Weak persistence of a stochastic delayed competition system with telephone noise and Allee effect

Journal Pre-proof Weak persistence of a stochastic delayed competition system with telephone noise and Allee effect

Chao Liu, Haichang Li, Lora Cheung

PII: DOI: Reference:

S0893-9659(19)30512-9 https://doi.org/10.1016/j.aml.2019.106186 AML 106186

To appear in:

Applied Mathematics Letters

Received date : 2 November 2019 Revised date : 4 December 2019 Accepted date : 4 December 2019 Please cite this article as: C. Liu, H. Li and L. Cheung, Weak persistence of a stochastic delayed competition system with telephone noise and Allee effect, Applied Mathematics Letters (2019), doi: https://doi.org/10.1016/j.aml.2019.106186. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Elsevier Ltd. All rights reserved.

Highlights

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Highlights:  A stochastic dynamical system with Allee effect and distributed delay is proposed.  Combined dynamic effects of telephone noise and Lévy jumps are investigated.  Stochastically ultimate boundedness of solution to stochastic system is discussed.

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 Sufficient conditions for weak persistence of competitive population are studied.

*Manuscript Click here to view linked References

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Weak persistence of a stochastic delayed competition system with telephone noise and Allee effect Chao Liua,b,c,∗, Haichang Lia,b , Lora Cheungc a Institute

of Systems Science, Northeastern University, Shenyang, China Key Laboratory of Integrated Automation of Process Industry, Shenyang, China c Department of Mathematics and Statistics, School of Science, York University, Toronto, Canada b State

Abstract

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A stochastic competition system with Allee effect and distributed delay is established, which is utilized to investigate combined dynamic effects of telephone noise and L´evy jumps on stochastic population dynamics. Stochastically ultimate boundedness of the positive solution is studied. By constructing appropriate stochastic Lyapunov functions, sufficient conditions for weak persistence of the corresponding competitive population are discussed. Keywords: Telephone noise; Distributed delay; L´evy jumps; Weak persistence.

1. Introduction

It is well known that stochastic perturbations and Allee effect are two main important and wellestablished disciplines in population dynamics study [1, 2], which portray better understandings and vivid reflections from ecological systems in the real world. Recently, many applied mathematics and theoretical biology research reveal that Allee effect significantly affects corresponding population growth as well as stochastic population dynamics [3, 4]. Ergodicity and extinction of corresponding population in stochastic prey predator system with Allee effect are studied in [6, 13]. Sufficient criteria for permanence and existence of stationary distribution in stochastic single system with Allee effect are discussed in [15, 16]. Existence and asymptotic stability of stationary distribution to stochastic competition system with Allee effect are investigated in [5, 11, 18]. Chen et al. [11] established a stochastic competition system with Holling III functional response and Allee effect,  [ ( ) ] 2  dx1 (t) = rx1 (t) 1 − x1 (t) (x1 (t) − m) − αx1 (t)x22 (t) dt + ω1 x1 (t)dB1 (t), M 1+σ1 x1 (t) [ ] (1) 2  dx2 (t) = Λ − δx2 (t) − βx2 (t)x21 (t) dt + ω2 x2 (t)dB2 (t), 1+σ2 x (t) 2

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where xi (t) (i = 1, 2) represents population density of competitive population, respectively. r denotes intrinsic growth rate of population x1 and Λ stands for recruitment rate of population x2 . M denotes maximum capacity rate of population x1 . Allee effect expressed in (1) is assumed to be strong Allee effect as m > 0. α and β stand for inter-competition coefficient. σ1 and σ2 represent handling coefficients of inter-competition. δ is death rate of population x2 . Bi (t) (i = 1, 2) denote mutually independent and standard Brownian motions, which are defined on a complete probability space (Ω, F, {Ft }t≥0 , P) with a filtration {Ft }t≥0 . ωi2 > 0 (i = 1, 2) represent intensities of white noises. Generally, telephone noise [14] (also known as telegraph noise, or burst noise) can be regarded as a switching state, which is memoryless with exponentially distributed waiting times [9, 12]). It usually allows for instantaneous transitions of population growth rates switch among two or more environmental regimes [9, 12]. Evidences from real world observations point out that intrinsic growth rates of some population are usually subject to telephone noise in monsoon area, where intrinsic growth rates often show variations due to annual rainfall amount [14]. It should be noted that growth dynamics of corresponding competitive population are usually described by stochastic competition model with distributed delays [8, 17], which are utilized to depict population dynamics ∗ Corresponding

author: Prof. Chao Liu ([email protected]).

Preprint submitted to Elsevier

December 3, 2019

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with specific time lags under fluctuating surviving environment. Recent studies show that L´evy jumps can effectively depict an unexpected severe perturbations arising in the real world [7, 10, 19] that cannot be accurately portrayed by the standard Brownian motion. Based on the above analysis, a delayed stochastic competition system with telephone noise, L´evy jumps and distributed delay is constructed as follows, ) ] [ (  α(ξ(t))x21 (t)x2 (t) x1 (t)  dt (x (t) − m) − dx (t) = r(ξ(t))x (t) 1 − 2 1 1 1  1+σ1 x1 (t)   ∫M  e +ω ] [ 1 x1 (t)dB1 (t) + X λ1 (u)x∫1 (t−)Z(dt, du), (2) t x22 (τ )x1 (τ )  dτ dt dx (t) = Λ − δ(ξ(t))x (t) − β(ξ(t)) G(t − τ )  2 2 2  −∞ 1+σ2 x2 (τ )  ∫  e +ω2 x2 (t)dB2 (t) + X λ2 (u)x2 (t−)Z(dt, du),

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where a distributed delay kernel in system (2) (incorporated using kernel G : [0, ∞) → ∫ ∞[0, ∞)) representing a L1 − weak generic kernel function G(t) = ρe−ρt with ρ > 0 such that 0 G(τ )dτ = 1). {ξ(t), t ≥ 0} denotes an irreducible and continuous Markov chain with finite state space N = {1, 2, · · · , K}, which is utilized to depict telephone noise. ξ(t) is assumed to be generated by transition rate matrix (µij )K×K , which is as follows, { µij △τ + o(△τ ), i ̸= j, P{ξ(τ + △τ ) = j|ξ(τ ) = i} = 1 + µii △τ + o(△τ ), i = j,

where µij ≥ 0 denotes transition rate from state i to j and µii = −ΣK i̸=j,i=1 µij holds for i ̸= j. Based on irreducibility property of ξ(t), there exists a unique stationary probability distribution ϕ = (ϕ1 , ϕ2 , · · · , ϕK ) ∈ R1×K subject to ΣK n=1 ϕn = 1 and ϕn > 0 hold for any n ∈ N. xi (t−) (i = 1, 2) denotes left limit of xi (t). V denotes a measurable subset of R+ , Z stands for an independent e Poisson counting measure with a L´evy measure ψ on V with ψ(V) < +∞ such that Z(dt, du) = Z(dt, du) − ψ(du)dt. By assuming that λi (u) > −1 and νi > 0 (i = 1, 2) satisfying ∫ [(ln(1 + λi (u))) ∨ ln(1 + λi (u))2 ]ψdu ≤ νi . (3) V

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Remark 1. By incorporating distributed delay, telephone noise and L´evy jumps into the mathematical model, we will extend work done in [11] and aim to investigate combined dynamic effects of telephone noise and L´evy jumps on stochastic population dynamics. Although mathematical modelling and dynamical analysis of stochastic competition system with Allee effect and time delays have attracted great attentions, to authors’ best knowledge, combined dynamic effects of telephone noise and L´evy jumps on stochastic competition system with Allee effects and distributed delay have not been investigated in the related work [5, 6, 11, 13, 15, 16, 18, 19]. ∫t x22 (τ )x1 (τ ) In order to facilitate the following proof, let x3 (t) = −∞ G(t − τ ) 1+σ dτ . For every finite 2 2 x2 (τ ) state space n ∈ N, it follows from the Markov chain law [2] that system (2) can be studied as a stochastic system switching among the following subsystems ) ] [ (  α(n)x21 (t)x2 (t) x1 (t)  (x (t) − m) − dt dx (t) = r(n)x (t) 1 − 2 1 1 1  M 1+σ1 x1 (t)   ∫  e  +ω1 x1 (t)dB1 (t) + V λ1 (u)x1 (t−)Z(dt, du),  dx2 (t) = [Λ − δ(n)x2 (t) − β(n)x3 (t)] dt (4) ∫   e  +ω x (t)dB (t) + λ (u)x (t−) Z(dt, du), 2 2 2 2 2  ]V [    dx (t) = ρ x1 (t)x22 (t) − x (t) dt. 3

1+σ2 x22 (t)

3

2. Main results

Lemma 2.1. For any initial value (x1 (0), x2 (0), x3 (0), n) ∈ R3+ × N, system (4) has a unique global positive solution for all t ≥ 0 almost surely. Proof. It is easy to show Lemma 2.1 based on similar arguments in [14, 16]. Hence, the proofs are standard and omitted here.

2

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Lemma 2.2. For any initial value (x1 (0), x2 (0), x3 (0), n) ∈ R3+ × N, the solution of system (4) are stochastically ultimate bounded. Proof. The proof of Lemma 2.2 can be found in Appendix A. Lemma 2.3. The following equation (5) has a solution (φ(1), φ(2), · · · , φ(K))T , (

where θ1 =

Π2i=1 θi β(n)M M +m

) 14

( )4 1 4 M 2 ΣK n=1 ϕn β (n) σ2 (M +m)2 +M 2

−

θ2 [(M + m)2 − σ2 ] − r(n)φ(n) + ΣK j=1 µnj φ(j) = 0, M (M + m) ( )4 1 4 M (M +m) ΣK n=1 ϕn β (n)

and θ2 =

σ2 (M +m)2 +M 2

(5)

.

Proof. The proof of Lemma 2.3 can be found in Appendix B. Theorem 2.4. When

( )4 1 4 M 2 ΣK n=1 ϕn β (n) ∑ [σ12 (M +m)2 +M 2 ] K n=1 ϕn δ(n)

> 1 and Γ∗ (ω, ν, n) > 0, Γ∗ (ω, ν, n) is defined in

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(10), if there exists a positive constant ζ such that (6) holds, { {( ) 14 }} 2 1 2σ2 θ2 M 2 Π θ β(n)M i i=1 ˘ ζ> + 2ˇ r max{ϕ(n)} + 6 max , n∈N n∈N 2ˆ r Q3 (ε)(M + m)2 M +m then for system (4) with any initial value (x1 (0), x2 (0), x3 (0), n) ∈ R3+ × N,  ) ˆ 2 ∫ ( Γ(ω,ν,n)[(M S(ε)+mQ(ε)) min{rˆ, ρ  2 ,δ }]  lim inf t→∞ 1t 0t P x1 (s) ≥ Γ(ω,ν,n) ds ≥ , 2 2C1 2C1 (˘ r M S(ε)) ( ) ρ ˆ 2 ∫ , δ ] Γ(ω,ν,n)[(M S(ε)+mQ(ε)) min r ˆ , } { t Γ(ω,ν,n) 2  ds ≥ ,  lim inf t→∞ 1t 0 P x2 (s) ≥ 2C2 2 32C2 (β˘r˘M S(ε))

(6)

a.s. a.s.

where Γ(ω, ν, n) is defined in (9) and Ci (i = 1, 2) are defined in (12).

Proof. Firstly, we define stochastic Lyapunov functions Wi (x1 , x2 , x3 , n) (i = 1, 2), W1 (x1 , x2 , x3 , n)

=

W2 (x1 , x2 , x3 , n)

=

φ(n)m(M + m) ( x1 ) ζm(M + m) ( x1 x1 ) 1− + − 1 − ln M m M m m ( ˇ 3) ζS 2 (ε) βx + x2 + , 2 (ε) + σ ) b ρ δ(Q(ε)S 2 − ln x2 −

θ1 ln x3 + χ(n), ρ

where ζ satisfies condition defined in (6) and χ(n) satisfies the following condition µnj χ(j) =

j=1

K ∑

ϕn δ(n) −

ur

K ∑

n=1

4ΣK n=1 ϕn β

1 4

(n)

(∏

2 i=1 θi (M

+ m)

M

) 14

− δ(n) + 4

(

Π2i=1 θi β(n)M M +m

) 14

.

According to Itˆo’s formula, it follows from Lemma 2.2 and Lemma 2.3 that

≤

Jo

LW1 (x1 , x2 , x3 , n)

( ) ( ) K ∑ (M + m)2 r(n)φ(n)(m − x1 )2 Q(ε) m(M + m)µkj φ(j) S(ε) − 1 (x1 − m) + 1 − M M M2 M j=1 ] [ m(M + m)x1 x2 (M + m)2 r(n)(m − x1 )2 S 3 (ε)Λ + −ζ + 2 (ε) + σ ) b M2 M (1 + σ1 x21 ) δ(Q(ε)S 2 ( 2 ) ) ( ζS 2 (ε) ζ(S(ε) + ϕ(n))(M + m) ω1 ω22 + ν1 + + ν2 + 2 (ε) + σ ) b M 2 2 δ(Q(ε)S 2 α ˇ (M + m)φ(n)x21 x2 ζS 3 (ε)x1 x22 + + . (7) 2 2 (ε) + σ )(1 + σ x2 ) b M (1 + σ1 x1 ) δ(Q(ε)S 2 2 2

3

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According to Itˆo’s formula and Lemma 2.2, it follows from simple computations that LW2 (x1 , x2 , x3 , n) ( 2 )1 ( 2 ) K Πi=1 θi β(n)M 4 Λ θ2 [σ2 (M + m)2 + M 2 ] ∑ 1 ω2 ≤ −4 − + θ1 + + µnj χ(j) − + ν2 M +m x2 M (M + m) S(ε) 2 j=1 +δ(n) + θ2

[

1

] )1 [ ( 2 ( x ) 14 ] σ2 + S 2 (ε) σ2 (M + m)2 + M 2 Πi=1 θi β(n)M 4 1 − 1− . +4 Q(ε) M (M + m) M +m m

Since 4(1 − u 4 ) ≤ 3(u − 1)2 − (u − 1) holds for any u > 0, it yields LW2 (x1 , x2 , x3 , n)

[

] ( 2 )1 x 1 )2 σ2 + S 2 (ε) σ2 (M + m)2 + M 2 Πi=1 θi β(n)M 4 ( ≤ θ2 − +3 1− Q(ε) M (M + m) M +m m ( 2 ) 14 ( ( 2 ) K ) ∑ Πi=1 θi β(n)M x1 1 ω2 + 1− + ν2 + µnj χ(j) − M +m m S(ε) 2 j=1 Π2i=1 θi β(n)M M +m

) 14

+ θ1 +

θ2 [σ2 (M + m)2 + M 2 ] + δ(n). M (M + m)

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−4

(

(8)

Let W (x1 , x2 , x3 , n) = W1 (x1 , x2 , x3 , n) + W2 (x1 , x2 , x3 , n). It follows from (7), (8) and Lemma 2.3 that

≤

LW (x1 , x2 , x3 , n) [ ]1 K ( )4 ∏2 θ (M + m) 4 ∑ 1 θ2 (σ2 + S 2 (ε)) i K i=1 −4 Σn=1 ϕn β 4 (n) + ϕn δ(n) + M Q(ε) n=1 [ ( ) ( 2 )1 ] 3 Πi=1 θi β(n)M 4 ζ(M + m)2 r(n) (M + m)2 r(n)ϕ(n) Q(ε) − + −1 − 2 (x1 − m)2 M2 M2 M m M +m ( ) ( 2 )1 K ) ∑ Πi=1 θi β(n)M 4 ( x1 m(M + m)µkj φ(j) S(ε) + − 1 (x1 − m) − −1 M M M +m m j=1 S 3 (ε)x1 x22 S 3 (ε)Λ α ˇ (M + m)φ(n)x21 x2 + + θ − 1 2 (ε) + σ )(1 + σ x2 ) 2 (ε) + σ ) b b M (1 + σ1 x21 ) δ(Q(ε)S δ(Q(ε)S 2 2 2 2 ( ) ( 2 ) ( 2 ) 2 ζ(S(ε) + ϕ(n))(M + m) ω1 ζS (ε) 1 ω2 + + ν1 + − + ν2 2 (ε) + σ ) b M 2 S(ε) 2 δ(Q(ε)S 2 +

−Γ(ω, ν, n) + H(x1 , n) +

where Γ(ω, ν, n) = ∗

∑K

ur

:=

n=1

Jo

Γ (ω, ν, n) =

H(x1 , n) =

[

ϕn δ(n)

α ˇ (M + m)φ(n)x21 x2 S 3 (ε)x1 x22 + , 2 (ε) + σ )(1 + σ x2 ) b M (1 + σ1 x21 ) δ(Q(ε)S 2 2 2 ( ) ( )4 1 4 M 2 ΣK n=1 ϕn β (n) ∑K

[σ12 (M +m)2 +M 2 ]

n=1

ϕn δ(n)

−1

(9)

+ Γ∗ (ω, ν, n) and

( ) ζ(S(ε) + ϕ(n))(M + m) ω12 + ν1 − 2 (ε) + σ ) b M 2 δ(Q(ε)S 2 ( )( ) ζS 2 (ε) 1 ω22 θ2 (σ2 + S 2 (ε)) − − + ν2 . − 2 (ε) + σ ) b Q(ε) S(ε) 2 δ(Q(ε)S 2 S 3 (ε)Λ

(10)

( ) ( 2 )1 ] ζ(M + m)2 r(n) (M + m)2 r(n)ϕ(n) Q(ε) 3 Πi=1 θi β(n)M 4 − + −1 − 2 (x1 − m)2 M2 M2 M m M +m ( ) ( 2 )1 K ) ∑ m(M + m)µkj φ(j) S(ε) Πi=1 θi β(n)M 4 ( x1 + − 1 (x1 − m) − −1 . (11) M M M +m m j=1 4

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It follows from simple computations that

∂H(x1 ,n) |{x1 = M } ∂x1 M +m

= 0, which yields Eq. (5) holds ) ( and < 0. Hence, it can be concluded that H(x1 , n) ≤ H MM +m , n = 0 provided that sufficient condition (6) holds. According to Eq. (9), it can be concluded that { ∫ ∫ C1 t 1 t t ∫0 E(LW (x1 , x2 , x3 , n))ds ≤ −Γ(ω, ν, n) + t ∫0 E(x1 (s))ds, (12) t C2 t 1 E(x2 (s))ds, t 0 E(LW (x1 , x2 , x3 , n))ds ≤ −Γ(ω, ν, n) + t 0 ∂ 2 H(x1 ,n) ∂x21

( ˇ ˇ +m) maxn∈N {ϕ(n)} C1 = S 2 (ε) α(M + M (1+σ1 Q2 (ε)) Simple computations show that

S 3 (ε) 2 (ε)+σ ) ˆ δ(Q(ε)S 2

)

ˇ α(M ˆ +m) maxn∈N {ϕ(n)} M

, C2 =

+

S 5 (ε) . 2 (ε)+σ ) ˆ δ(Q(ε)S 2

x3 x2 r˘M S(ε) { }. + < ˇ 2ρ 4β (M S(ε) + mQ(ε)) min rˆ, ρ2 , δˆ

x1 +

(13)

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Based on mathematical formulation of W (x1 , x2 , x3 , n) and Lemma 2.2, it is easy to show 2 ,x3 ,n) lim inf t→∞ ln W (x1 ,x ≥ 0 almost surely. Consequently, it follows from (12) that t { ∫t lim inf t→∞ 1t 0 E(x1 (s))ds ≥ Γ(ω,ν,n) , C1 ∫t (14) lim inf t→∞ 1t 0 E(x2 (s))ds ≥ Γ(ω,ν,n) . C2

Furthermore, it is easy to show that ( ) ( ) ∫ ∫ ∫ 1 t 1 t 1 t } } { { lim inf E(x1 (s))ds ≤ lim inf E x1 (s)I x (s)≥ Γ(ω,ν,n) ds + lim sup E x1 (s)I x (s)< Γ(ω,ν,n) ds 1 1 t→∞ t 0 t→∞ t 0 2C1 2C1 t→∞ t 0 ( ) ∫ 1 t Γ(ω, ν, n) ≤ lim inf E x1 (s)I{x (s)≥ Γ(ω,ν,n) } ds + , 1 t→∞ t 0 2C1 2C1 which yields that

lim inf t→∞

1 t

∫

t

0

( E x1 (s)I{x

Γ(ω,ν,n) 1 (s)≥ 2C1

}

)

ds ≥

Γ(ω, ν, n) . a.s. 2C1

(15)

On the other hand, according to (13), it can be obtained that ∫

t

0

( E x1 (s)I{x

1 (s)≥

Γ(ω,ν,n) 2C1

}

ur

1 t

)

ds

which follows from (15) that

t→∞

1 t

t

( E I{ x

Jo

lim inf

∫

0

1 (s)≥

Γ(ω,ν,n) 2C1

}

)

( ∫ t ( ) )( ∫ t ) 12 ( 2 ) 1 1 { } E I x (s)≥ Γ(ω,ν,n) E x1 (s) ds ≤ ds 1 t 0 t 0 2C1  2 ( ∫ t ( ) ) 1 r ˘ M S(ε) { } , ≤ E I{x (s)≥ Γ(ω,ν,n) } ds  1 t 0 2C1 (M S(ε) + mQ(ε)) min rˆ, ρ , δˆ

ds ≥

2

[ { }]2 Γ(ω, ν, n) (M S(ε) + mQ(ε)) min rˆ, ρ2 , δˆ 2C1 (˘ rM S(ε))

2

Consequently, it follows from (13), (14) and (16) that ) ˆ 2 ∫t ( Γ(ω,ν,n)[(M S(ε)+mQ(ε)) min{rˆ, ρ 2 ,δ }] ds ≥ lim inf t→∞ 1t 0 P x1 (s) ≥ Γ(ω,ν,n) , 2C1 2C (˘ r M S(ε))2 1

By utilizing similar arguments utilized in (15), it can be also concluded that ) ( ∫ Γ(ω, ν, n) 1 t lim inf E x2 (s)I{x (s)≥ Γ(ω,ν,n) } ds ≥ , a.s. 2 t→∞ t 0 2C2 2C2 5

a.s.

(16)

a.s.

(17)

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and it follows from (13) and similar arguments in (16), it yields

lim inf t→∞

1 t

∫

t

0

( E I{ x

2 (s)≥

Γ(ω,ν,n) 2C2

}

)

[ { }]2 Γ(ω, ν, n) (M S(ε) + mQ(ε)) min rˆ, ρ2 , δˆ ds ≥ ( )2 ˘rM S(ε) 32C2 β˘

a.s.

(18)

Hence, it follows from (13), (14) and (18) that lim inf t→∞

1 t

∫t ( P x2 (s) ≥ 0

Γ(ω,ν,n) 2C2

)

ds ≥

ˆ 2 Γ(ω,ν,n)[(M S(ε)+mQ(ε)) min{rˆ, ρ 2 ,δ }] . 2 32C2 (β˘r˘M S(ε))

a.s.

Remark 2. Sufficient conditions for weak persistence of two corresponding competitive population x1 and x2 are discussed in Theorem 2.4 by virtue of some appropriate stochastic Lyapunov functions. Along this line of research, existence of positive recurrence for the process (x1 (t), x2 (t), x3 (t)) can be further investigated in the case of weak persistence by establishing appropriate stochastic Lyapunov functions with regime switching.

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Acknowledgements

Authors would like to express their gratitude to editor and anonymous reviewers for valuable comments and suggestions, and the time and efforts they have spent in the review. This work is supported by the Natural Sciences and Engineering Research Council of Canada (NSERC). National Natural Science Foundation of China, grant No. 61673099 and grant No. 61702088. China Scholarship Council Programme, grant No. 201906085046. Appendix A. Proof of Lemma 2.2.

Proof. Let y1 (t) = xη1 (t), 0 < η < 1. By applying Itˆo’s formula [2] to et y1 (t) and integrating from 0 to t, expectations on both sides can be obtained as follows, ] ∫ t [ (m + M )x1 (s) E(et y1 (t)) ≤ y1 (0) + E es 1 + r(n)η − η(1 − η)ω12 y1 (s)ds M 0 ] ∫ t [∫ s η +E e [(1 + λ1 (u)) − 1 − ηλ1 (u)]ψdu y1 (s)ds. (19) 0

V

According to the following fundamental inequality xη1 ≤ 1 + η(x1 − 1) holds for x1 ≥ 0 and 0 < η < 1, if sufficient condition (3) holds, then it can be obtained that [ ] ∫ (m + M )x1 y1 (t) 1 + r(n)η − η(1 − η)ω12 + [(1 + λ1 (u))η − 1 − ηλ1 (u)]ψdu ≤ S1 (η), M V

Jo

ur

where S1 (η) is a positive constant with respect to η. Consequently, it can be derived that E(et y1 (t)) ≤ ∫t y1 (0) + E 0 es S1 (η)ds, and it is easy to show lim supt→∞ E(xη1 (t)) ≤ S1 (η). Based on similar arguments, if sufficient condition (3) holds, then it is easy to show there exists a positive constant S2 (η) with respect to η satisfying lim supt→∞ E(xη2 (t)) ≤ S2 (η). By formulation of x3 , it is easy to show lim supt→∞ E(xη3 (t)) ≤ S1 (η)S22 (η) := S3 (η). η Let X(t) = (x1 (t), x2 (t), x3 (t), n)T , it yields 2(1− 2 )∧0 |X(t)|η ≤ xη1 (t) + xη2 (t) + xη3 (t) and lim sup E|X(t)|η t→∞

η

≤ 0.5(1− 2 )∧0 lim sup E[xη1 (t) + xη2 (t) + xη3 (t)] t→∞

(1− η2 )∧0

≤ 0.5

[S1 (η) + S2 (η) + S3 (η)] := S(η).

For any arbitrarily small 0 < ε < 1 and let S(ε) =

(

S(η) ε η

) η1

(20)

, it follows from Chebyshev’s

inequality and simple computations that P[X(t) < S(ε)] ≤ S(ε) P[X −η (t)] and lim inf P[X(t) ≤ S(ε)] ≥ 1 − ε. t→∞

6

(21)

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Furthermore, by utilizing Chebyshev’s inequality and (21), it is easy to show there exists Q(ε) > 0 such that lim inf P[X(t) ≥ Q(ε)] ≥ 1 − ε. (22) t→∞

Based on (21), (22) and the above analysis, it can be concluded that solution of system (4) is stochastically ultimate bounded. Appendix B. Proof of Lemma 2.3. to facilitate the proof, we consider the following equation ΞΦ Proof. In order  r(1)(M  = L, +m) − µ11 −µ12 ··· −µ1K M   r(2)(M +m) − µ · · · −µ2K −µ21   22 M  and where Ξ =  .. .. .. ..   .   . . . +m) −µK1 −µK2 · · · r(n)(M − µ KK M [( ]T )1 ( 2 )1 Πi=1 θi β(K)M 4 Π2i=1 θi β(1)M 4 θ2 [(M +m)2 −σ2 ] θ2 [(M +m)2 −σ2 ] . − M (M +m) , · · · , − M (M +m) L= M +m M +m

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For each leading principal submatrix Ξn of matrix Ξ, it is easy to show that sum of each row in Ξn can be obtained as follows, r(l) + ΣK j=n+1 µlj ≥ r(l) > 0,

(l = 1, 2, · · · , n),

which derives all leading principal minors detΞn > 0 hold for each n = 1, 2, · · · , K. Hence, it can be concluded that the leading principal minors of Ξ are all positive. It follows from Lemma 5.3 and Theorem 2.10 [2] that Ξ is a nonsingular matrix and the equation ΞΦ = L has a solution Φ = (φ(1), φ(2), · · · , φ(K))T . References

[1] J. Hofbauer, K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge, 1998. [2] X. Mao, C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College Press, London, 2006. [3] L. Berec, E. Angulo, F. Courchamp, Multiple Allee effects and population management, Trends in Ecology and Evolution, 22 (2006) 185-191. [4] F. Courchamp, L. Berec, J. Gascoigne, Allee Effect in Ecology and Conservation, Oxford University Press, New York, 2009.

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[5] M. Krstic, M. Jovanovic, On stochastic population model with the Allee effect, Mathematical and Computer Modelling, 52 (2010) 370-379. [6] P. Aguirre, D.G. Olivares, S. Torres, Stochastic predator prey model with Allee effect on prey, Nonlinear Analysis: Real World Applications, 14 (2013) 768-779.

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[7] Q. Liu, Asymptotic properties of a stochastic n-species Gilpin-Ayala competitive model with L´evy jumps and Markovian switching, Communications in Nonlinear Science and Numerical Simulation, 26 (2015) 1-10. [8] M. Jovanovic, M. Krstic, The influence of time dependent delay on behavior of stochastic population model with the Allee effect, Applied Mathematical Modelling, 39 (2015) 733-746. [9] L. Liu, Y. Shen, New criteria on persistence in mean and extinction for stochastic competitive Lotka Volterra systems with regime switching, Journal of Mathematical Analysis and Applications, 430 (2015) 306-323. [10] Y. Zhao, S.L. Yuan, Stability in distribution of a stochastic hybrid competitive Lotka Volterra model with L´evy jumps, Chaos Solitons and Fractals, 85 (2016) 98-109. 7

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[11] L.S. Chen, X.Z. Meng, J.J. Jiao, Biological Dynamics, China Science Publishing Press, Beijing, 2017. [12] M. Liu, X. He, J.Y. Yu, Dynamics of a stochastic regime switching predator prey model with harvesting and distributed delays, Nonlinear Analysis: Hybrid Systems, 28 (2018) 87-104. [13] B.B. Zhang, H.Y. Wang, G.Y. Lv, Exponential extinction of a stochastic predator prey model with Allee effect, Physica A: Statistical Mechanics and its Applications, 507 (2018) 192-204. [14] M. Liu, Y. Zhu, Stationary distribution and ergodicity of a stochastic hybrid competition model with L´evy jumps, Nonlinear Analysis: Hybrid Systems, 30 (2018) 225-239. [15] X.W. Yu, S.L. Yuan, T.H. Zhang, Persistence and ergodicity of a stochastic single species model with Allee effect under regime switching, Communications in Nonlinear Science and Numerical Simulation, 59 (2018) 359-374. [16] M.L. Deng, Dynamics of a stochastic population model with Allee effect and L´evy jumps, Physica A: Statistical Mechanics and its Applications, 531 (2019) Article No. 121745.

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[17] D. Li, Y. Yang, Impact of time delay on population model with Allee effect, Communications in Nonlinear Science and Numerical Simulation, 72 (2019) 282-293. [18] M.C. Koehnke, H. Malchow, Disease-induced chaos, coexistence, oscillations and invasion failure in a competition model with strong Allee effect, Mathematical Bioscience, (2019) Article No. 108267.

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[19] T.T. Ma, X.Z. Meng, Z.B. Chang, Dynamics and optimal harvesting control for a stochastic one-predator-two-prey time delay system with jumps. Complexity, 2019 (2019) Article No. 2187274.

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*Author Contributions Section

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Authors’ Contributions C. Liu wrote the manuscript, Introduction section, established mathematical model, developed and showed detailed proof of Theorem 2.4, i.e. main results in this manuscript. H. Li developed and showed the detailed proof of Lemma 2.1 and Lemma 2.2 of this manuscript.

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L. Cheung developed and showed the detailed proof of Lemma 2.3 of this manuscript.

All three authors checked English writing and reviewed the final manuscript.

The authors declare that they have no competing interests. All authors of this article declare that there is no conflict of interests regarding the publication of this article. We have no proprietary, financial, professional, or other personal interest of

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any nature or kind in any product, service, and/or company that could be construed as influencing the position presented in, or

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review of this article.