Stationary Distribution and Periodic Solution of Stochastic Toxin-Producing Phytoplankton–Zooplankton Systems

In this paper, we investigate the dynamics of autonomous and nonautonomous stochastic toxin-producing phytoplankton– zooplankton system. For the autonomous system, we establish the sucient conditions for the existence of the globally positive solution as well as the solution of population extinction and persistence in the mean. Furthermore, by constructing some suitable Lyapunov functions, we also prove that there exists a single stationary distribution which is ergodic, what is more important is that Lyapunov function does not depend on existence and stability of equilibrium. For the nonautonomous periodic system, we prove that there exists at least one nontrivial positive periodic solution according to the theory of Khasminskii. Finally, some numerical simulations are introduced to illustrate our theoretical results. e results show that weaker white noise and/or toxicity will strengthen the stability of system, while stronger white noise and/or toxicity will result in the extinction of one or two populations.


Introduction
As well known, mathematical models describing the plankton dynamics have played an important role in understanding the various mechanisms involved in toxin-producing phytoplankton. ere are many scienti c works have been carried out to investigate the e ects of toxin-producing phytoplankton on plankton ecosystems [1][2][3][4][5][6][7]. For example, Upadhyay and Craniopathy [1] proposed three species food chain model with di erent functional forms to describe the liberation of toxin. e obtained results show that the increase of toxic substances released by toxic-producing phytoplankton has a stabilizing e ect. In particular, according to eld observations, Chattopadhyay et al. [5] formulated the following toxin-producing phytoplankton-zooplankton model: where ( ) and ( ) denote the density of toxin-producing plankton (TPP) population and the zooplankton population at time , respectively, subject to the nonnegative initial condition (0) = 0 ≥ 0 and (0) = 0 ≥ 0. and represent the intrinsic growth rate and the environmental carrying capacity of TPP population, respectively. is the rate of predation of zooplankton on TPP population, is the ratio of biomass consumed by zooplankton for its growth (satisfying the obvious restriction 0 < < ), and denotes the mortality rate of zooplankton due to nature death, denotes the rate of toxin liberation by TPP population.
represents the predational response function and g describes the distribution of toxic substances. All parameters above are positive. As liberation of toxin reduces the growth of zooplankton, causes substantial mortality of zooplankton and in this period toxin-producing phytoplankton population is not easily accessible, hence a more common and intuitively obvious choice is of the saturation functional form to describe the grazing phenomena. For instance, Tapan et al. [8] studies the system (1) when = g = / + (where > 0 denotes the half-saturation constant). e obtained result indicates that there is a threshold limit of toxin liberation by the phytoplankton species below which the system does not have any excitable nature and above which the system shows excitability.
Clearly, these important and useful works on deterministic phytoplankton-zooplankton model provide a great insight into the dynamics of plankton ecosystems. However, in the real world, the dynamics of plankton ecosystems are inevitably perturbed by various types of environment noises. Development from the deterministic models to the stochastic models can give us new insights into the dynamics of plankton ecosystems [9]. May [10] pointed out that the birth rates, carrying capacity, competition coe cients and other parameters involved in the system can be a ected by environmental noise. is may be especially true for plankton ecosystems due to unpredictability of photosynthetically active radiation, nutrient availability, water temperature, water depth, light, eutrophication, acidity, salinity, wind and many other physical factors embedded in aquatic ecosystems. Several scholars have studied the e ect of environmental uctuations on aquatic ecosystems [11][12][13]. Indeed, stochastic models could be more appropriate way of modeling in comparison with their deterministic counterparts, since they can provide some additional degree of realism. By introducing (stochastic) environmental noise, many investigators have studied stochastic epidemic models [14][15][16][17][18][19][20][21][22][23] and stochastic population models [24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42]. ey focus on the e ect of environmental uctuations on the dynamic behavior of these models. For instance, Silva [11] investigated a stochastic model of phytoplankton-zooplankton interactions with toxin-producing phytoplankton. eoretical results show that for certain values of the system parameters, the system posses asymptotic stability around the positive interior equilibrium which depicts the coexistence of all the species. erefore, it is meaningful to further incorporate the environmental stochasticity into the model (1) with = g = / + , which could provide us a deeper understanding for real aquatic ecosystems [43]. On the other hand, periodic behavior arises naturally in many real world problems, such as in biological, environmental and economic systems [44]. e phenomenon of periodic oscillations has been observed in the growth of populations, such as, a seasonal occurrence of Hematodinium perezi was reported by Newman and Johnson (1975) in the parasitic dinoagellate in blue crabs on the east coast of the United States from late spring to early winter [45]. Seasonal changes are cyclic, largely predictable, and arguably represent the strongest and most ubiquitous source of external variation in uencing human and natural systems [46]. However, to the best of our knowledge, there is little work on the existence of stochastic periodic solution for nonautonomous toxin-producing phytoplankton zooplankton model. Based on the aforementioned, we intend to study toxin-producing phytoplankton-zooplankton model with environmental uctuations, and then we extend this model into a nonautonomous stochastic model by taking into account seasonal variation in the next section. en in Section 3, we show the existence and uniqueness of the global positive solution. In Section 4, we obtain the sucient conditions for the solution of population extinction and persistence in the mean. In Section 5 and Section 6, using Khasminskii's methods and Lyapunov functions, we derive su cient conditions for the existence of the single ergodic stationary distribution of the autonomous system (2) and the existence of the nontrivial positive stochastic periodic solution of nonautonomous system (3). In Section 7, some numerical simulations are provided to demonstrate the analytical ndings. Finally, some conclusions are given in Section 8.

Model Formulation
ere are many kinds of approaches to introduce the white noise into the population models. For model (1), we assume that the growth rate of phytoplankton and the death rate of zooplankton are subjected to the Gaussian white noise (we follow the way used in [38]), then we can obtain the following stochastic model: where 1 ( ), 2 ( ) are independent Brownian motions de ned on the complete probability space Ω, F, F ≥0 , with a ltration F ≥0 satisfying the usual conditions (i.e. it is right continuous and increasing while F 0 contains all P-null sets). 2 represent the intensities of the white noise. Meanwhile, there are evidences suggesting that the toxic substances released by TPP do not remain constant but change over time, which is related to the seasonal changes. erefore, for better understanding the toxin-producing phytoplankton-zooplankton sustained oscillatory patterns, we further consider the following periodic system with stochastic perturbation by the method of Khasminskii [47].

Existence and Uniqueness of the Global Positive Solution
As we know, in order for a stochastic di erential equation to have a single global solution (i.e. no explosion in a nite time) for any given initial value, the functions involved with stochastic system are generally required to satisfy the linear growth condition and local Lipschitz condition [48,49]. However, the functions of system (2) do not satisfy the linear growth condition, so the solution of system (2) may explode at a nite time.
In this section, we show that there exists a single positive local solution of system (2), then using the Lyapunov analysis method, we prove that this solution is global. Explanation for "explosion time" used in following lemma.
with the initial value (0) = ln 0 , v(0) = ln 0 . e functions involved with dri part of above stochastic di erential system satisfy the linear condition and locally Lipschitz condition.
is the solution of the following system: By the same arguments as above, we have Similarly, we can obtain )+ 2 2 ( ) is the single solution of the following system: (11), (12), (15), we can get that It follows from [34] that Ψ( ), Φ( ), ( ) and ( ) will not be exploded at any nite time, then by the comparison theorem of stochastic di erential equations, we can derive that ( ), ( ) will globally exist. is completes the proof.
Proof. e proof is similar to eorem 2, so we omit it.
Proof. e proof is similar to eorem 2, so we omit it.

Extinction and Persistence in Mean
In this section, we will investigate the persistence and extinction of the system (2) under certain conditions. We give the de nition and lemma which can be used for our main results.
Remark 8. From eorem 7, we can see, when 2 is a constant, if 2 1 < 2 , then the population ( ) will be persistent in mean, but when we choose 1 large enough such that 2 1 > 2 , then the population ( ) will go to extinct. From an ecological point of views, the intensity of white noise has a negative e ect on the survival of ( ) population, which imply that weaker white noise will strengthen the stability of the system, while stronger white noise will lead to population ( ) extinct.

Stationary Distribution and Ergodicity
In this section, we shall consider whether there exists a single stationary distribution of system (2), which means that the zooplankton population can persist and not die out.
Let ( ) be a regular time-homogeneous Markov process in + described by the following stochastic di erential equation e di usion matrix is de ned as follows Next, we shall introduce a lemma which guarantees the existence and uniqueness of a stationary distribution and ergodicity (see Khasminskii [47]).
Let be a given open set in the -dimensional Euclidean space and 2 denotes the class of functions in which are twice continuously di erentiable with respect to .
By the rst equation of system (2), we have By Lemma 6, for ∀ → 0, we have lim inf To verify condition (B.2), it is su cient to prove that exists a nonnegative 2 −function ( ) and a neighborhood such that for some positive constant , ( ) ≤ − for any ∈ \ .
System (2) can be written into the following form: Suppose the following condition (H) holds: Theorem 11. Assume the condition (H) holds and 2 > 0( = 1, 2), then for any initial value 0 , 0 ∈ 2 + , system (2) has a single stationary distribution and it has ergodic property.

Complexity
Theorem 15. Assume the following conditions hold: then for any initial value 0 , 0 ∈ 2 + , the system (3) has a positive 1 -periodic solution.
Proof. By the same way as in eorem 2, we can obtain that, for any initial value 0 , 0 ∈ 2 + , the system 3 has a single global positive solution. Next, we only need to verify the conditions (B.1), (B.2) of Lemma 14.
De ne a 2,1 -function at is, the condition (B.1) is satis ed. erefore, according to Lemma 9, we know that the system (2) has a single stationary distribution which is ergodic.
Remark 12. From eorem 11, we can see that system (2) exists a single stationary distribution provided that the e ects of both environment noise ( = 1, 2) and the rate of release of toxic substances are not too large such that > 0; e ergodic property re ects the solution of system (2) converges to the single stationary distribution.

The Existence of Periodic Solution of Nonautonomous System
For convenience, we denote where ( ) is a continuous 1 -periodic function.
In this section, we will recall a basic de nition and introduce a lemma which gives a criteria for the existence of a periodic Markov process (see Khasminskii [47]).
Let be a given open set in the d-dimensional Euclidean space . = × [0, ∞), 2,1 is the family functions on which are twice continuously di erentiable with respect to ∈ and continuously di erentiable with respect to ∈ [0, ∞).

Complexity 8
Using the above two inequations, 1 can be estimated as follows: where ℎ 1 = / 1 + / Hence (60) ln We can obtain the following result according to eorem 15.

Numerical Simulation
In this section, we present some numerical simulations to illustrate our theoretical results obtained in previous sections. To this end, based on the method mentioned in Higham [50], we consider the following discretization equations in view of Eq. (64). Consequently, we can deduce that us, the condition (B.2) of Lemma 14 is satis ed. By Lemma 14, the system (3) exists a periodic Markov process. is completes the proof.
Considering the corresponding deterministic system of system (3) Next we use di erent values of 1 , 2 to see the e ect of the noise strength on the dynamics of the system (2).
We choose the parameters by with the initial value 0 , 0 = (80, 35). with the initial value 0 , 0 = (8, 33). en we use di erent values of , 1 , 2 to explore how the environmental noise and toxicity a ect the dynamics of system (3).
(iii) We rstly adopt ( ) = 0.25 + 0.005sin and let 1 ( ) = 2 ( ) = 0. It is easy to verify that the condition of Corollary 16 holds and deterministic nonautonomous system (74) also exists a periodic solution, shown in the Figure 4(a), and the corresponding solutions of system (74) shown in the Figure 4(b). Next, we increase strengths of environmental forcing to 1 ( ) = 0.2 + 0.5 sin , 2 ( ) = 0.12 + 0.1 sin , then Lyapunov function does not depend on existence and stability of equilibrium. For the nonautonomous periodic system, we mainly study the existence of positive periodic solution.
e theoretical results and numerical simulations show that the population would extinct as the enhancing of noise and/or toxic intensity, while the reductive speed of the population would slow down as the weakening of noise and/or toxic intensity, and the population would be persistent. at is, toxic intensity and environmental uctuations have great in uence on plankton ecosystems. From eorems 7 and 11, we know that the survival of plankton can be signi cantly a ected by the white noise densities ( = 1, 2) and the release rate of toxic substances . at is, if 1 is su ciently large, the phytoplankton ( ) may su er the danger of extinction. On the contrary, if the intensities of environment noise ( = 1, 2) and the rate of release of toxic substances are not too large such that > 0, then the system (2) exists a single ergodic stationary distribution. e obtained results also implies that the TPP may provide a possible biological strategy to control the plankton ecosystems.
Data Availability e data used to support the ndings of this study are included within the article. (iv) Now, we select ( ) = 1.2 + 0.005 sin , 1 ( ) = 2 ( ) = 0, then the population ( ) will be extinct, shown in Figure 6. By comparing Figure 4 with Figure 6, one can observe that the stronger toxicity will result in the extinction of zooplankton population.

Conclusion
e eld studies and laboratory studies point out that the toxic substance plays one of the important role on the growth of the zooplankton population and have a great impact on phytoplankton-zooplankton intersection. It is well established that large number of phytoplankton species produce toxic, such as Gymnodinium breve, Gymodinium catenatum, Pyrodinium bahamense, P esteria piscicida, Chrysochromulina polylepis, Prymnesium patelliferum, Noctiluca scintillans and so on. Zooplankton species such as Paracalanus will be greatly a ected by harmful phytoplankton species [1][2][3][4][5]. In this paper, we propose autonomous and nonautonomous stochastic toxin-producing phytoplankton-zooplankton model with Holling II functional response and made an attempt to reveal the e ects of toxic intensity and environmental uctuations on the plankton ecosystems. For the autonomous system, we establish su cient conditions for the existence of the globally positive solution, and obtain su cient conditions for the solution of population extinction and persistence in the mean. Furthermore, by using Khasminskii's method and technique of Lyapunov functions, we also prove that there exists a single ergodic stationary distribution, what is important is that