DSpace at VNU: Asymptotic Behavior of Predator-Prey Systems Perturbed by White Noise

Acta Appl Math (2011) 115:351–370 DOI 10.1007/s10440-011-9628-4 Asymptotic Behavior of Predator-Prey Systems Perturbed by White Noise Nguyen Hai Dang · Nguyen Huu Du · Ta Viet Ton Received: November 2010 / Accepted: July 2011 / Published online: 21 July 2011 © Springer Science+Business Media B.V 2011 Abstract In this paper, we develop the results in Rudnicki (Stoch Process Appl 108:93– 107, 2003) to a stochastic predator-prey system where the random factor acts on the coefficients of environment We show that there exists the density functions of the solutions and then, study the asymptotic behavior of these densities It is proved that the densities either converges in L1 to an invariant density or converges weakly to a singular measure on the boundary Keywords Prey-predator model · Diffusion process · Markov semigroups · Asymptotic stability Mathematics Subject Classification (2000) 34C12 · 47D07 · 60H10 · 92D25 Introduction The stable predator-prey Lotka-Volterra equation X˙ t = (αXt − βXt Yt − μXt2 ), Y˙t = (−γ Yt + δXt Yt − νYt2 ), (1.1) where Xt and Yt represent, respectively, the quantities of prey and the predator populations; α, β, γ , δ, μ and ν are positive constants, has attracted a lot of attention It has been proved that the solution of (1.1) is asymptotically stable For the stochastic predator-prey Lotka-Volterra equation, we have to mention one of the first attempts in this direction, the very interesting paper of Arnold et al [1] where the N.H Dang · N.H Du ( ) Faculty of Mathematics, Mechanics and Informatics, Hanoi University of Science, Hanoi, Vietnam e-mail: dunh@vnu.edu.vn T.V Ton Department of Applied Physics, Graduate School of Engineering, Osaka University, Osaka, Japan 352 N.H Dang et al authors used the theory of Brownian motion and the related white noise models to study the sample paths of the equation dXt = (αXt − βXt Yt − μXt2 )dt + σ Xt dWt , dYt = (−γ Yt + δXt Yt − νYt2 )dt + ρYt dWt , (1.2) where Wt is the one dimensional Brownian motion defined on the complete probability space ( , F , {Ft }t≥0 , P) with the filtration {Ft }t≥0 satisfying the usual conditions, i.e., it is increasing and right continuous while F0 contains all P−null sets The positive numbers σ, ρ are the coefficients of the effect of environmental stochastic perturbation on the prey and on the predator population respectively In this model, the random factor makes influences on the intrinsic growth rates of prey and predator In continuing this study, in [10–12], the authors showed that the distribution of each solution starting at a point in intR2+ has the density which either converges in L1 to an invariant density or converges weakly to a singular measure on the boundary (0, ∞) × {0} This paper studies a stochastic predator-prey system where the intrinsic growth rate of the prey, μ, and the one of the predator, ν, are perturbed stochastically μ → μ + σ W˙ t and ν → ν + ρ W˙ t This means that we consider the following stochastic equation dXt = (αXt − βXt Yt − μXt2 )dt + σ Xt2 dWt , dYt = (−γ Yt + δXt Yt − νYt2 )dt + ρYt2 dWt , (1.3) where σ and ρ are positive constants The existence and uniqueness of the positive solution of (1.3) has been considered by X Mao et al [7] Moreover, the estimates of upper growth and lower growth of the sample paths of its solution were also given in [7] and [2] The aim of this paper is to study further the asymptotic behavior of the system (1.3) by considering the convergence of the density of the solution Meanwhile the models (1.2) and (1.3) are rather similar but the calculation is much more complicated and the dynamic behavior of two these systems are different Moreover, it seems that the Khasminskii function method dealt with in [11] is not suitable to this model (see the comment in Sect 4) Therefore, in order to study the existence of a stationary distribution and show its attractivity, we have to use the method of analyzing the boundary distributions The organization of the paper is as follows In Sect we get an outline of the singular cases where either prey or predator is absent Section deals with asymptotic properties of Markov operators related to the dynamics of the solutions The theorem concerning with asymptotic stability and sweeping allows us to formulate the Foguel alternative This alternative says that under suitable conditions a Markov semigroup is either asymptotically stable or sweeping The last section gives some comments on the construction of Khasminskii functions Singular Cases By putting Xt = eξt ; Yt = eηt and substituting this transformation into (1.3) we obtain dξt = (α − βeηt − μeξt − σ 2ξt e )dt dηt = (−γ + δeξt − νeηt − + σ eξt dWt , ρ 2ηt e )dt + ρeηt dWt (2.1) Asymptotic Behavior of Predator-Prey Systems Perturbed by White 353 Denoting f1 (x, y) = α − μex − βey − σ 2x e , f2 (x, y) = −γ + δex − νey − ρ 2y e , we can rewrite (2.1) under the form dξt = f1 (ξ1 , ηt )dt + σ eξt dWt , dηt = f2 (ξ1 , ηt )dt + ρeηt dWt , or, under the Stratonovich equation dξt = (α − βeηt − μeξt − σ e2ξt )dt + σ eξt ◦ dWt , dηt = (−γ + δeξt − νeηt − ρ e2ηt )dt + ρeηt ◦ dWt (2.2) The infinitesimal operator of (2.1) is ∂ 2v ∂ 2v ∂ 2v ∂v ∂v + f1 + f2 · Lv = σ e2x + ρ e2y + σρex+y ∂x ∂y ∂x∂y ∂x ∂y The density of the random variable (ξt , ηt ), if it exists and is smooth, can be found from the Fokker-Planck equation: ∂ (ex+y v) ∂(f1 v) ∂(f2 v) ∂u ∂ (e2x v) ∂ (e2y v) = σ − − · + ρ + σρ 2 ∂t ∂x ∂y ∂x∂y ∂x ∂y (2.3) The behavior of the solutions for two boundary equations is easy to study If the prey is absent, the quantity Yt = eηt of the predator at the time t satisfies the equation dηt = −γ − νeηt − ρ 2ηt e dt + ρeηt dWt (2.4) Let x s1 (x) = y exp − 0 x = exp 2(−γ − νeu − ρ e2u ρ 2u e ) du dy 2ν + γ γ 2ν − 2y − y + y dy ρ2 ρ e ρ e We see that limx→∞ s1 (x) = ∞ and limx→−∞ s2 (x) > −∞ By [3, Chap 5, Theorem 3.1], we get lim ηt = −∞ or equivalently t→∞ lim Yt = a.s t→∞ This means that without the preys, the predators die with probability one Similarly, in the absence of the predators, the quantity Xt = eξt of prey at the time t satisfies the equation dξt = α − μeξt − σ 2ξt e dt + σ eξt dWt (2.5) 354 N.H Dang et al Let x s2 (x) = y exp − 0 x = exp 2(α − μeu − σ e2u σ 2u e ) du dy 2μ − α α 2μ + 2y − y + y dy σ σ e σ e It is easy to see that limx→∞ s1 (x) = ∞ and limx→−∞ s2 (x) = −∞ Then, by [3, Theorem 3.1] we also get lim supt→∞ ξt = ∞; lim inft→∞ ξt = −∞ or equivalently lim sup Xt = ∞, t→∞ lim inf Xt = a.s (2.6) t→∞ This means that, without the predators, the quantity of the preys oscillates between and ∞ Further, (2.5) has a unique stationary distribution with the density f∗ (x) satisfying the Fokker-Planck equation d σ 2x d2 e )f∗ (x) = σ e2x f∗ (x) − (α − μex − 2 dx dx (2.7) The general solution of (2.7) is y(x) = exp −3x + +d 2μ −x α e − e−2x σ σ exp x − c α 2μ −x e + e−2x dx , σ2 σ where c, d are two constants The conditions y(x) ≥ and ∞ −∞ y(x) dx = imply d = and = c ∞ −∞ ∞ = exp − 3x + y exp 2μ −x α e − e−2x dx σ2 σ 2μ α y − y dy σ2 σ It is easy to see that √ ∞ μ2 2μ σ = exp exp − σ u + √ √ √ c ασ 2α 2α − √2μ α σ α √ √ − 2μ πσ (2μ2 + ασ ) μσ + 1− exp = √ √ 2α 2α α σ α where u2 du μ2 , ασ (x) is the distribution function of a standard normal random variable N (0, 1) Thus, f∗ (x) = c exp −3x + 2μ −x α e − e−2x σ σ Asymptotic Behavior of Predator-Prey Systems Perturbed by White 355 If ξt is a solution of (2.5) then by ergodic theorem ∞ t t→∞ t g(ξs ) ds = lim −∞ g(x)f∗ (x) dx (2.8) a.s., for any g that is f∗ -integrable Moreover, ξt converges in distribution to f∗ as t → ∞ (see [13, Theorems 16 and 17].) Let m= +∞ −∞ ∞ ex f∗ (x)dx = c y exp 2μ α y − y dy σ2 σ (2.9) It is obvious that √ √ μ2 πcμσ 2μ cσ + 1− − √ exp m= √ 2α ασ α α σ α √ √ √ μ2 α ασ + παμσ − (− σ √2μα ) exp ασ √ =√ √ − √2μ 2 αμσ + πσ (2μ + ασ ) − exp σ α μ2 ασ Lemma 2.1 Denote by ξ¯t the solution of ¯ d ξ¯t = α − μeξt − σ 2ξ¯t ¯ e dt + σ eξt dWt with the initial condition ξ¯0 = ξ0 and denote by η¯t the solution of d η¯ t = −γ − νeη¯ t − ρ 2η¯t e dt + ρeη¯ t dWt with the initial condition η¯0 = η0 Then with probability 1, ξ¯t ≥ ξt and η¯t ≤ ηt for all t ≥ Proof Put Zt = e−ξt , Z¯t = e−ξt By Itô formula, ¯ dZt = (μ − αZt + βZt eηt + σ Zt−1 )dt − σ dWt , −1 d Z¯t = (μ − α Z¯t + σ Z¯t )dt − σ dWt Hence, by the comparison theorem (see Theorem 1.1: p 352 in [3]), Zt ≥ Z¯t for all t ≥ a.s It means that P{ξ¯t ≥ ξt } = for all t ≥ The second assertion η¯t ≤ ηt a.s for all t ≥ can be proved by a similar way Theorem 2.2 The following assertions are true: (a) lim sup t→∞ t (b) lim inf t→∞ t (c) lim inf t→∞ t t δeξs − νeηs − ρ 2ηs e ds ≤ γ a.s δeξs − νeηs − ρ 2ηs e ds ≥ a.s βeηs + μeξs + σ 2ξs e ds ≥ α a.s t t 356 N.H Dang et al ηt ≤ a.s Therefore, ln t Proof From [2, Theorem 2.4] we get lim supt→∞ t ρ2 δeξs − νeηs − e2ηs ds + ρ t→∞ t ηt y = lim sup ≤ t t→∞ t −γ + lim sup eηs dWs (2.10) For any ε > 0, applying the exponential martingale inequality [6, Theorem 7.4, p 44] it yields, t −ρeηs dWs − P sup 0≤t≤k t ε ρ e2ηs ds > ln k ≤ ε k By Borel-Cantelli lemma, for almost all ω, there exists a number k = k(ω) such that for all n > k and ≤ t ≤ n, t ε −ρeηs dWs − t ρ e2ηs ds < ln k , ε which implies that for k − ≤ t ≤ k t t −ρeηs dWs − ε t ρ e2ηs ds ≤ ln k (k − 1)ε As a result, t t lim inf t→∞ ρeηs dW (s) + t ε ρ e2ηs ds ≥ 0 From (2.10) and (2.11), it follows that lim sup t→∞ t t δeξs − νeηs − ρ (1 + ε) 2ηs e ds ≤ γ On the other hand, it follows from Lemma 2.1 that, lim sup t→∞ t t δeξs ds ≤ δ · lim sup t→∞ t t ¯ eξs ds = δm < ∞ a.s Therefore, lim sup t→∞ ≤ t t δeξs − νeηs − ρ 2ηs e ds t 1 lim sup + ε t→∞ t δeξs − νeηs − ε lim sup + + ε t→∞ t ρ (1 + ε) 2ηs e ds t δeξs − νeηs ds ≤ ε γ+ δm 1+ε 1+ε Letting ε → gets lim sup t→∞ t t δeξs − νeηs − ρ 2ηs e ds ≤ γ (2.11) Asymptotic Behavior of Predator-Prey Systems Perturbed by White 357 Thus, item a) has been proved Further, by Lemma 2.1 t ηt − η¯ t = t νeη¯ s + + t ρ 2ηs e ds + ρ δeξs − νeηs − ρ 2η¯s e ds − ρ t eηs dWs eη¯ s dWs ≥ (2.12) By the same argument as the proof of the first assertion, lim sup t→∞ t t t ε ρeηs dWs − ρ e2ηs ds ≤ 0, and t t lim sup t→∞ −ρeη¯ s dWs − ε t ρ e2η¯s ds ≤ 0 Combining these inequalities and (2.12) gets lim inf t→∞ t t δeξs − νeηs − Since η¯ t → −∞ as t → ∞, limt→∞ lim inf t→∞ Note that if t lim supt→∞ 1t ≤ lim inf t→∞ t→∞ t→∞ ≤ lim inf t→∞ t t t δeξs − νeηs − ρ(1+ε) 2η¯ s e )ds = Hence, ds ≥ ρ (1 − ε) 2ηs e ds ≥ ρ (1 − ε) 2ηs e ds t δeξs ds + lim inf t→∞ ρ (1 − ε) · lim sup t→∞ t lim supt→∞ 1t t t t − ρ (1 − ε) 2ηs e ds t e2ηs ds = −∞ e 2ηs < ∞ Hence, δeξs − νeηs − ρ (1 − ε) 2ηs e ds δeξs − νeηs − ρ 2ηs e ds + ε · lim sup t→∞ t t t ρ(1 + ε) 2η¯s e e2ηs = ∞ then for ε < 1, That is a contradiction Thus, t + ≤ δm − ≤ lim inf t η¯ s (νe δeξs − νeηs − t ≤ lim sup t t t − νeη¯ s + ρ (1 − ε) 2ηs e t ρ 2ηs e ds Let ε → 0, we obtain item (b) The proof of item (c) is similar to the one of item (a) Asymptotic Stability Let the triple (X, , m) be a σ -finite measure space Denote by D the subset of the space L1 = L1 (X, , m), consisting of the densities, i.e D = {f ∈ L1 : f ≥ 0, f = 1} A linear 358 N.H Dang et al mapping P : L1 → L1 is called a Markov operator if P (D) ⊂ D The Markov operator P is called an integral or kernel operator if there exists a measurable function k : X × X → [0, ∞) such that Pf (x) = X k(x, y)f (y)m(dy) for every density f ∈ D Let {P (t)}t≥0 be a semigroup of linear operators on L1 The family {P (t)}t≥0 is said to be a Markov semigroup if P (t) is a semigroup and for every t ≥ and P (t) is a Markov operator {P (t)}t≥0 is called integral, if for each t > 0, the operator P (t) is an integral Markov operator The semigroup {P (t)}t≥0 is called asymptotically stable if there is an invariant density f∗ such that limt→∞ P (t)f − f∗ = for all f ∈ D (a density f∗ is called invariant under the semigroup {P (t)}t≥0 if P (t)f∗ = f∗ for each t ≥ 0) The Markov semigroup {P (t)}t≥0 is called sweeping with respect to a set A ∈ if for any f ∈ D, lim t→∞ A P (t)f (x)m(dx) = If the semigroup is either asymptotically stable or sweeping with respect to compact sets then we say that the semigroup has the “Foguel alternative” It is clear that if the Markov semigroup {P (t)}t≥0 has an invariant density then it is not sweeping with respect to sets of finite measure We will use the nice property to exclude the sweeping and obtain asymptotic stability in Theorem 3.10 For the more details, we can refer to [11] We now consider the space (R2 , B(R2 ), m) where B(R2 ) is the σ -algebra of Borel subsets of R2 and m is the Lebesgue measure on (R2 , B(R2 )) We shall prove the Foguel alternative property of the semigroup {P (t)}t≥0 by using the following lemmas Lemma 3.1 Let (ξt , ηt ) be the solution of (2.1) with the initial value (ξ0 , η0 ) = (x0 , y0 ) ∈ R2 Then, the transition probability function P (t, x0 , y0 , ·) of the Markov diffusion process (ξt , ηt ), i.e., P (t, x0 , y0 , A) = P{(ξt , ηt ) ∈ A} for any A ∈ B(R2 ), has a density k(t, x, y, x0 , y0 ) with respect to m and k ∈ C ∞ ((0, ∞) × R2 × R2 ) for all t > Proof The proof is somewhat similar to the one in [11] We apply the Hormander theorem on the existence of smooth densities of the transition probability function for the degenerate diffusion processes described by the Stratonovich equation (2.2) If X(x) = (X1 , X2 )T and Y (x) = (Y1 , Y2 )T are vector fields on R2 then the Lie bracket [X, Y ] is a vector field given by [X, Y ]j (x) = X1 ∂Yj ∂Xj ∂Yj ∂Xj (x) − Y1 (x) + X2 (x) − Y2 (x) , ∂x1 ∂x1 ∂x2 ∂x2 j = 1, Let a0 (x, y) = α − βey − μex − σ e2x −γ + δex − νey − ρ e2y and a1 (x, y) = σ ex ρey Putting a = 2ρ β > 0, b = 2σ δ > 0, it is easy to get that a2 = [a0 , a1 ] = σ ex (α − βey + σ e2x ) + ρβe2y , ρey (−γ + δex + ρ e2y ) − σ δe2x a3 = [a1 , [a0 , a1 ]] = 2σ e4x + ae3y , 2ρ e4y − be3x a4 = [a1 , [a1 , [a0 , a1 ]]] = 6σ e5x + 3ρae4y − σ aex+3y −3σ be4x + 6ρ e5y + bρe3x+y Asymptotic Behavior of Predator-Prey Systems Perturbed by White 359 Now we assume that there exists a point (x, y) such that three vectors a1 (x, y), a3 (x, y) and a4 (x, y) not span the space R2 In this case, two vectors a1 , a3 are parallel and so are a1 , a4 Hence, σ ex (2ρ e4y − be3x ) = ρey (2σ e4x + ae3y ), σ ex (−3bσ e4x + 6ρ e5y + bρe3x+y ) = ρey (6σ e5x + 3aρe4y − aσ ex+3y ) (3.1) The first equation of (3.1) gives that 2σρ ex+4y = 2ρσ e4x+y + aρe4y + bσ e4x (3.2) Thus, 6σρ ex+5y = 6ρ σ e4x+2y + 3aρ e5y + 3bσρe4x+y , σ ex < ρey (3.3) From the second equation of (3.1) we obtain 6σρ ex+5y + bσρe4x+y + aσρex+4y = 6σ ρe5x+y + 3aρ e5y + 3bσ e5x (3.4) By combining (3.3) and (3.4) it yields bσρe4x+y + aσρex+4y + 6σ ρ e4x+2y + 3bσρe4x+y = 6σ ρe5x+y + 3bσ e5x (3.5) The relations (3.3) imply 6σ ρ e4x+2y > 6σ ρe5x+y , 3bσρe4x+y > 3bσ e5x which says that the equality (3.5) is impossible Thus, we get the Hörmander condition: (H) For every (x, y) ∈ R2 vectors a1 (x, y), [ai , aj ](x, y)0≤i,j ≤1 , [ai , [aj , ak ]](x, y)0≤i,j,k≤1 , span the space R2 Under the condition (H), the transition probability function P (t, x0 , y0 , ·) has a density k(t, x, y, x0 , y0 ) and k ∈ C ∞ ((0, ∞) × R2 × R2 ) (see [5, 9]) The lemma is proved We now denote by (ξt , ηt ) the solution of (2.1) with the initial random variable (ξ0 , η0 ), where the distribution of (ξ0 , η0 ) is absolutely continuous with the density v(x, y) From the above lemma, it is known that for any t > 0, (ξt , ηt ) has the density u(t, x, y) satisfying the Fokker-Planck equation (2.3) Further, u(t, x, y) = ∞ ∞ −∞ −∞ k(t, x, y; x1 , y1 )v(x1 , y1 )dx1 dy1 For any t ≥ 0, we consider the operator P (t) defined by P (t)v(x, y) = u(t, x, y) = ∞ ∞ −∞ −∞ k(t, x, y; x1 , y1 )v(x1 , y1 )dx1 dy1 for v ∈ D By the analytic prolongation of the operator P (t) on L1 (R2 , B(R2 ), m) and Lemma 3.1, we now get that {P (t)}t≥0 is an integral Markov semigroup with a continuous kernel k 360 N.H Dang et al Now we follow the method in [11] to check the positivity of k Fix a point (x0 , y0 ) ∈ R2 and a function φ ∈ C([0, T ]; R) Consider the following system of differential equations: xφ = α − βeyφ + (σ φ(t) − μ)exφ − σ e2xφ , (3.6) yφ = −γ + δexφ + (ρφ(t) − ν)eyφ − ρ e2yφ Put f¯1 (x, y) = α − βey + (σ φ(t) − μ)ex − σ e2x , f¯2 (x, y) = −γ + δex + (ρφ(t) − ν)ey − ρ e2y , we can rewrite (3.6) under the form xφ (t) = f¯1 (xφ (t), yφ (t)), yφ (t) = f¯2 (xφ (t), yφ (t)) Let (xφ , yφ ) be the solution of (3.6) with the initial condition xφ (0) = x0 ; yφ (0) = y0 and F : C([0, T ], R) → R2 be a mapping defined by F (h) = (xφ+h (T ), yφ+h (T )) We calculate the Frechet derivative Dx0 ,y0 ,φ of F by using means of the perturbation method for ordinary differential equations Let f = (f¯1 , f¯2 ) and (t) = f (xφ (t), yφ (t)) Denote Q(t, s), for ≤ s ≤ t ≤ T , the fundamental matrix of solutions of the equation Y˙ = i.e., ∂Q(t,s) ∂t = (t)Y, (t)Q(t, s) and Q(s, s) = I Then, T Dx0 ,y0 ,φ h = where q(x, y) = σ ex ρey Q(T , s) q(xφ (s), yφ (s)) h(s)ds, Let ε ∈ (0, T ) and h(t) = if ≤ t ≤ T − ε and h(t) = 1ε (t − T + ε) if T − ε ≤ t ≤ T By Taylor formula we have Q(T , s) = I − s → T Thus, T Dx0 ,y0 ,φ h = T −ε (T )(T − s) + o(T − s) as s −T +ε q(xφ (s), yφ (s))ds ε T + (T ) T −ε s −T +ε (s − T ) q(xφ (s), yφ (s))ds + o(ε ) ε Put x¯ = xφ (T ), y¯ = yφ (T ), c = φ(T ) From Theorem of Mean Value Integration, we have ε→0 ε T lim T −ε s −T +ε q(xφ (s), yφ (s))ds = q(x, ¯ y) ¯ ε (T ) ε→0 ε T lim T −ε and s −T +ε (s − T )q(xφ (s), yφ (s))ds = − (T )q(x, ¯ y) ¯ ε By a direct calculation we obtain (T ) = (cσ − μ)ex¯ − 2σ e2x¯ δex¯ −βey¯ (cρ − ν)ey¯ − 2ρ e2y¯ Asymptotic Behavior of Predator-Prey Systems Perturbed by White 361 Therefore, (T )q(x, ¯ y) ¯ = (cσ − μ)σ e2x¯ − 2σ e3x¯ − ρβe2y¯ δσ e2x¯ + (cρ − ν)ρe2y¯ − 2ρ e3y¯ If the vectors q(x, ¯ y) ¯ and (T )q(x, ¯ y) ¯ are linearly independent then rank Dx0 ,y0 ,φ = Thus, with the assumption of linear dependence between q(x, ¯ y) ¯ and (T )q(x, ¯ y) ¯ we see that x, ¯ y¯ are the solution of the equation det σ ex¯ ρey¯ (cσ − μ)σ e2x¯ − 2σ e3x¯ − ρβe2y¯ = δσ e2x¯ + (cρ − ν)ρe2y¯ − 2ρ e3y¯ or, σ (δ + 2σρey¯ )e3x¯ − (cσ − μ)σρey¯ e2x¯ + (cρ − ν)ρσ e2y¯ − 2σρ e3y¯ ex¯ + βρ e3y¯ = (3.7) It is easy to see that (3.7) has at most three solution curves and each solution curve is a graph of a function x¯ = g(y) ¯ Summing up, we have, Lemma 3.2 There exists at most three curves having the form x = g(y) such that rank(Dx0 ,y0 ,φ ) = if xφ (T ) = g(yφ (T )) Using the substitution zφ (t) = e−xφ (t) − ρ −1 σ e−yφ (t) (3.8) the system (3.6) becomes yφ = ρeyφ φ + g1 (yφ , zφ ), zφ = g2 (yφ , zφ ), (3.9) where g1 (y, z) = −γ + δ − νey − ρ e2y , z + σρ −1 e−y and g2 (y, z) = −αz − + (α + γ )σ −y σ + δσρ −1 e−y e + + (βz − σρ)ey ρ z + σρ −1 e−y σβ + μρ − νσ ρ Let E = (y, z)|z > −σρ −1 e−y From [3, Theorem 8.1, Chap 5], the support of a diffusion process described by a Stratonovich equation is the controllable domain of a system of ODEs, i.e., the domain where starting a point, we can go to another point after a finite time by a suitable control φ Therefore, we consider the controllability of the domain E By (3.8) we see that (yφ , zφ ) ∈ E for any φ ∈ C([0, T ]; R) 362 N.H Dang et al Hypothesis 3.3 There exists a point z∗ such that g2 (y, z∗ ) ≤ for all y ∈ R and (y, z∗ ) ∈ E It is easy to see that z∗ , if it exists, is in the interval [0, β −1 σρ] and we denote a ∗ = inf{z∗ |g2 (y, z∗ ) ≤ for all y ∈ R, (y, z∗ ) ∈ E}, (3.10) and E1 = {(x, y)|e−x − ρ −1 σ e−y < a ∗ } = {(y, z) ∈ E|z ≤ a ∗ }, E2 = {(x, y)|e−x − ρ −1 σ e−y > β −1 σρ} = {(y, z)|z > β −1 σρ} We have the following claims Claim Let z0 > z1 Since limy→−∞ g2 (y, z) = −∞ uniformly in z ∈ [z1 , z0 ], it follows that there exists y0 such that g2 (y0 , z) ≤ −1 for any z ∈ [z1 , z0 ] We choose φ(t) = − g1 (y0 , z(t)) , ρey0 where z(t) is the solution of the following equation z (t) = g2 (y0 , z(t)), z(0) = z0 It means that system (3.9) has the solution (yφ (t), zφ (t)) = (y0 , z(t)) and zφ (0) = z0 Since zφ = g2 (y0 , zφ ) ≤ −1 whenever zφ ∈ [z1 , z0 ], we can find a T > such that zφ (T ) = z1 Claim Let β −1 ρσ < z0 < z1 Since limy→∞ g2 (y, z) = ∞ uniformly in z ∈ [z0 , z1 ], we can find y0 such that g2 (y0 , z) ≥ for any z ∈ [z0 , z1 ] It is similar to Claim that there exist a function φ(t) and T > such that (3.9) has a solution yφ ≡ y0 , zφ (0) = z0 and zφ (T ) = z1 Claim Given z0 < and we put z = ln −σ ρz0 By virtue of the property limy→z− g2 (y, z0 ) = ∞, it is easy to see that there exist constants c1 > and y0 ∈ R such that g2 (y0 , z) ≥ for any z ∈ [z0 − c1 , z0 + c1 ] and (y0 , z) ∈ E In this case we also find a control function φ such that yφ ≡ y0 , zφ (0) = z0 and zφ (T ) = z0 + c1 for some T > Claim Suppose that Hypothesis 3.3 holds It is obvious that z∗ ∈ [0, β −1 σρ] By the definition of a ∗ , for any ε > 0, there exist δ1 , δ2 > having the property: if (z0 , z1 ) such that z1 − δ1 < z0 < z1 < a ∗ − ε, we can find a y0 satisfying g2 (y0 , z) > δ2 for z ∈ [z0 , z1 ] and (y0 , z) ∈ E We therefore find a control function φ such that yφ ≡ y0 , zφ (0) = z0 and zφ (T ) = z1 for some T > Claim Suppose that Hypothesis 3.3 does not hold It then follows that there exist δ1 , δ2 > having the property: if (z0 , z1 ) such that z1 − δ1 < z0 < z1 and z0 , z1 ∈ [0, β −1 σρ], we can find a y0 satisfying g2 (y0 , z) > δ2 for z ∈ [z0 , z1 ] and (y0 , z) ∈ E By the same arguments as above, there exists a control function φ such that yφ ≡ y0 , zφ (0) = z0 and zφ (T ) = z1 for some T > 0 } and [y0 , y0 + Claim Fix y0 ∈ R, L > 0, A1 > A0 and ε > such that ε < min{ L4 , A1 −A L] × [A0 , A1 ] ⊂ E Let m∗ = max{|g1 (y, z)| + |g2 (y, z)||(y, z) ∈ [y0 , y0 + L] × [A0 , A1 ]}, Asymptotic Behavior of Predator-Prey Systems Perturbed by White 363 and t0 = mε∗ The function φ is a constant and will be chosen later For every z0 ∈ [A0 + ε, A1 − ε], we see that the solution of system (3.9) with yφ (0) = y0 , zφ (0) = z0 satisfies for all t ∈ [0, t0 ], yφ (t0 ) ∈ y0 + L , y0 + L and zφ (t) ∈ [z0 − ε, z0 + ε] (3.11) Indeed, it follows from (3.9) that ρφeyφ (t) − m∗ ≤ yφ (t) ≤ ρφeyφ (t) + m∗ Using the fact that the solution of the equation y = Aey + B is y = y0 + Bt + ln B , Aey0 + B − Aey0 +Bt and the comparison theorem, we get for any t ∈ [0, t0 ], ∗ yφ (t) ≥ y0 − m∗ t + ln m∗ − ln ρφey0 (e−m t − 1) + m∗ , and ∗ yφ (t) ≤ y0 + m∗ t + ln m∗ − ln ρφey0 (1 − em t ) + m∗ Then yφ (t0 ) ≥ y0 − ε + ln m∗ − ln ρφey0 (e−ε − 1) + m∗ , and yφ (t0 ) ≤ y0 + ε + ln m∗ − ln ρφey0 (1 − eε ) + m∗ The first assertion of (3.11) therefore holds if we choose the constant φ such that y0 − ε + ln m∗ − ln ρφey0 (e−ε − 1) + m∗ > y0 + L , (3.12) and y0 + ε + ln m∗ − ln ρφey0 (1 − eε ) + m∗ < y0 + L It is easy to see that (3.12) and (3.13) are equivalent respectively to L φ∈ m∗ m∗ (1 − e−ε− ) ; , ρey0 (1 − e−ε ) ρey0 (1 − e−ε ) and φ< m∗ (1 − eε−L ) ρey0 (eε − 1) By some calculations, it easy to see that for sufficiently small ε, L m∗ (1 − e−ε− ) m∗ (1 − eε−L ) < , ρey0 (1 − e−ε ) ρey0 (eε − 1) (3.13) 364 N.H Dang et al i.e., we can choose φ such that the first assertion of (3.11) holds For the second one, we have for all ≤ t ≤ t0 , t |zφ (t) − zφ (0)| = t0 g2 (yφ (s), zφ (s))ds ≤ m∗ ds = m∗ t0 = ε From (3.11) it follows that for (y1 , z1 ) ∈ (y0 , y0 + L2 ] × [A0 + 2ε, A1 − 2ε] there exists a z0 ∈ [z1 − ε, z1 + ε] and a T ∈ (0, t0 ) such that yφ (T ) = y1 and zφ (T ) = z1 The same proof works for y1 ∈ (y0 − L2 , y0 ] We then get that for any (y1 , z1 ) ∈ (y0 − L2 , y0 + L2 ) × [A0 + 2ε, A1 − 2ε], there exists a z0 ∈ [z1 − ε, z1 + ε] and a T ∈ (0, t0 ) such that yφ (T ) = y1 , zφ (T ) = z1 Summing up, we obtain Lemma 3.4 If Hypothesis 3.3 does not hold, system (3.9) is controllable in E by piecewisecontinuous controls If Hypothesis 3.3 holds, system (3.9) is controllable in Ei (i = 1, 2), i.e., for all (y0 , z0 ), (y1 , z1 ) ∈ Ei , there exists a piecewise-continuous control function φ and T > such that yφ (0) = y0 , zφ (0) = z0 , yφ (T ) = y1 , zφ (T ) = z1 By the continuity with respect to the initial condition, we conclude that instead of using piecewise-continuous controls we can use continuous controls, i.e., φ ∈ C(0, T ; R) It is known that (see [11]), if φ ∈ C([0, T ]; R) such that the derivative Dx0 ,y0 ,φ has the rank then k(T , x, ¯ y, ¯ x0 , y0 ) > for x¯ = xφ (T ), y¯ = yφ (T ) Therefore, we have Lemma 3.5 If Hypothesis 3.3 does not hold, then for each (x0 , y0 ) ∈ R2 and for almost every (x, y) ∈ R2 , there exists a T > such that k(T , x, y, x0 , y0 ) > If Hypothesis 3.3 holds, then for any point (x0 , y0 ) and for almost every (x, y) satisfying (y0 , z0 ) and (y, z) in Ei , there exists a T > such that k(T , x, y, x0 , y0 ) > 0, where z = e−x − ρ −1 σ e−y Let ζt = e−ξt − ρ −1 σ e−ηt Although, the kernel k(T , x, y, x0 , y0 ) > for some T > but E2 is a transient set as the following lemma shows Lemma 3.6 If Hypothesis 3.3 holds, then lim supt→∞ ζt ≤ a ∗ a.s where a ∗ is defined in (3.10) Proof By the properties of Stratonovich integral, system (2.2) can be replaced by dηt = g1 (ηt , ζt )dt + ρeηt ◦ dWt , dζt = g2 (ηt , ζt )dt (3.14) Firstly, we show that inft≥0 ζt < a ∗ a.s Suppose in the contrary that inft≥0 ζt (ω) ≥ a ∗ for all ω ∈ B with P(B) > Since ζt ≥ a ∗ for all t ≥ 0, we have g1 (y, ζt ) ≤ g1 (y, a ∗ ) for ω ∈ B and y ∈ R In general, P(B) < then we are unable to use the comparison theorem here because the diffusion coefficient is not constant By putting λ(t) = e−ηt it yields dλ(t) = −λ(t)g1 (− ln λ(t), ζt ) dt − ρdWt Asymptotic Behavior of Predator-Prey Systems Perturbed by White 365 Let λ¯ be the solution of the equation d λ¯ (t) = −λ¯ (t)g1 (− ln λ¯ (t), a ∗ ) dt − ρdWt with λ¯ (0) = λ(0) Since −yg1 (ln y, ζt ) ≥ −yg1 (ln y, a ∗ ) on B for all t > 0, we see that λ(t, ω) ≥ λ¯ (t, ω) Thus, ηt ≤ ηt∗ (ω) for ω ∈ B, where ηt∗ = − ln λ¯ (t) is the solution of the following equation ∗ dηt∗ = g1 (ηt∗ , a ∗ )dt + ρeηt ◦ dWt , η0∗ = η0 Let x s3 (x) = y exp − 0 x = exp 2(−γ − νeu − ρ 2u e 2 ρ e2u + δ ) a ∗ +σρ −1 e−u 2ν + γ γ 2ν − 2y − y + y − ρ2 ρ e ρ e y du dy 2δ du dy (a ∗ + σρ −1 e−u )ρ e2u It is easy to see that limx→∞ s3 (x) = ∞, limx→−∞ s3 (x) > −∞ Then by [3, Theorem 3.1] we get limt→∞ ηt∗ = −∞ a.s We thus have limt→∞ ηt = −∞ on B On the other hand, by the definition of g2 , there is a M1 > such that g2 (y, z) ≤ −1 for all y ≤ −M1 , z ≥ Then for any ω ∈ B, there exists M2 > satisfying ηt (ω) ≤ −M1 for t > M2 , which implies that g2 (ηt (ω), ζt (ω)) ≤ −1 It follows from the second equation of (3.14) that limt→∞ ζt (ω) = −∞, which contradicts our assumption that inft≥0 ζt ≥ a ∗ on B Next, we show that lim supt→∞ ζt ≤ a ∗ a.s We see that g2 (y, z) = G(y, z) e−y (z + σρ −1 e−y ) , where G(y, z) = z(βz − σρ) + [σρ −1 (βz − σρ) − αz2 + (σβ + μρ − νσ )ρ −1 z]e−y + [(σβ + μρ − νσ )σρ −2 − (2α + γ )σρ −1 z]e−2y − (α + γ )σ ρ −2 e−3y is a polynomial function of third degree of the variable e−y By the definition of a ∗ , there exists at most one point c0 ∈ R such that g2 (c0 , a ∗ ) = It then follows from g2 (y, a ∗ ) ≤ for all y ∈ R that for every τ > 0, g2 (y, a ∗ ) < for all y > c0 + τ or y < c0 − τ By the continuity, we then can find an ε > and a “rectangle” A = (−∞, c0 − τ ] × [a ∗ , a ∗ + κ] ∪ [c0 + τ, ∞) × [a ∗ , a ∗ + κ] with κ > such that g2 (y, z) < −ε for all (y, z) ∈ A Using the same argument as in the proof of Lemma in [11], it follows that lim supt→∞ ζt ≤ a ∗ a.s The proof is complete Lemma 3.7 If Hypothesis 3.3 holds, then E1 is an invariant set, i.e., for all t > (ηt , ζt ) ∈ E1 a.s if (η0 , ζ0 ) ∈ E1 366 N.H Dang et al Proof Assume that there exists a solution (ηt , ζt ) of the system (3.14) such that (η0 , ζ0 ) ∈ E1 and (ηt1 (ω), ζt1 (ω)) ∈ / E1 with some t1 > 0, ω ∈ By the continuity of the path ζt (ω), it then follows that there exist ≤ t0 < t¯0 < t1 such that ζt0 (ω) = a ∗ and ζs (ω) < ζt (ω) for all t0 < s < t < t¯0 (3.15) The following property is called (P) that is mentioned in proof of Lemma 3.6 For any τ > there exist two positive numbers κ and ε such that g2 (y, z) < −ε for all (y, z) ∈ A, where A = (−∞, c0 − τ ] × [a ∗ , a ∗ + κ] ∪ [c0 + τ, ∞) × [a ∗ , a ∗ + κ] It follows from the equation dζt (ω) = g2 (ηt (ω), ζt (ω))dt, (3.15) and the property (P) that t (ω) = g2 (c0 , ζt (ω)) By (3.15), there exists a deηt (ω) = c0 for any t ∈ [t0 , t¯0 ] Then dζdt ∞ ¯ creasing sequence {sn }1 ⊂ (t0 , t0 ] such that limn→∞ sn = t0 and g2 (c0 , ζsn (ω)) > for any n = 1, 2, It is easy to see that this result contradicts the property (P) The lemma therefore is proved The two following assertions give conditions under which the semigroup {P (t)}t≥0 sweeping or not sweeping Proposition 3.8 Let (ξt , ηt ) be a solution of (2.1) with the initial value (ξ0 , η0 ) ∈ R2 Also, let m be defined by (2.9) If mδ < γ then limt→∞ ηt = −∞ a.s and the distribution of the process ξt converges weakly to the measure which has the density f∗ (x) as t → ∞ Proof The proof is similar to one of the lemma in [11] By Lemma 2.1, ξs ≤ ξ¯s which implies t ηt ≤ η0 + ¯ −γ + δeξs − νeηs − ρ 2ηs e ds + t ρeηs dWs Therefore, ηt η0 ρ ≤ + t t t eηs dWs − t ρ2 t e2ηs ds −γ +δ t t ¯ eξs ds (3.16) By the way in the proof of Theorem 2.2, lim sup ρ t t→∞ eηs dWs − t ρ2 t e2ηs ds ≤0 a.s (3.17) On the other hand, from the ergodic theorem it follows that t→∞ t t ¯ eξs ds = lim +∞ −∞ ex f∗ (x)dx = m, (3.18) (see (2.8)) From (3.16), (3.17), (3.18), we get that lim supt→∞ ηtt ≤ −γ + δm < 0, then limt→∞ ηt = −∞ a.s The way to prove that the distribution of the process ξt converges weakly to the measure which has the density f∗ (x) is similar to one in [11] and we omit it here The proposition is proved Asymptotic Behavior of Predator-Prey Systems Perturbed by White 367 Proposition 3.9 If mδ > γ , then there exists a stationary distribution for the Markov process (ξt , ηt ) Proof To simplify the proof, we use the former variables Xt and Yt again It is known that (Xt , Yt ) is a Markov process on R2+ and α = mμ + m σ2 where m = R e2x f∗ (x)dx Moreover, lim sup t→∞ t t Xs2 ds = lim sup t→∞ t t t t→∞ t e2ξs ds ≤ lim ¯ e2ξs ds = m Therefore, by virtue of the item (c) of Theorem 2.2 it yields lim inf t→∞ t t Ys + μ μ Xs ds ≥ m β β (3.19) Further, by item (b) of Theorem 2.2, t μ lim inf δβ t→∞ t −δXs ds + νYs + ρ2 μ Y ds ≥ − γ s δβ (3.20) Adding (3.19) and (3.20) side by side, we have lim inf t→∞ μρ 2 μν mδ − γ Y ds ≥ μ > Ys + δβ 2δβ s δβ t t 1+ Also in view of Theorem 2.2, lim sup t→∞ t t νYs + t ρ2 Ys ds ≤ lim sup t→∞ t t δXs ds ≤ mδ t Moreover, it follows from the inequality Ys ds ≤ t Ys2 ds that there exist two positive constants (independent on the initial condition) M1 , M2 satisfying 2M1 ≤ lim inf t→∞ t t Ys2 ds ≤ lim sup From the relation 2M1 ≤ lim inft→∞ lim inf t→∞ t→∞ t t t t t Ys2 ds ≤ M2 Ys2 ds, it is seen that t 1{Ys2 >M1 } Ys2 > M1 > By Holder’s inequality, t t ≤ E1{Ys2 >M1 } Ys2 ds t t E(1{Ys2 >M1 } ) θ+1 θ θ θ+1 t t EYt2+2θ θ+1 By [7, Theorem 2], for < θ < 0.5, there exists a constant M4 such that t lim supt→∞ 1t EYt2+2θ < M4 Thus, 368 N.H Dang et al M1 ≤ lim inf t→∞ t t E1{Ys2 >M1 } Ys2 ds t ≤ lim inf t→∞ t P{Ys2 > M1 } θ θ+1 lim sup t→∞ t t EYt2+2θ θ+1 Or, lim inf t→∞ t θ+1 t P{Ys2 > M1 }ds ≥ ε0 = M1 θ M4θ Further, by virtue of [4, Theorem 3.3], there exists a constant H > satisfying lim inft→∞ P{Xt + Yt ≤ H } ≥ − ε20 , which implies lim inf t→∞ t t Qt (A)ds ≥ ε0 > 0, (3.21) where Qt is the semigroup corresponding to the Markov process (Xt , Yt ) and A = {y ≥ √ M1 , x + y ≤ H } From (3.21) and Theorem in [8, Appendix], there exists a stationary distribution λ in R2+ for the process (Xt , Yt ) such that λ(A) > Since the boundary A1 = {x = 0} × R+ is invariant under Qt and limt→∞ Yt = if X0 = 0, it follows that λ(A1 ) = Thus, λ is a stationary distribution on intR2+ , i.e., there exists a stationary distribution for Markov process (ξt , ηt ) The proof is complete In summary, we have Theorem 3.10 Let (ξt , ηt ) be a solution of the system (2.1) with the initial value (ξ0 , η0 ) ∈ R2 Then for every t > the distribution of (ξt , ηt ) has a density u(t, x, y) which satisfies (2.3) Further, (1) If mδ > γ and Hypothesis 3.3 does not hold, the semigroup {P (t)}t≥0 is asymptotically stable on R2 , i.e., there exist a unique stationary density u∗ (x, y) of (2.3) such that lim t→∞ R2 |u(t, x, y) − u∗ (x, y)|dxdy = 0, (2) If mδ > γ and Hypothesis 3.3 holds, E2 is a transient set, E1 is an invariant set and the semigroup {P (t)}t≥0 is asymptotically stable on E1 This means that support u∗ ⊂ E1 and |u(t, x, y) − u∗ (x, y)|dxdy = lim t→∞ E1 (3) In the case where mδ < γ , then limt→∞ ηt = −∞ a.s and the distribution of the process ξt converges weakly to the measure which has the density f∗ (x) as t → ∞ Proof By virtue of Lemma 3.1 it follows that {P (t)}t≥0 is an integral Markov semigroup with a continuous kernel k(t, x, y, x0 , y0 ) for t > Then, the distribution of (ξt , ηt ) has a density u(t, x, y) which satisfies (2.3) According to the Lemma 3.5, for every f ∈ D we have ∞ P (t)f dt > 0 a.e on R2 or on E1 (3.22) Asymptotic Behavior of Predator-Prey Systems Perturbed by White 369 Fig Numerical solution of System (1.3) with stepsize 0.00005 and n = 500000 Fig Numerical solution of System (1.3) with stepsize 0.00005 and n = 500000 Using Corollary in [11] it follows immediately that the semigroup {P (t)}t≥0 is asymptotically stable or is sweeping with respect to compact sets Item (1) and item (2) follow from Lemma 3.6, Lemma 3.7 and Theorem 3.9 Item (3) is the assertion of Proposition 3.8 We illustrate the above results by studying two numerical examples Consider the case where α := 4; μ := 1; γ := 1; δ := 3; ν := 2; β := 1.5; σ := 2; ρ := Figure suggests that y(t) does not tend to zero as t tends to +∞ On the other hand, by computing, the threshold mδ − γ ≈ 1.27 Therefore, it follows from Theorem 3.10 that System (1.2) is asymptotically stable If α := 3; β := 1.5; μ := 1; γ := 5; δ := 1; ν := 2; σ := 2; ρ := In this case, the threshold mδ − γ ≈ −4.152 Therefore, by Theorem 3.10, limt→∞ ηt = −∞ a.s and the distribution of the process ξt converges weakly to the measure which has the density f∗ (x) as t → ∞ This claim agrees with Fig 370 N.H Dang et al Some Comments We have considered the asymptotic behavior of stochastic prey-predator systems where the random factor acts on the coefficients of environment It seems that the argument dealt with in [11] does not work for our model Although with the condition μγ < αδ, two curves 2 h1 (x, y) =: f1 (x, y) + σ2 e2x = and h2 (x, y) =: f2 (x, y) + ρ2 e2y = intersect and they divide the plane into four parts, but we are unable to construct a Khasminskii function V (x, y) as was done in [11] Therefore, to solve the problem, we have used the way of analyzing the behavior of the system on the boundary We constructed a threshold value mδ − γ whose sign determines that the distribution of the solution weekly converges to a unique stationary distribution on intR2+ or a singular one on the boundary {y = 0} Acknowledgements The authors would like to thank Prof Atsushi Yagi for his comments on the manuscript of this paper They would like to thank the reviewers for their very valuable remarks and comments, which will certainly improve the presentation of the paper This work was done under the support of the Grand NAFOSTED, No 101.02-2011.21 References Arnold, L., Horsthemke, W., Stucki, J.W.: The influence of external real and white noise on the LotkaVolterra model Biom J 21(5), 451–471 (1979) Du, N.H., Sam, V.H.: Dynamics of a stochastic Lotka-Volterra model perturbed by white noise J Math Anal Appl 324(1), 82–97 (2006) Ikeda, N., Wantanabe, S.: Stochastic Differential Equations and Diffusion Processes North-Holland, Amsterdam (1981) Luo, Q., Mao, X.: Stochastic 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Dynamics of Dissipation Lecture Notes in Physics, vol 597, pp 215– 238 Springer, Berlin (2002) 13 Skorokhod, A.V.: Asymptotic Methods of the Theory of Stochastic Differential Equations American Mathematical Society, Providence (1989) ... σ Asymptotic Behavior of Predator-Prey Systems Perturbed by White 355 If ξt is a solution of (2.5) then by ergodic theorem ∞ t t→∞ t g(ξs ) ds = lim −∞ g(x)f∗ (x) dx (2.8) a.s., for any g that... constant By putting λ(t) = e−ηt it yields dλ(t) = −λ(t)g1 (− ln λ(t), ζt ) dt − ρdWt Asymptotic Behavior of Predator-Prey Systems Perturbed by White 365 Let λ¯ be the solution of the equation d... aex+3y −3σ be4x + 6ρ e5y + bρe3x+y Asymptotic Behavior of Predator-Prey Systems Perturbed by White 359 Now we assume that there exists a point (x, y) such that three vectors a1 (x, y), a3 (x,