# Asymptotic Behaviors for Delay Lotka–Volterra Model Disturbed byG-Brownian Motion

Authors: Ping He; Yong Ren; Defei Zhang
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

In this paper, we propose the stochastic Lotka–Volterra model with delay disturbed byG-Brownian motion dx=diagx1,x2,…,xnAxt−τ+bdBt+σxdBt. Under a natural assumption on noise, we study existence and uniqueness of the global positive solution for the system and its asymptotic pathwise moment behavior and prove that the solution does not explode to infinity in a finite time.


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## Body

## 1. Introduction

Since the Lotka–Volterra model (LVM in short) was provided by Lotka [1] and Volterra [2], there were extensive works concerned with the dynamics of this system and global stability and the stochastic Lotka–Volterra population model, and in here, we only mention [3, 4] (for deterministic situation) and [5–8] (for stochastic situation). The well-known two-dimensional delay Lotka–Volterra ecological population model driven by Brownian motion is(1)dx1=x1a1+b11x1t−τ+b12x2t−τdt+c11x1+c12x2dWt,dx2=x2a2+b21x1t−τ+b22x2t−τdt+c21x1+c22x2dWt.Bahar and Mao in [9] proved that the solution of (1) is almost surely nonnegative and finite. Wu and Xu in [10] investigated stochastic LVM with infinite delay. Global asymptotic stability for a stochastic delay LVM was obtained in [11].Peng first established the stochastic analysis theory under theG-expectation framework in references [12–14]. Peng’s G-expectation space is an essential extension for probability measure space. Since then, many important theoretical results in this field are obtained, for example, SLL for sublinear expectations are obtained in [15], capacity theory results are discussed in [16] and [17–19], and other related technologies in [20–22]. Inspired by these results, we investigate a stochastic delay Lotka–Volterra model disturbed by G-Brownian motion:(2)dx=diagx1,x2,…,xnb+Axt−τdBt+σxdBt,with xs:−τ≤s≤0∈C−τ,0;R+n, where x=x1,…,xnT is a n-dimensional vector, xi(t) is the population size of species i at time t(t ≥ 0), b=b1,b2,…,bnT, bi is the species i’s growth rate, A=aijn×n is a n × n community matrix, aij(I ≠ j) is the interspecific interaction effect, and aii is the intraspecific interaction effect. We assume that the interaction effect in this system was disturbed by a G-Brownian motion with E^Bt2=σ¯2t and E^−Bt2=−σ¯2t, where σ=σijn×n is a constant matrix, representing the total interference intensity matrix for the system; B(t) has a variance-uncertainty but not mean-uncertainty; 〈B〉(t) has a mean-uncertainty property. Therefore, (〈B〉, B) is used to characterise the disturbed growth rate, disturbed interspecific, or intraspecific interactions and interference intensity at the same time. We think the model (2) considers the stochastic interference from both mean-uncertainty and variance-uncertainty, but the traditional stochastic model cannot describe this property. Indeed, we prove the solution of (2) is quasi-surely nonnegative and finite. Some asymptotic pathwise moment estimations for the solutions of this system are presented.


## 2. Stochastic Delay Lotka–Volterra Model Driven byG-Brownian Motion

Definitions about sublinear expectations,G-Brownian motions, and quadratic variation process 〈B〉(t) and notations, as well as more details can also be found in [12–14]. For a matrix A, we denote A=ATA and ∥A∥ = sup{|Ax|: |x| = 1}. C−τ,0;R+n denotes the family of continuous functions from [−τ, 0] to R+n. We assume the matrix σ satisfies the following assumption:(3)Aσii>0,i∈1,n,σij≥0,i≠j∈1,n.The assumption (A) was first assumed by Mao et al. in [5], and it is also necessary in our framework.Theorem 1.
If the matrixσ in system (2) satisfies assumption (A), then ∀A ∈ Rn×n, b ∈ Rn and {x(s): s ∈ [−τ, 0]}, then there exists a unique solution x of equation (2). Furthermore, xs∈R+n for all s ≥ −τ quasi-surely, namely, vω:xs∈R+n,s∈−τ,∞=1.Proof.
Because the coefficients of equation (2) are locally Lipschitz continuous, there exist a unique local solution x(s) on s ∈ [−τ, τe), where τe is called explosion time. To see it is also global, we must show τ∞ = ∞ q.s. Suppose k0(k0 > 0) is large enough s.t. x(t)(t ∈ [−τ, 0]) satisfies 1/k0 < min |x(t)|, max |x(t)| < k0. For any k(k ≥ k0), set τk = inf{s ∈ [0, τe): xi(s)∉(1/k, k), 1 < i ≤ n}, where inf ∅ = ∞. Noting that τk is increasing when k ⟶ ∞, let τ∞=limk⟶∞τk, then τ∞ ≤ τe q.s. If we can prove τ∞ = ∞ q.s., then τe = ∞ q.s. and xt∈Rn+ q.s., t ≥ 0. If τ∞ ≠ ∞ q.s., then ∃ a constant T > 0 s.t. V(ω: τ∞(ω) ≤ T) ≥ ɛ for any ɛ > 0, namely, ∃ an integer k1(k1 ≥ k0) s.t. V(Ak) ≔ V(ω: τk(ω) ≤ T) ≥ ɛ for all k ≥ k1. Let U:Rn+⟶R+ be Ux=∑i=1nxi−0.5logxi−1. Set k ≥ k0 and T > 0. Using the G-Itô lemma for Vˇt,x=:Ux+∫t−τtxs2dBs, t ∈ [0, τk ∧ T], we get(4)dVˇx,t=∑i=1n0.5xi0.5−0.5∑j=1naijxjt−τ+bidBt+−0.125xi0.5+0.25∑j=1nσijxj2dBt+∑i=1n0.5xi0.5−0.5∑j=1nσijxjdBt+x2−xt−τ2dBt,and noting that(5)∑i=1n0.5xi0.5−0.5bi+∑j=1naijxjt−τ≤12∑i=1nxi0.5−1bi+xt−τ2+∑i=1n∑j=1nnaij216xi0.5−12,and∑i=1n∑j=1nσijxj2≤∑i=1nσ2xi2=σ2x2, as well as ∑i=1nxi0.5∑j=1nσijxj2≥∑i=1nσii2xi2.5by the assumption (A). Thus,(6)dVˇt,xt≤∑i=1n12xi0.5−1bi+xt2+∑i=1n∑j=1nnaij216xi0.5−12dBt+−0.125∑i=1nσii2xi2.5+0.25σ2x2dBt+∑i=1n0.5xi0.5−0.5∑j=1nσijxjdBt.
Denote(7)fx,a,b,σ=∑i=1n12xi0.5−1bi+x21+14σ2+∑i=1n∑j=1nnaij216xi0.5−12−18∑i=1nσii2xi2.5,since we note that there is K s.t. f(x, a, b, σ) is bounded, namely, f(x, a, b, σ) < K, then(8)E^∫τk∧T−τT∧τkx2dBs+UxT∧τk≤E^∫−τ0x2dBs+Ux0+Kσ¯2E^T∧τk≤σ¯2∫−τ0E^x2ds+Ux0+Kσ¯2E^T∧τk,then E^Uxτk∧T≤Ux0+σ¯2∫−τ0E^xs2ds+Kσ¯2T<∞. From the definition of τk, we know ∀ω ∈ Ak, ∃ some i s.t. xi(τk, ω) ∉ (1/k, k), namely, xi(τk) ≤ 1/k, or xi(τk) ≥ k < ∞. Noting that the function U(xi) is decreasing when 0 < xi ≤ 1 and is increasing when xi > 1, hence U(x(τk)) ≥ U(1/k, …, 1/k) and U(x(τk)) ≥ U(k, …, k), namely, Uxτk≥max1/k−0.5log1/k−1,k−0.5logk−1. Therefore, we have(9)εCk≤CkVAk≤E^IAkUxτk≤E^UxT∧τk≤Ux0+σ¯2∫−τ0E^x2dt+Kσ¯2T.
Settingk ⟶ ∞, we have the contradiction ∞≤E^Uxτk∧T<∞; therefore, we have τ∞ = ∞ q.s., namely, τe = ∞ q.s., so vω:xt∈R+n,t≥0=1.


## 3. Asymptotic Behaviors of the Solution

Theorem 2.
Under the assumption (A), ifE^B12=σ¯2≤1, for any β ∈ (0, 1) and δ ∈ (0, 1), ∃C0 = C(δ) > 0 s.t. the solution x(t) of equation (2) is as follows:(10)limsupt⟶∞VeBt−σ¯2t/βxt≤C0≥1−δ.Proof.
Let(11)Ux=∑i=1nxiβ.
Using the G-Itô lemma forU(x) and noting(12)dxit=xi∑j=1naijxjt−τ+bidBt+∑j=1nσijxjdBt,we have(13)dUx=∑i=1nβxiβbi+∑j=1naijxjt−τdBt+∑j=1nσijxjdBt+12ββ−1∑i=1nxiβ∑j=1nσijxj2dBt.
Since(14)∑i=1n∑j=1nβxiβaijxjt−τ≤∑i=1n∑j=1nn4βaijxiβ2+xjt−τ2n,and from equation (13), we get(15)dUx≤∑i=1nβbixiβ+n4∑i=1n∑j=1nβaijxiβ2dBt+12ββ−1∑i=1nxiβ+2σii2+xt−τ2dBt+∑i=1nβxiβ∑j=1nσijxjdBt,then(16)deBtUxt=eBtUxtdBt+eBtdUxt≤eBt∑i=1nβbi+1xiβ+n4∑i=1n∑j=1nβaijxiβ2dBt+expBtxt−τ2+0.5ββ−1∑i=1nxiβ+2σii2dBt+eBt∑i=1nβxiβ∑j=1nσijxjdBt.
We set(17)F1x=∑i=1n1+βbixiβ+n4∑i=1n∑j=1nβaijxiβ2+eτx2−12β1−β∑i=1nxiβ+2σii2,then F1(x) is bounded in R+n , say K1, from (16),(18)deBtUxt≤eBtK1−eτxt2+xt−τ2dBt+eBt∑i=1nβxiβ∑j=1nσijxjdBt,namely,(19)E^eBtUxt≤U0+E^∫0teBsK1−eτxs2+xs−τ2dBs≤U0+E^∫0teσ¯2sK1−eτxs2+xs−τ2dBs≤U0+σ¯2E^∫0teσ¯2sK1−eτxs2+xs−τ2ds≤U0+K1eσ¯2t+σ¯2E^∫0teσ¯2sxs−τ2−eσ¯2s+τxs2ds=U0+K1eσ¯2t+σ¯2E^∫−τt−τeσ¯2s+τxs2−eσ¯2s+τxs2ds,where U0 = U(x(0)), and noting that σ¯2 satisfies σ¯2≤1, by (19), then(20)E^eBtUxt≤U0+K1eσ¯2t+σ¯2E^∫−τ0eσ¯2s+τxs2ds,therefore,(21)limsupt⟶∞E^eBt−σ¯2tUxt≤K1.
In addition, we note(22)xβ=∑i=1nxi2β/2≤nβ/2max1≤i≤nxiβ≤nβ/2Ux,so(23)limsupt⟶∞E^eBt−σ¯2txβ≤K0,∀δ > 0, let C0=K0/δ1/β, then(24)vexpBt−σ¯2tβx>C0≤E^expBt−σ¯2txβC0β≤K0C0β=δ.
Hence,(25)limsupt⟶∞VeBt−σ¯2t/βx≤C0≥1−δ.Theorem 3.
Suppose the (A) is true, and there existsK(K > 0) is independent of {x(s): s ∈ [−τ, 0]}, then(26)limsupT⟶∞−1TE^∫0T−x2dBs≤Kσ¯2.Proof.
Write (7) as gx,a,b,σ = g1x,a,b,σ − |x|2, where(27)g1x,a,b,σ≔∑i=1n0.5x0.5−1bi+∑i=1n∑j=1nnaij216x0.5−12−0.125∑i=1nσii2xi2.5+x22+0.25σ2,then gx,a,b,σ ≤ K − |x|2, where K:=maxx∈R+ng1x<∞. Taking expectation from 0 to τk ∧ T on both sides of equation (6), we have(28)0≤E^∫−τ0x2dBs+Ux0+E^∫0τk∧TK−x2dBs≤σ¯2∫−τ0E^x2ds+Ux0+Kσ¯2E^T∧τk+E^∫0T∧τk−x2dBs.
Lettingk ⟶ ∞ yields(29)−E^∫0T−x2dBs≤σ¯2∫−τ0E^x2ds+Ux0+Kσ¯2T.
Therefore, settingT ⟶ ∞,(30)limsupT⟶∞−1TE^∫0T−x2dBs≤Kσ¯2.


## 4. Asymptotic Moment Estimations

Theorem 4.
If condition (A) is true, then ∀ {x(s): s ∈ [−τ, 0]}, x(t) in (2) satisfies(31)limsupt⟶∞1tE^logxtn+σ^24n∫0txs2dBs≤σ¯2K,where σ^=min1≤i≤nσii.Proof.
LetV˜x=∑i=1nxitforx∈R+n, then by G-Itô’s lemma of Reference [13], we have(32)logV˜x=C0+∫0txTsVxsb+Axs−τdBs+∫0txTsσxsV˜xsdBs−∫0txTsσxs22V˜2xdBs,where C0 = log(Ṽ(x(0))). Noting that(33)∫0txTσxV˜xdBs,∫0txTσx2V˜2xdBs=∫0txTσx2V˜2xdBs,∀ε∈0,1/2, from Lemma 3.1 in reference [19], for any integer k ≥ 1, we have(34)Vsup0≤t≤k∫0txTσxV˜xdBs−ε2∫0txTσx2V˜2xdBs>2εlnk≤1k2,so(35)∑k=1∞Vsup0≤t≤k∫0txTσxV˜xdBs−ε2∫0txTσx2V˜2xdBs>2εlnk<∞,applying Lemma 2 in [15], we know for all but finitely many k,(36)sup0≤t≤k∫0txTσxV˜xdBs−ε2∫0txTσx2V˜2xdBs≤lnk2ε,quasi-surely true, i.e., ∃Ωi ⊂ Ω (vΩi = 1) s.t. ∀ω ∈ Ωi and ki = ki(ω) s.t.(37)∫0txTsσxV˜xdBs−ε2∫0txTσx2V˜2xdBs≤lnk2ε,0≤t≤k,k ≥ ki(ω). From equation (32) and inequality (37),(38)logV˜x≤C0+lnk2ε+∫0tnxs−τA+b−σ^2x21−ε2ndBs,t ∈ [0, ki(ω)], k ≥ ki(ω), in other words,(39)logV˜x+σ^21−2ε4n∫0tx2dBs≤C0+lnk2ε+∫0tnxs−τA+b−x2σ^24ndBs,where σ^=minσiii∈1,n. Taking G-expectation E^ for (39), and then ∀ω∈∩i=1nΩi, from (39), we get(40)E^σ^21−2ε4n∫0tx2dBs+logV˜x≤lnk2ε+σ¯2E^∫0txs−τA+bn−σ^24nx2ds+C0≤C0+lnk2ε+nAσ¯2E^∫−τ0xds+σ¯2E^∫0tnb+Ax−σ^24nx2ds≤C0+2lnkε+nAσ¯2E^∫−τ0xds+σ¯2Kt,where nAx+b−σ^2/4nxs2≤K. Set max{ki(ω), i ∈ [1, n]} = k0(ω), then ∀ω∈∩i=1nΩi, t ∈ [k − 1, k], k ≥  k0(ω), it gets from (40):(41)limsupt⟶∞1tE^logV˜x+σ^21−2ε4n∫0tx2dBs≤σ¯2K.
Lettingɛ tend to zero and noting that x≤nVx yield(42)limsupt⟶∞1tE^logxtn+σ^24n∫0txs2dBs≤σ¯2K.
The proof is complete.



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