Dynamics of a Stochastic Delayed Competitive Model with Impulsive Toxicant Input in Polluted Environments

Abstract

In the world today, with the rapid development of modern agriculture and industry, a large quantity of pollutants enter into ecosystems one by one, which is a threat to the persistence of the exposed populations. This paper investigates a stochastic delayed competitive system with impulsive toxicant input in a polluted environment. Under a simple condition, sufficient and necessary conditions for stability in the mean and extinction of each species are established. Some recent works are improved and extended greatly. Some numerical simulations are also included to illustrate and support the findings.

1. Introduction

In this paper, we consider the following stochastic delay competitive model in polluted environments with impulsive toxicant input: 𝑑𝑁₁(𝑡)=𝑁₁(𝑡)𝑟₁₀−𝑟₁₁𝐶₀(𝑡)−𝑎₁₁𝑁₁(𝑡)−𝑎₁₂𝑁₂𝑡−𝜏₁𝑑𝑡+𝜆₁𝑁₁(𝑡)𝑑𝑊₁(𝑡),𝑑𝑁₂(𝑡)=𝑁₂(𝑡)𝑟₂₀−𝑟₂₁𝐶₀(𝑡)−𝑎₂₂𝑁₂(𝑡)−𝑎₂₁𝑁₁𝑡−𝜏₂𝑑𝑡+𝜆₂𝑁₂(𝑡)𝑑𝑊₂(𝑡),𝑑𝐶₀(𝑡)𝑑𝑡=𝑘𝐶_𝑒(𝑡)−𝑔+𝑚𝐶₀(𝑡),𝑑𝐶_𝑒(𝑡)𝑑𝑡=−ℎ𝐶_𝑒(𝑡),𝑡≠𝑛𝛾,𝑛∈𝑍⁺,Δ𝑁_𝑖(𝑡)=0,Δ𝐶₀(𝑡)=0,Δ𝐶_𝑒(𝑡)=𝑏,𝑡=𝑛𝛾,𝑛∈𝑍⁺,𝑖=1,2, (1) with initial data 𝑁_𝑖(𝑡)=𝜓_𝑖(𝑡)[]>0,𝑡∈−𝜏,0;𝜓_𝑖(0)>0,𝑖=1,2, (2) where 𝜏_𝑖≥0, 𝜏=max{𝜏₁,𝜏₂}, and 𝜓_𝑖(𝑡) is a continuous function on [−𝜏,0]. All coefficients in model (1) are positive. Δ𝑓(𝑡)=𝑓(𝑡⁺)−𝑓(𝑡) and 𝑍⁺={1,2,…}; 𝑁_𝑖(𝑡) is the size of the 𝑖th population, 𝑖=1,2; 𝑟_𝑖0 is the growth rate of the 𝑖th population; 𝑟_𝑖1 is the response to the pollutant present in the organism of the 𝑖th population; 𝐶₀(𝑡) is the toxicant concentration in the organism; 𝐶_𝑒(𝑡) is the toxicant concentration in the environment; 𝑘𝐶_𝑒(𝑡) is the organism's net uptake of toxicant from the environment; 𝑔𝐶₀(𝑡)+𝑚𝐶₀(𝑡) is the egestion and depuration rates of the toxicant in the organism; ℎ𝐶_𝑒(𝑡) is the toxicant loss from the environment itself; 𝛾 is the period of the impulsive effect about the exogenous input of toxicant; 𝑏 is the amount of toxicant input at every time. 𝑊_𝑖(𝑡) is a standard Brownian motion defined on a complete probability space (Ω,ℱ,𝒫); 𝜆²_𝑖 is the intensity of the environmental noise.

Recently, population models with toxicant effect have received great attention; see, for example, [1–18]. Liu and Zhang [15] considered the following competitive model in polluted environments with impulsive toxicant input: 𝑑𝑁₁(𝑡)𝑑𝑡=𝑁₁(𝑡)𝑟₁₀−𝑟₁₁𝐶₀(𝑡)−𝑎₁₁𝑁₁(𝑡)−𝑎₁₂𝑁₂(𝑡),𝑑𝑁₂(𝑡)𝑑𝑡=𝑁₂(𝑡)𝑟₂₀−𝑟₂₁𝐶₀(𝑡)−𝑎₂₁𝑁₁(𝑡)−𝑎₂₂𝑁₂(𝑡),𝑑𝐶₀(𝑡)𝑑𝑡=𝑘𝐶_𝑒(𝑡)−𝑔+𝑚𝐶₀(𝑡),𝑑𝐶_𝑒(𝑡)𝑑𝑡=−ℎ𝐶_𝑒(𝑡),𝑡≠𝑛𝛾,𝑛∈𝑍⁺.Δ𝑁_𝑖(𝑡)=0,Δ𝐶₀(𝑡)=0,Δ𝐶_𝑒(𝑡)=𝑏,𝑡=𝑛𝛾,𝑛∈𝑍⁺,𝑖=1,2, (3) For model (3), the authors [15] proved the following.

Lemma 1 (see [15]). Define Γ=𝑎₁₁𝑎₂₂−𝑎₁₂𝑎₂₁,Γ₁=𝑟₁₀𝑎₂₂−𝑎₁₂𝑟₂₀,Γ₂=𝑟₂₀𝑎₁₁−𝑟₁₀𝑎₂₁,Γ₁∶=𝑎₂₂𝑟₁₁−𝑎₁₂𝑟₂₁,Γ₂∶=𝑎₁₁𝑟₂₁−𝑎₂₁𝑟₁₁,𝐾=𝑘𝑏.𝛾ℎ𝑔+𝑚 (4) Suppose that Γ>0, Γ₁>0, Γ₂>0.(A)If Γ₁>Γ₁𝐾 and Γ₂<Γ₂𝐾, then lim_𝑡→+∞𝑁₂(𝑡)=0 and limsup_𝑡→+∞𝑡⁻¹∫^𝑡₀𝑁₁(𝑠)𝑑𝑠>0.(B)If Γ₁<Γ₁𝐾 and Γ₂>Γ₂𝐾, then lim_𝑡→+∞𝑁₁(𝑡)=0 and limsup_𝑡→+∞𝑡⁻¹∫^𝑡₀𝑁₂(𝑠)𝑑𝑠>0.(C)If Γ₁>Γ₁𝐾 and Γ₂>Γ₂𝐾, then limsup_𝑡→+∞𝑡⁻¹∫^𝑡₀𝑁_𝑖(𝑠)𝑑𝑠>0, 𝑖=1,2.

From the work of Liu and Zhang [15], some important and interesting questions arise naturally.(Q1)In the real world, the growth of population is inevitably affected by random environmental fluctuations. May [19] have claimed that population systems should be stochastic. Therefore, what happens if system (3) is affected by environmental fluctuations?(Q2)Gopalsamy [20] have pointed out that, in order to be reality, time delays should not be ignored. Hence, what happens if system (3) incorporates with time delays?(Q3)Can we improve the results obtained in Lemma 1?The aims of this paper are to investigate the above questions. Recall that 𝑟_𝑖0 stands for the growth rate. In practice, we often estimate it by an average value plus an error term. Generally, by the famous central limit theorem, the error term follows a normal distribution. Hence, for short correlation time, we can replace 𝑟_𝑖0 with 𝑟_𝑖0+𝜆_𝑖𝑊̇_𝑖(𝑡) (see, e.g., [21–29]), where 𝑊̇_𝑖(𝑡) is white noise and 𝜆²_𝑖 is the intensity of the noise. At the same time, incorporating with time delays, we get model (1). For model (1), we will show the following.

Theorem 2. Suppose that Γ>0. Define 𝜂_𝑖=𝑟_𝑖0−𝑟_𝑖1𝐾−0.5𝜆²_𝑖Γ,𝑖=1,2,₁=0.5𝑎₂₂𝜆²₁−0.5𝑎₁₂𝜆²₂,Γ₂=0.5𝑎₁₁𝜆²₂−0.5𝑎₂₁𝜆²₁. (5)(I)If 𝜂₁<0 and 𝜂₂<0, then both 𝑁₁ and 𝑁₂ go to extinction almost surely; that is, lim_𝑡→+∞𝑁_𝑖(𝑡)=0𝑎.𝑠.,𝑖=1,2; (6)(II)if 𝜂₁>0 and 𝜂₂<0, then 𝑁₂ goes to extinction a.s. and 𝑁₁ is stable in the mean a.s.; that is, lim_𝑡→+∞𝑡^−1𝑡₀𝑁₁(𝑠)𝜂𝑑𝑠=¹𝑎₁₁,𝑎.𝑠.; (7)(III)if 𝜂₁<0 and 𝜂₂>0, then 𝑁₁ goes to extinction a.s. and 𝑁₂ is stable in the mean a.s.; that is, lim_𝑡→+∞𝑡^−1𝑡₀𝑁₂(𝑠)𝜂𝑑𝑠=²𝑎₂₂,𝑎.𝑠.; (8)(IV)if 𝜂₁>0, 𝜂₂>0,(A)if Γ₁>Γ₁𝐾+Γ₁ and Γ₂<Γ₂𝐾+Γ₂, then 𝑁₂ goes to extinction a.s. and 𝑁₁ is stable in the mean a.s.lim_𝑡→+∞𝑡^−1𝑡₀𝑁₁(𝑠)𝜂𝑑𝑠=¹𝑎₁₁,𝑎.𝑠.; (9)(B)if Γ₁<Γ₁𝐾+Γ₁ and Γ₂>Γ₂𝐾+Γ₂, then 𝑁₁ goes to extinction a.s. and 𝑁₂ is stable in the mean a.s.lim_𝑡→+∞𝑡^−1𝑡₀𝑁₂(𝑠)𝜂𝑑𝑠=²𝑎₂₂,𝑎.𝑠.; (10)(C)if Γ₁>Γ₁𝐾+Γ₁ and Γ₂>Γ₂𝐾+Γ₂, then both 𝑁₁ and 𝑁₂ are stable in the mean a.s.lim_𝑡→+∞𝑡^−1𝑡₀𝑁₁(𝑠)Γ𝑑𝑠=¹−Γ₁𝐾−Γ₁Γ,𝑎.𝑠.,lim_𝑡→+∞𝑡^−1𝑡₀𝑁₂(𝑠)Γ𝑑𝑠=²−Γ₁𝐾−Γ₂Γ,𝑎.𝑠. (11)

Remark 3. It is useful to point out that if 𝜂₁>0, 𝜂₂>0, and Γ>0, then Γ₁<Γ₁+Γ₁ and Γ₂<Γ₂+Γ₂ will not hold simultaneously.

Remark 4. In comparison with most of the existing results, our key contributions in this paper are as follows.(i)To the best of our knowledge, this paper is the first attempt to consider stochastic delay competitive model in polluted environments.(ii)Our conditions are much weaker. For example, the authors [15] supposed Γ₁>0 and Γ₂>0 which are dropped in this paper.(iii)Our results improve some recent works. For example, Lemma 1 indicates that the superior limit is positive while Theorem 2 proves that the limit exists and establishes the explicit form of the limit.

2. Proof

For simplicity, define 𝑅²₊=𝑎=𝑎₁,𝑎₂∈𝑅²∣𝑎_𝑖,𝑔(𝑡>0,𝑖=1,2)=𝑡^−1𝑡₀𝑔(𝑠)𝑑𝑠. (12)

Lemma 5. For any given initial data 𝜓(𝜃)=(𝜓₁(𝜃),𝜓₂(𝜃))∈𝐶([−𝜏,0],𝑅²₊), there is a unique global positive solution 𝑁(𝑡)=(𝑁₁(𝑡),𝑁₂(𝑡)) to model (1) on 𝑡≥−𝜏 a.s. and limsup_𝑡→+∞ln𝑁_𝑖(𝑡)ln𝑡≤1𝑎.𝑠.,𝑖=1,2. (13)

Proof. The proof is a special case of Theorems 5.1 and 5.2 in Liu and Wang [25] and hence is omitted.

Lemma 6 (see [26]). Suppose that 𝑥(𝑡)∈𝐶[Ω×[0,+∞),𝑅₊].(I)If there exist 𝛿 and positive constants 𝛿₀, 𝑇 such that ln𝑥(𝑡)≤𝛿𝑡−𝛿₀^𝑡₀𝑥(𝑠)𝑑𝑠+^𝑛_𝑖=1𝛽_𝑖𝑊_𝑖(𝑡) (14) for 𝑡≥𝑇, where 𝑊_𝑖(𝑡) are independent standard Brownian motions and 𝛽_𝑖 are constants, 1≤𝑖≤𝑛, then we have the following: if 𝛿≥0, then limsup_𝑡→+∞⟨𝑥(𝑡)⟩≤𝛿/𝛿₀ a.s.; if 𝛿<0, then lim_𝑡→+∞𝑥(𝑡)=0 a.s.(II)If there exist positive constants 𝛿₀, 𝑇, and 𝛿 such that ln𝑥(𝑡)≥𝛿𝑡−𝛿₀^𝑡₀𝑥(𝑠)𝑑𝑠+^𝑛_𝑖=1𝛽_𝑖𝑊_𝑖(𝑡) (15) for 𝑡≥𝑇, then liminf_𝑡→+∞⟨𝑥(𝑡)⟩≥𝛿/𝛿₀ a.s.

To begin with, let us consider the following subsystem of (1): 𝑑𝐶₀(𝑡)𝑑𝑡=𝑘𝐶_𝑒(𝑡)−𝑔+𝑚𝐶₀(𝑡),𝑑𝐶_𝑒(𝑡)𝑑𝑡=−ℎ𝐶_𝑒(𝑡),𝑡≠𝑛𝛾,𝑛∈𝑍⁺,Δ𝐶₀(𝑡)=0,Δ𝐶_𝑒(𝑡)=𝑏,𝑡=𝑛𝛾,𝑛∈𝑍⁺0≤𝐶₀(0)≤1,0≤𝐶_𝑒(0)≤1. (16)

Lemma 7 (see [12, 13]). System (16) has a unique positive 𝛾-periodic solution (𝐶₀(𝑡),𝐶_𝑒(𝑡))^𝑇 and for each solution (𝐶₀(𝑡),𝐶_𝑒(𝑡))^𝑇 of (16), 𝐶₀(𝑡)→𝐶₀(𝑡) and 𝐶_𝑒(𝑡)→𝐶_𝑒(𝑡) as 𝑡→∞. Moreover, 𝐶₀(𝑡)>𝐶₀(𝑡) and 𝐶_𝑒(𝑡)>𝐶_𝑒(𝑡) for all 𝑡≥0 if 𝐶₀(0)>𝐶₀(0) and 𝐶_𝑒(0)>𝐶_𝑒(0), 𝑖=1,2, where 𝐶₀(𝑡)=𝐶₀(0)𝑒^{−(𝑔+𝑚)(𝑡−𝑛𝛾)}+𝑘𝑏𝑒^{−(𝑔+𝑚)(𝑡−𝑛𝛾)}−𝑒^{−ℎ(𝑡−𝑛𝛾)}ℎ−𝑔−𝑚1−𝑒^−ℎ𝛾,𝐶_𝑒(𝑡)=𝑏𝑒^{−ℎ(𝑡−𝑛𝛾)}1−𝑒^−ℎ𝛾,𝐶₀(0)=𝑘𝑏𝑒^{−(𝑔+𝑚)𝛾}−𝑒^−ℎ𝛾ℎ−𝑔−𝑚1−𝑒^{−(𝑔+𝑚)𝛾}1−𝑒^−ℎ𝛾,𝐶_𝑒(0)=𝑏1−𝑒^−ℎ𝛾 (17) for 𝑡∈(𝑛𝛾,(𝑛+1)𝛾] and 𝑛∈𝑍⁺. In addition, lim_𝑡→+∞𝑡^−1𝑡₀𝐶₀(𝑠)𝑑𝑠=𝑘𝑏ℎ𝑔+𝑚𝛾=𝐾. (18)

Proof of Theorem 2. It follows from Lemma 7 that lim_𝑡→+∞𝑡^−1𝑡₀𝐶₀(𝑠)𝑑𝑠=lim_𝑡→+∞𝑡^−1𝑡₀𝐶₀(𝑠)𝑑𝑠=𝐾. (19) That is to say, for all 𝜀>0, there is 𝑇>0 such that 𝐾−𝜀≤𝐶₀(𝑡)≤𝐾+𝜀,𝑡>𝑇. (20) Applying Itô's formula to (1) leads to 𝑑ln𝑁₁(𝑡)=𝑑𝑁₁(𝑡)𝑁₁(𝑡)−𝑑𝑁₁(𝑡)²2𝑁²₁(𝑡)=𝑟₁₀−0.5𝜆²₁−𝑟₁₁𝐶₀(𝑡)−𝑎₁₁𝑁₁(𝑡)−𝑎₁₂𝑁₂𝑡−𝜏₁𝑑𝑡+𝜆₁𝑑𝑊₁(𝑡),𝑑ln𝑁₂(𝑡)=𝑑𝑁₂(𝑡)𝑁₂(𝑡)−𝑑𝑁₂(𝑡)²2𝑁²₂(𝑡)=𝑟₂₀−0.5𝜆²₂−𝑟₂₁𝐶₀(𝑡)−𝑎₂₁𝑁₁𝑡−𝜏₂−𝑎₂₂𝑁₂(𝑡)𝑑𝑡+𝜆₂𝑑𝑊₂(𝑡). (21) In other words, we have shown that ln𝑁₁(𝑡)−ln𝑁₁(0)=𝑟₁₀−0.5𝜆²₁𝑡−𝑟₁₁^𝑡₀𝐶₀(𝑠)𝑑𝑠−𝑎₁₁^𝑡₀𝑁₁(𝑠)𝑑𝑠−𝑎₁₂^𝑡₀𝑁₂𝑠−𝜏₁𝑑𝑠+𝜆₁𝑊₁(𝑡)=𝑟₁₀−0.5𝜆²₁𝑡−𝑎₁₂^𝑡₀𝑁₂(𝑠)𝑑𝑠+𝑎₁₂^𝑡_𝑡−𝜏1𝑁₂(𝑠)𝑑𝑠−⁰_−𝜏1𝑁₂(𝑠)𝑑𝑠−𝑟₁₁^𝑡₀𝐶₀(𝑠)𝑑𝑠−𝑎₁₁^𝑡₀𝑁₁(𝑠)𝑑𝑠+𝜆₁𝑊₁(𝑡),ln𝑁₂(𝑡)−ln𝑁₂(0)=𝑟₂₀−0.5𝜆²₂𝑡−𝑟₂₁^𝑡₀𝐶₀(𝑠)𝑑𝑠−𝑎₂₁^𝑡₀𝑁₁𝑠−𝜏₂𝑑𝑠−𝑎₂₂^𝑡₀𝑁₂(𝑠)𝑑𝑠+𝜆₂𝑊₂(𝑡)=𝑟₂₀−0.5𝜆²₂𝑡−𝑎₂₁^𝑡₀𝑁₁(𝑠)𝑑𝑠+𝑎₂₁^𝑡_𝑡−𝜏2𝑁₁(𝑠)𝑑𝑠−⁰_−𝜏2𝑁₁(𝑠)𝑑𝑠−𝑟₂₁^𝑡₀𝐶₀(𝑠)𝑑𝑠−𝑎₂₂^𝑡₀𝑁₂(𝑠)𝑑𝑠+𝜆₂𝑊₂(𝑡). (22)
(I) Assume that 𝜂₁=𝑟₁₀−0.5𝜆²₁−𝑟₁₁𝐾<0 and 𝜂₂=𝑟₂₀−0.5𝜆²₂−𝑟₂₁𝐾<0. In view of (22), 𝑡⁻¹𝑁ln¹(𝑡)𝑁₁(0)≤𝑟₁₀−𝑟₁₁^𝑡₀𝐶₀(𝑠)𝑑𝑠−0.5𝜆²₁+𝑡⁻¹𝜆₁𝑊₁(𝑡). (23) By lim_𝑡→+∞𝑊_𝑖(𝑡)/𝑡=0 a.s., 𝑖=1,2 (19) and 𝑟₁₀<0.5𝜆²₁+𝑟₁₁𝐾 we have that limsup_𝑡→+∞𝑡⁻¹ln𝑁₁(𝑡)≤𝑟₁₀−0.5𝜆²₁−𝑟₁₁𝐾=𝜂₁<0. (24) Therefore, lim_𝑡→+∞𝑁₁(𝑡)=0, a.s. Similarly, it follows from (9) that if 𝜂₂<0, then lim_𝑡→+∞𝑁₂(𝑡)=0, a.s.
(II) Assume that 𝜂₁>0 and 𝜂₂<0. Since 𝜂₂<0, then, by (I), lim_𝑡→+∞𝑁₂(𝑡)=0, a.s. Hence, for arbitrary 𝜀>0, there exists 𝑇>0 such that for 𝑡≥𝑇−𝜀2≤𝑎₁₂𝑡^−1𝑡₀𝑁₂𝑠−𝜏₁𝜀𝑑𝑠≤2,−𝜀2≤𝑡⁻¹ln𝑁₁(0)≤𝜀2. (25) When the above inequalities and (20) are used in (22), we can obtain that, for 𝑡≥𝑇, ln𝑁₁(𝑡)≤𝜂₁+2𝜀𝑡−𝑎₁₁^𝑡₀𝑁₁(𝑠)𝑑𝑠+𝜆₁𝑊₁(𝑡),ln𝑁₁(𝑡)≥𝜂₁−2𝜀𝑡−𝑎₁₁^𝑡₀𝑁₁(𝑠)𝑑𝑠+𝜆₁𝑊₁(𝑡). (26)(27) Note that 𝜂₁>0; we can let 𝜀 be sufficiently small such that 𝜂₁−2𝜀>0. Applying (i) and (ii) in Lemma 6 to (26) and (27), respectively, one can see that 𝜂₁−2𝜀𝑎₁₁≤liminf_𝑡→+∞𝑁₁(𝑡)≤limsup_𝑡→+∞𝑁₁(𝑡≤𝜂₁)+2𝜀𝑎₁₁,a.s. (28) It therefore follows from the arbitrariness of 𝜀 that lim_𝑡→+∞𝑁₁(𝑡=𝜂)¹𝑎₁₁,a.s. (29)
The proof of (III) can be obtained similarly and hence is omitted.
Now, we are in the position to prove (IV). Assume that 𝜂₁>0 and 𝜂₂>0. For 𝑖=1,2, consider the following stochastic equation: 𝑑𝑦_𝑖(𝑡)=𝑦_𝑖(𝑡)𝑟_𝑖0−𝑟_𝑖1𝐶₀(𝑡)−𝑎_𝑖𝑖𝑦_𝑖(𝑡)𝑑𝑡+𝜆_𝑖𝑦_𝑖(𝑡)𝑑𝑊_𝑖(𝑡),𝑦_𝑖(𝜃)=𝜓(𝜃),𝜃∈[−𝜏,0]. (30) By the classical stochastic comparison theorem [30], one can see that 𝑁₁(𝑡)≤𝑦₁(𝑡),𝑁₂(𝑡)≤𝑦₂(𝑡). (31) Note that 𝑟_𝑖0>0.5𝜆²_𝑖+𝑟_𝑖1𝐾, 𝑖=1,2; an argument, identical to the argument used in the proof of (II), shows that lim_𝑡→+∞𝑦_𝑖(𝑡)=lim_𝑡→+∞𝑡^−1𝑡₀𝑦_𝑖(𝑠)𝜂𝑑𝑠=^𝑖𝑎_𝑖𝑖a.s.,𝑖=1,2. (32) Consequently, lim_𝑡→+∞𝑡^−1𝑡_𝑡−𝜏1𝑦₂(𝑠)𝑑𝑠=lim_𝑡→+∞𝑡^−1𝑡₀𝑦₂(𝑠)𝑑𝑠−𝑡^{−1𝑡−𝜏1}₀𝑦₂(𝑠)𝑑𝑠=0,lim_𝑡→+∞𝑡^−1𝑡_𝑡−𝜏2𝑦₁(𝑠)𝑑𝑠=0,a.s. (33) This, together with (31), implies that lim_𝑡→+∞𝑡^−1𝑡_𝑡−𝜏1𝑁₂(𝑠)𝑑𝑠=0,lim_𝑡→+∞𝑡^−1𝑡_𝑡−𝜏2𝑁₁(𝑠)𝑑𝑠=0,a.s. (34) On the other hand, computing (9) ×𝑎₁₁ − (22) ×𝑎₂₁ gives 𝑎₁₁𝑁ln²(𝑡)𝑁₂(0)=𝑎₁₁𝑎₂₁^𝑡_𝑡−𝜏2𝑁₁(𝑠)𝑑𝑠−⁰_−𝜏2𝑁₁(𝑠)𝑑𝑠−𝑎₂₁𝑎₁₂^𝑡_𝑡−𝜏1𝑁₂(𝑠)𝑑𝑠−⁰_−𝜏1𝑁₂(𝑠)𝑑𝑠+𝑎₂₁𝑁ln¹(𝑡)𝑁₁(0)+Γ₂−Γ₂𝑡−Γ₂^𝑡₀𝐶₀(𝑠)𝑑𝑠−Γ^𝑡₀𝑁₂(𝑠)𝑑𝑠−𝑎₂₁𝜆₁𝑊₁(𝑡)+𝑎₁₁𝜆₂𝑊₂(𝑡). (35) In view of (13) and (34), for arbitrary 𝜀>0, there exits 𝑇>0 such that, for 𝑡≥𝑇, 𝑡⁻¹𝑎₂₁𝑁ln¹(𝑡)𝑁₁(0)<𝜀3,𝑡⁻¹𝑎₁₁ln𝑁₂(0)<𝜀3,𝑡⁻¹𝑎₁₁𝑎₂₁^𝑡_𝑡−𝜏2𝑁₁(𝑠)𝑑𝑠−⁰_−𝜏2𝑁₁(𝑠)𝑑𝑠−𝑡⁻¹𝑎₂₁𝑎₁₂^𝑡_𝑡−𝜏1𝑁₂(𝑠)𝑑𝑠−⁰_−𝜏1𝑁₂(𝑠)≤𝜀𝑑𝑠3. (36) When the above inequalities and (20) are used in (35), we can see that for 𝑡>𝑇𝑎₁₁ln𝑁₂(𝑡)≤Γ₂−Γ₂𝐾−Γ₂+2𝜀𝑡−Γ^𝑡₀𝑁₂(𝑠)𝑑𝑠−𝑎₂₁𝜆₁𝑊₁(𝑡)+𝑎₁₁𝜆₂𝑊₂(𝑡). (37) Similarly, computing (22) ×𝑎₂₂ − (9) ×𝑎₁₂ gives 𝑎₂₂𝑁ln¹(𝑡)𝑁₁(0)=𝑎₂₂𝑎₁₂^𝑡_𝑡−𝜏1𝑁₂(𝑠)𝑑𝑠−⁰_−𝜏1𝑁₂(𝑠)𝑑𝑠−𝑎₁₂𝑎₂₁^𝑡_𝑡−𝜏2𝑁₁(𝑠)𝑑𝑠−⁰_−𝜏2𝑁₁(𝑠)𝑑𝑠+𝑎₁₂𝑁ln²(𝑡)𝑁₂(0)+Γ₁−Γ₁𝑡−Γ₁^𝑡₀𝐶₀(𝑠)𝑑𝑠−Γ^𝑡₀𝑁₁(𝑠)𝑑𝑠+𝑎₂₂𝜆₁𝑊₁(𝑡)−𝑎₁₂𝜆₂𝑊₂(𝑡). (38) Hence, for sufficiently large 𝑡, 𝑎₂₂ln𝑁₁(𝑡)≤Γ₁−Γ₁𝐾−Γ₁+2𝜀𝑡−Γ^𝑡₀𝑁₁(𝑠)𝑑𝑠+𝑎₂₂𝜆₁𝑊₁(𝑡)−𝑎₁₂𝜆₂𝑊₂(𝑡). (39)
(A) Assume that Γ₁>Γ₁𝐾+Γ₁ and Γ₂<Γ₂𝐾+Γ₂. Since Γ₂<Γ₂𝐾+Γ₂, then we can let 𝜀 be sufficiently small such that Γ₂−Γ₂𝐾−Γ₂+2𝜀<0. Applying (i) in Lemma 6 to (37) results in lim_𝑡→+∞𝑁₂(𝑡)=0, a.s. The proof of lim_𝑡→+∞⟨𝑁₁(𝑡)⟩=𝜂₁/𝑎₁₁ a.s. is the same as that in (II) and hence is omitted. The proof of (B) is similar to that of (A) and hence is omitted.
(C) Assume that Γ₁>Γ₁𝐾+Γ₁ and Γ₂>Γ₂𝐾+Γ₂. Notice that Γ₂>Γ₂; then, by (37) and Lemma 6, limsup_𝑡→+∞𝑁₂(𝑡≤Γ)²−Γ₂𝐾−Γ₂+2𝜀Γ,a.s. (40) It then follows from the arbitrariness of 𝜀 that limsup_𝑡→+∞𝑁₂(𝑡≤Γ)²Γ₂𝐾−Γ₂Γ,a.s. (41) Similarly, by (39), Lemma 6, and the arbitrariness of 𝜀, we have limsup_𝑡→+∞𝑁₁(𝑡≤Γ)¹−Γ₁𝐾−Γ₁Γ,a.s. (42) Let 𝜀 be sufficiently small satisfying 𝑎₁₁((Γ₁−Γ₁𝐾−Γ₁)/Γ)−𝜀>0. Substituting (34), (41), and (20) into (22) yields 𝑡⁻¹ln𝑁₁(𝑡)=𝑡⁻¹ln𝑁₁(0)+𝑟₁₀−𝑟₁₁𝐶₀(𝑡)−0.5𝜆²₁−𝑎₁₁𝑁₁(𝑡)−𝑎₁₂𝑁₂(𝑡)+𝜆₁𝑊₁(𝑡)𝑡+𝑎₁₂𝑡^−1𝑡_𝑡−𝜏1𝑁₂(𝑠)𝑑𝑠−⁰_−𝜏1𝑁₂(𝑠)𝑑𝑠≥𝜂₁−2𝜀−𝑎₁₁𝑁₁(𝑡)−𝑎₁₂limsup_𝑡→+∞𝑁₂(𝑡)+𝜆₁𝑊₁(𝑡)𝑡≥𝜂₁−2𝜀−𝑎₁₁𝑁₁(𝑡)−𝑎₁₂Γ₂−Γ₂Γ+𝜆₁𝑊₁(𝑡)𝑡=𝑎₁₁Γ₁−Γ₁𝐾−Γ₁Γ−2𝜀−𝑎₁₁𝑁₁(𝑡)+𝜆₁𝑊₁(𝑡)𝑡, (43) for sufficiently large 𝑡. By (ii) in Lemma 6 and the arbitrariness of 𝜀, one can observe that liminf_𝑡→+∞𝑁₁(𝑡≥Γ)¹−Γ₁𝐾−Γ₁Γ,a.s. (44) Similarly, when (34) and (42) and (20) are used in (9), we can see that liminf_𝑡→+∞⟨𝑁₂(𝑡)⟩≥(Γ₂−Γ₂𝐾−Γ₂)/Γ, a.s. This, together with (41), (42), and (44), indicates that lim_𝑡→+∞𝑁₁(𝑡=Γ)¹−Γ₁𝐾−Γ₁Γ,lim_𝑡→+∞𝑁₂(𝑡=Γ)²−Γ₂𝐾−Γ₂Γ,a.s. (45)

3. Numerical Simulations

In this section, using the classical Milstein method (see, e.g., [31]), we work out some numerical figures to support the analytical results. In Figure 1, we choose 𝑟₁₀=0.7, 𝑟₂₀=0.5, 𝑎₁₁=0.8, 𝑎₁₂=0.5, 𝑎₂₁=0.4, 𝑎₂₂=0.7, 𝜏₁=10, 𝜏₂=12, 𝑘=𝑔=𝑚=0.2, ℎ=0.5, 𝑏=0.1, and 𝛾=1. Then, Γ=0.36, Γ₁=0.24, Γ₂=0.12,Γ₁=0.2, Γ₂=0.4, and 𝐾=𝑘𝑏/𝛾ℎ(𝑔+𝑚)=0.1. The only difference between conditions of Figures 1(a), 1(b), 1(c), 1(d), 1(e), and 1(f) is that the values of 𝜆₁ and 𝜆₂ are different.(a)In Figure 1(a), we choose 𝜆₁=1.1, 𝜆₂=1. Then, 𝜂₁=𝑟₁₀−𝑟₁₁𝐾−0.5𝜆²₁=−0.12, 𝜂₂=𝑟₂₀−𝑟₂₁𝐾−0.5𝜆²₂=−0.1. By virtue of (I) in Theorem 2, both 𝑁₁ and 𝑁₂ are extinct; see Figure 1(a).(b)In Figure 1(b), we set 𝜆₁=1, 𝜆₂=1. Then, 𝜂₁=0.1, 𝜂₂=−0.1. In view of (II) in Theorem 2, 𝑁₂ is extinct and lim_𝑡→+∞𝑡⁻¹∫^𝑡₀𝑁₁(𝑠)𝑑𝑠=𝜂₁/𝑎₁₁=0.125. Figure 1(b) confirms this.(c)In Figure 1(c), we let 𝜆₁=1.2, 𝜆₂=0.8. Then, 𝜂₁=−0.12, 𝜂₂=0.08. It follows from (III) in Theorem 2 that 𝑁₁ is extinct and lim_𝑡→+∞𝑡^−1𝑡₀𝑁₂(𝑠)𝜂𝑑𝑠=²𝑎₂₂=0.1143. (46) See Figure 1(c).(d)In Figure 1(d), we set 𝜆₁=1, 𝜆₂=0.8484. Then, 𝜂₁=0.1,𝜂₂=0.04, Γ₁=0.24>Γ₁𝐾+Γ₁=0.19, and Γ₂=0.12<Γ₂𝐾+Γ₂=0.128. According to (A) in Theorem 2, 𝑁₂ is extinct and lim_𝑡→+∞𝑡^−1𝑡₀𝑁₁(𝑠)𝜂𝑑𝑠=¹𝑎₁₁=0.10.8=0.125. (47) Figure 1(d) confirms this.(e)In Figure 1(e), we choose 𝜆₁=1.058, 𝜆₂=0.8. Then, 𝜂₁=0.04, 𝜂₂=0.08, Γ₁=0.24<Γ₁𝐾+Γ₁=0.252, and Γ₂=0.12>Γ₂𝐾+Γ₂=0.072. By (B) in Theorem 2, 𝑁₁ is extinct and lim_𝑡→+∞𝑡^−1𝑡₀𝑁₂(𝑠)𝜂𝑑𝑠=²𝑎₂₂=0.080.7=0.1143. (48) See Figure 1(e).(f)In Figure 1(f), we let 𝜆₁=1, 𝜆₂=0.8. Then, Γ₁=0.24>Γ₁𝐾+Γ₁=0.21 and Γ₂=0.12>Γ₂𝐾+Γ₂=0.096. It follows from (C) in Theorem 2 that lim_𝑡→+∞𝑡^−1𝑡₀𝑁₁(𝑠)Γ𝑑𝑠=¹−Γ₁𝐾−Γ₁Γ=0.030.36=0.0833,lim_𝑡→+∞𝑡^−1𝑡₀𝑁₂(𝑠)Γ𝑑𝑠=²−Γ₂𝐾−Γ₂Γ=0.0667. (49) See Figure 1(f).

(a)

(b)

(c)

(d)

(e)

(f)

Figure 1 

Solutions of system (1) for 𝑟10=0.7, 𝑟20=0.5, 𝑎11=0.8, 𝑎12=0.5, 𝑎21=0.4, 𝑎22=0.7, 𝜏1=10, 𝜏2=12, 𝑘=𝑔=𝑚=0.2, ℎ=0.5, 𝑏=0.1, 𝛾=1, 𝐶0(0)=𝐶𝑒(0)=0.1, and step size Δ𝑡=0.001. (a) is with 𝜆1=1.1,𝜆2=1; (b) is with 𝜆1=1, 𝜆2=1; (c) is with 𝜆1=1.2, 𝜆2=0.8; (d) is with 𝜆1=1, 𝜆2=0.8484; (e) is with 𝜆1=1.058, 𝜆2=0.8; (f) is with 𝜆1=1, 𝜆2=0.8.

4. Conclusions and Future Directions

This paper investigates a stochastic delay competitive model in a polluted environment with impulsive toxicant input. For each population, the critical value between stability in the mean and extinction is obtained. Some recent works are extended and improved. Our Theorem 2 has some important and interesting interpretation.(1)Time delay is harmless for stability in the mean and extinction of the stochastic model (1).(2)White noises can change the dynamics of the population model greatly.

Some interesting problems deserve further study. One can consider some more realistic systems, for example, stochastic delayed population model with the Markov switching (see, e.g., [22, 23, 29]). It is also interesting to extend Theorem 2 to 𝑛-species case.

Conflict of Interests

The authors declare that they have no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors thank the reviewers for their valuable comments. The authors also thank Dr. H. Qiu and Dr. C. Zhang for helping them to improve the English exposition. This research is supported by National Natural Science Foundation of PR China (nos. 11301207, 11171081, 11271364, and 11301112), Natural Science Foundation of Jiangsu Province (nos. BK2011407 and BK20130411), and Natural Science Research Project of Ordinary Universities in Jiangsu Province (no. 13KJB110002).