Abstract
In this paper, a Holling type IV nutrient-plankton model with time delay and linear plankton harvesting is investigated. The existence and local stability of all equilibria of model without time delay are given. Regarding time delay as bifurcation parameter, such system around the interior equilibrium loses its local stability, and Hopf bifurcation occurs when time delay crosses its critical value. In addition, the properties of the bifurcating periodic solutions are investigated based on normal form theory and center manifold theorem. What is more, the global continuation of the local Hopf bifurcation is discussed by using a global Hopf bifurcation result. Furthermore, the optimal harvesting is obtained by the Pontryagin’s Maximum Principle. Finally, some numerical simulations are given to confirm our theoretical analysis.
1. Introduction
In the aquatic ecosystem, all aquatic food chains are based on plankton, which are composed of phytoplankton and zooplankton. As pointed out in [1], phytoplankton in particular have great contribution for our earth; for example, they provide food for marine life and oxygen for human life, and they also absorb more than half the carbon dioxide which may be contributing to global warming.
In the past decade, the issues of phytoplankton-zooplankton models have been investigated by many experts [2–13]. Chattopadhy et al. [2] showed the effect of the delay forming the period of toxic substances on blooms. Wang et al. [4] considered a toxic phytoplankton-zooplankton model with delay-dependent coefficient and investigated influence of delay and harvesting for the system. Yang et al. [8] discussed the impact of reaction-diffusion on the dynamic behaviours of plankton. Sharma et al. [11] studied the bifurcation behaviour analysis of a plankton model with multiple delays. Nevertheless, these models mainly included interactions of phytoplankton-zooplankton.
However, nutrients play an significant role in the growth of plankton. The concentration of nutrients can better explain their mechanism in the phytoplankton-zooplankton system. As far as we know, the nutrient-plankton systems have been considered by a number of scholars [14–21]. Chakrabortyour et al. [14] demonstrated that the toxin producing phytoplankton (TPP) species control the bloom of other nontoxic phytoplankton species when nutrient-deficient conditions are favorable for TPP species to release toxic chemicals. Zhang and Wang [15] discussed Hopf bifurcation and bistability of nutrient-phytoplankton-zooplankton model. Fan et al. [16] proposed a nutrient model for a water ecosystem. Chakraborty et al. [17] investigated spatial dynamics of a nutrient-phytoplankton system with toxic effect on phytoplankton. Wang et al. [18] analyzed a nutrient-plankton model with delayed nutrient cycling. Jang and Allen [19] studied deterministic and stochastic nutrient-phytoplankton-zooplankton models with periodic toxin producing phytoplankton. Das and Ray [20] debated the effect of delay on nutrient cycling in phytoplankton-zooplankton interactions in estuarine system. Sharma et al. [21] discussed the effect of the discrete time delay on the nutrient-plankton interaction and proposed the model as follows:where , , and denote the concentration of nutrient, phytoplankton, and zooplankton population at time , respectively. is the time delay which is incorporated with the assumption that the liberation of toxin is not instantaneous; rather it is mediated by some time lag. Li and Xiao [22] showed a predator-prey model with Holling type IV: , where the predator is specifically not taking nutrient, and the function form of biomass conversion of herbivore is Holling type IV function. They also assumed that the extra mortality in zooplankton due to toxic substance term is also expressed by Holling type IV function.
Generally speaking, there are some economic benefits for some plankton in aquatic food chain. Some phytoplankton such as phylum Diatom, phylum Pyrrophyta, and phylum Chrysophyta can be harvested as bait for aquaculture, and some zooplankton such as jellyfish, krill, and Acetes can be harvested for human food. Thus, these plankton play a crucial role in marine conservation and fishery management. As we all know, harvesting is an essential element in our life. On the one hand, the harvesting is one of the economic sources of human economy. On the other hand, it is a method to control the balance of ecosystem. Thus, zooplankton-phytoplankton system with harvesting has attracted the attention of scholars [23–29]. Chakraborty and Das [23] proposed a two-zooplankton one-phytoplankton system with harvesting and analyzed the impact of harvesting on the coexistence and competitive exclusion of competitive predators. Lv et al. [24] proposed a phytoplankton-zooplankton system with harvesting and pointed out that overharvesting could lead to population extinction and appropriate harvesting should guarantee the sustainability of the population types in an aquatic ecosystem.
Some works have already been made to understand the nutrient-plankton system with time delay and toxic phytoplankton, but the study of nutrient-plankton system which contains Holling type IV function, time delay, toxic phytoplankton, and linear harvesting is rarely done so far. Taking account of the effect of the time delay in maturation of zooplankton into nutrient-plankton model can make the model more realistic. The main aim of the present study is to see whether the time delay in maturation of zooplankton and harvesting of plankton can exert an impact on the nutrient-plankton system.
Based on the works [21, 22] and the above discussions, the following model is proposed by introducing harvesting and time delaywhere , , and denote the number of nutrient, phytoplankton, and zooplankton population at time , respectively. is the constant input of nutrient concentration and is washout rate. and are the nutrient uptake rate and conversion rate of nutrient for growth of the phytoplankton, respectively. and are the mortality rates of the phytoplankton and zooplankton population, respectively. and are the maximal ingestion rate and conversion rate of zooplankton. denotes the rate of toxin liberation. is the nutrient recycle rate after death of phytoplankton population. The parameter is the half saturation constant for a Holling type IV function. denotes the time delay in maturation of zooplankton. The constants and are the catchability coefficients of the phytoplankton and zooplankton population, and is the harvesting effort.
The initial conditions of system (2) take the following formwhere , the Banach space of continuous functions mapping the interval into , where .
The organisation of this paper is as follows: in Section 2, preliminary dynamical results of system without delay are analyzed, such as the existence of equilibria, boundness, and the local stability of equilibria. In Section 3, with the presence of time delay, the local stability and the existence of Hopf bifurcation of system (2) are investigated. Based on normal form theory and center manifold theorem, direction of Hopf bifurcation and stability of the bifurcating periodic solutions are also discussed. What is more, the global existence of bifurcating periodic solution is given by using a global Hopf bifurcation result due to Wu [30]. In Section 4, the optimal control problem is formulated by Pontryagin’s Maximum Principle. In Section 5, numerical simulations are given to illustrate the results.
2. Preliminary Results of System without Delay
2.1. Positivity and Boundedness
When , system (2) isThe initial conditions for system (4) take the following form
Lemma 1. If , solutions of system (4) with initial conditions are defined on and remain positive for all .
Proof. From the first equation of system (4), we obtainHence, we obtainFrom the second equation of system (4), we haveSince , the third equation of system (4) shows that by a standard comparison argument
Theorem 2. All solutions of system (4) with initial conditions are bounded for all .
Proof. In order to check the boundedness of solutions of system (4), we define the function As derivative of with respect to system (4), we obtainwhere . Therefore, by the comparison theory [31], we haveThus, , and are bounded.
2.2. Equilibria and Their Local Stability
In this subsection, we will discuss the existence of equilibria of system (4) and their local asymptotical stability.
The given system (4) has three nonnegative equilibria, defined as the plankton extinction equilibrium , the zooplankton free equilibrium , and the coexistence equilibrium .
The zooplankton free equilibrium is , whereSince , if , then there is only one possible equilibrium involving phytoplankton but not zooplankton.
The coexistence equilibrium is , where , , and satisfy the following equationsFrom the third equation of (14), we haveIf , and , then (15) has exactly two positive real rootsAnd if , and , then (15) has two equal positive real rootsAgain from the first and second equations of (14), we haveIf , , then . If , , then . And if , , then .
In order to discuss the existence and the local stability of equilibria of system (4), some assumptions are given.
: , , and − (or − ).
: , and − , / − .
: , and .
: /, /, < < /, /, and .
: , /, and .
and .
From the above analysis, we obtain the following results.
Theorem 3. (i) The plankton extinction equilibrium always exists.
(ii) The zooplankton free equilibrium exists if .
(iii) The coexistence equilibrium (or ) exists if holds; the coexistence equilibria and exist if holds; the coexistence equilibrium exists if () holds.
According to the results of Theorem 3 and the above assumptions, we have the results as follows.
Theorem 4. (1) If , then the plankton extinction equilibrium is locally asymptotically stable.
(2) If and , then the zooplankton free equilibrium is locally asymptotically stable.
(3) The coexistence equilibrium (or ) is locally asymptotically stable if conditions and hold. The coexistence equilibria and are locally asymptotically stable if conditions and hold. The coexistence equilibrium is locally asymptotically stable if conditions and hold.
Proof. (1) The characteristic equation on is givenIt is clear that (19) has negative roots , and . So if , thenThus, the plankton extinction equilibrium is locally asymptotically stable. (2) The characteristic equation about is given byIf , thenFurther, the other roots and satisfy that . If , then . Therefore, if and , the zooplankton free equilibrium is locally asymptotically stable.
(3) The characteristic equation about is givenwhereSince , if () and () hold, then , . Therefore all roots of (23) have negative real parts. Based on Routh-Hurwitz criterion we obtain that the coexistence equilibrium (or ) is locally asymptotically stable.
Similarly, if and hold, the coexistence equilibria and are locally asymptotically stable; if and hold, the coexistence equilibrium is locally asymptotically stable.
Remark 5. From biological point of view, phytoplankton and zooplankton cannot survive, and the state is stable if the assumption holds; phytoplankton survive but zooplankton cannot survive, and the state is stable if the assumptions and hold; both phytoplankton and zooplankton survive, and the state is stable if the assumptions hold.
3. Hopf Bifurcation and Analysis of System with Delay
3.1. Hopf Bifurcation
In this part, we will only investigate the effect of time delay on the dynamics of system (2) about the coexistence equilibrium . For convenience, we denote the coexistence equilibrium = . We first translate the coexistence equilibrium to the origin by setting , and and drop the bar for convenience of notation. Then, the linearized system of system (2) can be obtainedwhere
Obviously, the characteristic equation of system (25) around takes the following formwhere
For , according to Corollary 2.4 [32], the sum of orders of the zeros of (27) in the open right half-plane can change only when a zero appears on or crosses the imaginary axis. It is obvious that is not a root of (27) when , and then the imaginary axis cannot be crossed by for any . And due to Corollary 2.4 [32], a stability switch (or a cross of the imaginary axis) occurs with for some real .
Now, we assume that is a root of (27). Separating real and imaginary parts, we haveAdding up the squares of both equations, we obtainBy denoting , (30) becomeswhere .
Let . According to the distribution of the roots of (31) and the results of [33], we can easily get the following lemmas.
Lemma 6. (i) If , (31) has at least one positive root.
(ii) If and , (31) does not have any positive root.
(iii) If and , (31) has positive roots if and only if and .
Thus, if the condition holds, then . Equation (31) has at least one positive root. Without loss of generality, we assume that it has three real positive roots, which are defined by , respectively. For convenience, we assume . Then (30) has three positive roots From (29), we have
Then is a pair of purely imaginary roots of (27) with . DefineAnd we have the following transversality condition.
Lemma 7. Let be the root of (27) satisfying , , , . Then
Proof. Differentiating both sides of (27) with respect to , we obtainThenFrom (30), we haveThus we haveSince , , we obtain , , . This ends the proof.
To summarize the above discussions, we have the following result.
Theorem 8. Suppose that and hold. System (2) at is locally asymptotically stable whenever and unstable when . System (2) undergoes a Hopf bifurcation for .
3.2. Properties of Hopf Bifurcation
From Theorem 8, we have obtained the conditions for the existence of Hopf bifurcation when . In this subsection, we shall consider the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions of system (2) by using the center manifold and normal form theory presented by Hassard [34].
Let , , and for . Using Taylor series expansion about , system (2) can be rewritten aswhere andwhereLinearization of the delayed system (41) around the origin is givenDefinewhere .
By the Riesz representation theorem [35], there exists matrix-valued function of bounded variation for such thatIn fact, we can choosewhere is a Dirac delta function.
Next, for , we defineandand then system (43) can be rewritten aswhere and for .
For , the adjoint operator of A is defined asand a bilinear inner productwhere . From the above discussion, we can know that are the eigenvalues of , and thus are also the eigenvalues of .
Let be the eigenvector of corresponding to the eigenvalue , and be the eigenvector of corresponding to the eigenvalue . With the conditions and , we obtainandwhere I is a third order identity matrix. Hence, we can obtain , .
In order to satisfy that , we need to choose the value of in the following part. For convenience, we denote . By virtue of (52) and (53), it derives thatHence, we can choose .
Nextly, we will compute the coordinates to detail center manifold at . Let be solution of (49) when . We define
On the center manifold , it derives thatwhere and are local coordinates for in the direction of and , respectively.
We only consider real solutions as is real. If is real, then we getFor the solution of (49), since , from (49) and (51), we havewhereIt follows from (55) and (56) that
From (41) and (59), it derives thatHere , , , , .
According to comparing its coefficients with (59), it derives thatSince and are still unknown, we still compute them next. By virtue of (49) and (55), we haveHere,
By computing and comparing coefficients, we can obtainFrom (63), we know thatUsing (59) in (66) and comparing coefficient with (64), it derives thatFrom the definition of , based on (65) and (67), we can obtain Because of , we haveSimilarly, it derives that from (65) and (66)Here , are three dimensional constant vectors and can be determined by setting in .
By above definitions and conditions, we can computewherewhere
Based on above analysis, the following values can be givenBy computing (75), we have the following theorem.
Theorem 9. Properties of bifurcating periodic solution in the center manifold at critical value are determined as follows:
(i) If , then Hopf bifurcation is supercritical (subcritical) and the bifurcating periodic solutions exist for .
(ii) Bifurcating periodic solutions are stable (unstable) if .
(iii) Period of the periodic solution increases (decreases) if .
3.3. Global Hopf Bifurcation
In the above parts, we have obtained that system (2) undergoes Hopf bifurcation at when . We will study the global continuation of these local bifurcating periodic solutions by using the global Hopf bifurcation theorem for delay differential equations [30], and state that system (2) admits periodic solutions globally for delay in a finite interval This includes the values that are not necessarily near the local Hopf bifurcation values.
For convenience, we give some notations, , , and . We can rewrite system (2) as the following functionwhere .
According to the work of Wu [30], we need to introduce the following notionsLet be the connected component of in , and denote as its projection on component. is a pair purely imaginary roots of (27) with . By Theorem 8, we know that is nonempty.
Lemma 10. Whenever the equilibrium exists, all the nontrivial solutions of system (2) are uniformly bounded.
Proof. We assume that is an arbitrary positive periodic solution of system (2) subject to initial condition (3). Let and be real quantities defined by , , , , , and . From the first and second equations of system (2), we havewhich clearly implies thatSince , .
Similarly, from the second equation of system (2), we haveThus, we obtainWe defined the function , for all . Then it follows from system (2) thatwhere . Therefore, by comparison with theory [31], we haveThus, all the nontrivial solutions of system (2) are uniformly bounded for .
Lemma 11. System (2) has no nonconstant -periodic solution if the condition andare satisfied.
Proof. In order to explain that system (2) has no nonconstant periodic solution of periodic , we only prove that system (4) has no nonconstant periodic solution (see [36], Lemma 5.5). We rewrite system (4) as . Suppose that is nonconstant periodic solution for system (4); then is nonconstant periodic solution of the following equationsFrom the Lemma 10, we know that the periodic orbits of system (85) corresponding to are included in the following areawhere and are the boundedness of the periodic solutions of system (2) in the .
The Jacobian matrix of system (85) isBy using the method of [37], its second additive compound matrix isThe second compound system isWe select vector module for as followsThen, the Lozinski measure of second additive compound matrix corresponding to a vector module above isThus, if holds, there exists an such that for all .
This shows that the conclusion of Lemma 11 follows from Corollary 3.5 in [38].
Theorem 12. Assume and are fulfilled; then there exists , such that system (2) has at least one positive periodic solution when varies among , , and , , where
Proof. The characteristic matrix of system (2) at an equilibrium is givenwhere is called a center if and . A center is said to be isolated if it is the only center in some neighbourhood of . It is obvious that (76) has an equilibrium in under the condition . It follows that is an isolated center from the discussion about the local Hopf bifurcation theorem in Section 3.1. By Lemma 7 and Theorem 8, there exist constants and , as well as a smooth curve such thatwhen holds, andDenotes , andApparently, if , such that if and only if . Hence all the assumptions of Theorem 3.2 in [30] are satisfied for . Moreover, if we definethen we have the crossing numberBy Theorem 3.2 of Wu [30], we conclude that the connected component through in is nonempty. Meanwhile, we have that
(i) is unbounded or
(ii) is bounded, is finite, and .
By expression (32), we all know that there exists an integer , such that , , which implies that .
Therefore, we have that if . This fact and Lemma 11 show that the projection of onto the -space is bounded. On the flip side, Lemma 10 implies that the projection of onto the x-space is uniformly bounded for . Otherwise, is bounded and , which contradicts the fact that nonconstant periodic solution and are both uniformly bounded for . The proof is completed.
4. Optimal Harvesting
The emphasis of this section is on the profit making aspect of plankton; optimal control addresses the problem of finding control law for a given system such that a certain optimality criterion is satisfied [25]. In the economic perspective, the commercial development of recyclable resources is a critical issue and it is necessary to determine the optimal tradeoff between present and future harvests.
The harvesting problem covers cost of harvesting comprising a set of differential equations that describe the paths of the control variables that minimize the cost functional. It is assumed that price is a function that decreases with increasing biomass. Thus, to maximize the total discounted net revenues from the plankton, the optimal control problem can be formulated aswhere is the constant harvesting cost per unit effort, and are the constant price per unit biomass of the phytoplankton and zooplankton, and , are the economic constants. The convexity of the objective function with respect to and the compactness of range values of the state variables can be combined to give the existence of the optimal control.
This problem is subject to system (2) and the control constrains ; here, is the maximum effect capacity of the harvesting industry. We want to find the optimal harvesting effect such thatwhere is the control set defined by
To find the maximum value of , we set and , where . Now, the Hamiltonian function of this optimal control problem is givenwhere , are adjoint variables. Transversality conditions are .
Pontryagin’s Maximum Principle with delay given in [28] is used to derive the necessary conditions for the optimal control. If we consider and defined above, then there exists a continuous function on satisfying the following three functions. The state equation isthe optimality condition isand the adjoint equation iswhere , , , and denote the derivative with respect to , , , , respectively. Now we use the necessary conditions to Hamiltonian .
Then there exist adjoint equations
The condition for the optimal harvesting can be obtained from (105), which is
Based on the above analysis, we have the following theorem.
Theorem 13. There exist an optimal control and corresponding solutions to system (2) that maximin over . Furthermore, there exist adjoint variables , and which satisfy (107) with the transversality conditions . Moreover the optimal control is given by expression (108).
5. Numerical Simulation
In this section, numerical simulations are carried out to verify the theoretical findings by the use of MATLAB. Most values of parameters in system (2) are taken from [21]. , , , , , , , , , , , , , .
When , we observe that holds and the coexisting equilibrium is . Condition is also valid. Therefore, the equilibrium is locally asymptotically stable (see Figure 1). Then, when , we obtained . Hence, (30) has at least one positive root. We obtain , and the critical value . According to Theorem 8, system (2) remains stable when , but system (2) is unstable when . The stability behaviour of system (2) around for is shown in Figure 2. Hopf bifurcation occurs when , shown in Figure 3.
(a) Waveform plot
(b) Phase diagram
(a) Waveform plot
(b) Phase diagram
(a) Waveform plot
(b) Phase diagram
Again, through the above parameter values, the properties of bifurcating periodic solutions at are given
Therefore, Hopf bifurcation is supercritical, bifurcating periodic solutions are stable (see Figure 4), and period of the periodic solution increases according to Theorem 9.
(a) Regular oscillations
(b) Phase diagram
Nextly, we consider a set of parametric values to illustrate optimal control problem as follows: , , , , , , , , , We first use forward forth order Runge-Kutta method for the state variables of (4) with nonnegative initial conditions. Then we solve the adjoint variables from (107) by using backward forth order Runge-Kutta method. It is observed that as the constant input of nutrient concentration or the half saturation constant increases, the optimal harvesting effort increases (see (a,b) of Figure 5). However, as the rate of toxin liberation increases, the optimal harvesting effort decreases (see (c) of Figure 5). We observe that the optimal harvesting effort tends to be stable. We see that the nutrient increases firstly and then tends to be stable as time varies in the presence of harvesting (see (a) of Figure 6), and the phytoplankton and zooplankton population decrease and even tend to be stable as time changes in the presence of harvesting (see (b,c) of Figure 6). All the final values of adjoint variables are zero (see Figure 7). Finally, using the same method in subject (100), we research optimal harvesting policy for model (2). Optimal harvesting effort is mainly dependent on gestation delay of zooplankton displayed in Figure 8. We can observe that the optimal harvesting effort increases monotonically.
(a)
(b)
(c)
(a) The nutrient
(b) The phytoplankton
(c) The zooplankton
(a)
(b)
(c)
6. Conclusion and Discussion
Sharma et al. [21] discussed the effect of the time delay about the liberation of toxin on nutrient-plankton interaction. In this paper, the dynamic of nutrient-plankton interaction was considered with the gestation delay and linear harvesting. It is assumed that the mortality caused by toxin to zooplankton population is described as Holling type IV function. We obtained that the system lost local stability around and Hopf bifurcation occurred when time delay crossed its corresponding critical value. By using normal form theory and center manifold theorem, we obtained the properties of Hopf bifurcation. Next, as time delay varies in some regions, the global existence result of the periodic solution of Hopf bifurcation was given by a global Hopf bifurcation result due to Wu [30]. Pontryagin’s Maximum Principle was used to obtain the optimal harvesting policy. We analyzed the effect of different parameter variables on the harvesting effort (see of Figure 5) and obtained the impact of time delay on the harvesting effort (see of Figure 8). In the sense of biology, optimal harvesting effort increased with increasing of the constant input of nutrient concentration; optimal harvesting effort stayed the same as the half saturation constant changes; optimal harvesting effort decreased with the increasing toxin liberation of plankton. And optimal harvesting effort increased with the longer time delay in maturation of zooplankton.
In the future research, in order to make the model more practical, harvesting might be delayed because zooplankton avoid harvesting, so this factor can be considered into our nutrient-plankton system, which iswhere and are the time delays in maturation of zooplankton and harvesting because zooplankton avoid harvesting, respectively.
Data Availability
The data used to support the findings of this study are included within the article. The reader can find the corresponding values of some parameters in the beginning of “Numerical Simulation”. Most parameter values and data utilized in the numerical simulation section of this paper are taken from the numerical simulation section in [21]. Of course, the reader can freely access the parameter values and data supporting the conclusions of the study in the corresponding article [21].
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work is supported by the National Natural Science Foundation of China (Grant No. 11661050, 11461041), the Development Program for HongLiu Outstanding Young Teachers in Lanzhou University of Technology (Q201308), and the Development Program for HongLiu Distinguished Young Scholars in Lanzhou University of Technology.