Dynamics of a delayed phytoplanktonzooplankton system with CrowleyMartin functional response
 Tiancai Liao^{1, 2},
 Hengguo Yu^{1, 2} and
 Min Zhao^{2, 3}Email author
https://doi.org/10.1186/s1366201610554
© The Author(s) 2017
Received: 15 June 2016
Accepted: 6 December 2016
Published: 4 January 2017
Abstract
In this paper, a delayed phytoplanktonzooplankton system with CrowleyMartin functional response is investigated analytically. We study the permanence and analyze the stability of the both boundary and positive equilibrium points for the system with delay as well as the system without delay. The global asymptotic stability is discussed by constructing a suitable Lyapunov functional. Numerical analysis indicates that the delay does not change the stability of the positive equilibrium point. Furthermore, we also show that due to the increase of the delay there occurs a Hopf bifurcation of periodic solutions. It is found that population fluctuations will not appear under the condition of certain parameters. In addition, we determine the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions by applying a normal form method and center manifold theory. Finally, some numerical simulations are carried out to support our theoretical analysis results.
Keywords
1 Introduction
Plankton plays an important role in the ocean and the climate because of their participation in the global carbon cycle at the base of the food chain [1]. There are two forms of plankton, the plant forms of the plankton community are known as phytoplankton and the animals in the plankton community are known as zooplankton. Phytoplankton biomass has the characteristics of rapid proliferation, this change is called bloom [2]. Under some circumstances, however, phytoplankton bloom can appear and affect the ecological balance, and it even can endanger human life and health. Therefore, a better understanding of mechanisms that determine the plankton dynamics is of considerable interest [3].
In recent years, there were many experimental ecologists as well as mathematical ecologists who have paid more attention to the phenomena of phytoplankton blooms. And many scholars proposed different deterministic models to study the dynamical behavior of the plankton system and tried to explain the dynamic mechanism of phytoplankton in different ways [4–11]. Abbas et al. [12] considered the two species competitive delay plankton allelopathy stimulatory deterministic model and investigated the existence and uniqueness of the solution as well as the persistence and the stability properties of the model, which is very useful and meaningful for the study of plankton bloom. Chakraborty and Das [13] investigated some properties of a twozooplankton onephytoplankton system that exhibits a Holling type II functional response in the presence of toxicity, the results they obtained can provide great help for researching the dynamic complexity of plankton systems. Kartal et al. [14] proposed a phytoplanktonzooplankton system via a new approach by using a system of differential equations with piecewise constant arguments and studied the biological dynamics of the bloom in the plankton system. It is significant that they explained the plankton bloom depended on three different parameters, namely θ (rate of toxin production per phytoplankton), β (zooplankton growth efficiency) and K (environmental carrying capacity of phytoplankton), which can effectively promote the research process of the plankton bloom. However, a clear understanding of the mechanisms that cause the plankton blooms is still lacking. Hence, by establishing a differential equation model to study the ecological problems and a suitable model to study the dynamic relationship between the phytoplankton and zooplankton is still very important. In the work of [15] and [16], Stomp et al. and Rhee et al. have indicated that there exist many factors affecting the dynamics of plankton.
 (I_{1}):

It is assumed that \(P(t)\) and \(Z(t)\) are the concentrations of phytoplankton and zooplankton populations, respectively.
 (I_{2}):

It is assumed that \(k(\mathit{ugl}^{1})\) is the carrying capacity, \(r(\mathit{day}^{1}) \) is the maximum growth rate, \(\mu(\mathit{day}^{1})\) is the zooplankton death rate and γ is the conversion efficiency [23].
 (I_{3}):

It is assumed that ω is the effects of the capture rate, \(k_{1}\) is the handing time and \(k_{2}\) is the magnitude of interference among predator (zooplankton) [18].
 (I_{4}):

From a biological point of view, all the parameters in (2) assume only positive values and will be considered as constants throughout our discussion.
With \(P(0)=P_{0}>0\) and \(Z(0)=Z_{0}>0\), where \(a=\frac{1}{k^{2}}\), \(b=\frac{\omega}{r}\), \(c=\frac {k_{1}}{k}\), \(d=k_{2}\), \(e=\frac{k_{1}k_{2}}{k}\), \(h=\frac{\mu}{r}\), \(g=\frac {\gamma\omega}{k}\).
The main purpose of this paper is to study the dynamic behaviors of system (3) and (4), especially, how the time delay affects the plankton system. We shall also compare all possible dynamics between the nondelayed system (3) and its corresponding delayed system (4). The remainder of the paper is organized as follows. In Section 2, we consider results for the nondelayed system (3), and we discuss the properties of positivity and boundedness, conditions for the existence of a positive equilibrium as well as the condition for the persistence, and the stability of various equilibrium points. Section 3 presents analogous results for the delayed system (4) and considers the stability and direction of the Hopf bifurcation. In Section 4, we present some numerical results supporting our analytical findings. Section 5 discusses the results and draws some conclusions.
2 Nondelayed system
In this section, our analysis shows the positivity and boundedness of the solutions, dissipativity, the persistence, local and global stability of both zooplanktonfree equilibrium and positive equilibrium points for system (3).
2.1 Positivity and boundedness of solutions
Let \(R_{+}\) denote the set of all nonnegative real numbers and \(R_{+}^{n}=\{P \in R^{n}:P=(P_{1}\cdots P_{n})\}\), where \(\{P_{i}\in R_{+},\forall i=1,2,\ldots, n\}\). If we denote the function on the righthand side of (3), by \(F=(F_{1},F_{2})\), it is clear that \(F\in C^{1}(R_{+}^{2})\). Thus, \(F:R_{+}^{2}\rightarrow R^{2}\) is locally Lipschitz on \(R_{+}^{2}=((P,Z):P\geq0,Z\geq0)\). Hence, the fundamental theorem of the existence and uniqueness ensures the existence and uniqueness of a solution for (3) with the given initial conditions. The state space for the system is a nonnegative cone \(R_{+}^{2}=\{(P,Z):P\geq0,Z\geq0\}\).
Lemma 2.1
The positive quadrant \((\operatorname{Int}(R_{+}^{2}))\) is invariant for system (3).
Now we consider the conservation of overall energy or biomass flow, and we search for the existence of some region in the dynamical space within which system (3) is bounded.
Lemma 2.2
[37]
Theorem 2.3
Proof
Using Lemma 2.2, we have \(\psi(t)\leq\frac{(h+1)^{2}}{4ah}\) as \(t\rightarrow+\infty\). Therefore, all the solutions for (3) are uniformly bounded with an ultimate bound. □
2.2 Dissipativity and permanence
In this section, we analyze the dissipativity, persistence, and permanence behavior of system (3). To prove our results, we first present the following definitions.
Definition 2.4
[38]
 (1)
\(P(t)\geq0\), \(Z(t)\geq0\), \(\forall t\geq0\).
 (2)
\(\limsup_{t\rightarrow+\infty}P(t)>0\), \(\limsup_{t\rightarrow +\infty}Z(t)>0 \).
System (3) is said to be strongly persistent if every solution \((P(t),Z(t))\) satisfies the following condition along with the first condition for the weak persistence: \(\liminf_{t\rightarrow+\infty}P(t)>0\), \(\liminf_{t\rightarrow+\infty}Z(t)>0\).
Definition 2.5
[38]
Lemma 2.6
[39]
The above results can be summarized in the following theorem.
Theorem 2.7
Theorem 2.8
Remark 2.9
Under Definition 2.4, Theorem 2.7 results, whereby system (3) is weakly persistent (dissipative), holding provided \(h<\frac{m_{1}g}{1+cm_{2}}\). Theorem 2.8 along with Definition 2.4 ensures that system (3) is strongly persistent provided the conditions \(d>b \) and \(h<\frac{gm_{2}}{1+cm_{1}}\) hold.
The conditions \(d>b \), \(h<\frac{gm_{1}}{1+cm_{2}}\), and \(h<\frac {gm_{2}}{1+cm_{1}}\) ensure that \(m_{2}>0\), \(Q_{1}>0\), and \(Q_{2}>0\), respectively. Since \(m_{1}>0\) and \(\frac{gm_{1}}{1+cm_{2}}>\frac {gm_{2}}{1+cm_{2}}\), when \(h< \frac{gm_{2}}{1+cm_{1}}\) holds, this ensures that \(Q_{1}>0\) and \(Q_{2}>0\). Thus, we arrive at the following result.
Theorem 2.10
System (3) is permanent if it satisfies any of the following two conditions: (H_{1}) \(d>b\), (H_{2}) \(h<\frac{gm_{2}}{1+cm_{1}}\).
2.3 Biomass equilibrium
 (i)
The trivial equilibrium \(E^{0}=(0,0)\).
 (ii)
The boundary equilibrium \(E^{1}=(\frac{1}{a},0)\).
 (iii)
The coexistence equilibrium (interior equilibrium) \(E^{*}=(P^{*},Z^{*})\).
From Lemma 2.2 of Song et al. [40], since \(A_{3}<0\), we see that equation (6) has at least one positive root. We only consider that equation (6) has one positive root denoted by \(P^{*}\). Furthermore, for this value of \(P^{*}\), the corresponding value of \(Z^{*}\) is given by \(Z^{*}=\frac{gP^{*}(1aP^{*})}{bh}\).
2.4 Dynamical behavior: stability analysis
In this section, we deal with the local stability, global stability, and bifurcation of system (3).
2.4.1 Behavior of system (3) around \(E^{0}=(0,0)\)
We observe that the Jacobian matrix of system (3) at \(E^{0}\) has eigenvalues 1 and −h. Therefore, system (3) is always unstable around \(E_{0} \), which is in fact a saddle point.
2.4.2 Behavior of system (3) around \(E^{1}=(\frac{1}{a},0)\)
The eigenvalues of the variational matrix \(J(E^{1})\) at the equilibrium solution \(E^{1}\) are 1 and \(\frac{g}{a+c}h \). Hence, for \(h>\frac{g}{a+c}\) system (3) is stable around \(E_{1}\), for which the PZ plane is at stable manifold. Conversely, for \(h<\frac{g}{a+c}\), system (3) is always unstable around \(E^{1}\), which is, in fact, a saddle point.
From the above analysis, we have the following result.
Theorem 2.11
 (i)
\(E^{0}\) is a saddle point.
 (ii)
\(E^{1}\) is a saddle point if \(h<\frac{g}{a+c}\), and it is a stable node if \(h>\frac{g}{a+c}\).
Theorem 2.12
The equilibrium solution \(E^{1}=(\frac{1}{a},0)\) is globally asymptotically stable if \(h\geq \frac{g}{a}\).
We prove the result for the delayed system. The proof for system (3) is similar.
2.4.3 Behavior of system (3) around \(E^{*}=(P^{*},Z^{*})\)
Hence, if \(b>d \) and \(ab>cbcd+e \), then (14) is satisfied. Thus, we have the following result.
Theorem 2.13
If \(b>d \) and \(ab>cbcd+e \) hold, then the positive equilibrium is locally asymptotically stable.
2.4.4 Global stability of the interior equilibrium \(E^{*}=(P^{*},Z^{*})\) of system (3)
Theorem 2.14
If (H_{1})(H_{2}) hold and \((1+eQ_{1})(1aP^{*})< a \), then the interior equilibrium solution \(E^{*}\) is globally asymptotically stable.
Proof
We conclude that if the hypotheses of Theorem 2.14 are satisfied, then \(\frac{dV}{dt}<0 \) along all the trajectories in \(R_{+}^{2} \) except \(E^{*}=(P^{*},Z^{*})\). Thus, the functional \(V(P,Z) \) satisfies all the properties of a Lyapunov functional. Therefore, \(E^{*}=(P^{*},Z^{*})\) is globally asymptotically stable under the conditions of Theorem 2.14. This completes the proof. □
3 Delayed system
In this section, we analytically prove positivity and boundedness of the solutions, the persistence, local and global stability, existence of the Hopf bifurcation by analyzing the associated characteristic transcendental equation for the model with a delay. Furthermore, we also discuss stability and direction of the Hopf bifurcation by means of the normal form method and center manifold theory.
3.1 Positivity and boundedness of system (4)
Theorem 3.1
All solutions for system (4) starting in \(R_{+}^{2} \) are confined to the region \(D^{*}=\{ (P,Z)\in R_{+}^{2}: 0\leq\phi(t)\leq\frac{(h+1)^{2}}{4ah}\} \) as \(t\rightarrow+\infty\) for all positive initial values \((P_{0}(\theta ),Z_{0}(\theta))\in R_{+}^{2}\), where \(\phi(t)= P(t\tau)+\frac {b}{g}Z(t) \).
Proof
Using Lemma 2.2, we have \(\phi(t)\leq\frac{(h+1)^{2}}{4ah} \). Thus all the solutions for system (4) are uniformly bounded with an ultimate bound. □
3.2 Permanence
Now we establish the persistence and permanence behavior of the delayed model using the positivity of the dependent variables.
Because \(\frac{gm_{1}}{(d+em_{2})h}>\frac {gm_{1}hcm_{2}h}{(d+em_{2})h}\equiv Q_{1} \), we obtain \(\limsup_{t\rightarrow+\infty}Z(t)\leq Q_{1} \).
The above results can be summarized as follows.
Theorem 3.2
If we define \(N^{*}=\min\{ m_{2}, Q_{2}\}\), \(L^{*}=\max\{ m_{1},Q_{1}\}\), and \(T^{*}=\max\{ t_{1},t_{2},t_{3},t_{4}\} \), then we have \(N^{*}<\{ P(t),Z(t)\}<L^{*}\) for \(t>T^{*}\), and we can state the permanence result for system (3).
Theorem 3.3
The delayed system (4) is permanent if \(d>b \) and \(h< \frac{gm_{2}}{1+cm_{1}} \).
3.3 Local stability and Hopfbifurcation analysis
In this section we discuss the local stability of the positive equilibrium \(E^{*} \) and the boundary equilibrium \(E^{1} \) for system (4) and establish the existence of the Hopf bifurcation at \(E^{*}\). We recall the following result, which provides the conditions for the absence of a delayinduced switch from stability to instability.
Theorem 3.4
[41]
 (i)
The real parts of all the roots of \(D(\lambda,0)=0 \) are negative.
 (ii)
For any real Z and any \(\tau\geq0 \), \(D(iZ,0)\neq0 \), where \(i=\sqrt{1} \), and \(D(\lambda,\tau)\) denotes the characteristic equation associated with (4).
Equation (22) has a positive root \(\omega_{+} \) if \(\frac{g}{a+c}>h \). Thus, there is a positive constant τ̂ such that, for \(\tau>\hat{\tau} \), \(E^{1} \) becomes unstable.
From Lemma 2.2 of Wang et al. [44] we know that: (s_{1}) If \(e_{2}<0 \), equation (29) has one positive root. (s_{2}) If \(e_{2}>0 \) and \(e_{1}<0 \), equation (29) has two positive roots when \(e_{2}^{2}+4e_{1}^{3}<0 \) and has a positive root of multiplicity two when \(e_{2}^{2}+4e_{1}^{2}=0 \). (s_{3}) If \(e_{2}=0 \) and \(e_{1}<0 \), equation (29) has one positive root.
From (30) it is obvious that the transversality condition \([\frac {d\mu}{d\tau}]_{\tau=\hat{\tau}}>0 \) for the occurrence of the Hopf bifurcation at \(\tau=\hat{\tau} \) is well satisfied provided \(\beta_{2}\beta_{1}+\alpha_{2}\alpha_{1}>0 \). Summarizing, we have established the following result for switching the stability behavior of system (4) around \(E^{*} \).
Theorem 3.5
 (i)
Let \(E^{*} \) be locally asymptotically stable for system (4) with \(\tau=0 \) and let \(hj_{11}j_{12}(j_{21}+\frac{g}{b}j_{11})<0 \). Then there exists \(\tau=\hat{\tau} \) such that \(E^{*} \) is locally asymptotically stable for \(\tau<\hat{\tau} \) and unstable for \(\tau>\hat{\tau} \), where \(\hat{\tau}>0 \) is the smallest value for which there is a solution to equation (23) for which the real part is zero. Furthermore, as τ exceeds τ̂, \(E^{*} \) bifurcates into periodic solutions, provided \(\beta_{2}\beta _{1}+\alpha_{2}\alpha_{1}>0 \).
 (ii)
If \(h+j_{12}\frac{g}{b}>j_{11} \), \(j_{12}(j_{21}+\frac {g}{b}j_{11})+hj_{11}<0 \), and (s_{1}) or (s_{2}) or (s_{3}) hold, then \(E^{*} \) for system (4) is locally asymptotically stable for all \(\tau>0 \).
3.4 Global stability of system (4) around \(E^{1} \) and \(E^{*} \)
If \(h>\frac{g}{a} \) or \(h=\frac{g}{a}\), then from (35) we have \(\frac{dV_{2}}{dt}<0\) except at \((P(t),Z(t))=(\bar{P},0)\) and also \(\frac{dV_{2}}{dt}=0\) if and only if \((P(t),Z(t))=(\bar{P},0)\) (see the equation of (34)). Hence LyapunovLasalle’s invariance principle implies the global asymptotic stability of \(E^{1} \).
We summarize the result in the following theorem.
Theorem 3.6
The equilibrium solution \(E^{1}=(\frac{1}{a},0) \) for system (4) is globally asymptotically stable if \(h\geq\frac{g}{a} \).
Theorem 3.7
Proof
Consider the set \(\eta=\{ (P,Z):m_{2}< P< m_{1},Q_{2}< Z< Q_{1}\} \). It is clear that η is a compact and bounded region in \(R_{+}^{2} \) and is at a positive distance from the coordinate axes. There exists T, such that, for \(t>T \), every solution for system (4) with \(\tau>0 \) enters and remains in region η.
We define \(W=\frac{g Q_{2}P^{*}(d+em_{2})}{Q_{1}\varphi_{1}}\frac {Q_{1}(1+g \, dZ^{*})}{2Q_{2}\varphi_{2}}\frac {gm_{1}(1+cP^{*})}{2Q_{2}\varphi_{2}} \) and \(K=\frac{\varphi _{2}gm_{1}(1+cP^{*})}{Q_{2}\varphi_{1}(abcZ^{*}b e Q_{1}Z^{*})} \). If we choose \(u_{1}=Ku_{2} \), then the matrix G is positive definite if \(a>bZ^{*}(c+eQ_{1})\), \(W>0 \), and \(4W\{ \frac{K(abcZ^{*}beQ_{1}Z^{*})}{\varphi_{2}}\frac {gm_{1}(1+cP^{*})}{2Q_{2}\varphi_{1}}\}\geq\frac {b(1+cP^{*})^{2}}{\varphi_{1}^{2}} \), that is, \(Q_{2}\varphi _{2}(b(1+cP^{*}))^{2}K^{2}4Q_{2}W\varphi _{1}^{2}(abcZ^{*}beQ_{1}Z^{*})K2W\varphi_{1}\varphi _{2}gm_{1}(1+cP^{*})\leq0 \). The result follows. □
Remark 3.8
If \(h\geq gP^{*} \), the sufficient conditions for global asymptotic stability of the positive solution \(E^{*} \) for nondelayed system (3) imply that the interior equilibrium solution \(E^{*} \) for the delayed system is globally asymptotic stable if \(E^{*} \) for nondelayed system (3) is globally asymptotic stable and conditions (i) and (ii) of Theorem 3.7 hold.
3.5 Stability and direction of Hopf bifurcation
Here we describe the process for computing the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions using the normal form method and center manifold theory introduced by Hassard et al. [46]. Without loss of generality, we always assume that system (4) undergoes a Hopf bifurcation in the state \(E^{*} \) for one of the critical values \(\tau=\hat{\tau}_{j}=\hat {\tau} \), where \(j=1,2,3,\ldots\) and \(\pm i\omega\) are the only purely imaginary roots of the characteristic equation at \(E^{*} \).
According to Hassard et al. [46] and using the quantity analysis above, the Hopfbifurcation properties are determined using the following theorem.
Theorem 3.9
 (i)
If \(\mu_{2}>0\) (<0), the Hopf bifurcation is supercritical (subcritical).
 (ii)
If \(\beta_{2}<0\) (>0), the bifurcated periodic solutions are stable (unstable).
 (iii)
If \(T_{2}>0\) (<0), period of the bifurcating periodic solution increases (decreases).
4 Numerical simulations
In the previous sections, a detailed theoretical analysis has been carried out and some interesting results of the research of system have been obtained. In order to prove the feasibility and effectiveness of the theoretical analysis results as well as to provide a more indepth understanding of the results of the analysis, in particular, the time delay influencing the complex dynamics of phytoplanktonzooplankton system, we perform some numerical simulations.
5 Discussion and conclusion
Over the two decades, a great deal of research has been devoted to the dynamics of the plankton ecosystem, however, a clear understanding of the mechanisms that cause the plankton blooms is still lacking and, therefore, it has been remained an interesting area of research for many ecologists and mathematical biologists [30]. In this paper, an attempt has been made to study the dynamic behaviors of a phytoplanktonzooplankton system with a CrowleyMartin functional response and its corresponding delayed version. In order to see how the length of gestation delay affects the dynamical behavior, we have first analyzed the system without delay and obtained the parameters conditions for the permanence (Theorem 2.10). That is, if (H_{1}) and (H_{2}) hold, the phytoplankton and zooplankton populations of the system will have lasting coexistence. It is also derived that the system in the absence of delay remains locally asymptotically stable when \(b>d\) and \(ab>cbcd+e\) (Theorem 2.13), which means that phytoplankton blooms will not occur. More interesting, it can be concluded that the positive interior equilibrium of the system without delay is globally asymptotically stable under the conditions of Theorem 2.14 (see Figure 1). From the biological point of view, it leads the phytoplankton and zooplankton populations to coexist.
Next we have presented analogous results for the delayed system and considered the stability and direction of the Hopf bifurcation. It is should be noted that the system in the presence of delay is also globally asymptotically stable if (H_{1})(H_{2}), \(a > bZ^{*}(c + eQ_{1})\), and (i)(ii) hold (see Figure 2). In the work of [2, 12, 34, 35, 42, 48], one showed that the time delay can cause a stable equilibrium to become unstable and even a switching of stabilities in their system, in other words, a time delay which incorporates in a biological ecosystem can lead to the ecosystem’s steady state switch from stable to unstable. In particular, in [34], the authors also investigated the dynamical behavior of a phytoplanktonzooplankton system with a gestation delay, they observed that the gestation delay has a destabilizing effect on the system dynamics. However, in the present paper, based on the comparative analysis of the global stability of the numerical simulation results, it was found that the delay (gestation) did not change the stability of the system (Figure 1(a) and (b) and Figure 2(a) and (b), Figures 3 and 4). In [4], where the global stability and the Hopf bifurcation in a zooplanktonphytoplankton system was studied, the obtained results also showed that the time delay did not change the stability of the system. Additionally, our results also indicated that the delay can accelerate the process of its stability (Figures 1(c) and 2(c)). Although the delay cannot change the stability of the system, it should be emphasized that the delays were bound to influence the process. These results may be very meaningful to study the dynamics of phytoplanktonzooplankton interaction and may have great importance for research on plankton bloom.
In addition, it has also been shown that the time delay can induce instability and oscillations via a Hopf bifurcation in the system in the case of the presence of delay, and thereafter, switching of stability occurs. In other words, the stability of system (4) can be changed by the delay. More specifically, we have established that when \(\tau<\hat{\tau}=0.4422694125\), the positive interior equilibrium \(E^{*} \) is stable under certain parametric restrictions mentioned in Theorem 3.5. However, when the time delay τ exceeds the threshold value \(\hat{\tau}=0.4422694125 \), the delayed phytoplanktonzooplankton system will undergo a Hopf bifurcation and exhibit a periodic orbit around the coexisting equilibrium point \(E^{*} \) (see Figure 6). It should be noted that gestation delay can enhance the population fluctuations when the delay is long enough, however, our analytical results demonstrated that population fluctuations will not appear if \(h+j_{12}\frac{g}{b}>j_{11} \), \(j_{12}(j_{21}+\frac{g}{b}j_{11})+h j_{11}<0 \) and any of (s_{1}), (s_{2}), (s_{3}) hold (Theorem 3.5). From the biological point of view, it implies the disappearance of bloom of plankton populations. Consequently, the gestation delay on CrowleyMartin functional response of the zooplankton can ensure some mechanism for controlling the plankton bloom. This may be helpful to study the problem of plankton bloom. Furthermore, by virtue of the normal form method and center manifold theory, we have derived that a Hopf bifurcation is supercritical (\(\mu_{2}>0\)) and the bifurcation periodic solutions are stable with decreasing period (\(\beta_{2}<0\) and \(T_{2}<0\)).
Finally, although the study of the problems of the limit in theory study, the data of numerical simulations are not based on real world survey, it can be seen that numerical simulation results support our analytical findings. Hence, our work may be helpful to field investigation and experimental research in the real situation, as well as may also be helpful for qualitative research into similar real systems in nature. Nevertheless, in the real world, the environment of the planktonic creature is random, we also believe that our theoretical results will be useful in the study of delayed CrowleyMartintype phytoplanktonzooplankton model systems in a stochastic environment, which we leave for future work.
Declarations
Acknowledgements
This work was supported by the National Natural Science Foundation of China (Grant Nos. 31370381 and 31570364).
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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