# Periodic Solution of a Neutral Delay Leslie Predator-Prey Model and the Effect of Random Perturbation on the Smith Growth Model

Authors: Tongtong Li; Wencai Zhao
Journal: Complexity
 (2020)

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## Abstract

This paper puts forward a class of ratio-dependent Leslie predator-prey models. Firstly, a neutral delay predator-prey model with ratio dependence and impulse control is established and the existence of positive periodic solutions is proved by the coincidence degree theory. Secondly, a stochastic disturbance Leslie model of Smith growth is obtained when the interference of white noise is taken into consideration and the impact of delay is ignored. Applying Ito^’s formula, we get the conditions of system persistence and extinction. Finally we verify the correctness of theoretical analysis with numerical simulations.


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## Body

## 1. Introduction

In the population dynamic systems, the relationship between population growth rate and population density is complex and diverse. The Verhulst–Pearl logistic equation [1](1)x˙t=rxt1−xtK,is widely used to describe population growth for its concise mathematical expression and clear biological meaning, where xt denotes the population density at time t, r is the intrinsic growth rate, and K is the environmental capacity. However, in 1963, Smith [2] studied the population dynamics of Daphnia magna and found that the growth of Daphnia magna population in the laboratory did not satisfy the above growth model, and he proposed the following growth equation:(2)x˙t=rxt1−xtK−x˙tcK,which is usually called the Smith growth equation, where c is a positive constant. Then, in 1991, Kuang [3] considered a neutral predator-prey model with time delays and the Smith growth as follows:(3)x˙t=rxt1−xt−τ+ρx˙t−τK−ytpxt,y˙t=yt−α+βpxt−σ,where τ and σ are the recovery time delay of the prey population and the digestion time delay of the predator population, respectively, α is the death rate of the predator, and β is the conversion factor of prey into the predator. ρx˙t−τ/K is the neutral delay term. px indicates the functional response of predator y to prey x and depends only on the density of the prey population. However, when the predators have to search for food, there will be competition among individual predators for limited prey. The amount of food the predator obtains depends not only on the density of the prey population but also on the density of the predator itself, which is influenced by the relative ratio of the number of predators and prey populations, reflecting the ratio-dependent characteristics. Leslie proposed the following Leslie model [4, 5] in studying the relationship between the predator and prey:(4)x˙t=xtr1−a1xt−c1yt,y˙t=ytr2−a2ytxt,where xtandyt stand for prey and predator densities at time t, respectively. r1 and r2 are the intrinsic growth rates of the prey and predator, a1 denotes the density-dependent coefficient of the prey, the parameter c1 is the capturing rate, and a2 is the conversion factor of prey into predators. Motivated by the works [3–5], we propose a ratio-dependent neutral delay Leslie predator-prey model as follows:(5)x˙t=rxt1−xt−τ1K−εx˙t−τ1−extytnyt+xt,y˙t=ayt1−byt−τ2xt−τ2,where all of the parameters are positive and have the biological meanings listed in Table 1.Table 1
Biological meanings of parameters for model (5).
Parameters
Description
τ
1
,
τ
2
The discrete delays
r
,
a
The intrinsic growth rates of prey and predator
K
The environmental capacity of the prey
ε
The neutral coefficient of the prey
e
The conversion rate
n
The capture saturation rate
b
The conversion factor of prey into predatorThe first equation of the system states that the growth rate of prey depends not only on its own density but also on the neutral term. If there is no predator, the prey population will grow in Smith mode. The Smith model mainly describes the growth rates of the population under the condition of limited food. The relative growth rate of population sizex˙t/xt at time t is proportional to the amount of food left before the time τ1. The food consumed by the population is mainly used for two purposes: the food needed to maintain the organism’s own survival xt−τ1/K and the food needed for population reproduction εx˙t−τ1. The denominator of functional response function p=ex/ny+x contains y, which reflects the interaction among individuals within the predator population. The second equation implies that the predator population grow in Leslie mode and the environmental capacity is proportional to prey population density.Developing and utilizing biological populations in a proper way is an important issue facing humanity, which can make biological resources work for human beings and ensure the sustainability of ecosystems [6–12]. People’s harvesting behavior is often not continuous, which may be once or several times at a certain time. For example, in fishery production, fishermen harvest fishes in few months and in agricultural production farmers spray pesticides at regular intervals to harvest (kill) pests. This harvesting method makes the population quantity or density be changed drastically in a short period of time, so it is more suitable to describe with impulsive differential equations. In the past 20 years, impulsive differential equations have been widely used in various biological models, for example, Zhao and Tang [13], Liu et al. [14], Li et al. [15], and Zhang et al. [16] introduced impulsive effects into the ordinary differential equation model. Li and Meng [17] and Qi et al. [18] established a class of impulsive stochastic differential equation models based on ordinary differential equations. Du and Feng [19] and Chen and Du [20] cooperated the impulsive control to the neutral predator-prey model, by using the coincidence degree theory, and they obtained the conditions for the existence of periodic solution. Then, based on [19, 20] and assuming that some parameters in system (5) are changed periodically and the population is pulse controlled, we will obtain the following neutral delay system with pulse effects:(6)x˙t=xtrt−rtxt−τ1tKt−ρtx˙t−τ1t−etytnyt+xt,y˙t=atyt1−btyt−τ2txt−τ2t,t≠tk,k=1,2,…,Δxtk=−θ1kxtk,Δytk=−θ2kytk,t=tk,k=1,2,…,with initial conditions(7)xt=pt,p∈C1−τ,0,0,+∞,p0>0,yt=qt,q∈C1−τ,0,0,+∞,q0>0,where Δxtk=xtk+−xtk, Δytk=ytk+−ytk, tk is the pulse harvesting time, and θiki=1,2 is the pulse harvesting ratio. This implies that the harvest yield is in proportion to the biomass at that time. rt,Kt,τ1t,τ2t,et,at,bt,andρt are continuous nonnegative T-periodic functions. n is positive constant and τ≔maxt∈0,Tτ1t,τ2t.On the contrary, biological populations multiply and thrive in nature, which will inevitably be affected by various environmental noise, so it is important to consider the effects of random disturbances on population dynamics [21–29]. In [30], the asymptotic stability of a stochastic May mutualism system was studied. In [31], Markov semigroup was used to study a stochastic ecological model of plants with infectious diseases. Authors in [32–34] focused on the long-term dynamic behavior of ecosystems affected by environmental noise. In particular, conditions for the existence of stationary distribution were obtained. Zhao et al. [35] presented an algal population growth model, where the authors not only gave the threshold conditions of permanence and extinction but also discussed the influence of environmental noise on the periodic blooms. Ignoring the influence of time delays and considering the disturbance of white noise, we construct the following stochastic Leslie predator-prey model with Smith growth:(8)dxt=rxt1+ρxt1−xtK−extytnyt+xt1+ρxtdt+σ1xtdB1t,dyt=ayt1−bytxtdt+σ2ytdB2t,where B1t and B2t are independent standard Brownian motions. σ12andσ22 denote the intensity of white noise. r,K,ρ,n,a,b,e,σ1, and σ2 are all positive and the biological significances are the same as that in model (5).To study the existence of periodic solutions for ordinary differential equations and delay differential equations, Gaines and Mawhin’s continuation theorem based on coincidence degree theory [36] is an important tool. However, it is seldom applied in the study of the existence of positive periodic solutions of neutral delay system with pulse effects. It is generally considered difficult to prove that the nonlinear operator H is compact on set Ω¯ and the solutions of the equation have prior bound. In this work, by using analytical methods such as the mean value theorem of integral and inequality techniques, we obtain the sufficient conditions for the existence of positive periodic solutions of system (6). In addition, considering the influence of environmental disturbance on the population, we include the white noise into the Leslie predator-prey system, establish a new stochastic Smith growth model (8), and discuss the conditions for the persistence and extinction of the system.


## 2. Preliminaries

First, we convert the impulse system (6) into a nonimpulsive form. In system (6), we assume thatI10<t1<t2<⋯<tk<⋯ are fixed points and limk⟶∞tk=+∞I2θik are real sequences such that θik<1 and ∏0<tk<t1−θik i=1,2 are T-periodic functionsAssumingI1 and I2, we convert system (6) to the following system:(9)D˙1t=D1trt−AtD1t−τ1t−ηtD˙1t−τ1t−EtD2tNtD2t+θ1tD1t,D˙2t=D2tat−BtD2t−τ2tθ2tD1t−τ2t,with initial conditions(10)D1t=pt,p∈C1−τ,0,0,+∞,p0>0,D2t=qt,q∈C1−τ,0,0,+∞,q0>0,where(11)D1t=∏0<tk<t1−θ1k−1xt,D2t=∏0<tk<t1−θ2k−1yt,At=rtKt∏0<tk<t−τ1t1−θ1k,ηt=ρt∏0<tk<t−τ1t1−θ1k,Et=et∏0<tk<t1−θ2k,Nt=n∏0<tk<t1−θ2k,θ1t=∏0<tk<t1−θ1k,θ2t=∏0<tk<t−τ2t1−θ1k,Bt=atbt∏0<tk<t−τ2t1−θ2k.Lemma 1.
Suppose thatI1 and I2 hold, then(1)
IfD1t,D2tT is a solution of (9), then xt,ytT is a solution of (6)(2)
Ifxt,ytT is a solution of (6), then D1t,D2tT is a solution of (9)Proof.
(1) SupposeD1t,D2tT is a solution of system (9). Next, we prove that xt=∏0<tk<t1−θ1kD1t,yt=∏0<tk<t1−θ2kD2t is the solution of system (6). When t≠tk, we obtain(12)x′t−xtrt−rtxt−τ1tKt−ρtx′t−τ1t−etytnyt+xt=∏0<tk<t1−θ1kD1′t−∏0<tk<t1−θ1kD1trt−rt∏0<tk<t−τ1t1−θ1kD1t−τ1tKt−ρt∏0<tk<t−τ1t1−θ1kD1′t−τ1t−et∏0<tk<t1−θ2kD2tn∏0<tk<t1−θ2kD2t+∏0<tk<t1−θ1kD1t=∏0<tk<t1−θ1kD1′t−D1trt−AtD1t−τt−ηtD1′t−τ1t−EtD2tNtD2t+θ1tD1t=0,y′t−ytat−atbtyt−τ2txt−τ2t=∏0<tk<t1−θ2kD2′t−D2tat−BtD2t−τ2tθ2tD1t−τ2t=0.
Whent=tk, one has(13)xtk+−xtk=∏0<tj≤tk1−θ1jD1tk−∏0<tj<tk1−θ1jD1tk=∏0<tj<tk1−θ1jD1tk−θ1k=−θ1kxtk,ytk+−ytk=∏0<tj≤tk1−θ2jD2tk−∏0<tj<tk1−θ2jD2tk=∏0<tj<tk1−θ2jD2tk−θ2k=−θ2kytk.
Therefore,xt,ytT is the solution of system (6).
(2) Similar to the proof of case (1), we could show that ifxt,ytT is a solution of (6), then D1t,D2tT is the solution of system (9). The proof of Lemma 1 is completed.
For convenience, throughout this paper we will use the symbols:(14)v¯=1T∫0Tvtdt,v^=mint∈0,Tvt,v⌢=maxt∈0,Tvt,where v=vt is a periodic continuous function with period T.


## 3. Existence of Positive Periodic Solution of System (6)

Before discussing the existence of periodic solution of system (6), we introduce the coincidence degree theory firstly.LetU and V be two Banach spaces, G: DomG⊂U⟶V is a linear mapping and H:U⟶V is a continuous mapping. If dim KerG = codim ImG<+∞ and ImG⊂V is closed, then we call the operator G is a Fredholm operator with index zero. If G is a Fredholm operator with index zero and there exist continuous projections P:U⟶U and Q:V⟶V such that ImP=KerG and ImG=kerQ=ImI−Q; then, G|DomG∩KerP:I−PU⟶ImG has an inverse function, and we set it as Kp. Assume that Ω⊂U is any open set, if QHΩ¯ is bounded and KpI−QH:Ω¯⟶U is relative compact, and then we say H is G-compact on Ω¯.Lemma 2 (see [36]).
LetG:Dom G⊂U⟶V be a Fredholm mapping with index zero and H is a G-compact on Ω¯. Furthermore, assume that(i)
Gx≠λHx, where λ∈0,1, x∈∂Ω∩ DomG(ii)
QHx≠0, for each x∈∂Ω∩kerG(iii)
degJQH,Ω∩KerG,0≠0, where J:ImQ⟶KerG is an isomorphism
Then, the operator equationGx=Hx has at least one solution in DomG∩Ω¯.Lemma 3 (see [37]).
Ifτ∈C1R,R with τt+T≡τt and τ′t<1 for t∈0,T, then function βt=t−τt has a unique inverse β−1t satisfying β−1∈C1R,R with β−1s+T≡β−1s+T for s∈0,T.
We are now in a position to state and prove our main result.Theorem 1.
Assume that the following conditions hold:H1τ1′t<1, τ2′t<1 for any t∈RH2μ′t<At for any t∈0,T, where μt=ηt/1−τ1′tH3η⌣eW<1, r¯>M¯, where WandM are defined in the proofH4 The algebraic equations(15)r¯−A¯x−1T∫0TEtyNty+θ1txdt=0,a¯−Bθ2¯yx=0,have finite solutions. Then, system (6) has at least one T-periodic solution.In the following proof, we first transform (9) into equivalent system. Then, we construct the mappings and open set Ω on Banach space and prove that the mappings satisfy Lemma 2 on Ω¯. Therefore, we obtain Theorem 1.Proof.
(1) Transform (9) into an equivalent system.
LetD1t=ewtandD2t=evt. Then, we can translate system (9) into(16)w′t=rt−Atewt−τ1t−ηtw′t−τ1tewt−τ1t−EtevtNtevt+θ1tewt,v′t=at−Btevt−τ2tθ2tewt−τ2t,where all coefficient functions are defined in system (9). It is easy to see that if system (16) has one T-periodic solution w∗t,v∗tT, then D1∗t,D2∗tT=expw∗t,expv∗tT is a positive T-periodic solution of system (9). So, system (6) has a T-periodic solution x∗t,y∗tT=∏0<tk<t1−θ1kD1∗t,∏0<tk<t1−θ2kD2∗tT. Therefore, we just have to prove system (16)s has one T-periodic solution.
(2) Construct Banach space and define mappings.
TakeU=V=ut=wt,vtT∈C1R,R2:ut+T=ut and denote u∞=maxt∈0,Twt+vt,u=u∞+u′∞. Then, U and V are both Banach spaces with the norm ⋅ and ⋅∞, respectively. Define operators G, P, Q, and H in the following form, respectively, G:U⟶V, Gut=w′t,v′tT; Pu=u¯, for u∈U; Qu=u¯, for u∈V; and(17)Hut=rt−Atewt−τ1t−ηtw′t−τ1tewt−τ1t−EtevtNtevt+θ1tewtat−Btevt−τ2tθ2tewt−τ2t.
Then, KerG=R2, ImG=u∈V:∫0Tutdt=0 is closed in V, dimKerG= codimImG. P and Q are continuous projectors such that ImP= KerG, KerQ=ImG=ImI−Q. Therefore, the Fredholm mapping G has a unique inverse. The generalized inverse(to G)Kp:ImG⟶ KerP∩DomG is given by the following form:(18)Kpu=∫0tusds−1T∫0T∫0tusdsdt.
Then,QH:U⟶V and KpI−QH:U⟶U read(19)QHut=1T∫0Trt−Atewt−τ1t−ηtw′t−τ1tewt−τ1t−EtevtNtevt+θ1tewtdt1T∫0Tat−Btevt−τ2tθ2tewt−τ2tdt,KpI−QHut=A−B−C,where(20)A=∫0trs−Asews−τ1s−ηsw′s−τ1sews−τ1s−EsevsNsevs+θ1tewsds∫0tas−Bsevs−τ2sθ2tews−τ2sds,B=1T∫0T∫0trs−Asews−τ1s−ηsws−τ1sews−τ1s−EsevsNsevs+θ1tewsdsdt1T∫0T∫0tas−Bsevs−τ2sθ2tews−τ2sdsdt,C=tT−12∫0Trs−Asews−τ1s−ηsw′s−τ1sews−τ1s−EsevsNsevs+θ1tewsdstT−12∫0Tas−Bsevs−τ2sθ2tews−τ2sds.
Distinctly, it is easy to know thatQH and KpI−QH are both continuous by the Lebesgue theorem. And by using Arzela–Ascoli theorem, we know that KpI−QHΩ¯ is compact for any open bounded set Ω⊂U. Hence, H is G-compact on Ω¯.
(3) Construct an open setΩ.
For the sake of using Lemma2, we need to look for an appropriate open-bounded subset Ω⊂U. First, we prove that the periodic solution of equation Gu=λHu is bounded. Corresponding to the operator equation Gu=λHu for λ∈0,1, we have(21)w′t=λrt−Atewt−τ1t−ηtw′t−τ1tewt−τ1t−EtevtNtevt+θ1tewt,v′t=λat−Btevt−τ2tθ2tewt−τ2t.Suppose that u=wt,vtT∈U is a solution of (21). Now we prove that u≤S∗, S∗ is a constant.Step 1.
Prove thatwt≤S1 and w′t≤S2.
Integrating (21) over the interval 0,T, we obtain(22)r¯T=∫0TAtewt−τ1t+ηtw′t−τ1tewt−τ1t+EtevtNtevt+θ1tewtdt,and(23)a¯T=∫0TBtevt−τ2tθ2tewt−τ2tdt.
In consideration ofH1 and H2, we obtain(24)∫0Tηtw′t−τ1tewt−τ1tdt=∫0Tηt1−τ1′tewt−τ1tdt=∫0Tηt1−τ1′tdewt−τ1t=−∫0Tμ′tewt−τ1tdt,which together with (22) gives(25)r¯T=∫0TAt−μ′tewt−τ1t+EtevtNtevt+θ1tewtdt.
In view of Lemma3 and H1, we obtain(26)∫0TAt−μ′tewt−τ1tdt=∫0TAβ1−1t−μ′β1−1tewt1−τ1′β1−1tdt,where β1t=t−τ1t. From (25), we obtain(27)r¯T=∫0TAβ1−1t−μ′β1−1tewt1−τ1′β1−1tdt+∫0TEtevtNtevt+θ1tewtdt,which implies(28)r¯T≥∫0TAβ1−1t−μ′β1−1tewt1−τ1′β1−1tdt.
Therefore,(29)1T∫0Tewtdt≤r¯R^,where Rt=Aβ1−1t−μ′β1−1t/1−τ1′β1−1t. According to (25), we obtain(30)r¯T≤∫0TAt−μ′tewt−τ1tdt+∫0TEtNtdt,together with H3, which implies(31)1T∫0Tewtdt≥r¯−M¯Rˇ,where Mt=Et/Nt.
According to (29) and (31), we can get that there exists φ1∈0,T such that(32)wφ1≤maxlnr¯R^,lnr¯−M¯Rˇ≔H1.
By means of (25) and (29), we have(33)∫0TAt−μ′tewt−τ1t+Rtewtdt≤2r¯T.
On the basis of the mean value theorem of integral calculus, we know that there existsξ∈0,T, such that(34)Aξ−μ′ξewξ−τ1ξ+Rξewξ≤2r¯.
Sincer¯>0, we have(35)wξ≤ln2r¯R^,ewξ−τ1ξ≤2r¯U^,where U=At−μ′t.
Through the first equation of (21) and (25), we obtain(36)∫0Tddtwt+λμtewt−τ1tdt=λ∫0Trt−At−μ′tewt−τ1t−EtevtNtevt+θ1tewtdt≤2r¯T.
Therefore,(37)wt+λμtewt−τ1t≤wξ+λμξewξ−τ1ξ+∫0Tddtwt+λμtewt−τ1tdt≤ln2r¯R^+2r¯μˇU^+2r¯T≔W.
So, we obtain(38)wt≤W.
From (36), we know(39)∫0Twtdt−∫0Tddtλμtewt−τ1tdt≤2r¯T.Consequently,(40)∫0Tw′tdt≤2r¯T+∫0Tμ′t+ηtw′t−τ1tewt−τ1tdt≤2r¯T+w′ˇeWT+ηˇeW∫0Tμ′tdt.
Through the transpose, we can obtain(41)∫0Tw′tdt≤2r¯T+μ′ˇeWT1−ηˇeW≔K.
Together with (32), we obtain(42)wt≤wφ1+∫0Tw′tdt≤H1+K≔S1.
Through the first equation of (21) and (38), we obtain(43)w′t=λrt−Atewt−τ1t−ηtw′tewt−τ1t−EtevtNtevt+θ1ewt,so we obtain(44)maxw′t≤rˇ+AˇeW+ηˇeWmaxw′t+EˇN^.
UtilizingH3, we have(45)w′t≤11−ηˇeWrˇ+AˇeW+EˇN^≔S2.Step 2.
Prove thatvt≤S3 and v′t≤S4.
By means of (23) and (38), we obtain(46)a¯T=∫0TBtevt−τ2tθ2tewt−τ2tdt≥B^θˇ2∫0Tevt1−τ2′β2−1tewtdt≥B^θˇ2σˇeW∫0Tevtdt,where β2t=t−τ2tandσt=1−τ2′β2−1t, which implies that(47)1T∫0Tevtdt≤a¯θˇ2σˇeWB^.
According to (23) and (42), we obtain(48)a¯T=∫0TBtevt−τ2tθ2ewt−τ2tdt≤Bˇθ^2∫0Tevt1−τ2′β2−1tewtdt≤Bˇθ^2σ^e−S1∫0Tevtdt.
Thus,(49)1T∫0Tevtdt≥a¯θ^2σ^e−S1Bˇ.
From (47) and (49), there exists φ2∈0,T such that(50)vφ2≤maxlna¯θˇ2σˇeWB^,lna¯θ^2σ^e−S1Bˇ≔H2.
From (21) and (23), it is easy to see that(51)∫0Tv′tdt≤2a¯T.
Then,(52)vt≤vφ2+∫0Tv′tdt≤H2+2a¯T≔S3.
Through the second equation of (21), (42), and (52), we have(53)v′t=λat−Btevt−τ2tθ2ewt−τ2t≤aˇ+BˇeS3θ^2e−S1≔S4.
From (42), (45), (52), and (53), we obtain(54)u=u∞+u′∞≤S1+S2+S3+S4≔S∗.
At this point, we have proved that the periodic solution of equationGu=λHu for λ∈0,1 is bounded.
In the following part, we will prove that the constant solution of equationQHu=0 is bounded. According to condition H4, the algebraic equation set(55)r¯−A¯x−1T∫0TEtyNty+θ1txdt=0,a¯−Bθ2¯yx=0,has finite positive solutions xi∗,yi∗i=1,2,…k. Therefore, for u∈KerG, the operator equation QHu=0 has finite positive solutions ui∗=wi∗,vi∗i=1,2,…k. Suppose that S0=max1≤i≤kwi∗+vi∗, then ui∗≤S0i=1,2,…k. Therefore, the solution of equation QHu=0 for u∈KerG is bounded.
LetS=S∗+S0, and take Ω=u=w,vT:u<S. It is easy to see that Ω verifies condition i in Lemma 2. When u∈∂Ω∩KerG=∂Ω∩R2, u is a constant vector in R2 with u=S, we obtain(56)QHut=r¯−A¯x−1T∫0TEtyNty+θ1txdta¯−Bθ2¯yx≠0.
Thus, conditionii of Lemma 2 holds.
(4) Calculate the degree of coincidence.
For eachu=w,vT∈ImQ, define isomorphism J:ImQ⟶ kerG so that Ju=u. By simply computing, we get the following Jacobian matrix:(57)J¯w,v=−A¯ew+1T∫0Tθ1tEtev+wNtev+θ1tew2dt1T∫0T−Etθ1tew+vNtev+θ1tew2dtBθ2¯evew−Bθ2¯evew.
Then, we have(58)detJ¯w,v=A¯ew−1T∫0Tθ1tEtev+wNtev+θ1tew2dtBθ2¯evew+Bθ2¯evew⋅1T∫0TEtθ1tew+vNtev+θ1tew2dt=Bθ2¯A¯ev>0,∀w,v∈R2,which implies(59)degJQH,Ω∩KerG,0=∑wi∗,vi∗∈QH−10sgndetJ¯,wi∗,vi∗>0.
By now, we have verified thatΩ satisfies all the requirements in Lemma 2. Thus, system (16) has at least one T-periodic solution, which implies that system (6) has at least one positive T-periodic solution. This completes the proof.Remark 1.
If the delayτ1tandτ2t both are positive constants, that is to say, condition H1 holds. At this time, the delay has no effect on the existence of periodic solutions.Remark 2.
If the neutral coefficientρt=0, that is, ηt=0, at this time, condition H2 holds. When H1,H3, and H4 hold, then the system has at least one positive periodic solution.
Next, we will study the predator-prey model with random disturbances.


## 4. Persistence in Mean and Extinction of System (8)

In this section, we discuss the properties of stochastic system (8) with Smith growth, the sufficient conditions for persistence and extinction are obtained.Lemma 4 (see [38]).
Letxt,yt be a solution of system (8) with initial value x0,y0, then limt⟶∞xt/t=0,limt⟶∞yt/t=0,limt⟶∞∫0txsdB1s/t=0,limt⟶∞∫0tysdB2s/t=0a.s.Theorem 2.
Letxt,yt be a solution of system (8) with initial value x0,y0. If(1)
r−σ12/2−e/n>0,a−σ22/2>0, then system (8) is persistent in mean(2)
r−σ12/2<0,a−σ22/2<0, then system (8) goes to extinction almost surelyProof.
(1) From the first equation of system (8) and using Ito^’s formula, we obtain(60)dlnx+ρx=1x+ρrx1+ρx1−xK−exyny+x1+ρx−σ122dt+σ11+ρxdB1t=r1−xK−eyny+x−σ122dt+σ11+ρxdB1t.
Integrating from 0 tot on both sides, we can obtain(61)lnxt+ρxt−lnx0−ρx0=r−σ122t−rK∫0txsds−∫0teysnys+xsds+∫0tσ11+ρxsdB1s.
Then,(62)lnxt=r−σ122t−∫0teysnys+xsds−rK∫0txsds−ρxt+lnx0+ρx0+Mt,where Mt=∫0tσ11+ρxsdB1s. Dividing t on both sides, we have that(63)1tlnxt≥r−σ122−en−rKt∫0txsds+Mtt−ρxtt+lnx0t+ρx0t.
According to Lemma4, we obtain(64)liminft⟶∞∫0txtdtt≥r−σ12/2−e/nKr>0.
From the second equation of system (8) and using Ito^’s formula, we can obtain(65)dlny=a−σ222−abyxdt+σ2dB2t.
Integrating from 0 tot on both sides, we obtain(66)lnyt−lny0=a−σ222t−ab∫0tysxsds+∫0tσ2dB2s.
Thus,(67)ab∫0tysxsds=−lnyt+lny0+a−σ222t+∫0tσ2dB2s.
Then, we can obtain(68)liminft⟶∞1t∫0tysxsds≥a−σ22/2ab>0.
Becausex is persistent, liminft⟶∞1/t∫0tysds>0. We complete the proof of (1).
(2) By (61) and (66), we can obtain(69)lnxt+ρxt−lnx0−ρx0=r−σ122t−rK∫0txsds−∫0teysnys+xsds+∫0tσ11+ρxsdB1s≤r−σ122t+Mt,lnyt−lny0=a−σ222t−ab∫0tysxsds+∫0tσ2dB2s≤a−σ222t+∫0tσ2dB2s.
Therefore,(70)limsupt⟶∞lnxtt≤r−σ122<0,limsupt⟶∞lnytt≤a−σ222<0.
So, system (8) goes to extinction almost surely.
Theorem2 shows that if the intensity of environmental noise is small, the prey and predator will survive for a long time, and vice versa, and they will become extinct. Therefore, environmental noise is not conducive to the survival of the population.


## 5. Conclusion and Numerical Simulation

This paper mainly solves the following problems:(1)
For system (6) with neutral time delay and impulse control, first we transform it into nonpulse equivalent system by transformation: xt=∏0<tk<t1−θ1kD1tandyt=∏0<tk<t1−θ2kD2t. Then, by using the coincidence degree theory, the sufficient conditions H1−H4 for the existence of periodic solution of the system is obtained.(2)
For the stochastic perturbation system (8) with Smith growth, the persistence and extinction of the system are discussed.In the following part, we introduce numerical simulations to illustrate our main theoretical results. Firstly, we verify the existence of positive periodic solution of the neutral delay Leslie predator-prey model (16).Example 1.
(1)
In system (16), we choose(71)rt=4+sin10πt,At=0.4−cos10πt,Et=0.2−0.5cos10πt,at=3+sin10πt,Bt=0.5+cos10πt,ηt=0.1+0.01sin10πt,Nt=4+sin10πt,θ1t=0.8+sin10πt,θ2t=0.5+sin10πt,τ1t=0.2+0.01cos10πt,τ2t=0.2+0.01sin10πt.By simple calculation, we know that conditionsH1 and H2 hold, and ηˇeW≈0.0987<1, r¯=4>M¯≈0.175. Obviously H3 holds. Moreover, the algebraic equations in H4 have only one positive solution, which together with Theorem 1 yields that Example 1 has at least one periodic solution (see Figure 1).(2)
We chooseτ1t=0.3andτ2t=0.3 in system (16), and other parameters are the same as those in Figure 1.In this case, conditionH1 is true because τ1t and τ2t are constants. Thus, there is at least one periodic solution as long as condition H2−H4 are true (see Figure 2). This shows that the existence of periodic solutions is not affected by the constant delays.(3)
We chooseηt=0 in system (16), and other parameters are the same as those in Figure 1.
Because the neutral coefficientρt=0, condition H2 must be true. The system has at least one periodic solution as long as other conditions are true. In this case, the prey population grows in logistic mode, which indicates that there is at least one periodic solution in general impulsive system (see Figure 3).Figure 1
Computer simulation of the pathswtandvt for system (16) with the initial value w0,v0=0.5,0.5.
(a)
(b)
(c)Figure 2
Computer simulation of the pathswtandvt for system (16) with the initial value w0,v0=0.5,0.5 and τ1t=τ2t=0.3
(a)
(b)
(c)Figure 3
Computer simulation of the pathswtandvt for system (16) with the initial value w0,v0=0.5,0.5 and ηt=0.
(a)
(b)
(c)Figure 4
Computer simulations of the pathsxtandyt for system (8) with the initial value x0,y0=0.03,0.04 and σ1=0.1, σ2=0.1, and its corresponding deterministic model with initial value x0,y0=0.03,0.04.
(a)
(b)Figure 5
Computer simulations of the pathsxtandyt for system (8) with the initial value x0,y0=0.03,0.04 and σ1=0.9, σ2=0.8, and its corresponding deterministic model with initial value x0,y0=0.03,0.04.
(a)
(b)Example 2.
Next we conduct simulations about the persistence and extinction of the stochastic Smith growth model (8). Obviously, when r>e/n+b, a+aρK/r+1r−e/n+b>be/n+b2, the deterministic model corresponding to model (8) has a unique locally asymptotically stable positive equilibrium P∗=x∗,y∗=K/rr−e/n+b,K/rbr−e/n+b. In system (8), we choose(72)r=0.4,a=0.2,e=0.4,n=2,ρ=0.1,b=0.5,K=0.8.(1)
Letσ1=0.1, σ2=0.1, then we can know that r>e/n+b,a+aρK/r+1r−e/n+b>be/n+b2,r−σ12/2−e/n>0,a−σ22/2>0. Both stochastic system and deterministic system exist for a long time. Thus, conclusion (1) of Theorem 2 holds (see Figure 4).(2)
Letσ1=0.9andσ2=0.8. By simple calculation, we can know that r>e/n+b, a+aρK/r+1r−e/n+b>be/n+b2, and then the deterministic system is persistent. However, r−σ12/2<0,a−σ22/2<0, and then the random system is extinct. Thus, conclusion (2) of Theorem 2 holds. This shows that environmental noise is harmful to the long-term survival of the population (see Figure 5).



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