Abstract
In this paper, we investigate the dynamics of autonomous and nonautonomous stochastic toxin-producing phytoplankton–zooplankton system. For the autonomous system, we establish the sufficient 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. The 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.
1. 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. There are many scientific works have been carried out to investigate the effects of toxin-producing phytoplankton on plankton ecosystems [1–7]. For example, Upadhyay and Craniopathy [1] proposed three species food chain model with different functional forms to describe the liberation of toxin. The obtained results show that the increase of toxic substances released by toxic-producing phytoplankton has a stabilizing effect. In particular, according to field 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 and . 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 ), 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 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 (where denotes the half-saturation constant). The 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 coefficients and other parameters involved in the system can be affected by environmental noise. This 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 effect of environmental fluctuations on aquatic ecosystems [11–13]. 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 [14–23] and stochastic population models [24–42]. They focus on the effect of environmental fluctuations on the dynamic behavior of these models. For instance, Silva [11] investigated a stochastic model of phytoplankton–zooplankton interactions with toxin-producing phytoplankton. Theoretical 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. Therefore, it is meaningful to further incorporate the environmental stochasticity into the model (1) with , 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]. The 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 dinoflagellate 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 influencing 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 fluctuations, and then we extend this model into a nonautonomous stochastic model by taking into account seasonal variation in the next section. Then in Section 3, we show the existence and uniqueness of the global positive solution. In Section 4, we obtain the sufficient 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 sufficient 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 findings. Finally, some conclusions are given in Section 8.
2. Model Formulation
There 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 are independent Brownian motions defined on the complete probability space. with a filtration satisfying the usual conditions (i.e. it is right continuous and increasing while contains all P-null sets). 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. Therefore, 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].
where are all positive -periodic continuous functions. For biological significance, we always assume that . Throughout this paper, let , a.s. means almost surely.
3. Existence and Uniqueness of the Global Positive Solution
As we know, in order for a stochastic differential equation to have a single global solution (i.e. no explosion in a finite 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 finite 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.
Lemma 1. For , there exists a single positive local solution of system (2) for almost surely, where is the explosion time.
Proof. Using the transformation of variables and applying Itô’s formula, for system (2), we have.
with the initial value . The functions involved with drift part of above stochastic differential system satisfy the linear condition and locally Lipschitz condition. Hence there exists a single local solution , for , where is any finite positive real number. Clearly, is the single positive local solution of stochastic differential system (2) starting from an interior point of the first quadrant.
Now we are in a position to show that this single solution is a global solution. To prove this we only need to show that a.s.
Theorem 2. For any initial value , the system (2) has a single global positive solution for all , and the solution will remain in with probability one.
Proof. By Lemma 1, we only need to prove that a.s. From the first equation of model (2), we have
Let
then is the single solution of the following auxiliary system:
Then by the comparison theorem for stochastic equations, we have
On the other hand, from the second equation of model (2), we have
Consider the following auxiliary system:
Then is the single solution of above system. Similarly by the comparison theorem for stochastic differential equations, we have
Similarly, we have
where is the solution of the following system:
By the same arguments as above, we have
Similarly, we can obtain
where is the single solution of the following system:
By (8), (11), (12), (15), we can get that
It follows from [34] that and will not be exploded at any finite time, then by the comparison theorem of stochastic differential equations, we can derive that will globally exist. This completes the proof.
Remark 3. For any initial value , the system (3) has a single global positive solution , for all , and the solution will remain in with probability one.
Proof. The proof is similar to Theorem 2, so we omit it.
Remark 4. When we choose , then the stochastic system (2) has a single global positive solution for all , and the solution will remain in with probability one.
Proof. The proof is similar to Theorem 2, so we omit it.
4. 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 definition and lemma which can be used for our main results.
Definition 5 [38]. (1) System (2) is said to be extinct if (2) System (2) is said to be persistent in mean if
Let be the family functions on which are continuously differentiable with respect to and continuously differentiable with respect to .
Lemma 6 [35]. Suppose that , if there are positive constant such that for .(1), then a.s.;(2), then a.s.,where is a constant.
Theorem 7. Let be the solution of the system (2), then the following statements hold:(i)If , then a.s.;(ii)If , then a.s.
Proof. (i) By the positiveness of the solution of system (2), we have
Consider the following system
We have
By the comparison principle of the stochastic differential equation and the theory of diffusion process (see [33] and Lemma A.2 in [34]), we can easily check that
Since , we can obtain that , that is a.s. For any , there exisit and a set such that and for .
By the first equation of system (2), we have
By Lemma 6, for , we have , and , that is to say(ii) Similar to the proof of (i), we obtain that a.s. for . In addition, by the first equation of system (2) and , we obtain . This completes the proof.
Remark 8. From Theorem 7, we can see, when is a constant, if , then the population will be persistent in mean, but when we choose large enough such that , then the population will go to extinct. From an ecological point of views, the intensity of white noise has a negative effect 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.
5. 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 differential equation
The diffusion matrix is defined 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 denotes the class of functions in which are twice continuously differentiable with respect to .
Lemma 9. The Markov process has a unique stationary distribution , if there exists a bounded domain with regular boundary such that its closure , having the following properties:(B.1) In the open domain and some neighborhood thereof, the smallest eigenvalue of the diffusion matrix is bounded away from zero.(B.2) If , the mean time at which a path issuing from reaches the set is finite, and for every compact subset . Moreover, if is a function integrable with respect to the measure , then
Remark 10. To prove condition (B.1), it suffices to verify that is uniformly elliptical in , where , there exists a positive number such that . To verify condition (B.2), it is sufficient to prove that exists a nonnegative 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 , then for any initial value , system (2) has a single stationary distribution and it has ergodic property.
Proof. By Theorem 2, we have obtained that for any initial value , the system (2) has a single global positive solution .
The following proof is motivated by Liu et al. [38], if we substitute into system (2) and then by Itô’s formula, we get
Define a -function
where , q is a constant satisfying , , . Clearly, .
An application of the operator to and leads to
and
Using the above two inequations, can be estimated as follows
where .
where . Hence
To confirm the condition (B.2) of Lemma 9, we consider the bounded open subset
where is a sufficiently small number. In the set , let us choose sufficiently small such that
For convenience, we divide into four domains,
Clearly, . Next, we will prove for any .
Case 1. In domain , owing to , then , we have
since , combing with the definition of and Eqs. (37) and (38), we get
Case 2. Similarly, for any , owing to , we have
by Eqs. (37) and (39), we also obtain
Case 3. In domain we have
where , which gives in view of Eq. (40).
Case 4. In domain we have
where , which gives according to Eq. (41). Consequently, we can deduce that
That is, the condition (B.2) holds.
On the other hand, there exists such that
That is, the condition (B.1) is satisfied. Therefore, according to Lemma 9, we know that the system (2) has a single stationary distribution which is ergodic.
Remark 12. From Theorem 11, we can see that system (2) exists a single stationary distribution provided that the effects of both environment noise and the rate of release of toxic substances are not too large such that ; The ergodic property reflects the solution of system (2) converges to the single stationary distribution.
6. The Existence of Periodic Solution of Nonautonomous System
For convenience, we denote
where is a continuous -periodic function.
In this section, we will recall a basic definition and introduce a lemma which gives a criteria for the existence of a periodic Markov process (see Khasminskii [47]).
Definition 13 [47]. A stochastic process is said to be -periodic if for every finite sequence of numbers , the joint distribution of random variables is independent of .
Consider the integral equation
where are continuous functions of and for some constant , the following conditions hold.
Let be a given open set in the d-dimensional Euclidean space . , is the family functions on which are twice continuously differentiable with respect to and continuously differentiable with respect to .
Lemma 14 [47]. Suppose that the coefficients of Equation (52) are -periodic in and satisfy the conditions Equation (53) in every cylinder , and assume further there exists a function , which is -periodic in and satisfies:(B.1) (B.2) outside some compact set,then the system (52) exists at least a -periodic Markov process.
Theorem 15. Assume the following conditions hold:(H.1) (H.2) then for any initial value , the system (3) has a positive -periodic solution.
Proof. By the same way as in Theorem 2, we can obtain that, for any initial value , the system 3 has a single global positive solution. Next, we only need to verify the conditions (B.1), (B.2) of Lemma 14.
Define a -function by
where , is a constant, , . Clearly, . Here, let
which are defined by . It is not difficult to prove that are all -periodic functions, and is a bounded function. That is, there exists such that
In order to confirm the condition (B.1) of Lemma 14, it is sufficient to verify that
where , which is clearly established since the coefficients of the term of are all positive. Next, we only verify the condition (B.2) of Lemma 14.
By Ito’s formula, we have
and
Using the above two inequations, can be estimated as follows:
where , and
where .
Hence
To confirm the condition (B.2) of Lemma 14, we choose small enough such that
Consider the bounded open subset
Denote
Clearly, . Next, we will prove for any , respectively.
Case 1. In domain , owing to , we have
since , combined with the definition of and Eq. (63), we get
Case 2. Similarly, for any , owing to , we have
by Eq. (63), we also obtain
Case 3. In domain we have
where , which gives in domain in view of Eq. (64).
Case 4. In domain we have
where , which gives in domain in view of Eq. (64). Consequently, we can deduce that
Thus, the condition (B.2) of Lemma 14 is satisfied. By Lemma 14, the system (3) exists a periodic Markov process. This completes the proof.
Considering the corresponding deterministic system of system (3)
We can obtain the following result according to Theorem 15.
Corollary 16. Assume are all positive -periodic continuous functions and , then for any initial value, the system (74) exists a positive -periodic solution.
7. 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
where and are independent Gaussian random variables which follow standard normal distribution .
We choose the parameters by
with the initial value .
Next we use different values of to see the effect of the noise strength on the dynamics of the system (2).(i)Fix , and let vary to see the effect of noise on the dynamics of the system (2). We firstly take , then , by Theorem 7 (i) the population will be persistent in mean and population will be extinct, see Figure 1; Now let vary from to , then , by Theorem 7 (ii) the population and will be extinct, see Figure 2. From Figure 1 and Figure 2, we can see that under small noise, system (2) preserves some stability and large noise may lead to the extinction of population.(ii)Let , then , by Theorem 11, the system (2) exists a single ergodic stationary distribution, see Figure 3. This implies that system (2) still keep some stability and exhibits oscillation around the interior equilibrium of the corresponding deterministic system.
(a)
(b)
(c)
(d)
Next, we choose the 2 -periodic functions
with the initial value . Then we use different values of to explore how the environmental noise and toxicity affect the dynamics of system (3).(iii)We firstly adopt and let . 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 , then the conditions (H.1), (H.2) hold. By Theorem 15, we know the system (3) exists a positive 2 -periodic Markov process, shown in the Figure 5(a), and the corresponding solutions of system (3) shown in the Figure 5(b).(iv)Now, we select , 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.
(a)
(b)
(a)
(b)
(a)
(b)
8. Conclusion
The field 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, Pfiesteria piscicida, Chrysochromulina polylepis, Prymnesium patelliferum, Noctiluca scintillans and so on. Zooplankton species such as Paracalanus will be greatly affected by harmful phytoplankton species [1–5]. 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 effects of toxic intensity and environmental fluctuations on the plankton ecosystems. For the autonomous system, we establish sufficient conditions for the existence of the globally positive solution, and obtain sufficient 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 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.
The 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. That is, toxic intensity and environmental fluctuations have great influence on plankton ecosystems. From Theorems 7 and 5.3, we know that the survival of plankton can be significantly affected by the white noise densities and the release rate of toxic substances . That is, if is sufficiently large, the phytoplankton may suffer the danger of extinction. On the contrary, if the intensities of environment noise and the rate of release of toxic substances are not too large such that , then the system (2) exists a single ergodic stationary distribution. The obtained results also implies that the TPP may provide a possible biological strategy to control the plankton ecosystems.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work is supported by Fujian Provincial Natural Science of China (2018J01418) and the Natural Science Foundation of China (11301216).