- Research
- Open Access
Nonlinear dynamics of a marine phytoplankton-zooplankton system
- Pengfei Wang^{1, 2},
- Min Zhao^{2, 3}Email author,
- Hengguo Yu^{2, 3},
- Chuanjun Dai^{2, 3},
- Nan Wang^{1, 2} and
- Beibei Wang^{1, 2}
https://doi.org/10.1186/s13662-016-0935-y
© Wang et al. 2016
- Received: 15 July 2015
- Accepted: 8 August 2016
- Published: 23 August 2016
Abstract
In the paper, the nonlinear dynamics of a marine phytoplankton-zooplankton system is discussed in detail. The aim is to deeply expound how to use the dynamical laws of marine ecological system to perfectly reflect the operating rules of marine ecosystems. The mathematical work has been pursuing the investigation of some conditions for the local and global stability of the equilibrium, as well as Turing instability, which can deduce a key parameter control relationship and in turn can provide a theoretical basis for the numerical simulation. The simulation work indicates that the theoretical results are correct, and the nonlinear dynamics of the marine system mainly depend on some critical parameters. The results further suggested that the diffusion and the environmental carrying capacity have an important role to play in the interaction between phytoplankton and zooplankton. All these results are expected to be of use in the study of the internal operating rules and characteristics of marine ecosystems.
Keywords
- dynamics
- pattern formation
- reaction-diffusion
- stability
1 Introduction
It is well known that plankton systems are abundant in marine ecosystems, where they comprise phytoplankton and zooplankton. Phytoplankton are primary producers in the marine ecosystem, where they form the basis of the food chain. Phytoplankton systems also play very important roles in chemical cycles and energy conversion marine ecosystems. Zooplankton can control the amount of phytoplankton via their consumption, where the zooplankton systems then provide a resource that is utilized by consumers at higher trophic levels. Thus, plankton research is highly significant for fishery production and basic theoretical marine science, further the importance of plankton for ecosystems is gradually recognized by contemporary researchers [1].
In marine ecosystems, animal populations often migrate to other habitats to search for food, widen their gene pool, or extend their territory. Factors such as latitude, salinity, ocean currents, and light can also affect their distribution. Thus, the dynamic behaviors present in the trophic interactions of aquatic ecosystems are highly complex. However, it is difficult to describe the dynamics of plankton ecosystems. In recent years, major progress has been made in the mathematical modeling of ecosystems and many studies have described the different dynamical behaviors of marine ecosystems [2–5]. In general, these systems are all predator-prey systems. Some previous studies have provided detailed analyses of these predator-prey systems [6–10]. There have been numerous investigations of the dynamics of phytoplankton-zooplankton models [11–13]. For example, Steele [13] considered the interactions among phytoplankton and herbivore. References [12, 14–16] incorporated nutrients into the interactions among plankton. The toxic substances produced by phytoplankton have a major impact on phytoplankton-zooplankton interactions, there have been several analyses of toxic-phytoplankton systems [17–21]. In [22], a mathematical system was proposed to describe the interaction between non-toxic and toxic phytoplankton in the presence of a single nutrient. In many ways research into plankton is still highly active.
Mathematical systems and methods are often used to study ecological problems [6, 7, 23, 24]. However, some of the mathematical systems used to describe ecological phenomena cannot be applied to planktonic systems. Thus, we need to select specific functions and functional responses to build a model of a planktonic system. In 1952, Turing [25] proposed the reaction-diffusion equation and described the concept of Turing instability. The importance of pattern formation was studied subsequently by Segal and Jackson [26]. Reaction-diffusion processes are now very important for describing the dynamic behavior of phytoplankton-zooplankton systems. In 1965, Holling [27] proposed three types of biological functional responses, called Holling types I, II, and III. In [28], analysis was provided of reaction-diffusion toxin-phytoplankton-zooplankton systems with either Holling-type II or Holling-type III functional responses. In [29], an analysis was presented of a predator-prey system with a Holling-type IV functional response. In 1965, Ivlev [30] proposed the Ivlev-type functional response. Subsequent studies also considered a grazing function as Ivlev-type function in a prey-predator system [31, 32].
The equilibrium states of system (1.2) are the same as those of the homogeneous system (1.1).
Based on the above considerations, the remainder of this paper is organized as follows. In Section 2, we study the equilibrium of the system and the stability of the system. In Section 3, we consider the system with diffusion. We derive the criteria for diffusive stability and the Turing instability region of the system. Finally, simulations and discussions are presented.
2 Equilibrium and the stability of system (1.1)
2.1 Equilibrium of the system
- (i)
If \(K_{1}>m\), \(\frac{r_{1} m}{\beta_{1}}\leq K_{2} \leq\frac{r_{1} r_{2} (K_{1}+m)^{3}}{4K_{1} \beta_{1} [r_{2} (K_{1}+m)+\beta_{2} (K_{1}-m)]}\), then there exist two positive equilibria \(E_{e1}=(P_{e1}, Z_{e1})\) and \(E_{e2}=(P_{e2},Z_{e2})\) in system (1.1), where \(P_{e1}<\frac {K_{1}-m}{2}\), \(P_{e2}>\frac{K_{1}-m}{2}\).
- (ii)
If \(K_{1}>m\), \(\frac{r_{1} r_{2} (K_{1}+m)^{3}}{4K_{1} \beta_{1} [r_{2} (K_{1}+m)+\beta_{2} (K_{1}-m)]}< K_{2}<\frac{r_{1} m}{\beta_{1}}\), then there exists one positive equilibrium \(E_{e3}=(P_{e3}, Z_{e3})\) in system (1.1), where \(P_{e3}<\frac{K_{1}-m}{2}\).
- (iii)
If ① \(K_{1}>m\), \(K_{2}<\min\{\frac{r_{1} m}{\beta _{1}},\frac{r_{1} r_{2} (K_{1}+m)^{3}}{4K_{1} \beta_{1} [r_{2} (K_{1}+m)+\beta_{2} (K_{1}-m)]}\}\), or ② \(K_{1}< m\), \(K_{2}<\frac{r_{1} m}{\beta _{1}} \), then there at least exists one positive equilibrium \(E_{e4}=(P_{e4},Z_{e4})\) in system (1.1), where \(P_{e4}>\frac {K_{1}-m}{2}\).
2.2 Stability of the equilibrium
It is obvious that \(E_{0}=(0,0)\) is unstable and \(E_{1}=(K_{1},0)\) is a saddle point. \(E_{2}=(0,K_{2})\) is a saddle point if \(r_{1}>\frac{\beta_{1} K_{2}}{m}\), and is locally asymptotically stable when \(r_{1}<\frac{\beta _{1} K_{2}}{m}\).
Lemma 2.1
If there exists one positive equilibrium \(E_{e4}=(P_{e4},Z_{e4})\) in system (1.1) and \(P_{e4}>\frac{K_{1}-m}{2}\), then the equilibrium is always locally asymptotically stable.
Lemma 2.2
If there exist two positive equilibria \(E_{e1}=(P_{e1},Z_{e1})\) and \(E_{e2}=(P_{e2},Z_{e2})\) in system (1.1), we find that \(E_{e2}=(P_{e2},Z_{e2})\) is always locally asymptotically stable, and \(E_{e1}=(P_{e1},Z_{e1})\) is a saddle point if \(K_{2}\leq m\).
Proof
Define \(F(P)=-\frac{r_{1}}{K_{1} \beta _{1}}[P^{2}+(m-K_{1})P-K_{1} m]-[K_{2}+\frac{\beta_{2} K_{2} P}{r_{2} (m+P)}]\) where \(0< P<\frac{K_{1}-m}{2}\). By combining this with the conditions in (i) we find that \(F'(P)>0\) when \(0< P<\frac{K_{1}-m}{2}\).
Hence, \(E_{e1}=(P_{e1},Z_{e1})\) is a saddle point if \(K_{2}\leq m \). □
Lemma 2.3
If there exists one positive equilibrium \(E_{e3}=(P_{e3},Z_{e3})\) for system (1.1) and \(P_{e3}<\frac {K_{1}-m}{2}\), then the equilibrium \(E_{e3}=(P_{e3},Z_{e3})\) is locally asymptotically stable when \(K_{2}\geq m\) and \(\frac{r_{1}}{r_{2}+\beta _{2}}<\frac{K_{1}}{K_{1}-m}\), or \(K_{2}\geq m\) and \((r_{1} K_{1}-r_{1} m-r_{2}K_{1}-K_{1} \beta_{2})^{2}-8r_{1} r_{2} K_{1} m<0\).
Proof
Hence, according to the above conditions, \(E_{e3}=(P_{e3},Z_{e3})\) is locally asymptotically stable. □
Theorem 2.1
The positive equilibrium \(E_{e4}=(P_{e4},Z_{e4})\) (\(P_{e4}>\frac{K_{1}-m}{2}\)) is globally asymptotically stable in the presence and the absence of diffusion if \(0< K_{1}\leq\frac{r_{1} m(m+P_{e4})}{\beta_{1} Z_{e4}}\) holds.
Proof
3 Stability and instability of the system with diffusion
In this section, we consider how the diffusion affects the stability of system (1.2). The stability and instability of system (1.2) are considered as follows.
Theorem 3.1
The positive equilibrium \(E_{e2}=(P_{e2},Z_{e2})\) in system (1.2) is uniformly asymptotically stable in the presence of diffusion.
Proof
Thus, a negative constant C exists, which is independent of i, such that \(\operatorname{Re}(\lambda^{\pm}_{i})< C\) for any i. By referring to [35], we find that the positive equilibrium \(E_{e2}=(P_{e2},Z_{e2})\) of system (1.2) is uniformly asymptotically stable in the presence of diffusion. The proof is complete. □
Theorem 3.2
If there exists one positive equilibrium \(E_{e4}=(P_{e4},Z_{e4})\) in system (1.2) and \(P_{e4}>\frac {K_{1}-m}{2}\), then \(E_{e4}=(P_{e4},Z_{e4})\) is globally asymptotically stable in the presence of diffusion if \(0< K_{1}\leq\frac{r_{1} m(m+P_{e4})}{\beta_{1} Z_{e4}}\) holds.
Proof
This completes the proof. □
Next, we analyze the formation of patterns. Based on the analysis above, we know that the positive equilibrium point \(E_{e3}=(P_{e3},Z_{e3})\) of system (1.2) is locally asymptotically stable when \(K_{2}\geq m\) and \(\frac{r_{1}}{r_{2}+\beta_{2}}<\frac {K_{1}}{K_{1}-m}\), or \(K_{2}\geq m\) and \((r_{1} K_{1}-r_{1} m-r_{2} K_{1}-K_{1} \beta _{2})^{2}-8r_{1} r_{2} K_{1} m<0\) holds. In this case, we can reach the following conclusion.
Theorem 3.3
Proof
In Theorem 3.3, we found that (3.7) is a sufficient condition for the diffusive instability of system (1.2). If criterion (3.7) is satisfied, the diffusion of the plankton population can cause the system to shift from a stable state to an unstable state. In ecological terms, this means that if the zooplankton diffusion rate is higher than the phytoplankton diffusion rate and (3.7) is satisfied, then the instability will continue to prevail. At this time point, we can use the system parameters to obtain the patterns of the plankton populations. The patterns of the plankton populations are considered in Section 4.
4 Numerical results
4.1 Numerical analysis
In this section, we will explore the carrying capacity of zooplankton \(K_{2}\) and the half saturation constant m and the influence on the steady-state transformation problem of the proposed system (1.1) and (1.2) using a numerical simulation analysis; thus, some parameters are fixed as follows: \(r_{1}=0.8\), \(K_{1}=1\), \(\beta_{1}=0.8\), \(\beta_{2}=0.7\), \(r_{2}=0.7\).
4.2 Numerical simulations
In order to further expound the environmental carrying capacity \(K_{2}\) and the diffusion \(D_{1}\) affecting the spatial distributions of phytoplankton population, some kinds of patterns will be given using the numerical simulations. Here, we perform extensive numerical simulations of system (1.2) in two dimensional space with the same parameter values as given in Figure 1B. Meanwhile, the value range of the parameters \(K_{2}\) and \(D_{1}\) can be chosen in the area III in Figure 1B.
The above numerical results have shown some dynamics in system (1.2), and it is found that there exist abundant patterns in system (1.2). In order to further reveal the spatial distribution of phytoplankton, another parameter set is employed, as follows: \(r_{1}=2.425\), \(K_{1}=1\), \(\beta _{1}=1.35\) [38], \(\beta_{2}=1.0125\), \(r_{2}=0.97 \), \(m=0.35\), where the parameter \(\beta_{2}\) was derived from the literature [38, 39], and the parameters \(r_{1}\) and \(r_{2}\) were derived from the literature [40]. The other parameters (i.e., \(k_{2}\), \(D_{1}\), and \(D_{2}\)) were chosen as controlling parameters.
5 Discussion
In this study, we studied a nonlinear dynamics of a phytoplankton-zooplankton system with diffusion in detail. The structure of the system is very simple, but the study showed that the composition of the positive equilibrium was highly complex in the system. For system (1.1), our theoretical results show that if \(K_{1}>m\), the value of the environmental carrying capacity of the zooplankton will influence the dynamical behaviors of system (1.1). With the value of \(K_{2}\) changing, the stability of system (1.1) will change. Our numerical simulation leads to the same conclusion, through the results of Figure 1A; if the value of m is smaller, then it is obvious that the stability of system (1.1) may change with the increase of parameter \(K_{2}\). The result can suggest that the environmental carrying of the zooplankton is an important critical factor to influence the system dynamics.
The reaction-diffusion system exhibits some interesting dynamics depending on the parameters of the system. One of the main aims in our study is to analyze the effect of diffusion on the properties, especially the stability of the system. However, the results of Theorems 3.1 and 3.2 show that diffusion did not influence the stability of the system in some situations. The theoretical results show that only if the value of \(\frac{D_{2}}{D_{1}}\) grows large enough, then the equilibrium \(E_{e3}\) will shift from a stable state to an unstable state. We also show this result through a two dimensional bifurcation diagram (see Figure 1B). Our results show that the Turing instability will occur under certain conditions in the reaction-diffusion system. The occurrence of a Turing instability means that a solution which is stable in the absence of diffusion becomes unstable when diffusion is incorporated in the system. An effective way to study it is to give a small perturbation around the steady-state solutions and then obtain the conditions under which these fluctuations increase. In Figure 1B, our numerical results not only show that the diffusion can induce the occurrence of a Turing instability but also indicate that the environment carrying capacity of the zooplankton has an important role to play in the steady-state transformation.
Both theoretical and numerical analysis results have pointed out that the diffusion coefficient and the environment carrying capacity of the zooplankton have a significant impact on the dynamic properties. In 1993 [41], Pearson have presented 12 classic categories of patterns and these pattern formation have been studied further in recent years [42]. Our results show that the dynamics of system (1.2) comprised the diffusion controlled formation of stripes, hot/cold stripes/spots and some more interesting patterns. We aim to further demonstrate the influence of important parameters on the system by patterns formation; thus the results of Figure 4 are intuitive in reflecting the important role in these two parameters in the system. In general, the pattern formation can verify the rich dynamic properties around the Turing line of system, and the importance of the two parameters \(K_{2}\) and \(D_{1}\) is also deduced by the variation of the pattern.
Stated succinctly, the environment carrying capacity of the zooplankton and the diffusion term can seriously affect the dynamical behaviors of systems (1.1) and (1.2). Furthermore, it is our hope that these results can provide some reference value in the study of the internal operating rules and characteristics of marine ecosystems.
Declarations
Acknowledgements
This work was supported by the National Natural Science Foundation of China (Grant No. 31370381), the National Key Basic Research Program of China (973 Program, Grant No. 2012CB426510), and the Key Program of Zhejiang Provincial Natural Science Foundation of China (Grant No. LZ12C03001).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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