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Stochastic Dynamics of a Time-Delayed Ecosystem Driven by Poisson White Noise Excitation Wantao Jia 1, *, Yong Xu 1, * and Dongxi Li 2 1 2

*

Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an 710072, China College of Big Data Science, Taiyuan University of Technology, Taiyuan 030024, China; [email protected] Correspondence: [email protected] (W.J.); [email protected] (Y.X.); Tel.: +86-29-8843-1660 (W.J.)

Received: 16 December 2017; Accepted: 12 February 2018; Published: 23 February 2018

Abstract: We investigate the stochastic dynamics of a prey-predator type ecosystem with time delay and the discrete random environmental fluctuations. In this model, the delay effect is represented by a time delay parameter and the effect of the environmental randomness is modeled as Poisson white noise. The stochastic averaging method and the perturbation method are applied to calculate the approximate stationary probability density functions for both predator and prey populations. The influences of system parameters and the Poisson white noises are investigated in detail based on the approximate stationary probability density functions. It is found that, increasing time delay parameter as well as the mean arrival rate and the variance of the amplitude of the Poisson white noise will enhance the fluctuations of the prey and predator population. While the larger value of self-competition parameter will reduce the fluctuation of the system. Furthermore, the results from Monte Carlo simulation are also obtained to show the effectiveness of the results from averaging method. Keywords: predator-prey ecosystem; time delay; stochastic averaging; Poisson white noise

1. Introduction In the real-world ecosystem, a time delay effect in the ecosystems is inevitable. Such as, in the predator-prey type ecosystem, it takes time for the predator population to adjust any change of the prey population. Thus, the ecosystems with time delay effects have received more and more attentions in past decades. Several types of models considering the time delay effects are introduced. The dynamical properties of the ecosystem with time delay, such as stability and bifurcations, have been studied [1–7]. It indeed can be found that the effect of the time delay can change the behavior of the ecosystem substantially. However, the models in all these studies are deterministic. They are too idealistic since the changes in the environment are not taken into consideration. As a matter of fact, random perturbation is universal in the real world. Thus, the stochastic excitations describing the fluctuations in the environment are added to the deterministic models. The dynamics of the ecosystem under stochastic excitations have been widely investigated [8–20]. Especially, the stochastic dynamics of the noise-induced predator-prey type ecosystems with time delay, including stochastic responses [9,19,21,22], stochastic stability [23], stochastic resonance [16,24] and noise-delayed extinction [16], have been investigated by some numerical and analytical methods. It can be noted that the random perturbations in the environment in the previously mentioned works are usually modelled as continuous stochastic processes, such as the Gaussian white noise or Gaussian colored noise. However, the perturbations in the real-world environment are complicated. There are some unavoidable sparse, drastic changes in the environment, for instance sudden natural disasters, earthquakes, forest fires, floods [25–27]. These changes are pulse-type perturbations, which cannot be characterized by the continuous stochastic processes properly. Therefore, the stochastic jump

Entropy 2018, 20, 143; doi:10.3390/e20020143

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processes, such as Poisson white noise, are considered to model such pulse-type perturbations [28,29]. So far, studying the stochastic dynamics of ecosystem with stochastic jump process has attracted more and more attention [30–34]. For example, Zhu and his coauthors have investigated the Lotka–Volterra (LV) system under Poisson white noise by using the generalized cell mapping method [32] and the stochastic averaging method [33]. Duan and Xu have studied the stochastic stability of a logistic model subjected to Poisson white noise process via the Lyapunov exponent [34]. People can find that the influence of Poisson white noise on the ecosystem is different from those of continuous stochastic processes on ecosystem. However, the ecosystems studied in these works are all systems without time delay effect. Consequently, it is necessary to study the stochastic dynamics of time-delayed ecosystems driven by stochastic jump processes. Inspired by this, we study the stochastic behaviors of a time-delayed predator-prey ecosystem model under discrete random environmental fluctuations. The fluctuations are modeled as Poisson white noises. Due to the existence of the Poisson white noises, it is difficult to obtain the probability density functions (PDFs) of the species populations since the related dynamical equations contain infinite terms. Thus, it is necessary to develop some methods to study this problem. In the present paper, a procedure to calculate the stationary PDFs is formatted theoretically. The stochastic averaging procedure is first applied to deduce the averaged generalized Fokker-Planck-Kolmogorov (GFPK) equation governing the PDF of the first integral. Then, the stationary PDFs of prey and predator populations are obtained by using a perturbation method. Finally, the influences of the time delay, self-competition parameter and the Poisson white noises on the stochastic behaviors of the ecosystem are discussed based on the stationary PDFs. In addition, the results from Monte Carlo simulation are also carried out to show the effectiveness of proposed method. 2. Delayed-Type Predator-Prey System with Poisson White Noises 2.1. The Deterministic Model with Time Delay Terms Without considering the random fluctuations in the environment, the time-delayed predator-prey type ecosystem can be described by following integral-differential equation [9]: .

x1 = a1 x1 − sx12 − bx1 x2 , Rt . x2 = −cx2 + f x2 −∞ F (t − τ ) x1 (τ )dτ.

(1)

The second equation of Equation (1) can also be written as following equivalent form [3] .

x2 = −cx2 + f x2

Z ∞ 0

F (τ ) x1 (t − τ )dτ.

(2)

where x1 and x2 are the population densities of prey and predator, respectively. The parameters a1 , s, b, c and f are positive constants. a1 is the birth rate of the preys. c is the death rate of the predators. Rt The term −sx12 denotes the prey self-competition. The terms −bx1 x2 and f x2 −∞ F (t − τ ) x1 (τ )dτ provide a balance between the prey and predator populations. The integral term represents the time delay effect of the prey population on the predator population. It means that the change of the prey population affects the predator population after a time lag. And this time delay depends on an average over past populations, not only on the population at some particular instant in the past. F (·) in the integrand is called the delay function and normalized as follows: Z ∞ 0

F (t)dt = 1.

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Macdonald has presented two ways to reduce the system (1) in [3]. The first one is exact, but, it depends on the choice of a simple form of the time delay function F (·). The other one does not need to specify the time delay function, but needs its moments, i.e.: γ=

Z ∞ 0

τF (τ )dτ

(3)

Equation (3) can be viewed as a measure of the average delay time. Two reasonable choices for F (t) provieded by Cai and Lin [9] are: F (t) =

1 −t/γ 4 e and F (t) = 2 te−2t/γ . γ γ

To avoid specifying time delay function, the second way is used to reduce the system (1) in the present paper. As pointed out in [3], this method is valid for small γ. Substituting the first-order approximation of the Taylor expansion of x1 (t − τ ) about τ = 0 into Equation (2) and applying Equation (3), the first-order approximation of Equation (2) can be derived as: .

.

x2 = −cx2 + f x1 x2 − f γx2 x1

(4)

Together with the first equation of Equation (1), the original Equation (1) can be rewritten as: .

x1 = a1 x1 − sx12 − bx1 x2 . x2 = −cx2 + f (1 − γa1 ) x1 x2 + s f γx12 x2 + b f γx1 x22

(5)

Let a = a1 − sc/ f . Then the system can be converted to: .

x1 = x1 [ a − bx2 − sf ( f x1 − c)] .

x2 = x2 [(−c + f x1 )(1 + sγx1 ) + f γx1 (bx2 − a)]

(6)

The system (6) can be used to investigate the effect of the time delay on the populations of the prey and predator. It is easy to check that system (6) has the same equilibrium point x10 = c/ f , x20 = a/b as that of the Lotka–Volterra model without time-delay and stochastic excitations [8,9]. In addition, it is pointed in Cai and Lin’s work [9] that, for a physically meaningful and stable ecological system, the choice of the parameters must make the ecosystem stable. All the parameters in the present paper meet this requirement. 2.2. Stochastic Model In the present paper, we suppose that there are some unavoidable sparse, drastic changes in the environment. These changes are modeled as the Poisson white noises. They will cause random jump in the prey growth rate and the predator death rate. Thus, the stochastic model has following form: h i . X 1 = X1 a − bX2 − sf ( f X1 − c) + X1 W1 (t) .

X 2 = X2 [(−c + f X1 )(1 + sγX1 ) + f γX1 (bX2 − a)] + X2 W2 (t)

(7)

where W1 (t) and W2 (t) are two independent Poisson white noises, which can be defined as [35,36] Ni (t)

Wi (t) =

∑

Yik δ(t − tik ), i = 1, 2

(8)

k =1

where the δ(•) is the Dirac delta function; Ni (t) is a Poisson counting process with mean arrival rate λi > 0 and gives the number of the pulses arriving in the time interval [0, t]; Yik represents the

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amplitude of the Poisson white noises, which is independent of the pulse arrival time tik . In the present paper, Yik is Gaussian distribution with mean value 0 and variance E[Yi2 ] [32,33]. It is hard to obtain the exact solution of Equation (7) analytically, even for the Gaussian white noise case. Therefore, some methods have been developed to solve this problem. Among which, the stochastic averaging method is an analytical efficient method. It has been successfully applied to analyze the stochastic behaviors of the ecosystem under stochastic continuous excitations [18,19,22,37]. In recent years, the stochastic averaging method has been generalized to investigate the nonlinear system with random jump excitations [38–41]. In the following section, we will employ the stochastic averaging method to study stochastic dynamics of the ecosystem (7). To proceed further analysis, the following assumptions about the system are in order. First, the coefficient s in self-competition term is generally small. It means that the self-competition term has small influence when the prey population density is small. Second, the parameter γ is small, corresponding to shorter time-delay effect. Finally, the noise intensities of W 1 (t) and W 2 (t) are small, indicating small random fluctuations in the system dynamics. These assumptions are generally valid for the real-world ecosystem [8,9]. For the convenience of the further theoretical analysis, in the following part, the parameter s, γ and λi E[Yi2 ] in Equation (7) are replaced by: ε2 s, ε2 γ and λi ε2 E[Yi2 ], respectively, where ε is a small parameter. Thus, Equation (7) can be rewritten as: h i . 2 X 1 = X1 a − bX2 − ε f s ( f X1 − c) + εX1 W1 (t) . 
 X 2 = X2 (−c + f X1 ) 1 + ε4 sγX1 + ε2 f γX1 (bX2 − a) + εX2 W2 (t)

(9)

Equation (9) is usually modeled as the Stratonovich stochastic differential equation (SDE) [9,13,18]. By adding some correction terms [13,35,36], it can be converted to following equivalent Itô SDE: h dX1 = X1 a − bX2 −

ε2 s f ( f X1

i ∞ − c) dt + X1 dC1 (t) + X1 ∑


dX2 = X2 (−c + f X1 ) 1 + ε4 sγX1 + ε2 f γX1 (bX2 − ∞

The terms X1 ∑

i =2

i 1 i! ( εdC1 ( t ))

∞

and X2 ∑

i =2

i 1 i! ( εdC2 ( t ))

i 1 i! ( εdC1 ( t )) i =1 ∞  a) dt + X2 ∑ i!1 (εdC2 (t))i i =1

(10)

in Equation (10) are the correction terms from

Stratonovich SDE to Itô SDE. Then, Equation (10) can be converted to following equivalent stochastic integro-differential equation (SIDE) [42]: h i R 2 dX1 = X1 a − bX2 − ε f s ( f X1 − c) dt + Y h11 P1 (dt, dY1 ) 1 R 
 dX2 = X2 (−c + f X1 ) 1 + ε4 sγX1 + ε2 f γX1 (bX2 − a) dt + Y h22 P2 (dt, dY2 )

(11)

2

in which:

∞

∞ i ε εi (Y1 )i ; h22 = X2 ∑ (Y2 )i i! i! i =1 i =1

h11 = X1 ∑

(12)

and P1 (dt, dY1 ) (i = 1, 2) are the Poisson random measures; Yi (i = 1, 2) are Poisson mark spaces [42]. 2.3. Stochastic Averaging Approach In this subsection, a stochastic averaging procedure is derived to simplify the original system. For future use, we choose the following deterministic conservative system for the stochastic averaging [9,18]: .

x1 = x1 ( a − bx2 )
. x2 = x2 (−c + f x1 ) 1 + ε4 sγx1

(13)

x 1  x1  a  bx2 



x 2  x2  c  fx1  1  ε4 sγx1

(13)



Entropy 2018, 20, 143 (13) The system

5 of 15 has the same equilibrium point as that of system (9) without the Poisson white noises. System (13) possesses a first integral [9]

4 The system (13) has the sameequilibrium point asthat (9) εwithout the fx1  bx2 of  system ε4Poisson c 2 sγ white 4 2 r x x fx c c bx a a csγx fsγx ,    ln    ln  ε   (14)  integral  [9]   2 possesses 1 2 1 1 noises. System1 (13) a first





 c 

 a 

2f

2

    ε4 c2 sγ bx2integral ε4 fx This integral anbx extension of first for 1the f sγx12 +LV system [8,43]. (14) r ( x1 , first x2 ) = − cbe lnview1as + − ε4 csγx + standard f x1 − ccan 2 − a − a ln 2 f constant It can be seen that r  x 1 , x 2   0 cat the equilibrium pointa  c f , a b  . And2for each positive

K , rThis a periodic the for equilibrium point. illustrative can be view as antrajectory extensionsurrounding of first integral the standard LVFor system [8,43]. x 2  integral K represents  x1 ,first f , a/b). And It can be seen that r ( x1 , x2the 0 at the equilibrium point for each positive (c/different ) =equilibrium purpose, Figure 1 shows point O and three periodic trajectories forconstant system K, r ( x1 , x2 ) = K represents a periodic trajectory surrounding the equilibrium point. For illustrative 2 2 (13) with ε  0.1 , a  0.9 , b  1.0 , c  0.5 , f  0.5 , ε s  0.2 , ε γ  0.2 . In this figure, the purpose, Figure 1 shows the equilibrium point O and three different periodic trajectories for system (13) equilibrium point corresponds andε2the correspond to K the K f = 0 ,0.5,  0.892 ，1.146 , 1.565 with ε = 0.1, a = 0.9, b = 1.0, c =to0.5, s =trajectories 0.2, ε2 γ = 0.2. In this figure, equilibrium point . And the period trajectory can be determined corresponds to K of = the 0, and the trajectories correspondfrom to K = 0.892, 1.146, 1.565. And the period of the trajectory can be determined from

T K =



T (K ) =

I

K

dx1 4 dx2 K K x dx a 1 bx2  dt = x2  c  fx1  1  ε4 sγx1 = 1

dt   I

K

K

dx2

  x (−c + f x )(1 + ε sγx ) 2

1

1

 I  K

(15) (15)

x1 ( a − bx2 )





 K. integration symbol symbol means means integration integration along along the the trajectory trajectory r (rx1x in which the integration , x1 ,2 )x2= K. 3.5 3

r=1.565

2.5 r=1.146

x2

2 r=0.892

1.5 1

O

0.5 0 0

1

2

3

4

x1 (13). Figure 1. Equilibrium and periodic trajectories of the deterministic system (13).



Replace xx11 and processes X X11(tt) and stochastic and xx bythe thestochastic stochastic processes andXX , the stochasticcounterpart counterpart of 2 2by 2 (2t ),t the Equation (14) R(t) = R( X1 (t), X2 (t)) can be given as: Equation (14) R  t   R X 1  t  , X 2  t  can be given as:     bX2 ε44 ε44c22sγ f X1 4 2 − ε csγX + f sγX + (16) R(t) = f X1 − c − c ln c fX1+ bX2 − a − a ln  bX  ε ε c sγ 1 1 2 4 2 R t  fX1  c  c ln  (16)   bX2  a  a ln  a   ε csγX1  2 fsγX1  2 f







 c 

 a 

2

2f

Based on the stochastic jump-diffusion chain rule [42] and Equation (11), the SIDE for R(t) is: i Based on thehstochastic jump-diffusion chain rule [42] and Equation (11), the SIDE for
2 dR = − ε f s ( f X1 − c)2 1 + ε4 sγX1 + ε2 f γX1 (bX2 − a)2 dt  o R n

4 2 − c ln 1 + h11 + Y f − ε4 csγ h11 + ε2 f sγ 2X1 h11 + γ11 P1 (dt, dY1 ) X1 1  o R n + Y bh22 − a ln 1 + hX222 P2 (dt, dY2 ) 2

R t

is:

(17)

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In order to possess the stochastic averaging method, consider the Taylor expansion of ln(1 + h11 /X1 ) and ln(1 + h22 /X2 ) in Equation (17), and then substitute Equation (12) to Equation (17). After collecting the terms of same order of ε, Equation (17) can be written as: dR =

i
4 − ε f s ( f X1 − c)2 1 + ε4 sγX1 + ε2 f γX1 (bX2 − a)2 dt i R h + Y εA11 Y1 + ε2 A12 Y21 + ε3 A13 Y31 + ε4 A14 Y41 + · · · P1 (dt, dY1 ) 1 i R h + Y2 εA21 Y2 + ε2 A22 Y22 + ε3 A23 Y32 + ε4 A24 Y42 + · · · P2 (dt, dY2 ) h

where:

  A11 = ( f X1 − c) ε4 sγX1 + 1 ;    i 1h ( f X1 − c) ε4 sγX1 + 1 + ε4 f sγX12 + c ; 2   1  c  1 4 4 2 A13 = ε f sγX1 + c − ; ( f X1 − c) ε sγX1 + 1 + 6 2 3       7 4 c 1 4 2 ε f sγX1 + c − ; A14 = ( f X1 − c) ε sγX1 + 1 + 24 24 4 A12 =

1 [(bX2 − a) + a]; 2 1 1 = [(bX2 − a) + a]; A24 = [(bX2 − a) + a]. 6 24

A21 = (bX2 − a); A22 = A23

(18)

(19) (20) (21) (22) (23) (24)

It can be seen from Equation (18) that R(t) is a slowly varying stochastic process since ε is a small parameter. According to the stochastic averaging method for the nonlinear system under the Poisson white noise [33], the averaged GFPK equation can be derived as: ∂ ∂t p (r, t )





∂2 ε2 A11 (r ) + ε4 A12 (r ) p(r, t) + 2!1 ∂r ε2 A21 (r ) + ε4 A22 (r ) p(r, t) 2


3 4 − 3!1 ∂r∂ 3 ε4 A3 p(r, t) + 4!1 ∂r∂ 4 ε4 A4 p(r, t) + O ε6

= − ∂r∂

(25)


where O ε6 contains the term of the order of ε6 and higher, and the coefficients for the GFPK equation are given in Appendix A. Note that the SIDE (18) contains infinite terms. To get the closed form of averaged equations, truncation is needed during the averaging process. In the averaged GFPK Equation (25), only the terms up to the fourth order of ε are considered for simplicity. 2.4. Stationary Probability Density Functions The averaged GFPK equation can be solved with certain boundary and initial conditions. The stationary PDF of R(t), denoted by p(r ), can be obtained by solving the reduced GFPK equation: 0=





∂2 ε2 A11 (r ) + ε4 A12 (r ) p(r ) + 2!1 ∂r ε2 A21 (r ) + ε4 A22 (r ) p(r ) 2


3 4 − 3!1 ∂r∂ 3 ε4 A3 (r ) p(r ) + 4!1 ∂r∂ 4 ε4 A4 (r ) p(r ) + O ε6

− ∂r∂

(26)

with the following conditions: p(r )|r→∞ = 0 and

Z ∞ 0

p(r )dr = 1.

(27)

The perturbation method [33,44] is used to solve the reduced GFPK equation. The second order perturbation solution p(r ) = p0 (r ) + εp1 (r ) + ε2 p2 (r ) is adopted in our calculation. On substituting

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the second order perturbation solution into Equation (27) and grouping terms of the same order of ε, the following set of differential equations is obtained: In ε2 ,
1 d2
d − A11 p0 (r ) + A21 p0 (r ) = 0 (28) 2 dr 2 dr In ε3 ,


1 d2
d εA11 p1 (r ) + εA21 p1 (r ) = 0 dr 2 dr2

(29)

 1 d2  

d 2 1 d3 1 d4 2 ε A11 p2 (r ) + ε A p r = A p r A4 p0 (r ) − ( ) ( ) 2 3 0 21 2 3 4 dr 2 dr 3! dr 4! dr

(30)

− In ε4 ,

−

Equation (28) is a Fokker-Planck-Kolmogorov equation for Gaussian white noise excitation. The exact solution can be obtained if the FPK Equation (28) belongs to the class of generalized stationary potential [39]. The perturbation solutions p0 (r ), p1 (r ) and p2 (r ) can be obtained by solving Equations (28)–(30) successively. After getting p(r ), the stationary PDFs of X1 and X2 can be calculated as follows [9,13]: p X1 X2 ( x 1 , x 2 ) =

p (r ) , x1 x2 T (r )

p X2 ( x 2 ) =

p X1 ( x 1 ) =

R∞ 0

R∞ 0

p X1 X2 ( x1 , x2 )dx2 ,

(31)

p X1 X2 ( x1 , x2 )dx1

where r is the function of x1 and x2 given in Equation (14) and T (r ) is the quasi-period with the form in Equation (15). The other statics can be easily derived from Equation (31). 3. Results Some numerical results are presented in Figures 2 and 3 for ecosystem (7) with different noise intensities. The parameters are shown in the captions of these figures. In these figures, the solid lines are the results of ecosystem under Poisson white noises obtained by the proposed method. The dotted lines are the results from Monte Carlo simulation. A good agreement between the theoretical and simulation results in both figures shows the accuracy of the proposed method. For the purpose of comparison, the stationary PDFs for the time-delayed ecosystem under Gaussian white noises with the same intensity are also depicted in these figures. They are denoted by the dashed lines. One can see that the approximate stationary PDFs for time-delayed ecosystem under Poisson white noise are higher than those for system under Gaussian white noise. It means that, for the same noise intensity, the influence of the Gaussian white noises on the ecosystem is stronger than that of the Poisson white noises. In the Monte Carlo simulation for ecosystem (7), the Runge-Kutta method for the Poisson white noise excitation proposed by Di Paola and the triangular pulse model of the Poisson white noise excitation are adopted [35]. It is important to note that the numerical simulation time should be long enough to get the stationary responses. In present paper, for each set of system parameters, eight samples are simulated. The time of Monte Carlo simulation is 400,000,000 units for each sample, and the time step length is 0.001. Cai and Lin [8] show that the stability of system (6) depends on the prey self-competition parameter and the time-delay parameter. Thus, in the following sections, we will investigate the influences of the parameters ε2 s and ε2 γ on the ecosystem (7). The stationary PDFs and the statics of prey and predator populations for different values of these two parameters are given and discussed in following sections.

In theare Monte Carlo[35]. simulation for ecosystem method for theshould Poissonbewhite excitation adopted It is important to note(7), thatthe theRunge-Kutta numerical simulation time long noise excitation proposed by Di Paola and the triangular pulse model of the Poisson white noise enough to get the stationary responses. In present paper, for each set of system parameters, eight excitation aresimulated. adopted [35]. is important notesimulation that the numerical simulation time should be long samples are The It time of Monte to Carlo is 400,000,000 units for each sample, and enough to get the stationary responses. In present paper, for each set of system parameters, eight the time step length is 0.001. samples are20, simulated. The time of Monte Carlo simulation is 400,000,000 units for each sample,8 and Entropy 2018, 143 of 15 the time step length is 0.001.

Figure 2. Probability density functions of prey and predator populations for parameters: ε =0.1 , , b  1.0 , f  0.5 , c  0.5 , ε s  0.2 , ε γ  0.1 , λ1  λ2  0.3 , a  0.9 Probability Figure Figure 2. 2. Probability density density functions functions of of prey prey and and predator predator populations populations for for parameters: parameters: εε =0.1 = 0.1,, 2 2 2 2 2 2 2 2 2 2 2 . 2 λ ε E [ Y ]  λ ε E [ Y ]  0.015 aa = = 0.5,,2c =f 0.5, ε s =, 0.2, γ = 0.1, ε E[Y ε E[Yλ 1 0.9, 2 = 0.3,  0.5 , λε1 s=λ0.2 , λ 0.9 b 1=, 1.0, b 2f 1.0 c ε 0.5 ε1 γ 0.1 2] =  0.015. 0.3 , 1 ] =, λ2λ 2

2

2

1

2

2

2

2

λ1ε E[Y1 ]  λ 2 ε E[Y2 ]  0.015 .

2.5 Poisson result Gaussian result Monte Carlo Poisson result Gaussian result Monte Carlo

2.5 2 2 1.5 1.5 1 1 0.5 0.5 0 0

0

0.5

1

1.5

2

2.5

1.5

2

2.5

x2 0

0.5

1

Probability density predator populations populations for x2 parameters: Probability density functions functions of of prey prey and and predator for parameters: εε =0.1 = 0.1,, 2 2 2 2 2 2 2 2 =, 1.0, = 0.5, ,c =f 0.5, ε s =, 0.2, ε γ0.5 = 0.1, = 0.4, [Y2λ] =0.0075.  0.5 s λ20.2 , λε1 = , λε1 εγE[Y0.1 b f 1.0 c 0.4 , 1 ] = ,λ2 ελ1E 2 ε =0.1 , Probability density functions of prey and predator populations for parameters: 2 2 . 2 E[Y12 ], λb2ε2 E [Y22 ] , 0.0075 f Parameter  0.5 , cε2 aλ1εEffects 0.9 1.0 3.1. The of the Time Delay γ 0.5 , ε s  0.2 , ε γ  0.1 , λ1  λ2  0.4 , Figure 3. 3. Figure aa= 0.9, 0.9b Figure 3.

. λ1ε2this Eand [Y12subsection, ]Lin  λ2[8] ε2 E[show Y22 ]we  0.0075 In investigate the of influence time delay theself-competition ecosystem (7). Cai that the stability system of (6) the depends on theon prey Figures 4 and 5 the show the effectsparameter. of the timeThus, delayinparameter on the PDFs and relative fluctuations parameter and time-delay the following sections, we will investigate the p Cai [8]the show of system (6) depends on the prey self-competition 2 the stability preythat population and predator population. Xi ) of Var )/E(Lin ( Xiand influences of the parameters ε s and ε2 γ on the ecosystem (7). The stationary PDFs and the statics parameter time-delay parameter. thepopulation following sections, we predator will investigate the Figureand 4a,bthe depict the stationary PDFsThus, of thein prey p X ( x1 ) and population 1

2 γ, respectively. In these figures, the solid lines are the p X2 ( x2 ) forofdifferent values of ε2 stime influences the parameters anddelay the ecosystem (7). The stationary PDFs and the statics ε2 γ εon theoretical results obtained by the proposed method. It can be found that, with increasing the time delay ε2 γ from 0.05 to 0.12, the peak values of both p X1 ( x1 ) and p X2 ( x2 ) become lower, and the probabilities in both lower and higher population become higher. Besides, the results from Monte Carlo simulation are also plotted to show the effectiveness of the proposed method. The other parameters of the system are given as:ε = 0.1, b = 1.0, c = 0.5, f = 0.5, λ1 = λ2 = 1.0, ε2 λ1 E[Y12 ] = ε2 λ2 E[Y22 ] = 0.015. Figure 5 shows the relative fluctuations of the prey population and predator population changing versus the values of the time delay parameter ε2 γ for different self-completion parameter ε2 s. In this figure, the results obtained by the Monte Carlo simulation are represented by point lines, while the results obtained by the proposed method are shown by solid or dashed lines. Each curve of relative fluctuation increases monotonously with the increase of time delay. It means that the fluctuations of the prey population and predator population increase when the time delay is relatively large. Namely,

of prey prey and and predator predator populations populations for for different different values values of of these these two two parameters parameters are are given given and and discussed discussed of in following following sections. sections. in

3.1. The The Effects Effects of of the the Time Time Delay Delay Parameter Parameter εε22γγ 3.1.

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In this this subsection, subsection, we we investigate investigate the the influence influence of of the the time time delay delay on on the the ecosystem ecosystem (7). (7). Figures Figures 44 In and 55 show show the the effects effects of of the the time time delay delay parameter parameter on on the the PDFs PDFs and and relative relative fluctuations fluctuations and the longer time delay will lead to a wider range distribution of the population, indicating that the Var X Xii becomes Xii less of the the prey population population and and predator predator population. population. Var EE X of prey ecosystem stable.

11

pX (x 1) X 1

   

Figure 4. 4. Probability Probability density density functions functions of of prey prey population population and predator predator population population for for different different Figure Figure 4. Probability density functions of prey population and and predator population for different values 22 2 values of parameter parameter values of εε γγ.. of parameter ε γ.

22

εε2γγ 5. Relative Relativefluctuations fluctuationsversus versusthe thetime timedelay delay parameterε γ for fordifferent differentvalues valuesof ofparameter parameter Figure 5. Relative fluctuations versus the time delay parameter for different values of Figure parameter 2 ε 2s. The other parameters are the same as those in Figure 4. 2 The other other parameters parameters are are the the same same as as those those in in Figure Figure 4. 4. εε ss.. The 3.2. The Effects of the Self-Competition Parameter ε2 s Figure 4a,b 4a,b depict depict the the stationary stationary PDFs PDFs of of the the prey prey population population ppX xx11 and and predator predator population population Figure 11 In this subsection, we will investigate the influence of the Xself-competition parameter ε2 s. 2 forthe different values of time time delay respectively. In these these figures, the the solid lines are are the the ppXXFigure xx22 for In 6, stationary PDFs of prey population and predator population forsolid different values different values of delay In figures, lines εε2 γγ,, respectively. p 22 2 of ε s are given. Figure 7 shows the dependence of the relative fluctuations Var X /E X on the ( ) ( ) i ithe theoretical results results obtained obtained by by the the proposed proposed method. method. It It can can be be found found that, that, with with increasing increasing the time time theoretical 2 s for different values of time delay parameter ε2 γ. self-competition parameter ε 22 from 0.05 0.05 to to 0.12, 0.12, the the peak peak values values of both both pXX xx11 and and ppXX xx22 become become lower, lower, and and the the delay εε γγ from delay In Figure 6, it can be seen that, for fixed of value of pother parameters, 11 22 with the increasing value of 2 s, the PDFs become εprobabilities more and the peak value higher. of the PDFs become This in both both lower lowerconcentrated, and higher higher population population become higher. Besides, the higher. results from fromimplies Monte probabilities in and become Besides, the results Monte 2 that thesimulation system will become more stable whenthe increasing ε s from 0.15. In addition, Carlo simulation are also plotted plotted to show show the effectiveness of 0.08 the to proposed method. the Theresults other Carlo are also to effectiveness of the proposed method. The other parameters ofthe the system are given as: as: , , , , obtained from Monte Carloare simulation are εεalso calculated to show the validity of the proposed  0.1 b  1.0 parameters of the system given , , , , c  0.5 λ  λ  1.0 ,,  0.1 b  1.0 0.5 c  0.5 λ11  λ22  1.0 ff  0.5 method The system parameters are: ε = 0.1, a = 0.9, b = 1.0, f = 0.5, c = 0.5, 22 22 .. λ11EE[[Y Y1122]in ]the λ22figure. Y2222]] 0.015 0.015other εε λ εε λ EE[[Y λ1 = λ2 = 1.0, ε2 E[Y12 ] = εE[Y22 ] = 0.015. Figure 7 shows the relative fluctuations of the prey population and the predator population versus the self-competition parameter ε2 s for different values of time delay ε2 γ. Compared with Figure 5, the trends of the relative fluctuations curves are different. For each value of the time delay parameter ε2 γ, the curve of the relative fluctuations of both prey population and predator population decrease monotonously with the increase of the self-competition parameter ε2 s. It reveals that the fluctuations

 

 

 

 

becomes less stable. 2

3.2. ε2 ss 3.2. The The Effects Effects of of the the Self-Competition Self-Competition Parameter Parameter ε 2

In ε2 ss .. In In this this subsection, subsection, we we will will investigate investigate the the influence influence of of the the self-competition self-competition parameter parameter ε In

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Figure ε2 ss Figure 6, 6, the the stationary stationary PDFs PDFs of of prey prey population population and and predator predator population population for for different different values values of of ε

   

(a) (a)

22

2 2s=0.08,Theoretical s=0.08,Theoretical 2 2s=0.08,Monte Carlo s=0.08,Monte Carlo 2 2s=0.10,Theoretical s=0.10,Theoretical 2 2s=0.10,Monte Carlo s=0.10,Monte Carlo 2 2s=0.12,Theoretical s=0.12,Theoretical 2 2s=0.12,Monte Carlo s=0.12,Monte Carlo 2 2s=0.15,Theoretical s=0.15,Theoretical 2 2s=0.15,Monte Carlo

=0.08 =0.08

11 0.8 0.8

s=0.15,Monte Carlo

0.6 0.6

22

1.2 1.2

(x2)) ppXX (x 2

are Var X Xii E E X Xii on on the the selfselfare given. given. Figure Figure 77 shows shows the the dependence dependence of of the the relative relative fluctuations fluctuations Var of the system will decrease when increase ε2 s, which indicates that the system becomes more stable for 22 22 competition larger ε2 s. parameter competition parameter ε s for for different different values values of of time time delay delay parameter parameter ε γ ..

0.4 0.4 0.2 0.2 00

00

0.5 0.5

11

1.5 1.5

xx1 1

22

2.5 2.5

33

3.5 3.5

Figure 6. Probability Figure 6. Probability density functions of prey and predator populations for different values values of Probability density density functions functions of of prey prey and and predator predator populations populations for for different of 2 2 2 parameter parameter εεs.s .. 0.65 0.65

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

0.6 0.6 0.55 0.55 0.5 0.5 0.45 0.45

=0.05,Theoretical =0.05,Theoretical =0.05,Monte =0.05,Monte Carlo Carlo =0.07,Theoretical =0.07,Theoretical =0.07,Monte =0.07,Monte Carlo Carlo =0.09,Theoretical =0.09,Theoretical =0.09,Monte =0.09,Monte Carlo Carlo =0.11,Theoretical =0.11,Theoretical =0.11,Monte =0.11,Monte Carlo Carlo

0.4 0.4 0.35 0.35 0.3 0.3 0.25 0.25 0.08 0.08

0.09 0.09

0.1 0.1

0.11 0.11

0.12 0.12

0.13 0.13

0.14 0.14

2

22 different values Figure self-competition different values of for different values Figure 7. 7. Relative Relative fluctuations fluctuations versus versus the the time time self-competition self-competition parameter parameter εεs sforfor 2 2 parameter ε γ. The otherother parameters are the same same as thosethose in FigureFigure 6. 2 of The other parameters parameters are are the the same as as those in in Figure 6. 6. of parameter parameter ε γ .. The

3.3. The Effects of the Poisson White Noise In In Figure Figure 6, 6, it it can can be be seen seen that, that, for for fixed fixed value value of of other other parameters, parameters, with with the the increasing increasing value value of of 2 22 It is known that the mean arrival rate λ and the variance of the noise amplitude E [ Y ] are two ε s ,, the the PDFs PDFs become become more more concentrated, concentrated, and and the the peak peak value value of of the the PDFs PDFs become become higher. higher. This This key parameters for the Poisson white noise. In the following section, the influences of λ and E[Y2 ] are investigated respectively. Figure 8 shows the effects of the mean arrival rate λ1 = λ2 = λ on the stationary PDFs. The results are calculated with following parameters: ε = 0.1, a = 0.9, b = 1.0, c = 0.5, f = 0.5, ε2 s = 0.2, ε2 γ = 0.1. It can be seen that, when fix the value ε2 E[Y21 ] = ε2 E[Y22 ] = ε2 E[Y2 ] = 0.01, with increasing the mean arrival rate λ from 0.5 to 2.0, the range of the vibration of the predator population and prey population become larger, and the peak values of the PDFs become lower. And the probabilities in both lower and higher population become higher, indicating a less stable system. This is reasonable. A larger mean arrival rate of the Poisson white noise implies more pulses per unit time. The more the number of pulses is, the more unstable the ecosystem will be. The same conclusion can be obtained from Figure 9, which shows the relative fluctuation of the predator population and prey population versus the mean arrival rate λ. As shown in this figure, the curves of the relative fluctuations for both the predator population and prey population increase monotonously as increasing the mean arrival

in both lower and higher population become higher, indicating a less stable system. This is reasonable. A larger mean arrival rate of the Poisson white noise implies more pulses per unit time. The more the number of pulses is, the more unstable the ecosystem will be. The same conclusion can be obtained from Figure 9, which shows the relative fluctuation of the predator population and prey population versus the mean arrival rate λ . As shown in this figure, the curves of the relative Entropy 2018, 20, 143 11 of 15 fluctuations for both the predator population and prey population increase monotonously as increasing the mean arrival rate λ . This implies that the system has become unstable. The other parameters are the same as system those inhas Figure 8. Also depicted Figures 8 and 9are arethe results rate λ. This implies that the become unstable. The in other parameters sameobtained as those from the Monte simulation. in Figure 8. AlsoCarlo depicted in Figures 8 and 9 are results obtained from the Monte Carlo simulation.

Figure 8. Probability of Figure 8. Probability density density functions functions of of prey prey and and predator predator populations populations for for different different values values of λ . parameter parameter Entropy 2018, 20, xλ. 12 of 16 0.8 Theoretical Monte Carlo

0.7 0.6 0.5 0.4 0.3 0.2 0.1

0

2

4

6

8

10

Figure9. values of of Figure 9. Relative Relativefluctuations fluctuations of of the the predator predator population population and and prey prey population population for for different different values parameter λ.λ . parameter 2 2 2 2 2 In Figures Figures1010 11, influences the influences on the ecosystem are 2ε]2on E[Ythe ] ecosystem In andand 11, the of ε2 E[of Y 2 ] ε=Eε[2YE[]Y12 ]ε=E[εY21E][Y are calculated 2 2 2 2 with systemwith parameters: = 0.1, a = 0.9,ε b= 0.8., 0.5λ,1 =ε2λs 2=0.2 , ca= 0.5, calculated system ε parameters: 0.11.0, 0.9 f, b=0.5, 1.0ε ,s = c 0.2, 0.5ε ,γ f=0.1, Figures 10 and 11 depict the dependence of the stationary PDFs and relative fluctuations of the predator and 11 depict theconclusions dependenceobtained of the stationary PDFs and ε 2 γ  0.1 , and λ1 prey λ 2 population. 0.8 . Figures population We10 can arrive at similar from Figures 8 and 9. 2 2 relative fluctuations of the predator population and prey population. We can arrive at similar As increasing the variance of the impulses ε E[Y ], the ecosystem becomes less stable. This agrees with the intuitive consideration. When increase the value ofthe ε2 E [Y2 ], theoffluctuation of the the conclusions obtained from Figures 8 and 9. As increasing variance the impulses ε 2pulses E[Y 2 ] ,will become larger, which will result in bigger fluctuation of ecosystem. Besides, the results from Monte ecosystem becomes less stable. This agrees with the intuitive consideration. When increase the value Carlo representedofby dotted lines also shown these will two figures. the pulses willare become larger,inwhich result in bigger fluctuation of ε 2 Esimulation [Y 2 ] , the fluctuation Shown in Figure 12 are the stationary PDFs of the prey and predator populations for different of ecosystem. Besides, the results from Monte Carlo simulation represented by dotted lines are also mean arrival rates of the Poisson white noises, corresponding to the same noise intensity λi ε2 E[Yi2 ]. shown in these two figures. The results presented in this figure are all obtained by using the proposed method. It is seen that, as Shown in Figure 12 are the stationary PDFs of the prey and predator populations for different increasing the value of the mean arrival rate of the Poisson white noises from 0.3 to 5.0, the approximate λiε2 Enoises [Yi2 ] . mean arrival rates theand Poisson whitepopulations noises, corresponding the same under noise intensity stationary PDFs of of prey predator of proposedtoecosystem Poisson white The results presented figure are all Gaussian obtained white by using the with proposed method. It is seen that, as become closer to those in of this ecosystem under noises the same intensity. This implies increasing the value of behavior the meandepends arrival rate Poisson white 0.3 to 5.0, the approximate that the non-Gaussian on of thethe mean arrival ratenoises of thefrom Poisson white noise. The other stationary PDFs of prey and predator populations proposed ecosystem under Poisson white noises system parameters are given in the caption of this of figure. become closer to those of ecosystem under Gaussian white noises with the same intensity. This implies that the non-Gaussian behavior depends on the mean arrival rate of the Poisson white noise. The other system parameters are given in the caption of this figure.

The results presented in this figure are all obtained by using the proposed method. It is seen that, as increasing the value of the mean arrival rate of the Poisson white noises from 0.3 to 5.0, the approximate stationary PDFs of prey and predator populations of proposed ecosystem under Poisson white noises become closer to those of ecosystem under Gaussian white noises with the same intensity. This implies that the non-Gaussian behavior depends on the mean arrival rate of the Poisson white noise. The12 other Entropy 2018, 20, 143 of 15 system parameters are given in the caption of this figure.

Figure Figure 10. 10. Probability Probability density density functions functions of of prey prey and and predator predator populations populations for for different different values values of of 2 2 2 2 2 2 2 E [Y 2 ] = ε 2 E [Y 2 ] = ε 2 E [Y 2 ] . parameter Entropy 2018, 20, xεε 13 of 16 E[Y ] = ε E[1Y ] = ε E[2Y ] . 1

Entropy 2018, 20, x 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0

2

13 of 16

0.6 0.6

Theoretical Monte Carlo Theoretical Monte Carlo

0.5 0.5

Theoretical Monte Carlo Theoretical Monte Carlo

0.4 0.4 0.3 0.3 0.2 0.2 0.02 0.02

0.04 0.04

0.06 0.06

0.1 0.1 0 0

0.08 0.08

0.02 0.02

0.04 0.04

0.06 0.06

0.08 0.08

1.2 1.2 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 0 0

 =0.3 =0.6 =0.3 =5.0 =0.6

 =0.3 =0.3 =0.6 =0.6 =5.0 Gaussian =5.0

1.5 1.5

Gaussian =5.0

Gaussian

p(xp(x ) 2 2)

p(x1p(x ) ) 1

Figure 11. 11. Relative Relative fluctuations fluctuationsof ofthe thepredator predatorpopulation populationand andprey prey population different values Figure population forfor different values of Figure 11. Relative fluctuations population and prey population for different values 22 2 2 2 2 2 of 2the 2 predator 2 2 2 [ Y ] = ε E [ Y ] . of parameter parameter ε E[εY2 E][Y =2 ]ε =Eε[2YE ] = ε E [ Y ] . 1 12 22 2 2 of parameter ε E[Y ] = ε E[Y1 ] = ε E[Y2 ] .

Gaussian

1 1 0.5 0.5

0.5 0.5

1 1

1.5 x1 1.5 x1

2 2

2.5 2.5

3 3

0 0 0 0

0.5 0.5

1 1

1.5 x2 1.5 x2

2 2

2.5 2.5

Figure 12. Probability density functions of prey and predator populations for different values of mean Figure 12. Probability density functions of prey and predator populations for different values 2of mean Figure Probability density functions predator different values s mean 0.2 , c  0.5 , ε2of arrival 12. rates. The system parameters areofεprey =0.1and a 0.9 , bpopulations  1.0 , f for0.5 2 2 parameters are ε ε==0.1 0.1, a a=0.9, b ,=b1.0, f =, 0.5, c= 0.5, ε s= 0.2, ε γs  = 0.2 0.1. ε , , arrival rates. The system are 0.9  1.0 c 0.5 f  0.5 2 , ε γ  0.1 . , ε 2 γ  0.1 .

4. Conclusions 4. Conclusions This paper investigates the stochastic behaviors of a time-delayed predator-prey ecosystem 4. Conclusions This paper investigates the stochasticfluctuations. behaviors ofIn a this time-delayed predator-prey ecosystem under pulse-type stochastic environment model, the stochastic fluctuations This paper investigates the stochastic behaviors In of this a time-delayed predator-prey ecosystem under pulse-type stochastic environment fluctuations. model, the stochastic fluctuations are under pulse-type stochastic environment fluctuations. In this model, the stochastic fluctuations are characterized by the Poisson white noises, and the time delay in the interaction between the prey and characterized by the Poisson white noises, anddelay the time delay inThe theoriginal interaction between the prey and predator is described approximately by a time parameter. ecosystem is first modeled predator is describedSDE approximately by a time delay parameter. The some original ecosystemhypothesises, is first modeled as the Stratonovich and then transferred to Itô SDE. Under reasonable the as the Stratonovich SDE and then transferred to Itô SDE. Under some reasonable hypothesises, the stochastic averaging method is applied to simplify the original system. Finally, the approximate stationary

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are characterized by the Poisson white noises, and the time delay in the interaction between the prey and predator is described approximately by a time delay parameter. The original ecosystem is first modeled as the Stratonovich SDE and then transferred to Itô SDE. Under some reasonable hypothesises, the stochastic averaging method is applied to simplify the original system. Finally, the approximate stationary PDFs of the prey and predator populations are obtained by using the perturbation method. To verify the accuracy of the proposed method, the results from Monte Carlo simulation are also calculated. Based on the approximate stationary PDFs of prey and predator populations, the influences of the prey self-competition parameter, the time-delay parameter and the Poisson white noise excitation on the system are investigated in detail. The results show that, increasing time delay parameter as well as the mean arrival rate and the variance of the impulse of the Poisson white noise will enhance the fluctuations of the prey and predator population and make the ecosystem less stable. While the larger value of self-competition parameter will reduce the fluctuations of the system and increase the stability of the ecosystem. In our present paper, only the effects of some system parameters are studied. Similar approaches may be adopted to study the influences of other system parameters. In the present investigation, the amplitudes of the Poisson white noises are assumed to be Gaussian distribution. They can be other types of distributions depending on the realistic situation. Acknowledgments: This study was supported by the National Nature Science Foundation of China under NSFC Grant Nos. 11502199, 11402157. Author Contributions: Wantao Jia, Yong Xu and Dongxi Li collectively realized theoretical analyses; Wantao Jia computed the numerical simulation results; Wantao Jia wrote the paper; Yong Xu and Dongxi Li revised the paper. All authors have read and approved the final manuscript. Conflicts of Interest: The authors declare no conflict of interest.

Appendix The coefficients of GFPK equation are:  A11 (r ) =

s − ( f X1 − c)2 + f γX1 (bX2 − a)2 f  A12 (r ) =

f X1 24







+ t 4



λ1 E[Y1 ] + t

f X1 2

bX2 24



2

λ1 E[Y1 ] + t





bX2 2



λ2 E[Y2 2 ]; t

λ2 E[Y2 4 ]; t

D E D E A21 (r ) = ( f X1 − c)2 λ1 E[Y1 2 ] + (bX2 − a)2 λ2 E[Y2 2 ]; t





t



 f b X1 (7 f X1 − 4c) λ1 E[Y1 4 ] + X2 (7bX2 − 4a) λ2 E[Y2 4 ]; 12 12 t t D E D E 4 4 A4 (r ) = ε4 ( f X1 − c) λ1 E[Y1 4 ] + ε4 (bX2 − a) λ2 E[Y2 4 ];

A22 (r ) =

t

t

where h[]it denotes the time average in one quasiperiod, defined as

h[]it =

1 T

I r

[]dt =

1 T

I r

1 []dx2 = T x2 ( f x1 − c)(1 + ε4 sγx1 )

and T (•) is the period given in Equation (15).

I r

[]dx1 x1 ( a − bx2 )

(A1)

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