# Stochastic Analysis of a Hassell-Varley Type Predation Model

Authors: Feng Rao; Shunjun Jiang; Yanqiu Li; Hao Liu
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We investigate a Hassell-Varley type predator-prey model with stochastic perturbations. By perturbing the growth rate of prey population and death rate of predator population with white noise
terms, we construct a stochastic differential equation model to discuss the effects of the environmental noise on the dynamical behaviors. Applying the comparison theorem of stochastic equations and Itô’s formula, the unique positive global solution to the model for any positive initial value is obtained. We find out some sufficient conditions for stochastically asymptotically boundedness, permanence, persistence in mean and extinction of the solution. Furthermore, a series of numerical simulations to illustrate our mathematical findings are presented. The results indicate that the stochastic perturbations do not cause drastic changes of the dynamics in the deterministic model when the noise intensity is small under some conditions, but while the noise intensity is sufficiently large, the species may die out, which does not happen in the deterministic model.


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## Body

## 1. Introduction

It is well known that predator-prey interaction is one of basic interspecies relations for ecosystems, and it is also the basic block of more complicated food chain, food web, and biophysical network structure [1]. Because of the universal existence of predator and prey and their importance in ecology, the dynamical relationship between them has long been and will continue to be one of the dominant themes [2, 3].The classical predator-prey model has received extensive attentions from mathematicians as well as ecologists [4–7], and it can be expressed by a model of nonlinear ordinary differential equations as follows:
(1)dNdt=f(N)N-bg(N,P)P,dPdt=P(cg(N,P)-d),
where N=N(t) and P=P(t) denote the density of prey and predator population at time t, respectively. Parameters b, c, and d are positive constants. b stands for capturing rate of prey by predator, c is conversion rate of prey into predator, and d is the natural death rate of the predator. The function f(N) represents the density-dependent specific growth rate of prey in absence of predator. The amount of prey biomass consumed by each predator per unit of time is described by the functional response g(N,P).In this paper, we consider the usual logistic form of the growth function for prey in the absence of predator as(2)f(N)=r(1-NK),
where r(>0) is the natural growth rate of prey and K(>0) is the environmental carrying capacity. The functional response g(N,P) is taken as
(3)g(N,P)=NN+mPα,
which is called the Hassell-Varley type functional response and α∈(0,1) is the Hassell-Varley constant [8] and m(>0) stands for half capturing saturation constant. The predator-prey model with Hassell-Varley type functional response has been studied in the ecological literature [6, 9–11].For more biological motivation in population dynamics, we take into account the density-dependence of predator population. And the corresponding Hassell-Varley type predator-prey model is described by the following form:(4)dNdt=rN(1-NK)-bNPN+mPα,dPdt=P(cNN+mPα-d-hP),N(0)=N0>0,P(0)=P0>0,
where hP stands for the density-dependence of the predator population and h>0.On the other hand, most natural phenomena do not follow strictly deterministic laws but rather oscillate randomly about some average. So that the population density never attains a fixed value with the advancement of time but rather exhibits continuous oscillation around some average values [12, 13]. In fact, there are many benefits to be gained by using stochastic models because real life is full of random fluctuations (i.e., the effects of noise), which undeniably arise from either environmental variability or internal species. The basic mechanism and factors of population growth like resources and vital rates—birth, death, immigration, and emigration—change nondeterministically due to continuous fluctuations in the environment (e.g., variation in intensity of sunlight, temperature, water level, etc.) [2, 3, 14]. Recent advances in stochastic differential equations enable a lot of authors to introduce noise into the model of physical phenomena, whether it is a random noise in the system of differential equations or environmental fluctuations in parameters [15–31]. So far as our knowledge is concerned, the work of a modified Hassell-Varley type predator-prey model with stochastic perturbations seems rare. Motivated by these, we attempt to study the stochastic behaviors of the modified Hassell-Varley type predation model in a random fluctuating environment.The organization of this paper is as follows. In Section2, we present a stochastic model corresponding to the deterministic model (4) and discuss it in detail. In Section 3, we use numerical simulations to reveal the influence of noise on the dynamical behaviors of the model. A brief discussion is given in Section 4.


## 2. The Stochastic Model and Analysis

In this section, we investigate the effects of fluctuating environments on the dynamical behaviors of model (4). Assuming that random fluctuations in the environment would display themselves as fluctuations in the growth rate of prey population N and in the death rate of predator population P, then the parameters r and d in model (4) can be replaced by
(5)r⟶r+σ1B˙1(t),-d⟶-d+σ2B˙2(t).
In this way, model (4) will be reduced to the following form:
(6)dN=N(r-rKN-bPN+mPα)dt+σ1NdB1(t),dP=P(cNN+mPα-d-hP)dt+σ2PdB2(t),
where σ12 and σ22 are known as the intensities of environmental noise and B˙i(t)(i=1,2) is a standard white noise; that is, Bi(t)(i=1,2) is a Brownian motion defined in a complete probability space (Ω,ℱ,P) with a filtration {ℱt}t∈R+ satisfying the usual conditions (i.e., it is right continuous and increasing while ℱ0 contains all P-null sets) [14].


### 2.1. Positive and Global Solution

For model (6), there is a positive local solution.Lemma 1.
There is a unique local solution(N(t),P(t)) for t∈[0,τe) to model (6) almost surely for initial value (N0,P0)∈R+2, where τe is the explosion time.The proof of this lemma is rather standard and hence is omitted.Lemma1 only tells us that there is a unique positive local solution to model (6). Next, we show that this solution is global which is more interesting.In particular, let us consider the one-dimensional stochastic population model(7)dN¯(t)=rN¯(t)(1-N¯(t)K)dt+σ1N¯(t)dB1(t),t≥0,N¯(0)=N¯0;
there is an explicit solution
(8)N¯(t)=exp{(r-σ122)t+σ1B1(t)}×(1N¯0+rK∫0t‍exp{(r-σ122)shhhhhhhhhhhhh22hhhh+σ1B1(s)(r-σ122)(r-(σ12/2))}ds∫0t‍)-1.
From model (6), we have
(9)dN(t)≤rN(t)(1-N(t)K)dt+σ1N(t)dB1(t).
By the comparison theorem of stochastic equations [14], we have N(t)≤N¯(t) a.s. t∈[0,τe).Besides, for the following equation(10)dN_(t)=N_(t)(r-bm-rKN_(t))dt+σ1N_(t)dB1(t),N_(0)=N_0,
there is a unique solution as
(11)N_(t)=exp{(r-bm-σ122)t+σ1B1(t)}×(1N_0+rK∫0t‍exp{(r-bm-σ122)shhhhhhhhhhh22hhhhhh+σ1B1(s)(r-bm-σ122)}∫0t‍)-1.
In model (6), for α∈(0,1), we can get
(12)dN(t)≥N(t)(r-bm-rKN(t))dtdN(t)≥+σ1N(t)dB1(t);
then N(t)≥N_(t) a.s. t∈[0,τe).Consequently, we obtain(13)N_(t)≤N(t)≤N¯(t)a.s.t∈[0,τe).On the other hand, the equation(14)dP_(t)=P_(t)(c-d-cm-hP_(t))dt+σ2P_(t)dB2(t),hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhP_(0)=P_0,
has a unique solution as follows:
(15)P_(t)=exp{(c-d-cm-σ222)t+σ2B2(t)}×(1P_0+h∫0t‍exp{(c-d-cm-σ222)shhhhhhhhhhhhhhhhh+σ2B2(s)(c-d-cm-σ222)(c-d-cm-(σ22/2))}ds∫0t‍)-1.
Considering the predator population P(t) in model (6), we have
(16)dP(t)≤P(t)(c-d)dt+σ2P(t)dB2(t),dP(t)=P(t)(c-d-cmPα(t)N(t)+mPα(t)-hP(t))dtdP(t)=+σ2P(t)dB2(t)dP(t)≥P(t)(c-d-cm-hP(t))dt+σ2P(t)dB2(t).
By the comparison theorem, we obtain P(t)≥P_(t) a.s. t∈[0,τe); then
(17)P_(t)≤P(t)≤P0exp{(c-d-σ222)t+σ2B2(t)}=P¯(t)a.s.t∈[0,τe).From the representation of solutionsN¯(t), N_(t), P_(t), and P¯(t), we can see that they are all existence for t∈[0,∞); that is, τe=∞. Therefore, we have the following theorem to show that the positive solution of model (6) is global, which is essential for a population system.Theorem 2.
There is a unique positive solution(N(t),P(t)) of model (6) almost surely for any initial value (N0,P0)∈R+2. Moreover there exist functions N¯(t), N_(t), P_(t), and P¯(t) defined as (8), (11), (15), and (17) such that
(18)N_(t)≤N(t)≤N¯(t),P_(t)≤P(t)≤P¯(t)a.s.t≥0.


### 2.2. Stochastic Boundedness

In this subsection, we show that the solution(N(t),P(t)) of model (6) with any positive initial value is uniformly bounded in mean.Theorem 3.
The solution(N(t),P(t)) of model (6) with any positive initial value has the property that
(19)limsupt→∞E[N(t)]≤K,limsupt→∞E[P(t)]≤cK(r+d)24rbd.Proof.
From (7), we obtain
(20)limsupt→∞E[N¯(t)]≤K;
combining N(t)≤N¯(t), then
(21)limsupt→∞E[N(t)]≤K.
Set(22)M(t)=N(t)+bcP(t);
then
(23)dM(t)=(rN(t)(1-N(t)K)-bcP(t)(d+hP(t)))dtdM(t)=+σ1N(t)dB1(t)+σ2bcP(t)dB2(t)dM(t)=((r+d)N(t)-rKN2(t)-bhcP2(t)-dM(t))dtdM(t)=+σ1N(t)dB1(t)+σ2bcP(t)dB2(t).
Integrating the above equation from 0 to t, we obtain
(24)M(t)=M(0)+∫0t‍(bhc(r+d)N(s)M(t)=hhh5hhhhhhh-rKN2(s)-bhcP2(s)-dM(s))dsM(t)=+σ1∫0t‍N(s)dB1(s)+σ2bc∫0t‍P(s)dB2(s),
and taking expectations leads to
(25)E[M(t)]=M(0)+∫0t‍E[(r+d)N(s)-rKN2(s)E[M(t)]=5hhhhhhhhhhh-bhcP2(s)-dM(s)]ds;
then
(26)dE[M(t)]dt=(r+d)E[N(t)]dE[M(t)]dt=-rKE[N2(t)]-bhcE[P2(t)]-dE[M(t)]dE[M(t)]dt≤(r+d)E[N(t)]-rK(E[N(t)])2dE[M(t)]dt=-bhc(E[P(t)])2-dE[M(t)]dE[M(t)]dt≤K(r+d)24r-dE[M(t)].
By the comparison theorem, we can get
(27)limsupt→∞E[M(t)]=limsupt→∞(E[N(t)]+bcE[P(t)])≤K(r+d)24rd.
Therefore, we obtain
(28)limsupt→∞E[P(t)]≤cK(r+d)24rbd.
This completes the proof.


### 2.3.  The Long Time Behavior

It is well known that the property of permanence is more desirable since it means the long time survival in a population dynamics. Now, the definition of stochastic permanence will be given below [32, 33].Definition 4.
The solution(N(t),P(t)) of model (6) is said to be stochastically permanent, if, for any ɛ∈(0,1), there exists a pair of positive constants δ=δ(ɛ) and χ=χ(ɛ) such that, for any initial value (N0,P0)∈R+2, the solution (N(t),P(t)) to model (6) has the properties that
(29)liminft→∞P{|N(t),P(t)|≥δ}≥1-ɛ,liminft→∞P{|N(t),P(t)|≤χ}≥1-ɛ.Lemma 5.
For any initial value(N0,P0)∈R+2, the solution (N(t),P(t)) satisfies that
(30)limsupt→∞E[(N2+P2)-θ/2]≤Ck,
where C=C(θ) is a positive constant and θ,k are arbitrary positive constants satisfying
(31)θmin{r-bm,c-d}>θ(θ+1)2max{σ12,σ22}+k.Proof.
Set a function(32)V(N,P)=1N+P,
for (N(t),P(t))∈R+2; using Itô’s formula, we have
(33)dV=-V2[N(r-rKN-bPN+mPα)hhhhh+P(cNN+mPα-d-hP)N(r-rKN-bPN+mPα)]dt+V3[σ12N2+σ22P2]dt-V2[σ1NdB1+σ2PdB2].
Choosing a positive constant θ and by Itô’s formula, we get
(34)L(1+V)θ=θ(1+V)θ-1{-V2[N(cNN+mPα-d-hP)(r-rKN-bPN+mPα)hhhhhhhhhhhhhhh+P(cNN+mPα-d-hP)(r-rKN-bPN+mPα)]hhhhhhhhhh+V3[σ12N2+σ22P2][N(r-rKN-bPN+mPα)}+θ(θ-1)2V4(1+V)θ-2[σ12N2+σ22P2].
Letk>0 be sufficiently small such that it satisfies (31); by Itô’s formula, then
(35)Lekt(1+V)θ=ekt(1+V)θ-2{θ(θ-1)2k(1+V)2-θV2hhhhhhhhhhhhh×[N(cNN+mPα-d-hP)(r-rKN-bPN+mPα)hhhhhhhhhhhhhhhh+P(cNN+mPα-d-hP)(r-rKN-bPN+mPα)]hhhhhhhhhhhhhh-θV3[N(cNN+mPα-d-hP)(r-rKN-bPN+mPα)hhhhhhhhhhhhhhhhhhhh+P(cNN+mPα-d-hP)(r-rKN-bPN+mPα)]hhhhhhhhhhhhhh+θV3[σ12N2+σ22P2]hhhhhhhhhhhhhh+θ(θ-1)2V4[σ12N2+σ22P2]}.
Based on the following inequality,
(36)V3(σ12N2+σ22P2)≤max{σ12,σ22}V.
Therefore, we obtain
(37)Lekt(1+V)θ≤ekt(1+V)θ-2[θ(θ-1)2k(1+V)2hhhhhhhhhhhhhh-θV2(N(r-rKN-bPN+mPα)hhhhhhhhhhhhhhhhhhhh+P(cNN+mPα-d-hP))hhhhhhhhhhhhhh-θV3(N(r-rKN-bPN+mPα)hhhhhhhhhhhhhhhhhhhh+P(cNN+mPα-d-hP))hhhhhhhhhhhhhh+θVmax{σ12,σ22}hhhhhhhhhhhhhh+θ(θ-1)2V2max{σ12,σ22}]=ekt(1+V)θ-2[k+(-θ(θ-1)2max{σ12,σ22})2k-θV2hhhhhhhhhhhhhhhhh×(-θ(θ-1)2max{σ12,σ22})rN-rKN2-bNPN+mPαhhhhhhhhhhhhhhhhhhhh+cNPN+mPα-dP-hP2-θ(θ-1)2max{σ12,σ22}))hhhhhhhhhhhhhhhhh+θmax{σ12,σ22}-θ(θ-1)2max{σ12,σ22}))Vhhhhhhhhhhhhhh-(θ(-θ(θ-1)2max{σ12,σ22})rN-rKN2-bNPN+mPαhhhhhhhhhhhhhhhhhhh+cNPN+mPα-dP-hP2-θ(θ-1)2max{σ12,σ22}))hhhhhhhhhhhhhhhhh-θ(θ-1)2max{σ12,σ22})V2]≤ekt(1+V)θ-2×[-θ(θ-1)2max{σ12,σ22})k+θV2(rKN2+hP2)hh+(2k+θV2(rKN2+hP2)-θV2hhhhhh×(rN-bNPN+mPα+cNPN+mPα-dP)hhhhhh+θmax{σ12,σ22}(rKN2+hP2))Vhh-(θ(rN-bNPN+mPα+cNPN+mPα-dP)hhhhhhh-θ(θ-1)2max{σ12,σ22}-k)V2]≤ekt(1+V)θ-2[-θ(θ-1)2max{σ12,σ22})(k+θmax{rK,h})hhhhhhhhhhhhhh+(2k+θmax{rK,h}hhhhhhhhhhhhhhhhh-θmin{r-bm,c-d}hhhhhhhhhhhhhhhhh+θmax{σ12,σ22}{rK,h})Vhhhhhhhhhh-(θmin{r-bm,c-d}hhhhhhhhhhhhh-θ(θ-1)2max{σ12,σ22}-k)V2].
There exists a positive constant C0 such that Lekt(1+V)θ≤C0ekt; then
(38)E[ekt(1+V)θ]≤(1+V(0))θ+C0kekt.
So, we can get
(39)limsupt→∞EVθ(t)≤limsupt→∞E(1+V(t))θ≤C0k.
In addition, we know that(N+P)θ≤2θ(N2+P2)θ/2; consequently,
(40)limsupt→∞E[(N2+P2)-θ/2]≤2θlimsupt→∞EVθ(t)≤2θC0k≜Ck.
The proof is complete.Based on the results of Theorem3, Lemma 5, and the Chebyshev inequality [14], we can obtain the following theorem.Theorem 6.
Assume thatmax{σ12,σ22}<2min{r-b/m,c-d}; the solution of model (6) is stochastically permanent.In a view of ecology, the coexistence of species may be a good situation. In the following, we consider the stochastic persistence (i.e., stochastic persistence in mean) of the species.Theorem 7.
Assume thatr-σ12/2>b/m holds, for any initial value N0>0; then the solution N(t) to model (6) has the property
(41)K(r-b/m-σ12/2)r≤liminft→∞1t∫0t‍N(s)ds≤limsupt→∞1t∫0t‍N(s)ds≤K(r-σ12/2)ra.s.Proof.
DenotingV(N)=lnN and by Itô’s formula, we obtain
(42)dV=(r-rKN(t)-bP(t)N(t)+mPα(t)-σ122)dt+σ1dB1(t).
Then, we have
(43)lnN(t)=lnN0+(r-σ122)t-rK∫0t‍N(s)ds-b∫0t‍P(s)N(s)+mPα(s)ds+σ1B1(t).
And we get
(44)rK∫0t‍N(s)ds≤-lnN(t)+lnN0+(r-σ122)t+σ1B1(t).
Dividing t on both sides of the previously mentioned inequality yields
(45)rK1t∫0t‍N(s)ds≤-lnN(t)trK1t∫0t‍N(s)ds≤+lnN0t+(r-σ122)+σ1B1(t)t.
Lettingt→∞, we know that
(46)limt→∞lnN(t)t=0a.s.;
Then we obtain
(47)limsupt→∞1t∫0t‍N(s)ds≤K(r-σ12/2)ra.s.
On the other hand,(48)rK∫0t‍N(s)ds≥-lnN(t)+lnN0rK∫0t‍N(s)ds≥+(r-σ122)t-bmt+σ1B1(t);
dividing t on both sides and letting t→∞, then
(49)liminft→∞1t∫0t‍N(s)ds≥K(r-b/m-σ12/2)r.
From the above results, inequality (41) holds.Theorem 8.
Assume thatc-d-σ22/2>0 holds and that (N(t),P(t)) is the solution of model (6) for any initial value (N0,P0)∈R+2; then
(50)liminft→∞1t∫0t‍N(s)N(s)+mPα(s)ds≥d+σ22/2c,limsupt→∞1t∫0t‍Pα(s)N(s)+mPα(s)ds≤c-d-σ22/2cm,
which are stable in time average.Proof.
DenotingV(P)=lnP and by Itô’s formula, we have
(51)dV=(cN(t)N(t)+mPα(t)-d-hP(t)-σ222)dt+σ2dB2(t).
Then,
(52)c∫0t‍N(s)N(s)+mPα(s)ds=lnP(t)-lnP0+(d+σ222)t+h∫0t‍P(s)ds-σ2B2(t)≥lnP(t)-lnP0+(d+σ222)t-σ2B2(t).
Dividing t on both sides yields
(53)ct∫0t‍N(s)N(s)+mPα(s)ds≥lnP(t)t-lnP0t+d+σ222-σ2B2(t)t;
we obtain
(54)liminft→∞1t∫0t‍N(s)N(s)+mPα(s)ds≥d+σ22/2c.
Furthermore,(55)dV=(c-cmPα(t)N(t)+mPα(t)-d-hP(t)-σ222)dtdV=+σ2dB2(t);
then we can get
(56)cm∫0t‍Pα(s)N(s)+mPα(s)ds=-lnP(t)+lnP0+(c-d-σ222)t-h∫0t‍P(s)ds+σ2B2(t)≤-lnP(t)+lnP0+(c-d-σ222)t+σ2B2(t).
Dividing t on both sides, we obtain
(57)limsupt→∞1t∫0t‍Pα(s)N(s)+mPα(s)ds≤c-d-(σ22/2)cm.


### 2.4. Extinction

From (17), if c-d-σ22/2<0, then limt→∞P(t)=0 a.s. Moreover, from Theorem 7 and (49), we know that if r-b/m-σ12/2>0 holds, then
(58)liminft→∞1t∫0t‍N(s)ds≥K(r-b/m-σ12/2)ra.s.,
which implies that there are a T0>0 and a positive constant n0 such that N(t)>n0 a.s. for t≥T0. Besides, for all ɛ>0, there are T>T0 and Ωɛ such that P(Ωɛ)≥1-ɛ and bP(t)/N(t)≤ɛ for t≥T. Then we obtain
(59)dN(t)=N(t)(r-rKN(t)-bP(t)N(t)+mPα(t))dtdN(t)=+σ1N(t)dB1(t)dN(t)≥N(t)(r-rKN(t)-bP(t)N(t))dt+σ1N(t)dB1(t)dN(t)≥N(t)(r-rKN(t)-ɛ)dt+σ1N(t)dB1(t),
which implies that
(60)liminft→∞1t∫0t‍N(s)ds≥K(r-ɛ-σ12/2)r>0.From Theorem7 and (47), we have
(61)limsupt→∞1t∫0t‍N(s)ds≤K(r-σ12/2)ra.s.
Therefore, by the arbitrary of ɛ, we get
(62)limt→∞1t∫0t‍N(s)ds=K(r-σ12/2)ra.s.Combining the above arguments, we can get the theorem as follows.Theorem 9.
Let(N(t),P(t)) be the solution of model (6) with any initial value (N0,P0)∈R+2. If r-b/m-σ12/2>0 and c-d-σ22/2<0, then
(63)limt→∞1t∫0t‍N(s)ds=K(r-σ12/2)r,limt→∞P(t)=0.Furthermore, setu(t)=lnN(t) and v(t)=lnP(t); for the first equation of model (6) we have
(64)du(t)=(r-rKeu(t)-bev(t)eu(t)+meαv(t)-σ122)dt+σ1dB1(t)≤(r-σ122)dt+σ1dB1(t).
Taking the comparison theorem of stochastic equations and the theory of diffusion processes [14], then limt→∞u(t)=-∞ a.s.; that is,
(65)limt→∞N(t)=0a.s.Similarly, we obtain(66)limt→∞P(t)=0a.s.
If not, then there is a positive constant H such that
(67)limsupt→∞P(t)=H>0a.s.
Hence, for any given ɛ>0, there exist t0 and a set Ωɛ such that P(Ωɛ)≥1-ɛ and cN(t)/(N(t)+mPα(t))≤ɛ for t≥t0. Therefore,
(68)-P(t)(d+hP(t))dt+σ2P(t)dB2(t)≤dP(t)≤P(t)(-d+ɛ)dt+σ2P(t)dB2(t),-(d+hP(t)+σ222)dt+σ2dB2(t)≤dv(t)≤(-d+ɛ-σ222)dt+σ2dB2(t).
By the same reasoning as previously stated, we can get limt→∞v(t)=-∞ a.s.; that is,
(69)limt→∞P(t)=0a.s.
There is a contradiction; hence (66) is true.Based on the above, we obtain the following theorem which means that if the noise satisfies some conditions, then both speciesN and P of model (6) will die out.Theorem 10.
Let(N(t),P(t)) be the solution of model (6) with any initial value (N0,P0)∈R+2. If r-b/m-σ12/2<0, then
(70)limt→∞N(t)=0,limt→∞P(t)=0.


## 2.1. Positive and Global Solution

For model (6), there is a positive local solution.Lemma 1.
There is a unique local solution(N(t),P(t)) for t∈[0,τe) to model (6) almost surely for initial value (N0,P0)∈R+2, where τe is the explosion time.The proof of this lemma is rather standard and hence is omitted.Lemma1 only tells us that there is a unique positive local solution to model (6). Next, we show that this solution is global which is more interesting.In particular, let us consider the one-dimensional stochastic population model(7)dN¯(t)=rN¯(t)(1-N¯(t)K)dt+σ1N¯(t)dB1(t),t≥0,N¯(0)=N¯0;
there is an explicit solution
(8)N¯(t)=exp{(r-σ122)t+σ1B1(t)}×(1N¯0+rK∫0t‍exp{(r-σ122)shhhhhhhhhhhhh22hhhh+σ1B1(s)(r-σ122)(r-(σ12/2))}ds∫0t‍)-1.
From model (6), we have
(9)dN(t)≤rN(t)(1-N(t)K)dt+σ1N(t)dB1(t).
By the comparison theorem of stochastic equations [14], we have N(t)≤N¯(t) a.s. t∈[0,τe).Besides, for the following equation(10)dN_(t)=N_(t)(r-bm-rKN_(t))dt+σ1N_(t)dB1(t),N_(0)=N_0,
there is a unique solution as
(11)N_(t)=exp{(r-bm-σ122)t+σ1B1(t)}×(1N_0+rK∫0t‍exp{(r-bm-σ122)shhhhhhhhhhh22hhhhhh+σ1B1(s)(r-bm-σ122)}∫0t‍)-1.
In model (6), for α∈(0,1), we can get
(12)dN(t)≥N(t)(r-bm-rKN(t))dtdN(t)≥+σ1N(t)dB1(t);
then N(t)≥N_(t) a.s. t∈[0,τe).Consequently, we obtain(13)N_(t)≤N(t)≤N¯(t)a.s.t∈[0,τe).On the other hand, the equation(14)dP_(t)=P_(t)(c-d-cm-hP_(t))dt+σ2P_(t)dB2(t),hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhP_(0)=P_0,
has a unique solution as follows:
(15)P_(t)=exp{(c-d-cm-σ222)t+σ2B2(t)}×(1P_0+h∫0t‍exp{(c-d-cm-σ222)shhhhhhhhhhhhhhhhh+σ2B2(s)(c-d-cm-σ222)(c-d-cm-(σ22/2))}ds∫0t‍)-1.
Considering the predator population P(t) in model (6), we have
(16)dP(t)≤P(t)(c-d)dt+σ2P(t)dB2(t),dP(t)=P(t)(c-d-cmPα(t)N(t)+mPα(t)-hP(t))dtdP(t)=+σ2P(t)dB2(t)dP(t)≥P(t)(c-d-cm-hP(t))dt+σ2P(t)dB2(t).
By the comparison theorem, we obtain P(t)≥P_(t) a.s. t∈[0,τe); then
(17)P_(t)≤P(t)≤P0exp{(c-d-σ222)t+σ2B2(t)}=P¯(t)a.s.t∈[0,τe).From the representation of solutionsN¯(t), N_(t), P_(t), and P¯(t), we can see that they are all existence for t∈[0,∞); that is, τe=∞. Therefore, we have the following theorem to show that the positive solution of model (6) is global, which is essential for a population system.Theorem 2.
There is a unique positive solution(N(t),P(t)) of model (6) almost surely for any initial value (N0,P0)∈R+2. Moreover there exist functions N¯(t), N_(t), P_(t), and P¯(t) defined as (8), (11), (15), and (17) such that
(18)N_(t)≤N(t)≤N¯(t),P_(t)≤P(t)≤P¯(t)a.s.t≥0.


## 2.2. Stochastic Boundedness

In this subsection, we show that the solution(N(t),P(t)) of model (6) with any positive initial value is uniformly bounded in mean.Theorem 3.
The solution(N(t),P(t)) of model (6) with any positive initial value has the property that
(19)limsupt→∞E[N(t)]≤K,limsupt→∞E[P(t)]≤cK(r+d)24rbd.Proof.
From (7), we obtain
(20)limsupt→∞E[N¯(t)]≤K;
combining N(t)≤N¯(t), then
(21)limsupt→∞E[N(t)]≤K.
Set(22)M(t)=N(t)+bcP(t);
then
(23)dM(t)=(rN(t)(1-N(t)K)-bcP(t)(d+hP(t)))dtdM(t)=+σ1N(t)dB1(t)+σ2bcP(t)dB2(t)dM(t)=((r+d)N(t)-rKN2(t)-bhcP2(t)-dM(t))dtdM(t)=+σ1N(t)dB1(t)+σ2bcP(t)dB2(t).
Integrating the above equation from 0 to t, we obtain
(24)M(t)=M(0)+∫0t‍(bhc(r+d)N(s)M(t)=hhh5hhhhhhh-rKN2(s)-bhcP2(s)-dM(s))dsM(t)=+σ1∫0t‍N(s)dB1(s)+σ2bc∫0t‍P(s)dB2(s),
and taking expectations leads to
(25)E[M(t)]=M(0)+∫0t‍E[(r+d)N(s)-rKN2(s)E[M(t)]=5hhhhhhhhhhh-bhcP2(s)-dM(s)]ds;
then
(26)dE[M(t)]dt=(r+d)E[N(t)]dE[M(t)]dt=-rKE[N2(t)]-bhcE[P2(t)]-dE[M(t)]dE[M(t)]dt≤(r+d)E[N(t)]-rK(E[N(t)])2dE[M(t)]dt=-bhc(E[P(t)])2-dE[M(t)]dE[M(t)]dt≤K(r+d)24r-dE[M(t)].
By the comparison theorem, we can get
(27)limsupt→∞E[M(t)]=limsupt→∞(E[N(t)]+bcE[P(t)])≤K(r+d)24rd.
Therefore, we obtain
(28)limsupt→∞E[P(t)]≤cK(r+d)24rbd.
This completes the proof.


## 2.3.  The Long Time Behavior

It is well known that the property of permanence is more desirable since it means the long time survival in a population dynamics. Now, the definition of stochastic permanence will be given below [32, 33].Definition 4.
The solution(N(t),P(t)) of model (6) is said to be stochastically permanent, if, for any ɛ∈(0,1), there exists a pair of positive constants δ=δ(ɛ) and χ=χ(ɛ) such that, for any initial value (N0,P0)∈R+2, the solution (N(t),P(t)) to model (6) has the properties that
(29)liminft→∞P{|N(t),P(t)|≥δ}≥1-ɛ,liminft→∞P{|N(t),P(t)|≤χ}≥1-ɛ.Lemma 5.
For any initial value(N0,P0)∈R+2, the solution (N(t),P(t)) satisfies that
(30)limsupt→∞E[(N2+P2)-θ/2]≤Ck,
where C=C(θ) is a positive constant and θ,k are arbitrary positive constants satisfying
(31)θmin{r-bm,c-d}>θ(θ+1)2max{σ12,σ22}+k.Proof.
Set a function(32)V(N,P)=1N+P,
for (N(t),P(t))∈R+2; using Itô’s formula, we have
(33)dV=-V2[N(r-rKN-bPN+mPα)hhhhh+P(cNN+mPα-d-hP)N(r-rKN-bPN+mPα)]dt+V3[σ12N2+σ22P2]dt-V2[σ1NdB1+σ2PdB2].
Choosing a positive constant θ and by Itô’s formula, we get
(34)L(1+V)θ=θ(1+V)θ-1{-V2[N(cNN+mPα-d-hP)(r-rKN-bPN+mPα)hhhhhhhhhhhhhhh+P(cNN+mPα-d-hP)(r-rKN-bPN+mPα)]hhhhhhhhhh+V3[σ12N2+σ22P2][N(r-rKN-bPN+mPα)}+θ(θ-1)2V4(1+V)θ-2[σ12N2+σ22P2].
Letk>0 be sufficiently small such that it satisfies (31); by Itô’s formula, then
(35)Lekt(1+V)θ=ekt(1+V)θ-2{θ(θ-1)2k(1+V)2-θV2hhhhhhhhhhhhh×[N(cNN+mPα-d-hP)(r-rKN-bPN+mPα)hhhhhhhhhhhhhhhh+P(cNN+mPα-d-hP)(r-rKN-bPN+mPα)]hhhhhhhhhhhhhh-θV3[N(cNN+mPα-d-hP)(r-rKN-bPN+mPα)hhhhhhhhhhhhhhhhhhhh+P(cNN+mPα-d-hP)(r-rKN-bPN+mPα)]hhhhhhhhhhhhhh+θV3[σ12N2+σ22P2]hhhhhhhhhhhhhh+θ(θ-1)2V4[σ12N2+σ22P2]}.
Based on the following inequality,
(36)V3(σ12N2+σ22P2)≤max{σ12,σ22}V.
Therefore, we obtain
(37)Lekt(1+V)θ≤ekt(1+V)θ-2[θ(θ-1)2k(1+V)2hhhhhhhhhhhhhh-θV2(N(r-rKN-bPN+mPα)hhhhhhhhhhhhhhhhhhhh+P(cNN+mPα-d-hP))hhhhhhhhhhhhhh-θV3(N(r-rKN-bPN+mPα)hhhhhhhhhhhhhhhhhhhh+P(cNN+mPα-d-hP))hhhhhhhhhhhhhh+θVmax{σ12,σ22}hhhhhhhhhhhhhh+θ(θ-1)2V2max{σ12,σ22}]=ekt(1+V)θ-2[k+(-θ(θ-1)2max{σ12,σ22})2k-θV2hhhhhhhhhhhhhhhhh×(-θ(θ-1)2max{σ12,σ22})rN-rKN2-bNPN+mPαhhhhhhhhhhhhhhhhhhhh+cNPN+mPα-dP-hP2-θ(θ-1)2max{σ12,σ22}))hhhhhhhhhhhhhhhhh+θmax{σ12,σ22}-θ(θ-1)2max{σ12,σ22}))Vhhhhhhhhhhhhhh-(θ(-θ(θ-1)2max{σ12,σ22})rN-rKN2-bNPN+mPαhhhhhhhhhhhhhhhhhhh+cNPN+mPα-dP-hP2-θ(θ-1)2max{σ12,σ22}))hhhhhhhhhhhhhhhhh-θ(θ-1)2max{σ12,σ22})V2]≤ekt(1+V)θ-2×[-θ(θ-1)2max{σ12,σ22})k+θV2(rKN2+hP2)hh+(2k+θV2(rKN2+hP2)-θV2hhhhhh×(rN-bNPN+mPα+cNPN+mPα-dP)hhhhhh+θmax{σ12,σ22}(rKN2+hP2))Vhh-(θ(rN-bNPN+mPα+cNPN+mPα-dP)hhhhhhh-θ(θ-1)2max{σ12,σ22}-k)V2]≤ekt(1+V)θ-2[-θ(θ-1)2max{σ12,σ22})(k+θmax{rK,h})hhhhhhhhhhhhhh+(2k+θmax{rK,h}hhhhhhhhhhhhhhhhh-θmin{r-bm,c-d}hhhhhhhhhhhhhhhhh+θmax{σ12,σ22}{rK,h})Vhhhhhhhhhh-(θmin{r-bm,c-d}hhhhhhhhhhhhh-θ(θ-1)2max{σ12,σ22}-k)V2].
There exists a positive constant C0 such that Lekt(1+V)θ≤C0ekt; then
(38)E[ekt(1+V)θ]≤(1+V(0))θ+C0kekt.
So, we can get
(39)limsupt→∞EVθ(t)≤limsupt→∞E(1+V(t))θ≤C0k.
In addition, we know that(N+P)θ≤2θ(N2+P2)θ/2; consequently,
(40)limsupt→∞E[(N2+P2)-θ/2]≤2θlimsupt→∞EVθ(t)≤2θC0k≜Ck.
The proof is complete.Based on the results of Theorem3, Lemma 5, and the Chebyshev inequality [14], we can obtain the following theorem.Theorem 6.
Assume thatmax{σ12,σ22}<2min{r-b/m,c-d}; the solution of model (6) is stochastically permanent.In a view of ecology, the coexistence of species may be a good situation. In the following, we consider the stochastic persistence (i.e., stochastic persistence in mean) of the species.Theorem 7.
Assume thatr-σ12/2>b/m holds, for any initial value N0>0; then the solution N(t) to model (6) has the property
(41)K(r-b/m-σ12/2)r≤liminft→∞1t∫0t‍N(s)ds≤limsupt→∞1t∫0t‍N(s)ds≤K(r-σ12/2)ra.s.Proof.
DenotingV(N)=lnN and by Itô’s formula, we obtain
(42)dV=(r-rKN(t)-bP(t)N(t)+mPα(t)-σ122)dt+σ1dB1(t).
Then, we have
(43)lnN(t)=lnN0+(r-σ122)t-rK∫0t‍N(s)ds-b∫0t‍P(s)N(s)+mPα(s)ds+σ1B1(t).
And we get
(44)rK∫0t‍N(s)ds≤-lnN(t)+lnN0+(r-σ122)t+σ1B1(t).
Dividing t on both sides of the previously mentioned inequality yields
(45)rK1t∫0t‍N(s)ds≤-lnN(t)trK1t∫0t‍N(s)ds≤+lnN0t+(r-σ122)+σ1B1(t)t.
Lettingt→∞, we know that
(46)limt→∞lnN(t)t=0a.s.;
Then we obtain
(47)limsupt→∞1t∫0t‍N(s)ds≤K(r-σ12/2)ra.s.
On the other hand,(48)rK∫0t‍N(s)ds≥-lnN(t)+lnN0rK∫0t‍N(s)ds≥+(r-σ122)t-bmt+σ1B1(t);
dividing t on both sides and letting t→∞, then
(49)liminft→∞1t∫0t‍N(s)ds≥K(r-b/m-σ12/2)r.
From the above results, inequality (41) holds.Theorem 8.
Assume thatc-d-σ22/2>0 holds and that (N(t),P(t)) is the solution of model (6) for any initial value (N0,P0)∈R+2; then
(50)liminft→∞1t∫0t‍N(s)N(s)+mPα(s)ds≥d+σ22/2c,limsupt→∞1t∫0t‍Pα(s)N(s)+mPα(s)ds≤c-d-σ22/2cm,
which are stable in time average.Proof.
DenotingV(P)=lnP and by Itô’s formula, we have
(51)dV=(cN(t)N(t)+mPα(t)-d-hP(t)-σ222)dt+σ2dB2(t).
Then,
(52)c∫0t‍N(s)N(s)+mPα(s)ds=lnP(t)-lnP0+(d+σ222)t+h∫0t‍P(s)ds-σ2B2(t)≥lnP(t)-lnP0+(d+σ222)t-σ2B2(t).
Dividing t on both sides yields
(53)ct∫0t‍N(s)N(s)+mPα(s)ds≥lnP(t)t-lnP0t+d+σ222-σ2B2(t)t;
we obtain
(54)liminft→∞1t∫0t‍N(s)N(s)+mPα(s)ds≥d+σ22/2c.
Furthermore,(55)dV=(c-cmPα(t)N(t)+mPα(t)-d-hP(t)-σ222)dtdV=+σ2dB2(t);
then we can get
(56)cm∫0t‍Pα(s)N(s)+mPα(s)ds=-lnP(t)+lnP0+(c-d-σ222)t-h∫0t‍P(s)ds+σ2B2(t)≤-lnP(t)+lnP0+(c-d-σ222)t+σ2B2(t).
Dividing t on both sides, we obtain
(57)limsupt→∞1t∫0t‍Pα(s)N(s)+mPα(s)ds≤c-d-(σ22/2)cm.


## 2.4. Extinction

From (17), if c-d-σ22/2<0, then limt→∞P(t)=0 a.s. Moreover, from Theorem 7 and (49), we know that if r-b/m-σ12/2>0 holds, then
(58)liminft→∞1t∫0t‍N(s)ds≥K(r-b/m-σ12/2)ra.s.,
which implies that there are a T0>0 and a positive constant n0 such that N(t)>n0 a.s. for t≥T0. Besides, for all ɛ>0, there are T>T0 and Ωɛ such that P(Ωɛ)≥1-ɛ and bP(t)/N(t)≤ɛ for t≥T. Then we obtain
(59)dN(t)=N(t)(r-rKN(t)-bP(t)N(t)+mPα(t))dtdN(t)=+σ1N(t)dB1(t)dN(t)≥N(t)(r-rKN(t)-bP(t)N(t))dt+σ1N(t)dB1(t)dN(t)≥N(t)(r-rKN(t)-ɛ)dt+σ1N(t)dB1(t),
which implies that
(60)liminft→∞1t∫0t‍N(s)ds≥K(r-ɛ-σ12/2)r>0.From Theorem7 and (47), we have
(61)limsupt→∞1t∫0t‍N(s)ds≤K(r-σ12/2)ra.s.
Therefore, by the arbitrary of ɛ, we get
(62)limt→∞1t∫0t‍N(s)ds=K(r-σ12/2)ra.s.Combining the above arguments, we can get the theorem as follows.Theorem 9.
Let(N(t),P(t)) be the solution of model (6) with any initial value (N0,P0)∈R+2. If r-b/m-σ12/2>0 and c-d-σ22/2<0, then
(63)limt→∞1t∫0t‍N(s)ds=K(r-σ12/2)r,limt→∞P(t)=0.Furthermore, setu(t)=lnN(t) and v(t)=lnP(t); for the first equation of model (6) we have
(64)du(t)=(r-rKeu(t)-bev(t)eu(t)+meαv(t)-σ122)dt+σ1dB1(t)≤(r-σ122)dt+σ1dB1(t).
Taking the comparison theorem of stochastic equations and the theory of diffusion processes [14], then limt→∞u(t)=-∞ a.s.; that is,
(65)limt→∞N(t)=0a.s.Similarly, we obtain(66)limt→∞P(t)=0a.s.
If not, then there is a positive constant H such that
(67)limsupt→∞P(t)=H>0a.s.
Hence, for any given ɛ>0, there exist t0 and a set Ωɛ such that P(Ωɛ)≥1-ɛ and cN(t)/(N(t)+mPα(t))≤ɛ for t≥t0. Therefore,
(68)-P(t)(d+hP(t))dt+σ2P(t)dB2(t)≤dP(t)≤P(t)(-d+ɛ)dt+σ2P(t)dB2(t),-(d+hP(t)+σ222)dt+σ2dB2(t)≤dv(t)≤(-d+ɛ-σ222)dt+σ2dB2(t).
By the same reasoning as previously stated, we can get limt→∞v(t)=-∞ a.s.; that is,
(69)limt→∞P(t)=0a.s.
There is a contradiction; hence (66) is true.Based on the above, we obtain the following theorem which means that if the noise satisfies some conditions, then both speciesN and P of model (6) will die out.Theorem 10.
Let(N(t),P(t)) be the solution of model (6) with any initial value (N0,P0)∈R+2. If r-b/m-σ12/2<0, then
(70)limt→∞N(t)=0,limt→∞P(t)=0.


## 3. Numerical Simulations

In this section, we perform some numerical simulations for model (6) with environmental noise to illustrate the previously mentioned analytical findings by referring to the method mentioned in Higham [34]. Next, we consider the discretization equations
(71)Ni+1=Ni+Ni(r-rKNi-bPiNi+mPiα)Δt+αNiΔtξ1i+α22Ni2(ξ1i2-1)Δt,Pi+1=Pi+Pi(cNiNi+mPiα-d-hPi)Δt+βPiΔtξ2i+β22Pi2(ξ2i2-1)Δt,
where ξ1i and ξ2i(i=1,2,…,n) are the Gaussian random variables N(0,1).When choosing the values of parametersr=2, K=0.15, b=0.8, c=0.5, d=0.1, m=0.1, α=0.2, and h=0.1 for model (4), which has three equilibria in the positive quadrant, where E0=(0,0) (total extinct) and E1=(0.15,0) (extinct of the predator or prey only) are saddle points, E*=(0.023,0.2025) (coexistence of the prey and predator) is globally asymptotically stable. The trajectory of the prey and predator population of model (4) is shown in Figure 1.Figure 1
Phase portrait of model (4). Other parameters are taken as r=2, K=0.15, b=0.8, c=0.5, d=0.1, m=0.1, α=0.2, and h=0.1. The horizontal axis is prey population N and the vertical axis is predator population P. E0=(0,0) and E1=(0.15,0) are two saddle points; E*=(0.023,0.2025) is stable.Figure2 shows the time-series plots of model (6) with different noise intensities σ12 and σ22. When choosing σ1=σ2=0.045 (Figure 2(a)) and σ1=0.12, σ2=0.3 (Figure 2(b)), from Theorem 6, we know that the positive solution of model (6) is stochastically permanent, which means that stochastic perturbations do not change the permanence of the deterministic model (4). Moreover, from Theorem 7, the model will be stochastically persistent in mean. In other words, we can use the deterministic model (4) describing the dynamics of the stochastic model when the noise intensities σ12 and σ22 are small. From Figure 2(b), we can observe that the violent fluctuations appear as the noise intensities further increase. It means that noise has strong destabilizing effects on the model and the amplitude of the fluctuations in population density of prey and predator species increases obviously, implying instability of the coexisting equilibrium point in the stochastic environment.Solutions of model (6) with different noise and other parameters are the same as those of Figure 1 and initial condition (N0,P0)=(0.05,0.23). (a) σ1=σ2=0.045 and (b) σ1=0.12, σ2=0.3.
(a)
(b)In Figure3, we continue to choose different noise intensities σ12 and σ22 to consider the effects of noise to model (6). When choosing σ1=0.15 and σ2=0.9 (Figure 3(a)), the conditions of Theorem 9 are satisfied; then we can find that prey population N(t) of model (6) is permanent and predator population P(t) will die out. Choosing σ1=0.918 and σ2=0.6 in Figure 3(b), which satisfies the conditions of Theorem 10, both species N and P in model (6) become extinct. That is to say, big noise can make the two species die out. From the above numerical results and by Theorems 3, 6, 7, 9, and 10, we conclude that for some noise intensities σ12 and σ22 the dynamical behaviors of stochastically ultimately boundedness, permanence, persistence in mean, and extinction can be observed in model (6).Solutions of model (6) with different noise and other parameters are the same as those of Figure 1. (a) σ1=0.15, σ2=0.9 and (b) σ1=0.918, σ2=0.6.
(a)
(b)


## 4. Conclusions and Remarks

In this paper, we consider a stochastic Hassell-Varley type predator-prey model. The value of this study lies in twofolds. First, it verifies some relevant properties of the corresponding stochastic model (6), which shows the global existence, boundedness and stochastic permanence, persistence in mean, and extinction of the positive solution. Second, it illustrates the dynamics of the model via numerical simulations, which shows that if the noise is not large and satisfies some conditions, the stochastic perturbations do not cause drastic changes of the dynamics in the deterministic model (4), while if the noise is sufficiently large and satisfies some conditions, it will force two species in the model to die out.In order to study the stochastic model (6), we perturb the deterministic model (4) by incorporating white noise terms in the growth rate of prey population and in the death rate of predator population. Establishing a Lyapunov function, there is a unique positive solution to the model for any positive initial value. Applying Itô’s formula, we derive that, under some conditions, the solution of model (6) is stochastically bounded, permanent, and stochastic persistent in mean and extinct. For the fixed parameters r, K, b, c, d, m, α, and h, some conditions depend on the intensities of noise σ12 and noise σ22. When the noise intensities satisfy some conditions of Theorem 9, we can find that prey population N(t) of model (6) is permanent and predator population P(t) will die out (see Figure 3(a)), while with the noise intensities increasing which satisfy the conditions of Theorem 10, from Figure 3(b), two species N and P will die out. In other words, when the noise satisfies some conditions of Theorems 6 and 7 and is not sufficiently large, the populations N and P may be stochastically permanent and persistent in mean, while large noise satisfying the conditions of Theorems 9 and 10 will force the population to become extinct. Our complete analysis of the stochastic model will give some suggestions to the studies of the population dynamics.



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*Source: 738342-2013-12-29.xml*