θλ K (cid:0) e λt − (cid:1) . (31) From (31) we obtain lim sup t →∞ E "(cid:18) x ( t ) (cid:19) θ = lim sup t →∞ E (cid:2) U θ ( t ) (cid:3) ≤ lim sup t →∞ E (cid:2) (1 + U ( t )) θ (cid:3) ≤ θλ ( θ ) K, this implies (26) . ([9]) The solution X ( t ) to the system (3) is said to be stochas-tically ultimately bounded, if for any ε ∈ (0 , , there is a positive constant χ = χ ( ε ) > , such that for any initial value X ∈ R , the solution to thesystem (3) has the property that lim sup t →∞ P {| X ( t ) | > χ } < ε. In what follows in this section we will assume that Assumption 1 holds.
Theorem 2.
The solution X ( t ) to the system (3) with initial value X ∈ R is stochastically ultimately bounded. Proof.
From the Lemma 3 we have the estimate lim sup t →∞ E [ x i ( t )] ≤ K i , i = 1 , . (32)For X = ( x , x ) ∈ R we have | X | ≤ x + x , therefore, from (32)lim sup t →∞ E [ | X ( t ) | ] ≤ L = K + K . Let χ > L/ε , ∀ ε ∈ (0 , . Then ap-plying the Chebyshev inequality yields lim sup t →∞ P {| X ( t ) | > χ } ≤ χ lim sup t →∞ E [ | X ( t ) | ] ≤ Lχ < ε.
The property of stochastic permanence is important since it means thelong-time survival in a population dynamics.
Definition 2.
The population density x ( t ) is said to be stochastically perma-nent if for any ε > , there are positive constants H = H ( ε ) , h = h ( ε ) suchthat lim inf t →∞ P { x ( t ) ≤ H } ≥ − ε, lim inf t →∞ P { x ( t ) ≥ h } ≥ − ε, for any inial value x > . Theorem 3. If p > , where p ( t ) = a ( t ) − β ( t ) , then for any initial value x > , the predator population density x ( t ) is stochastically permanent. Proof.
From Lemma 3 we have estimate lim sup t →∞ E [ x ( t )] ≤ K. Thus for any given ε > , let H = K/ε , by virtue of Chebyshev’s inequality,we can derive that lim sup t →∞ P { x ( t ) ≥ H } ≤ H lim sup t →∞ E [ x ( t )] ≤ ε. Consequently lim inf t →∞ P { x ( t ) ≤ H } ≥ − ε .From Lemma 4 we have estimate lim sup t →∞ E "(cid:18) x ( t ) (cid:19) θ ≤ K ( θ ) , < θ < . For any given ε > , let h = ( ε/K ( θ )) /θ , then by Chebyshev’s inequality, wehave lim sup t →∞ P { x ( t ) < h } ≤ lim sup t →∞ P ((cid:18) x ( t ) (cid:19) θ > h − θ ) ≤ h θ lim sup t →∞ E "(cid:18) x ( t ) (cid:19) θ ≤ ε. Consequently lim inf t →∞ P { x ( t ) ≥ h } ≥ − ε . Theorem 4.
If the predator is absent, i.e. x ( t ) = 0 a.s., and p > , where p ( t ) = a ( t ) − β ( t ) , then for any initial value x > , the prey populationdensity x ( t ) is stochastically permanent. Proof.
From Lemma 3 we have estimate lim sup t →∞ E [ x ( t )] ≤ K. Thus for any given ε > , let H = K/ε , by virtue of Chebyshev’s inequality,we can derive that lim sup t →∞ P { x ( t ) ≥ H } ≤ H lim sup t →∞ E [ x ( t )] ≤ ε. Consequently lim inf t →∞ P { x ( t ) ≤ H } ≥ − ε . For the process U ( t ) = 1 /x ( t ) by the Itô formula we have U ( t ) = U (0) + t Z U ( s ) b ( s ) x ( s ) − a ( s ) + σ ( s )+ Z R γ ( s, z )1 + γ ( s, z ) Π ( dz ) ds − t Z U ( s ) σ ( s ) dw ( s ) − t Z Z R U ( s ) γ ( s, z )1 + γ ( s, z ) ˜ ν ( ds, dz ) − t Z Z R U ( s ) δ ( s, z )1 + δ ( s, z ) ν ( ds, dz ) . Then, using the same arguments as in the proof of Lemma 4 we can derive theestimate lim sup t →∞ E "(cid:18) x ( t ) (cid:19) θ ≤ K ( θ ) , < θ < , For any given ε > , let h = ( ε/K ( θ )) /θ , then by Chebyshev’s inequality, wehave lim sup t →∞ P { x ( t ) < h } = lim sup t →∞ P ((cid:18) x ( t ) (cid:19) θ > h − θ ) ≤ h θ lim sup t →∞ E "(cid:18) x ( t ) (cid:19) θ ≤ ε. Consequently lim inf t →∞ P { x ( t ) ≥ h } ≥ − ε . Remark 1.
If the predator is absent, i.e. x ( t ) = 0 a.s., then the equationfor the prey x ( t ) has the logistic form. So Theorem 4 gives us the suffi-cient conditions for the stochastic permanence of the solution to the stochas-tic non-autonomous logistic equation disturbed by white noise, centered andnon-centered Poisson noises. Definition 3.
The solution X ( t ) = ( x ( t ) , x ( t )) , t ≥ to the equation (3) will be said extinct if for every initial data X ∈ R , we have lim t →∞ x i ( t ) = 0 almost surely (a.s.), i = 1 , . Theorem 5. If ¯ p ∗ i = lim sup t →∞ t t Z p i ( s ) ds < , where p i ( t ) = a i ( t ) − β i ( t ) , i = 1 , , then the solution X ( t ) to the equation (3) with initial condition X ∈ R willbe extinct. Proof.
By the Itô formula, we have d ln x i ( t ) = (cid:20) a i ( t ) − b i ( t ) x i ( t ) − c i ( t ) x ( t ) m ( t ) + x ( t ) − β i ( t ) (cid:21) dt + dM i ( t ) ≤ [ a i ( t ) − β i ( t )] dt + dM i ( t ) , i = 1 , , (33)where the martingale M i ( t ) = t Z σ i ( s ) dw i ( s ) + t Z Z R ln(1 + γ i ( s, z ))˜ ν ( ds, dz )+ t Z Z R ln(1 + δ i ( s, z ))˜ ν ( ds, dz ) , i = 1 , , (34)has quadratic variation h M i , M i i ( t ) = t Z σ i ( s ) ds + t Z Z R ln (1 + γ i ( s, z ))Π ( dz ) ds + t Z Z R ln (1 + δ i ( s, z ))Π ( dz ) ds ≤ Kt, i = 1 , . Then the strong law of large numbers for local martingales ([10]) yields lim t →∞ M i ( t ) /t = 0 , i = 1 , a.s. Therefore, from (33) we obtain lim sup t →∞ ln x i ( t ) t ≤ lim sup t →∞ t t Z p i ( s ) ds < , a.s.So lim t →∞ x i ( t ) = 0 , i = 1 , a.s. Definition 4 ([11]) . The population density x ( t ) will be said non-persistentin the mean if lim t →∞ t Z t x ( s ) ds = 0 a.s. Theorem 6. If ¯ p ∗ = 0 , then the prey population density x ( t ) with initialcondition x > will be non-persistent in the mean. Proof.
From the first equality in (33) we have for i = 1ln x ( t ) ≤ ln x + t Z p ( s ) ds − b t Z x ( s ) ds + M ( t ) , (35) where martingale M ( t ) is defined in (34) . From the definition of ¯ p ∗ and thestrong law of large numbers for M ( t ) it follows, that ∀ ε > , ∃ t ≥ , and ∃ Ω ε ⊂ Ω , with P(Ω ε ) ≥ − ε , such that t t Z p ( s ) ds ≤ ¯ p ∗ + ε , M ( t ) t ≤ ε , ∀ t ≥ t , ω ∈ Ω ε . So, from (35) we derive ln x ( t ) − ln x ≤ t (¯ p ∗ + ε ) − b t Z x ( s ) ds = tε − b t Z x ( s ) ds, ∀ t ≥ t , ω ∈ Ω ε . (36)Let y ( t ) = R t x ( s ) ds , then from (36) we have ln (cid:18) dy ( t ) dt (cid:19) ≤ εt − b y ( t ) + ln x ⇒ e b y ( t ) dy ( t ) dt ≤ x e εt , ∀ t ≥ t , ω ∈ Ω ε . By integrating of last inequality from t to t we obtain e b y ( t ) ≤ b x ε (cid:0) e εt − e εt (cid:1) + e b y ( t ) , ∀ t ≥ t , ω ∈ Ω ε . So y ( t ) ≤ b ln (cid:20) e b y ( t ) + b x ε (cid:0) e εt − e εt (cid:1)(cid:21) , ∀ t ≥ t , ω ∈ Ω ε , and therefore lim sup t →∞ t t Z x ( s ) ds ≤ εb , ∀ ω ∈ Ω ε . Since ε > is arbitrary and x ( t ) > a.s., we have lim t →∞ t t Z x ( s ) ds = 0 a.s. Theorem 7. If ¯ p ∗ = 0 and ¯ p ∗ < , then the predator population density x ( t ) with initial condition x > will be non-persistent in the mean. Proof.
From the first equality in (33) with i = 2 we have for c = c /m sup ln x ( t ) ≤ ln x + t Z p ( s ) ds − c t Z x ( s ) m ( s ) + x ( s ) ds + M ( t )= ln x + t Z p ( s ) ds − c t Z m ( s ) (cid:20) x ( s ) − x ( s ) x ( s ) m ( s ) + x ( s ) (cid:21) ds + M ( t ) ≤ ln x + t Z p ( s ) ds − c t Z x ( s ) ds + c t Z x ( s ) x ( s ) m sup + x ( s ) ds + M ( t ) , (37)where martingale M ( t ) is defined in (34) . From Theorem 5, the definitionof ¯ p ∗ and the strong law of large numbers for M ( t ) it follows, that ∀ ε > , ∃ t ≥ , and ∃ Ω ε ⊂ Ω with P(Ω ε ) ≥ − ε , such that t t Z p ( s ) ds ≤ ¯ p ∗ + ε , M ( t ) t ≤ ε , x ( t ) m sup + x ( t ) ≤ ε, ∀ t ≥ t , ω ∈ Ω ε . So, from (37) we derive ln x ( t ) − ln x ≤ t (¯ p ∗ + ε ) − c (1 − ε ) t Z t x ( s ) ds = tε − c (1 − ε ) t Z t x ( s ) ds, ∀ t ≥ t , ω ∈ Ω ε . (38)Let y ( t ) = R tt x ( s ) ds , then from (38) we have ln (cid:18) dy ( t ) dt (cid:19) ≤ εt − c (1 − ε ) y ( t ) + ln x ⇒ e c (1 − ε ) y ( t ) dy ( t ) dt ≤ x e εt , ∀ t ≥ t , ω ∈ Ω ε . By integrating of last inequality from t to t we obtain e c (1 − ε ) y ( t ) ≤ c (1 − ε ) x ε (cid:0) e εt − e εt (cid:1) + 1 , ∀ t ≥ t , ω ∈ Ω ε . So y ( t ) ≤ c (1 − ε ) ln (cid:20) c (1 − ε ) x ε (cid:0) e εt − e εt (cid:1)(cid:21) , ∀ t ≥ t , ω ∈ Ω ε , and therefore lim sup t →∞ t t Z x ( s ) ds ≤ εc (1 − ε ) , ∀ ω ∈ Ω ε . Since ε > is arbitrary and x ( t ) > a.s., we have lim t →∞ t t Z x ( s ) ds = 0 a.s. Definition 5 ([11]) . The population density x ( t ) will be said weakly persistentin the mean if ¯ x ∗ = lim sup t →∞ t Z t x ( s ) ds > a.s. Theorem 8. If ¯ p ∗ > , then the predator population density x ( t ) with initialcondition x > will be weakly persistent in the mean. Proof.
If the assertion of theorem is not true, then P { ¯ x ∗ = 0 } > . From thefirst equality in (33) we get t (ln x ( t ) − ln x ) = 1 t Z t p ( s ) ds − t Z t c ( s ) x ( s ) m ( s ) + x ( s ) ds + M ( t ) t ≥ t Z t p ( s ) ds − c m inf t Z t x ( s ) ds + M ( t ) t , where martingale M ( t ) is defined in (34) . For ∀ ω ∈ { ω ∈ Ω | ¯ x ∗ = 0 } in virtuestrong law of large numbers for martingale M ( t ) we have lim sup t →∞ ln x ( t ) t ≥ ¯ p ∗ > . Therefore P (cid:26) ω ∈ Ω | lim sup t →∞ ln x ( t ) t > (cid:27) > . But from Lemma 2 we have P (cid:26) ω ∈ Ω | lim sup t →∞ ln x ( t ) t ≤ (cid:27) = 1 . This is a contradiction.
Theorem 9. If ¯ p ∗ > and ¯ p ∗ < , then the prey population density x ( t ) with initial condition x > will be weakly persistent in the mean. Proof.
Let P { ¯ x ∗ = 0 } > . From the first equality in (33) with i = 1 we get t (ln x ( t ) − ln x ) = 1 t Z t p ( s ) ds − t Z t b ( s ) x ( s ) ds − t Z t c ( s ) x ( s ) m ( s ) + x ( s ) ds + M ( t ) t ≥ t Z t p ( s ) ds − b t Z t x ( s ) ds − c m inf t Z t x ( s ) ds + M ( t ) t (39) where martingale M ( t ) is defined in (34) . From definition of ¯ p ∗ , strong law oflarge numbers for martingale M ( t ) and Theorem 2 for x ( t ) we have ∀ ε > , ∃ t ≥ , ∃ Ω ε ⊂ Ω with P(Ω ε ) ≥ − ε , such that t t Z p ( s ) ds ≥ ¯ p ∗ − ε , M ( t ) t ≥ − ε , t Z t x ( s ) ds ≤ εm inf c , ∀ t ≥ t , ω ∈ Ω ε . So, from (39) we get for ω ∈ { ω ∈ Ω | ¯ x ∗ = 0 } ∩ Ω ε lim sup t →∞ ln x ( t ) t ≥ ¯ p ∗ − ε > for sufficiently small ε > . Therefore P (cid:26) ω ∈ Ω | lim sup t →∞ ln x ( t ) t > (cid:27) > . But from Corollary 1 P (cid:26) ω ∈ Ω | lim sup t →∞ ln x ( t ) t ≤ (cid:27) = 1 . Therefore we have a contradiction.
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