DYNAMICS OF A STOCHASTIC THREE-SPECIES NUTRIENT-PHYTOPLANKTON-ZOOPLANKTON MODEL

We investigate a stochastic three-species nutrient-phytoplankton-zooplankton model in this paper. We verify that the system admits a unique positive global solution starting from any positive initial value firstly. Then the sufficient conditions that guarantee the population extinction and persistence in the mean are derived. The results show that weaker white noise will strengthen the stability of the system, while stronger white noise will result in extinction of population. We also show that around the positive equilibrium point of deterministic system, the stochastic system is weakly persistent in the mean under some conditions.

1. Introduction * Corresponding author E-mail addresses: chunjinwei92@163.comReceived March 19, 2018 S. Jang et al. [5] proposed periodic systems of phytoplankton-zooplankton interactions with toxin producing phytoplankton to study the effects of TPP upon extinction and persistence of the populations.More related literature can be seen in [6,7,8,9,10].
However, unlike the above models, in this paper, we model the harmful effect of phytoplankton by choosing the zooplankton grazing function as a type-IV function due to prey toxicity.This type of functional response implies that for large phytoplankton density, the predation rate decreased, which resembles a group defence mechanism for phytoplankton against zooplankton and hence a type-IV function would be an appropriate choice to model zooplankton predation.
Mukhopadhyay et al. [2] consider the following nutrient-phytoplankton-zooplankton model with holling type-IV : where N, P and Z denote the instantaneous concentrations of nutrient, phytoplankton and zooplankton at time t, respectively, subject to the non-negative initial condition N(0) = N 0 ≥ 0, P(0) = P 0 ≥ 0 and Z(0) = Z 0 ≥ 0. The system parameters are all non-negative and are interpreted as follows: a: maximal nutrient uptake rate of phytoplankton; i: the zooplankton's immunity from, or tolerance of, the phytoplankton; γ, ε: death rates for phytoplankton and zooplankton; γ 1 , ε 1 : nutrient recycling rates from dead phytoplankton and zooplankton, respectively with In [2], the authors have analyzed stability and bifurcation behavior of different equilibrium points of system (1.1), and considered diffusion-driven instability as well as stability.The result showed that, when ε + D 2 < c 1 , γ + D 1 < a 1 N * k+N * , the model (1.1) has a positive equilibrium point E * = (N * , P * , Z * ).In fact, since aquatic ecosystems are generally complex, open systems, the natural growth and death rate of populations are inevitably affected by environmental stochasticity.Therefore, it is meaningful to further incorporate the environmental stochasticity into the underlying system (1.1), which could provide us a deeper understanding for the real ecosystems.For the convenience of calculation, we assume a 1 = a, c 1 = c.So far as our knowledge is concerned, no results related to nutrient-phytoplankton-zooplankton model with stochastic perturbation have been reported.Therefore, we assume the fluctuation of uptake rates a and c are subjected to the Guassian white noise , that is, a → a + σ 1 Ḃ1 (t), c → c + σ 2 Ḃ1 (t), then we obtain the following stochastic model: where B 1 (t), B 2 (t) are independent Brownian motions defined on the complete probability space (Ω, F , {F t } t ≥0 , P) with a filtration {F t } t ≥0 satisfying the usual normal conditions (i.e. it is right continuous and increasing while F 0 contains all P-null sets).σ 2 1 , σ 2 2 represent the intensity of the white noise.
Scholars have pointed out that random fluctuations may change the dynamics of population models greatly and studied the effect of the noise on the dynamic behavior of the population models, see [11,12,14,13,15,16] and the references therein.For example, Zhao and Yuan [11] analyzed the stationary distribution and ergodicity of a stochastic phytoplankton allelopathy model under regime switching, which takes both white and colored noises into account.
Liu and Wang [12] investigate a double delayed (maturation delay for toxin producing phytoplankton and gestation delay for zooplankton) bioeconomic phytoplankton zooplankton system with commercial harvesting on zooplankton and environmental stochasticity, they studied the existence and uniqueness of the global positive solution as well as the stochastic stability and existence of stochastic Hopf bifurcation.This paper is organized as follows.In Section 2, we show the existence and uniqueness of the global positive solution.In Section 3, we establish the sufficient conditions of population extinction and persistence in the mean.We obtain the asymptotic behavior of stochastic system near the positive equilibrium point of deterministic system in Section 4. The paper ends with some conclusions in Section 5.

Existence and uniqueness of the global positive solution
As we know, in order for a stochastic differential equation to have a unique global solution (i.e.no explosion in a finite time) for any given initial value, the function involved with stochastic system are generally required to satisfy the linear growth condition and local Lipschitz condition [17,18].However, the function of system (1.2) do not satisfy the linear growth condition, though they are local Lipschitz condition, so for any given initial value (N(0), P(0), Z(0))∈R 3  + , there only a unique positive local solution (N(t), P(t), Z(t)) on t ∈ [0, τ e ) a.s., where τ e is the explosion time.To show this solution is global, we only to prove τ e → ∞ a.s..
To obtain the global positive solution of system (1.2) , we consider an auxiliary equation: clearly, we have dW (t) = dN(t) + dP(t) + dZ(t), using system (1.2) in the above expression, we obtain ) Now, applying the theorem of differential inequalities, we can get Theorem 2.1.For any initial value (N(0), P(0), Z(0))∈R 3 + , system (1.2) has a unique global positive solution (N(t), P(t), Z(t)) for all t > 0, and the solution will remain in R 3  + with probability one.
For each integer k ≥ k 0 , define the stopping time We set inf Φ = ∞ (Φ denotes the empty set).Obviously, τ k is increasing on k and τ k < τ e .
If the statement is false, then for any constant T > 0, there is an ε ∈ (0, 1) and an integer According to (2.2), we define a C 2 -function V (N, P, Z) by By Ito's formula, we have where Noting that Integrating and take the expectation of both sides, we have And set 3), we obtain P(Ω k ) ≥ ε.Noting that for every ω ∈ Ω k , there is at least one of P(τ k , ω), N(τ k , ω), Z(τ k , ω) equals 1 k , then Therefore we obtain that τ ∞ = ∞ a.s..This completes the proof.

Extinction and persistence in mean
In this section, we investigate persistence and extinction of system (1.2).For obtaining the main results, we need the following Definition and Lemma.≤ l < 0 a.s.; (2) P(t) is said to be weak persistent in mean if there exist a constant l > 0 such that lim sup t→∞ 1 t t 0 P(s)ds ≤ l < +∞ a.s.; (3) P(t) is said to be strong persistent in mean if there exist a constant l > 0 such that lim inf t→∞ [21]).Let (N(t), P(t), Z(t)) be the solution of system (1.2) with initial value (N(0), P(0), Z(t)) ∈ R 3 + .Then Theorem 3.1.Let (N(t), P(t), Z(t)) be the solution of system (1.2), then the following statements hold: , then P(t) and Z(t) are all extinct; (ii persistent in mean a.s., and Z(t) is weak persistent in mean a.s., where Proof.(i) By the positiveness of the solution (N(t), P(t), Z(t)) of system (1.2).
An application of It ô s formula to ln P(t) yields Integrating both sides of (3.1) from 0 to t yields where ) < 0.
which implies lim t→∞ P(t) = 0 a.s.. Then for any ε > 0, there exisit T > 0 and a set Ω ε such that An application of It ô s formula to the third equation of system (1.2), we have (3.4) Integrating both sides of (3.4) from 0 to t, and dividing by t on both sides, yields ln where M 3 (t) = t 0 σ 2 P (P 2 /i)+P+b dB 2 (s), analogously, by using of the Lemma 3.2, and combininbg with lim sup t→∞ ln Z(0) t = 0 a.s., and a random small constant ε such that c b ε − ε − D 2 < 0, we can get lim sup which implies lim t→∞ Z(t) = 0 a.s..
(ii) Combining with (2.2), and application of It ô s formula to ln P(t) yields Integrating both sides of (3.6) from 0 to t, and dividing by t on both sides, we have ln where t + ln P(0) t .Besides, by (2.1), we have Integrating both side of above inequation from 0 to t, we obtain According to system (1.2), (2.1)and (2.2), taking the limit superior of both side of (3.8), we have, for all t ≥ 0, D , so Z(t) is weak persistent in mean a.s., this completes the proof.

The asymptotic behavior of the positive equilibrium point
k+N * , then the corresponding deterministic system of (1.2) exist the interior equilibrium point E * = (N * , P * , Z * ), however, the point E * = (N * , P * , Z * ) is not the equilibrium point of system (1.2), therefore, it is necessary to consider the asymptotic behavior of stochastic system near the positive equilibrium point of deterministic system.Theorem 4.1.Suppose that (N(t), P(t), Z(t)) is any solution of system (1.2) with initial value (N(0), P(0), Z(0)), if the coefficients of system (1.2) k+N * and η 1 > 0, η 2 > 0, η 3 > 0, then the system will be weakly persistent in the mean under the expectation around the positive equilibrium point E * = (N * , P * , Z * ) in the long run, mathematically, it says that lim sup where Proof.Because of the condition of ε + D 2 < c, γ + D 1 < aN * k+N * , then the corresponding deterministic system of (1.2) exist the interior equilibrium point E * = (N * , P * , Z * ), and satisfy the following conditions Define a C 2 -function V : R 3 → R + by making use of (4.1), and the elementary inequalities we have Similarly, Consequently, we have by Young's inequality, we have where   This completes the proof.

Conclusion
In this paper, we consider a stochastic three-species nutrient-phytoplankton-zooplankton model.Firstly, We verify that the system admits a unique positive global solution starting from any positive initial value.Then we obtain the sufficient conditions of population extinction and persistence in the mean, the results show that if σ 1 > a √ 2(γ+D 1 ) , then P(t) and Z(t) are all extinct; if σ 1 , σ 2 are small enough such that DN 0 D > max{ Θ(k+C 1 ) a , b(ε+D 2 ) c }, then P(t) is strong persistent in mean and Z(t) is weak persistent in mean.Our results also imply that weaker white noise will strengthen stability of the system, while stronger white noise will result in extinction of population.Furthermore, we obtain the asymptotic behavior of stochastic system near the positive equilibrium point of deterministic system, that is, the stochastic system is weakly persistent in the mean under some conditions.