The complex dynamics of a stochastic toxic-phytoplankton-zooplankton model
© Rao; licensee Springer. 2014
Received: 21 August 2013
Accepted: 11 December 2013
Published: 16 January 2014
In this paper, an analytical study of a toxin-producing phytoplankton-zooplankton model with stochastic perturbation is performed. By constructing suitable Lyapunov functions, we investigate the global stability of a positive equilibrium and give the condition of the existence of Hopf bifurcation for the deterministic plankton model. Under the perturbation of environmental noise, there is a globally positive solution to the stochastic model and it is stochastically ultimately bounded. In addition, the stochastic model is stochastically permanent under some conditions. A series of numerical simulations to illustrate these mathematical findings are presented.
Phytoplankton are primary producers as the base of the aquatic food web, floating freely near the surfaces of all aquatic environments. Zooplankton are the animals in the plankton community and feed on other phytoplankton . Phytoplankton and zooplankton are subject to water movements to a large extent. They act as the basis of all food chains and webs in aquatic systems and play an important role in the ecology of the ocean [2, 3]. In the last two decades, there has been a global increase in harmful plankton blooms in aquatic ecosystem [4–6]. Bandyopadhyay et al.  pointed out that a broad classification of harmful plankton species distinguishes two groups: one is the toxin producers, which can contaminate seafood or kill fish, and the other is the high-biomass producers, which can cause anoxia and indiscriminate mortalities of marine life after reaching dense concentrations. The toxin-producing phytoplankton play an important role on the growth of the zooplankton population. And because of phytoplankton and zooplankton universal existence and importance, understanding of the dynamical behaviors of interacting species will continue to be a predominant topic. In recent years, a great deal of attention has been paid towards in toxin-producing plankton blooms and a lot of its extensions from several researchers [2, 3, 7–14].
In the real world, population dynamics is inevitably subjected to environmental noise, which is an important component in an ecosystem. Most natural phenomena do not follow strictly deterministic laws, but rather oscillate randomly about some average values. So that the population density never attains a fixed value with the advancement of time [15, 16]. The basic mechanism and factors of population growth like resources and vital rates - birth, death, immigration, and emigration - change non-deterministically due to continuous fluctuations in the environment (e.g. variation in intensity of sunlight, temperature, water level, etc.) . Recent advances in stochastic differential equations enable a lot of authors to introduce randomness into a deterministic model of physical phenomena to reveal the effect of environmental variability, whether it is a random noise in the system of differential equations or environmental fluctuations in parameters, see, e.g. [18–26]. Of them, Beddington and May  studied harvesting natural populations in a randomly fluctuating environment. In , Mao showed that the noise cannot only have a destabilising effect but can also have a stabilising effect in the control theory. The growth of populations in a random environment subjected to variable effort fishing policies was studied in . Braumann  generalized the previous results  to density-dependent positive noise intensities of general form so that they also become independent from the way environmental fluctuations affect population growth rates. Liu et al.  presented a spatial version of the phytoplankton-zooplankton model that includes some important factors such as external periodic forces, noise and diffusion processes. These important results reveal the significant effects of the environmental noise on some models.
To our knowledge, a toxin-producing phytoplankton-zooplankton model with a Holling type-II functional response has deserved a lot of attention, but mainly in deterministic case. The research on the dynamical behavior of the toxic-phytoplankton-zooplankton model with Holling type-II functional response under environmental noise seems rare. Based on the discussion above, in this paper, we focus on dynamical properties of a toxin-producing plankton model with Holling type-II functional response under stochastic perturbation. The organization of this paper is as follows. In the next section, a stochastic toxic-phytoplankton-zooplankton model is established. Firstly, we give a general survey of the stability analysis of a positive equilibrium of the model without noise. Then, we concentrate our attention on the stochastic version of the toxin producing phytoplankton-zooplankton model and discuss the existence of global positive solution, stochastic boundedness, and the global asymptotic stability of the stochastic model. In Section 3, we give some numerical examples and make a comparative analysis of the stability of the model system within deterministic and stochastic environments. Finally, we give a concluding remark section.
2 Model and dynamical analysis
In the above model, is the density of toxin producing phytoplankton population and is the density of zooplankton population at any instant of time t. represents the functional response for the grazing of phytoplankton by zooplankton and descries the distribution of toxin substance which ultimately contributes to the death of zooplankton populations.
where and are known as the intensities of environmental noise. () is a standard white noise, that is, () independent Brownian motion defined in a complete probability space with a filtration satisfying the usual conditions (i.e., it is right continuous and increasing while contains all P-null sets) . Parameters K, m, a, c, d are positive constants, where K is the environmental carrying capacity of toxin-producing phytoplankton population, m the half saturation constant for a Holling type-II functional response, a the maximum uptake rate for zooplankton species, c the rate of toxic substances produced by per unit biomass of phytoplankton and d the natural death rate of zooplankton.
2.1 Dynamical analysis of model (1)
(total extinct) is a saddle point.
(extinct of the zooplankton, or phytoplankton-only) is a stable node if and a saddle point if .
- (iii)(coexistence of the phytoplankton and zooplankton) is a positive interior equilibrium, where
with and .
By the Roth-Hurwitz criterion, we have a sufficient condition for the local stability of of model (1).
Theorem 2.1 If holds, then the positive equilibrium of model (1) is locally asymptotically stable.
Theorem 2.2 If and hold, then the positive equilibrium of model (1) is globally asymptotically stable in the interior of the first octant.
If the conditions and hold, then in , where the equality holds only at the equilibrium point . Hence the equilibrium is globally asymptotically stable. □
Referring to , we study the Hopf bifurcation of model (1), which is space-independent. It breaks the temporal symmetry of a system and gives rise to oscillations that are uniform in space and periodic in time. Here, taking K as the bifurcation parameter, we have the following theorem.
Theorem 2.3 If holds, then model (1) undergoes Hopf bifurcation around the positive equilibrium point . The Hopf bifurcation occurs at its critical value .
when exists, the characteristic equation is , whose roots are purely imaginary;
Therefore, all the conditions of Hopf bifurcation theorem are satisfied, thus there exists a small amplitude periodic solution near . This completes the proof. □
2.2 Dynamical analysis of model (2)
and the solution () is an Itô process.
For a stochastic differential equation, in order to have a unique global solution (that is, no explosion in a finite time) for any given initial value, the coefficients of the equation are generally required to satisfy the linear growth condition and local Lipschitz condition . To show that model (2) has a positive global solution, let us firstly prove that the model has a positive local solution by making a change of variables.
Theorem 2.4 There is a unique local solution for to model (2) almost surely for the initial value , , where is the explosion time.
on . The coefficients of (6) satisfy the local Lipschitz condition and there is a unique local solution on . By Itô’s formula, we can see that , is the unique positive local solution to model (2) with initial value , . □
In the following, we show this solution is global, i.e., .
Theorem 2.5 For model (2) and any given initial value , there is a unique solution on and the solution will remain in almost surely.
where we set (∅ represents the empty set). is increasing as .
where is the indicator function of . Letting leads to the contradiction . This completes the proof. □
From Theorem 2.5, model (2) has a positive global solution. By constructing some Lyapunov functions, we analyze the stability of the positive equilibrium of the stochastic model.
Proof By the stability theory of stochastic differential equations , we only need to establish a Lyapunov function satisfying and the identity holds if and only if , where is the solution of the stochastic differential equation (5) and is the equilibrium position of (5).
then we see that the above inequality implies along all trajectories in the first quadrant except . Hence, the theorem holds. □
This theorem shows that when environmental noises satisfy some conditions, the unique solution of model (2) in is a stochastically asymptotically stable. That’s to say, both species under the effect of noises can coexist in stable conditions and eventually tend to the equilibrium sate.
Theorem 2.5 shows that the solution to model (2) will remain in . The property lets us continue to discuss how the solution varies in in more detail. We first present the definition of stochastic ultimate boundedness [30, 31], one of the important topics in population dynamics and defined as follows.
Theorem 2.7 The solution of model (2) is stochastically ultimately bounded for any initial value .
and we have the required assertion by taking the Chebyshev inequality. □
Generally speaking, the non-explosion property, the existence and the uniqueness of the solution are not enough, but the property of permanence is more desirable since it means long time survival in a population dynamics. Now, the definition of stochastic permanence [30–32] will be given below.
The proof is complete. □
Based on the results of Theorems 2.7, 2.8 and the Chebyshev inequality, we can obtain the following theorem.
Theorem 2.9 Assume that , the solution of model (2) is stochastically permanent.
In the above discussion, we show that under certain conditions, the stochastic model (2) is ultimately bounded and stochastically permanent. Here, we will show that if the noise is sufficiently large, the solution to the stochastic model will become extinct. In other words, the following theorem reveals the important fact that the environmental noise may make the population extinct.
as required. □
Corollary 2.1 Assume and hold, then for any initial value , the solution to model (2) will be extinct.
3 Numerical analysis
In order to facilitate the interpretation of our mathematical results in the stochastic model (2), we proceed to investigate them by numerical simulations.
We note that, if a positive equilibrium of the deterministic model is globally stable, then the stochastic model preserves the property of the stochastic asymptotical stability when noise is not sufficiently large. In this case, we can ignore the noise and use the deterministic model to describe the population dynamics. However, when the intensity of noise is sufficiently large, the noise can force the population to give rise to drastically ruleless oscillations and to become extinct. In this case, we cannot ignore the effect of noise and only use the stochastic model to describe the population dynamics.
4 Conclusions and remarks
In reality, the varying environment is always fraught with all kinds of randomness, but the knowledge as regards the effects of environmental noise on the toxin-producing phytoplankton population and zooplankton population is limited. In this paper, we consider a stochastic toxic-phytoplankton-zooplankton model with Holling type-II functional response. The value of this study lies in two aspects. First, stability criteria of model (1) are analyzed both from a local and a global point of view. The existence of a Hopf bifurcation around an interior equilibrium is established. Second, it presents the complex dynamics of the plankton model (2) with the effect of environmental noise.
For the toxin-producing phytoplankton-zooplankton model (1), by analyzing the corresponding characteristic equation and constructing a Lyapunov function, we study both the local and the global stability of a positive equilibrium. Taking the carrying capacity K as the bifurcation parameter, when K crosses a threshold value the toxin-producing plankton model enters into a Hopf bifurcation and has a periodic orbit around the coexisting equilibrium .
In order to study the stochastic model (2), we perturb the deterministic plankton model with respect to environmental noise around the growth rates of toxic-phytoplankton and zooplankton. Applying a Lyapunov function, we show that there is a unique positive solution to the model for any positive initial value. By Itô’s formula, we derive that the solution is stochastically bounded and stochastically permanent under some conditions. These conditions depend on the intensities of noise, and . When the intensities of noise satisfy some conditions and are not sufficiently large, the population of the stochastic model may be stochastically permanent. Our complete analysis of the model will give some suggestions for the studies on the population dynamics of other models.
The author drafted the manuscript, read and approved the final manuscript.
The author would like to express her gratitude to the editor and referees for their careful reading of the manuscript and a number of excellent criticisms and suggestions.
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