DSpace at VNU: On the dynamics of a stochastic ratio-dependent predator–prey model with a specific functional response

DSpace at VNU: On the dynamics of a stochastic ratio-dependent predator–prey model with a specific functional response t...

J Appl Math Comput DOI 10.1007/s12190-014-0812-3 ORIGINAL RESEARCH On the dynamics of a stochastic ratio-dependent predator–prey model with a specific functional response Yan Zhang · Shujing Gao · Kuangang Fan · Yanfei Dai Received: 14 May 2014 © Korean Society for Computational and Applied Mathematics 2014 Abstract In this paper, a new stochastic two-species predator–prey model which is ratio-dependent and a specific functional response is considered in, is proposed The existence of a global positive solution to the model for any positive initial value is shown Stochastically ultimate boundedness and uniform continuity are derived Moreover, under some sufficient conditions, the stochastic permanence and extinction are established for the model At last, numerical simulations are carried out to support our results Keywords Itô’s formula · Stochastically permanent · Extinction · White noise Introduction Predator–prey relationship can be important in regulating the number of prey and predators And the dynamic relationship between predators and their preys has long been and will continue to be one of the dominant themes in both ecology and mathematical ecology due to its universal existence and importance [1] In the past decades, more and more mathematical models for predator–prey behavior are carried out, see [2–4] and the references cited therein Y Zhang (B) · S Gao · Y Dai Key Laboratory of Jiangxi Province for Numerical Simulation and Emulation Techniques, Gannan Normal University, Ganzhou 341000, People’s Republic of China e-mail: zhyan8401@163.com S Gao e-mail: gaosjmath@126.com K Fan School of Mechanical and Electrical Engineering, Jiangxi University of Science and Technology, Ganzhou 341000, People’s Republic of China e-mail: kuangangfriend@163.com 123 Y Zhang et al When investigating biological phenomena, there are many factors which affect dynamical properties of biological and mathematical models One of the familiar nonlinear factors is functional response [5] There are many significant functional responses in order to model various different situations and evidences show that when predators have to search or compete for food, a more suitable functional response depending on the densities of both prey and predator, which is called a ratio-dependent functional response, should be introduced in realistic models [6,7] In the past decades, many authors proposed different forms of ratio-dependent functional responses to model this process and the three classical predator-dependent functions are Crowley– Martin type [8], Beddington–DeAngelis type by Beddington [9] and DeAngelis et al [10], as well as Hassell–Varley type [11] In this paper, we consider the following model with a specific functional response: ⎧ ⎪ ⎪ ˙ = x(t) r − ax(t) − ⎨ x(t) ωy(t) , + α1 x(t) + α2 y(t) + α3 x(t)y(t) f ωx(t) ⎪ ⎪ ⎩ y˙ (t) = y(t) h − by(t) + + α1 x(t) + α2 y(t) + α3 x(t)y(t) (1.1) where x(t), y(t) stand for the population densities of prey and predator at time t, respectively These parameter are defined as follows: r is the growth rate of the prey, a measures the strength of competition among individuals of species x, ω is the capturing rate of predator, f is the rate of conversion of nutrients into the production of predator, h, b have the similar meaning to r, a, respectively 1+α1 x(t)+αωx(t) is the y(t)+α3 x(t)y(t) radio-dependent functional response, where α1 , α2 , α3 ≥ are constants It is very important to note that this functional response becomes a linear mass-action function response (or Holling type I functional response) if α1 = α2 = α3 = 0, the Holling type-II functional response if α2 = α3 = 0, the modified Holling type-II functional response proposed in [12,13] when α3 = 0, and Crowley–Martin functional response presented in [6,8,14] if α3 = α1 α2 As a matter of fact, population systems in the real world are often inevitably subject to environmental noises Many researchers pointed out the fact that due to environmental noise, the birth rate, carrying capacity and other parameters involved in the model exhibit random fluctuation to a greater or lesser extent Therefore, more and more interest is focused on stochastic systems and many research has been done in this field , see e.g., [15–20] In this paper, taking into the effect of randomly fluctuating environment, we incorporate white noise in each equations of the system (1.1) and we assume that fluctuations in environment will manifest themselves mainly as fluctuations in the growth rate and capturing rate of the prey population and the growth rate and conversion rate of the predator population, therefore, the corresponding stochastic system to Eq (1.1) can be described as follows: ωy(t) dt + α1 x(t) + α2 y(t) + α3 x(t)y(t) σ2 y(t) + x(t) σ1 + d B1 (t), + α1 x(t) + α2 y(t) + α3 x(t)y(t) d x = x(t) r − ax(t) − 123 On the dynamics of a stochastic ratio-dependent predator–prey model f ωx(t) + α1 x(t) + α2 y(t) + α3 x(t)y(t) δ2 x(t) + y(t) δ1 + d B2 (t) + α1 x(t) + α2 y(t) + α3 x(t)y(t) dy = y(t) h − by(t) + (1.2) where r, a, h, b, ω, f, αi are positive parameters,i = 1, 2, σi2 and δi2 represent the intensities of the white noises, i = 1, B1 (t), B2 (t) are independent Brownian motions defined on a complete probability space ( , F, P) with a filtration {Ft }t∈R+ satisfying the usual conditions(i.e it is right continuous and increasing while F0 con2 the positive cone in R , and also denote by tains all P−null sets) We denote by R+ X (t) = (x(t), y(t)) and |X (t)| = (x (t) + y (t)) As far as we know, there is no work has been done on the stochastic model (1.2), and the aim of this paper is to analyze the dynamical properties of the model and show the impact of environmental noise on the population system (1.2) The rest of the paper is arranged as follows In this paper, we firstly show that the system (1.2) has a unique global solution for any positive initial value in Sect 2.1 The stochastic boundedness of the positive solution is discussed in Sect 2.2 Moreover, we show that the solution is uniformly continuous in Sect 2.3 In Sect 3, by constructing a suitable Lyapunov function, we establish the sufficient conditions on the stochastic permanence and extinction of the model At last, in Sect 4, a numerical simulation which verifies our qualitative results is given and the paper is ended with a discussion Properties of the solution 2.1 Existence, uniqueness and global positive solution In this section, as x(t), y(t) in system (1.2) stand for the population densities of prey and predator at time t, respectively, we will only interested in the positive solutions of system (1.2) In order for a stochastic differential equation to have a unique global solution for any given initial value, the coefficients of the equation are generally required to satisfy the linear growth condition and local Lipschitz condition [21] However, the coefficients of system (1.2) neither satisfy the linear growth condition, nor local Lipschitz continuous In the following, by making the change of variables, existence, uniqueness of the positive solution will be shown Theorem For any initial value x0 > 0, y0 > 0, there is a unique positive local solution(x(t), y(t)) for t ∈ [0, τe ) of system (1.2) a.s Proof Let u(t) = lnx(t), v(t) = lny(t), then we obtain the following equations du(t) = r − aeu(t) − − 0.5 σ1 + + α1 eu(t) ωev(t) + α2 ev(t) + α3 eu(t) ev(t) σ2 ev(t) u(t) + α1 e + α2 ev(t) + α3 eu(t) ev(t) dt 123 Y Zhang et al + σ1 + σ2 ev(t) d B1 (t), + α1 eu(t) + α2 ev(t) + α3 eu(t) ev(t) dv(t) = h − bev(t) + − 0.5 δ1 + + δ1 + f ωeu(t) + α1 eu(t) + α2 ev(t) + α3 eu(t) ev(t) δ2 eu(t) + α1 eu(t) + α2 ev(t) + α3 eu(t) ev(t) dt δ2 eu(t) d B2 (t) + α1 eu(t) + α2 ev(t) + α3 eu(t) ev(t) (2.1) for t ≥ with initial value u = lnx0 , v0 = lny0 Obviously, the coefficients of model (2.1) satisfy the local Lipschitz condition Therefore, there exists a unique local solution u(t), v(t) on t ∈ [0, τe ), where τe is the explosion time So, by Itô’s formula, it is easy to obtain that x(t) = eu(t) , y(t) = ev(t) is the unique positive local solution to system (1.2) with initial value x0 > 0, y0 > Next, we will show the unique positive solution of model (2.1) is global, i.e., τe = ∞ Theorem For any given value (x(0), y(0)) = X ∈ R2+ , there is a unique solution with probability (x(t), y(t)) to Eq (1.2) on t ≥ and the solution will remain in R+ 1, namely (x(t), y(t)) in R+ for all t ≥ almost surely Proof Our Proof is motivated by the works of Mao et al [22] and Liu and Wang [23] Let k0 > be sufficiently large for X lying within the interval [1/k0 , k0 ] For each integer k > k0 , define the stopping times τk = inf{t ∈ [0, τe ] : x(t)∈(1/k, k), or y(t)∈(1/k, k)}, where throughout this paper we set in f ∅ = ∞ Obviously, τk is increasing as k → ∞ Let τ∞ = lim τk , then τ∞ ≤ τe If we can k→+∞ a.s for show τ∞ = ∞ a.s., then we can obtain that τe = ∞a.s and (x(t), y(t)) ∈ R+ all t ≥ In other words, to complete the proof all of we need to show is that τ∞ = ∞ a.s If this statement is false, then there exist constants T > and ε ∈ (0, 1) such that P{τ∞ ≤ T } > ε Hence, there exists an integer k1 ≥ k0 such that P{τk ≤ T } ≥ ε for all k ≥ k1 (2.2) → R by V (x, y) = (x − − lnx) + (y − − lny) Define a C -function V : R+ + Then the nonnegativity of the function V (x, y) can easily be seen from u − − lnu ≥ on u > ˆ formula, we obtain that If (x(t), y(t)) ∈ R2+ , in view of I t o’s d V (x, y) = Vx d x + 0.5Vx x (d x)2 + Vy dy + 0.5Vyy (dy)2 = (1 − 1/x)x r − ax(t) − 123 ωy(t) + α1 x(t) + α2 y(t) + α3 x(t)y(t) On the dynamics of a stochastic ratio-dependent predator–prey model + (1 − 1/y)y h − by(t) + + 0.5 σ1 + f ωx(t) + α1 x(t) + α2 y(t) + α3 x(t)y(t) σ2 y(t) + α1 x(t) + α2 y(t) + α3 x(t)y(t) dt dt δ2 x(t) dt + α1 x(t) + α2 y(t) + α3 x(t)y(t) σ2 y(t) d B1 (t) + (1 − 1/x)x σ1 + + α1 x(t) + α2 y(t) + α3 x(t)y(t) δ2 x(t) + (1 − 1/y)y δ1 + d B2 (t) + α1 x(t) + α2 y(t) + α3 x(t)y(t) + 0.5 δ1 + ≤ (r +a)x(t)−ax (t)+ + 0.5 δ1 + δ2 α1 ω fω σ2 + +(h +b)y −by −h +0.5(σ1 + )2 dt α2 α3 α2 dt +(x(t)−1) σ1 + σ2 y(t) d B1 (t) 1+α1 x(t)+α2 y(t)+α3 x(t)y(t) δ2 x(t) d B2 (t) + α1 x(t) + α2 y(t) + α3 x(t)y(t) + (y(t) − 1) δ1 + Thus, there exists a positive number G such that σ2 y(t) d B1 (t) + α1 x(t) + α2 y(t) + α3 x(t)y(t) δ2 x(t) d B2 (t) + (y(t) − 1) δ1 + + α1 x(t) + α2 y(t) + α3 x(t)y(t) d V (x, y) ≤ Gdt + (x(t) − 1) σ1 + Taking integral for each side of the above inequality from to τk T yields that τk T τk T d V (x, y) ≤ Gdt τk T + (x(t)−1) σ1 + + (y(t)−1) δ1 + σ2 y(t) d B1 (t) 1+α1 x(t)+α2 y(t)+α3 x(t)y(t) δ2 x(t) d B2 (t) 1+α1 x(t)+α2 y(t)+α3 x(t)y(t) where τk T = min{τk , T } Whence taking expectation, we have that E V (x(τk T ), y(τk T )) ≤ V (x(0), y(0)) + G E(τk T ) ≤ V (x(0), y(0)) + G1T (2.3) Set k = {τk ≤ T } for k ≥ k1 and by (2.2), P( k ) ≥ ε Note that for every ω ∈ k , there is x(τk , ω) or y(τk , ω) equals either k or 1/k, and hence V (x(τk , ω), y(τk , ω)) ≥ min{k − − lnk, 1/k − + lnk} 123 Y Zhang et al It then follows from (2.3) that V (x(0), y(0)) + G T ≥ E[1 k (ω)V (x(ω), y(ω))] ≥ ε min{k − − lnk, 1/k − + lnk} where k is the indicator function of k Letting k → ∞ leads to the contradiction ∞ > V (x(0), y(0)) + G T = ∞ So we must have τ∞ = ∞ a.s This completes the proof of Theorem 2.2 Stochastic boundedness Definition (See [24]) The solution X (t) = (x(t), y(t)) of Eq (1.2) is said to be stochastically ultimately bounded, if for any ε ∈ (0, 1), there is a positive constant δ = δ(ε), such that for any given initial value X ∈ R2+ , the solution X (t) to (1.2) has the property that lim sup P{|X (t)| > δ} < ε t→∞ Theorem The solutions of model (1.2) are stochastically ultimately bounded for any initial value X = (x0 , y0 ) ∈ R2+ Proof Define Lyapunov functions V1 = et x p and V2 = et y p respectively, for (x, y) ∈ R2+ and p > Then by the Itô formula, we compute d(et x p ) = et x p dt + et px p−1 d x + 0.5 p( p − 1)x p−2 (d x)2 = et x p (t)dt + pet x p (t) r −ax(t)− ωy(t) dt 1+α1 x(t)+α2 y(t)+α3 x(t)y(t) σ2 y(t) + α1 x(t) + α2 y(t) + α3 x(t)y(t) σ2 y(t) d B1 (t) + pet x p (t) σ1 + + α1 x(t) + α2 y(t) + α3 x(t)y(t) + 0.5 p( p − 1)et x p (t) σ1 + dt and d(et y p ) = et y p (t)dt + pet y p (t) h −by(t)+ f ωx(t) dt 1+α1 x(t)+α2 y(t)+α3 x(t)y(t) δ2 x(t) + α1 x(t) + α2 y(t) + α3 x(t)y(t) δ2 x(t) d B2 (t) + pet y p (t) δ1 + + α1 x(t) + α2 y(t) + α3 x(t)y(t) + 0.5 p( p − 1)et y p (t) δ1 + 123 dt On the dynamics of a stochastic ratio-dependent predator–prey model Therefore L V1 = et x p (t)dt + pet x p (t) r − ax(t) − + 0.5 p( p − 1)et x p (t) σ1 + ωy(t) + α1 x(t) + α2 y(t) + α3 x(t)y(t) σ2 y(t) + α1 x(t) + α2 y(t) + α3 x(t)y(t) ≤ et x p (t) + pr + 0.5 p( p − 1) σ1 + σ2 α2 2 − apx ≤ G ( p)et (2.4) and L V2 = et y p (t)dt + pet y p (t) h − by(t) + + 0.5 p( p − 1)et y p (t) δ1 + ≤ et y p (t) + ph + f ωx(t) + α1 x(t) + α2 y(t) + α3 x(t)y(t) δ2 x(t) + α1 x(t) + α2 y(t) + α3 x(t)y(t) 2 pf ω δ2 + 0.5 p( p − 1) δ1 + α1 α1 − pby ≤ G ( p)et (2.5) where + pr + 0.5 p( p − 1) σ1 + G ( p) = σ2 α2 p+1 , a p ( p + 1) p+1 + ph + G ( p) = pf ω α1 + 0.5 p( p − 1) δ1 + δ2 α1 p+1 b p ( p + 1) p+1 Thus, taking integral and expectations on both sides of (2.4) and (2.5), respectively, we have the following equalities lim sup E x p ≤ G ( p) < +∞, lim sup E y p ≤ G ( p) < +∞ t→∞ t→∞ On the other hand, for X (t) = (x(t), y(t)) ∈ R2+ , note that |X (t)| p ≤ max{x (t), y (t)} p p ≤ 2 (x p (t) + y p (t)) Consequently, lim sup E|X (t)| p ≤ G ( p) < +∞ t→∞ 123 Y Zhang et al p where G ( p) = 2 (G ( p) + G ( p)) By the Chebyshev inequality, the proof is completed 2.3 Uniform continuity In this section, we continue to show the positive solution X (t) = (x(t), y(t)) is uniformly Holder ă continuous Main tools are to use appropriate Lyapunov functions and fundamental inequalities Lemma (See [25,26]) Suppose that a n− dimensional stochastic process X (t) on t ≥ satisfies the condition E|X (t) − X (s)|α1 ≤ c|t − s|1+α2 , ≤ s, t < ∞, for some positive constants α1 , α2 and c Then there exists a continuous modification X˜ (t) of X (t) which has the property that for every υ ∈ 0, αα21 , there is a positive random variable ψ(ω) such that P ω: | X˜ (t, ω) − X (t, ω)| ≤ υ |t − s| − 2−υ 0 (3.2) , by the Itô formula, V (x, y) ωy + α1 x + α2 y + α3 x y f ωx dt + y h − by + + α1 x + α2 y + α3 x y σ2 x y d B1 (t) + σ1 x + + α1 x + α2 y + α3 x y δ2 x y d B2 (t) + δ1 y + + α1 x + α2 y + α3 x y d V (x, y) = x r − ax − and ωy + α1 x + α2 y + α3 x y f ωx dt + y h − by + + α1 x + α2 y + α3 x y dU (X ) = −U (X ) x r − ax − + U (X ) x σ1 + + y δ1 + σ2 y + α1 x + α2 y + α3 x y δ2 x + α1 x + α2 y + α3 x y 2 dt σ2 x y d B1 (t) + α1 x + α2 y + α3 x y δ2 x y d B2 (t) + δ1 y + + α1 x + α2 y + α3 x y σ2 x y )d B1 (t) = LU (X )dt − U (X )(σ1 x + + α1 x + α2 y + α3 x y δ2 x y )d B2 (t) − U (X )(δ1 y + + α1 x + α2 y + α3 x y − U (X ) σ1 x + 123 Y Zhang et al Under Assumption (H1), we can certainly choose a positive constant θ such that it obeys (3.1) Applying the Itô formula again, we get L(1 + U (X ))θ = θ (1 + U (X ))θ−1 LU (X ) + θ (θ − 1)(1 + U (X ))θ−2 U (X ) 2 σ2 y × x σ1 + + α1 x + α2 y + α3 x y + y δ1 + δ2 x + α1 x + α2 y + α3 x y Now, we can choose p > sufficiently small for (3.2) to hold Then, let W (X ) = e pt (1 + U (X ))θ and thus, L W (X ) = pe pt (1 + U (X ))θ + e pt L(1 + U (X ))θ = e pt L(1 + U (X ))θ−2 × P(1 + U (X ))2 − θU (X )x r − ax − ωy + α1 x + α2 y + α3 x y f ωx + α1 x + α2 y + α3 x y ωy − θU (X ) x r − ax − + α1 x + α2 y + α3 x y f ωx + y h − by + + α1 x + α2 y + α3 x y − θU (X )y h − by + + θU (X ) x σ1 + + y δ1 + + σ2 y + α1 x + α2 y + α3 x y δ2 x + α1 x + α2 y + α3 x y 2 σ2 y θ (θ + 1) U (X ) x σ1 + + α1 x + α2 y + α3 x y + y δ1 + δ2 x + α1 x + α2 y + α3 x y 2 It is easy to see that θU (X ) x σ1 + ≤ θU (X ) max σ1 , 123 σ2 y 1+α1 x +α2 y +α3 x y σ2 δ2 , δ1 , α1 α2 + y δ1 + ; δ2 x 1+α1 x +α2 y +α3 x y On the dynamics of a stochastic ratio-dependent predator–prey model θ (θ +1) σ2 y U (X ) x σ1 + 1+α1 x +α2 y +α3 x y + y δ1 + δ2 x 1+α1 x +α2 y +α3 x y 2 σ2 δ2 θ (θ + 1) ; U (X ) max σ1 , , δ1 , α1 α2 ωy f ωx + y h −by + θU (X ) x r −ax − 1+α1 x +α2 y +α3 x y 1+α1 x +α2 y +α3 x y ω ≥ −θ max{a, b} + θ min{r, h}U (X ) − θ U (X ) α3 ≤ and ωy f ωx + y h −by + 1+α1 x +α2 y +α3 x y 1+α1 x +α2 y +α3 x y ω ≥ −θ max{a, b}U (X ) + θ min{r − , h}U (X ) α2 θU (X ) x r −ax − Thus, L W (X ) ≤ e pt (1 + U (X ))θ−2 {( p + θ max{a, b}) + p − θ min{r, h} + θ max σ1 , + θ max{a, b}U (X ) + p + + σ2 δ2 , δ1 , α1 α2 U (X ) ω θω − θ r − , h α3 α2 θ (θ + 1) σ2 δ2 max σ1 , , δ1 , α1 α2 U (X ) U (X ) From (3.2), we know that there exists a positive constant S such that L W (X ) ≤ Se pt , which implies that E[e pt (1 + U (X ))θ ] ≤ (1 + U (0))θ + S(e pt − 1) p Therefore, lim sup E[U θ (X (t))] ≤ lim sup E[(1 + U (X (t))θ ] ≤ t→∞ t→∞ S p Note that θ (x + y)θ ≤ 2θ (x + y ) = 2θ |X |θ , where X = (x, y) ∈ R2+ Consequently, lim sup E t→∞ |X (t)|θ ≤ 2θ lim sup EU θ (X ) ≤ 2θ t→∞ S := M p 123 Y Zhang et al Thus, for any ε > 0, let δ = ( Mε ) θ , by Chebyshev inequality, we can obtain that P{|X (t)| < δ} = P{|X (t)|−θ > δ −θ } ≤ E[|X (t)|−θ ]/δ −θ = δ θ E[|X (t)|−θ ] that is, lim inf P{|X (t)| ≥ δ} ≥ − ε t→∞ Also, we can obtain that for any ε > 0, there exists χ > 0, such that lim inf P{|X (t)| ≤ χ } ≥ − ε Here we omit the proof Theorem is proved t→∞ 3.2 Extinction In the previous section, we have shown that the positive solutions of system (1.2) will not explode to infinity in a finite time and will be ultimately bounded and permanent However, in this subsection, we will show that if the noise is sufficiently large, the species will become extinct with probability one Theorem Assume (H2) holds, then for any given initial value (x0 , y0 ) ∈ R2+ , the solution (x(t), y(t)) of system (1.2) will be extinct with probability one Proof Let V3 = lnx, V4 = lny, then by the Itˆo’s formula, we can derive from (1.2) that r −ax − dlnx = + σ1 + σ2 y + α1 x + α2 y + α3 x y h −by + dlny = + δ1 + ωy σ2 y −0.5 σ1 + 1+α1 x +α2 y +α3 x y 1+α1 x +α2 y +α3 x y dt d B1 (t) f ωx δ2 x −0.5 δ1 + 1+α1 x +α2 y +α3 x y 1+α1 x +α2 y +α3 x y δ2 x + α1 x + α2 y + α3 x y 2 dt d B2 (t) Hence, integrating them from to t, yields lnx(t) = lnx0 + (r − 0.5σ12 )t − + ax + ωy + α1 x + α2 y + α3 x y σ1 σ2 y σ2 y + 0.5 + α1 x + α2 y + α3 x y + α1 x + α2 y + α3 x y t + 123 t σ1 d B1 (s) + M1 (t) ds (3.3) On the dynamics of a stochastic ratio-dependent predator–prey model t lny(t) = lny0 + (h − 0.5δ12 )t − + by − f ωx + α1 x + α2 y + α3 x y δ2 x δ1 δ2 x + 0.5 + α1 x + α2 y + α3 x y + α1 x + α2 y + α3 x y t + δ1 d B2 (s) + M2 (t) ds (3.4) σ2 y d B1 (s), M2 (t) = where M1 (t) = 1+α1 x+α y+α3 x y martingales and the quadratic variations are t t < M1 (t), M1 (t) > = t < M2 (t), M2 (t) > = t δ2 x 1+α1 x+α2 y+α3 x y d B2 (s) σ2 y + α1 x + α2 y + α3 x y δ2 x + α1 x + α2 y + α3 x y are ds, ds By Virtue of the exponential martingale inequality(see [20]), for any positive constants T, α and β, we have P sup 0≤t≤T M(t) − α < M(t), M(t) > > β ≤ e−αβ Choose T = n, α = 1, β = 2lnn, we get P sup 0≤t≤n P sup 0≤t≤n < M1 (t), M1 (t) > M2 (t) − < M2 (t), M2 (t) > M1 (t) − n2 > 2lnn ≤ n > 2lnn ≤ An application of Borel-Cantelli lemma (see [20]) yields that for almost all ω ∈ there is a random integer n = n (ω) such that for n ≥ n sup 0≤t≤n sup 0≤t≤n < M1 (t), M1 (t) > M2 (t) − < M2 (t), M2 (t) > M1 (t) − , ≤ 2lnn ≤ 2lnn That is to say, t M1 (t) ≤ 2lnn + 0.5 t M2 (t) ≤ 2lnn + 0.5 σ2 y + α1 x + α2 y + α3 x y δ2 x + α1 x + α2 y + α3 x y ds, ds 123 Y Zhang et al for all ≤ t ≤ n, n ≥ n a.s Substituting the above inequality into Eq (3.3) and (3.4),respectively, and we get that lnx(t) ≤ lnx0 + (r − 0.5σ12 )t + lny(t) ≤ lny0 + h + t σ1 d B1 (s) + 2lnn, fω − 0.5δ12 t + α1 t δ1 d B2 (s) + 2lnn It finally follows from (3.3) and (3.4) by dividing t (0 < n − ≤ t ≤ n) on the both side and letting t → ∞ that lim sup t→∞ fω lnx(t) lny(t) ≤ r − 0.5σ12 , lim sup ≤h+ − 0.5δ12 a.s t t α1 t→∞ By assumption (H2 ), the desired assertion is derived Numerical simulation and discussion Numerical verification of the results is necessary for completeness of the analytical study In this section, we present some numerical simulations to substantiate and augment our analytical findings of system (1.2) by means of the Milstein method mentioned in Higham [27] From model (1.2), we consider the discretization equations: xk+1 = xk + xk r − axk − + xk σ1 yk+1 √ tξk + ωyk + α1 xk + α2 yk + α3 xk yk t √ σ12 σ2 yk (ξk − 1) t + tξk + α1 xk + α2 yk + α3 xk yk yk (ξk2 − 1) t , + α1 xk + α2 yk + α3 xk yk f ωxk t = yk + yk h − byk + + α1 xk + α2 yk + α3 xk yk √ √ δ2 δ2 xk + yk δ1 tηk + (ηk2 − 1) t + tηk + α1 xk + α2 yk + α3 xk yk + σ22 + δ22 xk + α1 xk + α2 yk + α3 xk yk (ηk2 − 1) t (4.1) where time increment t > 0, and ξk , ηk , k = 1, 2, , n are independent Gaussian random variables N (0, 1) which can be generated numerically by pseudorandom number generators In order to understand their role on the dynamics, we use different values of σ1 , σ2 , δ1 and δ2 In all the following figures, the imaginary lines and the real lines in the figures represent solutions of the deterministic system (1.1) and the stochastic system (1.2), respectively 123 6 5 4 y(t) x(t) On the dynamics of a stochastic ratio-dependent predator–prey model 3 2 1 0 20 40 60 80 0 100 20 40 t 60 80 100 t 6 5 4 y(t) x(t) Fig The trajectories of deterministic system (1.1) and stochastic system (1.2) with σ1 = 0.3, σ2 = 0.2, δ1 = 0.2, δ2 = 0.3 3 2 1 0 20 40 60 80 100 20 40 t 60 80 100 t 6 5 4 y(t) x(t) Fig The trajectories of deterministic system (1.1) and stochastic system (1.2) with σ1 = 0.1, σ2 = 0.1, δ1 = 0.1, δ2 = 0.1 3 2 1 0 20 40 60 t 80 100 0 20 40 60 80 100 t Fig The trajectories of deterministic system (1.1) and stochastic system (1.2) with σ1 = 0.3, σ2 = 0.02, δ1 = 0.2, δ2 = 0.03 In Figs 1, 2, 3, 4, 5, 6, we always choose initial values x(0) = 0.3, y(0) = 0.2 and parameters r = 2, a = 1, ω = 1, f = 0.5, h = 1, b = 2, α1 = 4, α2 = 5, α3 = The only difference between Figs and is the different intensities of white 123 6 5 4 y(t) x(t) Y Zhang et al 3 2 1 0 20 40 60 80 100 20 40 t 60 80 100 t Fig The trajectories of deterministic system (1.1) and stochastic system (1.2) with σ1 = 0, σ2 = 0.2, δ1 = 0, δ2 = 0.3 y(t) 0 0.5 1.5 2.5 3.5 x(t) Fig The stationary distribution of stochastic population system (1.2) with σ1 = 0.3, σ2 = 0.2, δ1 = 0.2, δ2 = 0.3 noise which satisfy condition (H1) By Matlab software, we simulate the solution of model (1.2) with different values of σ1 , σ2 , δ1 and δ2 and the solution of model (1.1) In Fig 1, we choose σ1 = 0.3, σ2 = 0.2, δ1 = 0.2, δ2 = 0.3 and in Fig 2, we choose σ1 = 0.1, σ2 = 0.1, δ1 = 0.1, δ2 = 0.1 From the Figs and 2, we can see that the curves of model (1.2) always fluctuate around the curves of the deterministic system (1.1) and moreover, by comparisons of the two figures, we can also obtain that the larger the intensities of the white noises are, the larger the fluctuations of the solutions will be Furthermore, in Fig 3, we keep σ1 , δ1 the same value as in Fig 1, but take σ2 , δ2 smaller values σ2 = 0.02, δ2 = 0.03 Then we can observe that the fluctuations will be very smaller, which suggest that the intensities of σ2 , δ2 have little effect on the fluctuation Next, in Fig 4, we choose σ1 = δ1 = 0, and from Figs and 4, we can see 123 6 5 4 y(t) x (t) On the dynamics of a stochastic ratio-dependent predator–prey model 3 2 1 0 20 40 60 t 80 100 0 20 40 60 80 100 t Fig The solution of the stochastic system (1.2) with σ1 = 2.3, σ2 = 0.2, δ1 = 2, δ2 = 0.3 The system is stochastic extinction that the fluctuation for x(t) and y(t) almost can not be seen, then we can conjecture that the model (1.2) is stochastically stable in the large Moreover, the stationary distribution and solutions of stochastic population system (1.2) are also given in Fig 5, from which we can also obtain the stochastic permanence of system (1.2) At last, in Fig 6, we choose the intensities of white noise as σ1 = 2.3, σ2 = 0.2, δ1 = 2, δ2 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Zhang et al When investigating biological phenomena, there are many factors which affect dynamical properties of biological and mathematical models One of the familiar nonlinear factors is functional. .. incorporate white noise in each equations of the system (1.1) and we assume that fluctuations in environment will manifest themselves mainly as fluctuations in the growth rate and capturing rate of the. .. fluctuation for x(t) and y(t) almost can not be seen, then we can conjecture that the model (1.2) is stochastically stable in the large Moreover, the stationary distribution and solutions of stochastic