Survival analysis of a stochastic cooperation system with functional response in a polluted environment

In this paper, we propose and study a stochastic two-species cooperation model with functional response in a polluted environment. We first perform the survival analysis and establish sufficient conditions for extinction, weak persistence, and stochastic permanence. Then we further perform the survival analysis based on the temporal average of population size and derive sufficient conditions for the strong persistence in the mean and weak persistence in the mean. Finally, we present numerical simulations to justify the theoretical results.


Introduction
With rapid development of industries and agriculture, a mass of toxicants has been emitted into the environment, such as the industrial wastewater, domestic sewage, and other contaminants. The presence of a variety of toxicants in the environment not only seriously threatened the survival of the exposed populations but also affected the human life style. Therefore it is important to estimate the environmental toxicity, which requires quantitative estimates for the survival risk of species in a polluted environment. This motivates scholars to utilize the mathematical models to assess the effects of toxicants on various ecosystems. Hallam et al. [1][2][3] did pioneering works. It has been a significant topic of considerable researches, and more and more deterministic models have been proposed and analyzed (see [4][5][6][7][8][9][10][11][12][13][14][15][16][17]). They all provide a great insight into the effects of pollutants. In this paper, we mainly attempt to study a two-species cooperation model with functional response. We assume that the environment is of complete spatial homogeneity and there is no migration. Let x i (t) represent the population size (population density) of the ith species at time t. We consider the following cooperation model in a polluted environment: 1+b 2 x 2 (t) ], x 2 (t) = x 2 (t)[r 2a 2 x 2 (t) + c 1 x 1 (t) 1+b 1 x 1 (t) ], (1.1) where the positive coefficients r 1 , r 2 and a 1 , a 2 are the intrinsic growth rates and selfinhibition rates, respectively. In the classical cooperation model the mutualism effects are described by a bilinear function, that is, x i response to x j is assumed to be increasingly monotonic, an inherent assumption meaning that the more x j there exist in the environment, the better off the x i . But here the term c i x i (t)/(1 + b i x i (t)) represents the functional response function, and moreover, it is an increasing function with respect to x i and has a saturation value for large enough x i . The positive coefficients c j measure the mutualism effects of species x j on species x i , and b j , i, j = 1, 2, i = j, are positive control constants. For a relevant ecological model of system (1.1), we refer the readers to [18]. Now we are in the position to describe the dynamics of population in a polluted environment. We assume that the living organisms absorb part of toxicants into their bodies, the dynamics of the population is affected by internal toxicant, and the individuals in two species have identical concentration of organismal toxicant at time t (see [5]). To simplify the mathematical model, we assume that the capacity of the environment is so large that the change of toxicant in the environment that comes from uptake and egestion by the organisms can be neglected (see [3]). Let C 0 (t) represent the concentration of toxicant in the organism at time t, and let C E (t) represent the concentration of toxicant in the environment at time t. A coupling between species and toxicant is formulated by assuming that the intrinsic growth rate of the ith population, r i0r i1 C 0 (t), is a linear function of concentration of toxicant present in the organism. Then the population dynamics can be given as follows: where r i0 is the intrinsic growth rate of the ith species in the absence of toxicant, r i1 , i = 1, 2, is the dose-response rate of species i to the toxicant concentration, d represents the uptake rate of toxicant from environment per unit biomass, η represents the intake rate of toxicant from food per unit biomass, l 1 represents the organismal net ingestion rates of toxicant, l 2 represents the organismal excretion rates of toxicant, and h represents the loss rate of toxicant from the environment due to the processes such as biological transformation, chemical hydrolysis, volatilization, microbial degradation, and photosynthetic degradation. Here u(t) represents the input rate of exogenous toxin at time t, which is a nonnegative continuous function defined on [0, ∞) with u 1 := sup t≥0 u(t) > 0.
Considering that the fate of young recruits after reproduction is quite sensitive to the environmental fluctuations, so the growth rate of population is inevitably affected. May and Allen [19] have claimed that the growth rates in population systems should be fluctuating around some average values because of the environmental fluctuations. In this sense, it is necessary to incorporate the environmental fluctuations into our model. In practice, we usually estimate the growth rate r i0 by an average value plus an error term; by the central limit theorem the error term follows a normal distribution, and thus we can approximate the error term by σ iḂi (t), that is, whereḂ i (t) represents a white noise process (i.e., B i (t) is a standard Brownian motion), and σ 2 i is the intensity of the white noise. According to our discussion, a stochastic two-species cooperation model in polluted environment is derived as follows: (1. 3) The initial conditions satisfy the conditions In recent years, many interesting and important works about the stochastic model in polluted environment have been reported (see [20][21][22][23][24][25][26][27][28][29][30][31][32][33]). But to the best of our knowledge, there exist few published papers concerning system (1.3). Noting that the last two equations in system (1.3) can be explicitly solved, so we only need to consider the following subsystem: (1.4) Motivated by the existing results, the rest of this paper is arranged as follows. In Sect. 2, we introduce several commonly used basic lemmas and classical definitions. In Sect. 3, we perform the survival analysis and establish sufficient criteria for extinction, weak persistence, and stochastic permanence. In Sect. 4, we further discuss the survival analysis based on the temporal average of population size and derive sufficient conditions for the strong persistence in the mean and weak persistence in the mean. In Sect. 5, we present several numerical simulations to validate our theoretical results. The limitation of the model is also discussed in the last section.

Preliminaries
From now on, unless otherwise specified, we always work on the complete probability space (Ω, F, P) with filtration {F t } t≥0 satisfying the usual conditions, that is, it is right continuous, and F 0 contains all P-null sets. Note that both C 0 (t) and C E (t) represent the concentrations of toxicant, so to be realistic, we must have 0 ≤ C 0 (t) < 1 and 0 ≤ C E (t) < 1 for all t ≥ 0. In fact, we can prove that by solving the last two equations of system (1.3).
For convenience and simplicity, we introduce the following notations: Definition 2.1 (See [34]) To state and prove our main results, we recall some classical concepts.
• Population x i is said to be weakly persistent if lim sup t→∞ x i (t) > 0 a.s.
• Population x i is said to be weakly persistent in the mean if lim sup t→∞ x i (t) > 0 a.s.
• Population x i is said to be strongly persistent in the mean if lim inf t→∞ x i (t) > 0 a.s.

Survival analysis
In this section, we first study the existence and uniqueness of a globally positive solution for the biological significance and then perform the survival analysis of system (1.4).

) has a unique solution x(t), and x(t) remains in R 2
+ for all t ≥ 0 with probability one.
Proof The proof of the theorem is standard. Firstly, let us consider the following stochastic system: with initial condition y 1 (0) = ln x 1 (0), y 2 (0) = ln x 2 (0). It is easy to verify that the coefficients of system (3.1) satisfy the local Lipschitz condition, and thus system (3.1) has a unique solution y(t) on [0, τ e ), where τ e is the explosion time (see [35]). By applying Itô's formula and a simple calculation, it is easy to derive that x 1 (t) = e y 1 (t) , x 2 (t) = e y 2 (t) is the unique positive local solution to system (1.4) with initial value x(0) ∈ R 2 + . To show that this solution is global, we only need to show that τ e = ∞.
Let n 0 be sufficiently large such that every component of x(0) remains in the interval [ 1 n 0 , n 0 ]. For each integer n ≥ n 0 , we define the stopping time Here we set inf ∅ = +∞ (∅ denotes the empty set). Obviously, τ n is increasing as n → ∞. Let τ ∞ = lim n→∞ τ n , whence τ ∞ ≤ τ e almost surely; if we can show that τ ∞ = ∞ almost surely, then τ e = ∞ almost surely, and the proof is completed.
The nonnegativity of this function can be seen from v -1 -ln v ≥ 0 for v > 0. Let T > 0 be an arbitrary constant. Then for 0 ≤ t ≤ τ n ∧ T, by Itô's formula we obtain Integrating both sides from 0 to τ n ∧ T, we have Taking the expectations on both sides yields that Let Ω n = {τ n ≤ T}. For arbitrary ω ∈ Ω n , there exists some i such that x i (τ n , ω) equals either n or 1 n , and thus V (x(τ n , ω)) is not less than (n -1 -ln n) ∧ ( 1 n -1 + ln n) μ(n). It then follows from (3.6) that where 1 Ω n is the indicator function of Ω n . Let n → ∞. Then μ(n) → ∞, and (3.7) leads to P(τ ∞ ≤ T) = 0. By the arbitrariness of T we have P(τ ∞ = ∞) = 1 almost surely. This completes the proof of Theorem 3.1.
It is well known that the threshold is very important for assessing the risk of extinction for species exposed to toxicant from a biological point of view. In the following, we first show that the pth moment of the solution of system (1.4) is upper bounded and then establish the threshold between weak persistence and extinction for species x i modeled by (1.4). To begin with, we present the fundamental assumption that r i0 > 0.5σ 2 i . Unless otherwise stated, it is always assumed in this paper.  Obviously, L(x 1 , x 2 ) is upper bounded; we denote it by L, that is, Applying Itô's formula to e t V (x 1 , x 2 ) yields that Integrating from 0 to t and then taking the expectation of both sides yield that This gives that This completes the proof of Theorem 3.2.

(t). (3.22)
Integrating both sides and then taking the expectations, we obtain that On the other hand, For arbitrary ε ∈ (0, 1), let α = ( ε δ ) 1 m . By Chebyshev's inequality we have (3.26) which gives that In the following, we prove that for arbitrary ε ∈ (0, 1), there is a constant β > 0 such that  30) which implies that Consequently, This completes the proof of Theorem 3.3.
Remark 3.1 Theorem 3.3, which directly measures the population size x i (t), indicates that the population size will neither too small nor too large with large probability when the time is sufficiently large.
then population x i goes to extinction with probability one.
Proof Applying Itô's formula yields that Integrating both sides yields that Letting t → ∞ and applying the strong law of large numbers, we obtain that In other words, lim t→∞ x i (t) = 0. This completes the proof of Theorem 3.4.
Remark 3.2 Theorem 3.4 indicates the worst case that the population will go to extinction almost surely.

then population x i is weakly persistent almost surely.
Proof We denote S := {ω : lim sup t→∞ x i (t, ω) = 0} and assume that P(S) > 0. Then for all ω ∈ S, we have lim sup t→∞ x i (t, ω) = 0. For arbitrary small ε satisfying 0 < ε < 1, there exists T(ω) such that (3.36) It then follows that By the continuity of x i (t, ω) there must be a constantK such that x i (t, ω) ≤K for 0 ≤ t ≤ T(ω). On the other hand, for sufficient large t. Since ε is arbitrarily small, we obtain that This completes the proof of Theorem 3.5.
Remark 3.3 Theorem 3.5 admit the case that the population size is close to zero even if the time is sufficiently large. In this case the survival of species can be dangerous in reality. In addition, Theorems 3.4 and 3.5 reveal that C 0 * = (r i0 -0.5σ 2 i + c j b j )/r i1 is the threshold between extinction and weak persistence.

The estimation of temporal averages
In this section, we further discuss the survival analysis of system (1.4) based on the temporal average of the population size x i (t). Proof By Itô's formula, Integrating both sides yields that where N(t) = t 0 σ i e s dB i (s). Note that N(t) is a local martingale with quadratic variation By the exponential martingale inequality we have P sup 0≤t≤γ n N(t) -0.5e -γ n N, N > θ e γ n ln n ≤ n -θ , (4.5) where θ > 1, and γ > 0 is arbitrary. By the Borel-Cantelli lemma, for almost all ω ∈ Ω, there is a random integer n 0 (ω) such that for all n ≥ n 0 (ω) and 0 ≤ t ≤ γ n, Substituting (4.6) into (4.3) yields that Since 0 ≤ t ≤ γ n and x i (t) > 0, there is a constant such that that is, for 0 ≤ t ≤ γ n, we have e t ln x i (t) -ln x i (0) ≤ e t -1 + θ e γ n ln n. (4.9) Therefore, if γ (n -1) ≤ t ≤ γ n and n ≥ n 0 (ω), then we obtain that  In the following, we show that lim inf t→∞ = 0 a.s., for any ε ∈ (0, 1), there exists a positive constant T such that (4.14) for t > s ≥ T. From equations (3.38) and (4.14) we have This gives that that is, x -1 i (t) ≤Ke 2ε(t+T) almost surely. Therefore we obtain that that is, population x i is strongly persistent in the mean almost surely.
Proof Recalling (3.34), we obtain that Then from Theorem 4.2 it follows that On the other hand, Similarly, we obtain that In other words, population x i is weakly persistent in the mean almost surely.

Discussion
In this paper, considering the fact of polluted environment, we propose and study a stochastic two-species cooperation model with functional response. However, we only consider the case that a coupling between species and toxicant is a linear function of concentration of toxicant present in the organism, that is, r i0r i1 H i (C 0 ) = r i0r i1 C 0 (t), where H i (C 0 ) is the dose-response function of species x i to the toxin. However, someone suggested that H i (C 0 ) should be nonlinear in many cases, such as a sigmoid dose-response curve (see [37,38]). Liu and Wang [24] also introduced a more general case: H i (C 0 ) is a nondecreasing continuous function of C 0 with H i (0) = 0. Clearly, our assumption is a particular case of this generalized condition. Moreover, the exogenous toxin cannot be continuous but rather of the impulse form, the growth rate of population is also inevitably affected by other environment noise such as Lévy jumps, and the regime-switching is other common random perturbation in the environment (see [39][40][41]). All these questions associated with the polluted environment are interesting topics to deserve further investigation, and we leave them for our future works.