Dynamical Behaviour of a Nutrient-Plankton Model with Holling Type IV , Delay , and Harvesting

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’sMaximumPrinciple. Finally, some numerical simulations are given to confirm our theoretical analysis.


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 phytoplanktonzooplankton models have been investigated by many experts [2][3][4][5][6][7][8][9][10][11][12][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][15][16][17][18][19][20][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-phytoplanktonzooplankton 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: () = /( +  2 ), 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][24][25][26][27][28][29].Chakraborty and Das [23] proposed a twozooplankton 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 nutrientplankton system.
Based on the works [21,22] and the above discussions, the following model is proposed by introducing harvesting and time delay where (), (), and () denote the number of nutrient, phytoplankton, and zooplankton population at time , respectively. 0 is the constant input of nutrient concentration and  is washout rate. and  1 ( 1 ≤ ) 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  1 ( 1 ≤ ) are the maximal ingestion rate and conversion rate of zooplankton. denotes the rate of toxin liberation. 1 ( 1 ≤ ) 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  1 and  2 are the catchability coefficients of the phytoplankton and zooplankton population, and  is the harvesting effort.
The initial conditions of system (2) take the following form where  1 (),  2 (),  3 () ∈ ([−, 0],  3 + ), the Banach space of continuous functions mapping the interval [−, 0] into  3  + , where  3 + = {( 1 ,  2 ,  3 ) :   ≥ 0,  = 1, 2, 3}.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.

Preliminary Results of System without Delay
. .Positivity and Boundedness.When  = 0, system (2) is The initial conditions for system (4) take the following form (5) Lemma 1.If  ≥ 1, solutions of system ( ) with initial conditions are defined on [0, +∞) and remain positive for all  ≥ 0.
Proof.From the first equation of system (4), we obtain Hence, we obtain From the second equation of system (4), we have Since  3 (0) ≥ 0, the third equation of system (4) shows that by a standard comparison argument Theorem 2. All solutions of system ( ) with initial conditions are bounded for all  ≥ 0.
. .Equilibria and eir Local Stability.In this subsection, we will discuss the existence of equilibria of system (4) and their local asymptotical stability.
The zooplankton free equilibrium is  1 ( 1 ,  1 , 0), where Since ), then there is only one possible equilibrium involving phytoplankton but not zooplankton.The coexistence equilibrium is  * ( * ,  * ,  * ), where  * ,  * , and  * satisfy the following equations From the third equation of ( 14), we have If  1 −  > 0, and △ = ( 1 − ) 2 − 4( +  2 ) 2 > 0, then (15) has exactly two positive real roots And if  1 −  > 0, and △ = 0, then (15) has two equal positive real roots Again from the first and second equations of ( 14), we have If In order to discuss the existence and the local stability of equilibria of system (4), some assumptions are given.

Hopf Bifurcation and Analysis of System with Delay
. .Hopf Bifurcation.In this part, we will only investigate the effect of time delay on the dynamics of system (2) about the coexistence equilibrium  * 1 .For convenience, we denote the coexistence equilibrium  * 1 =  * .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 obtained where Obviously, the characteristic equation of system (25) around  * takes the following form where For  ̸ = 0, 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  = 0 is not a root of ( 27) when  1 ̸ = , and then the imaginary axis cannot be crossed by  = 0 for any  > 0. And due to Corollary 2.4 [32], a stability switch (or a cross of the imaginary axis) occurs with  = ± for some real  > 0.
Now, we assume that  =  ( > 0) is a root of ( 27).Separating real and imaginary parts, we have Adding up the squares of both equations, we obtain By denoting  =  2 , (30) becomes where 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  3 < 0, ( ) has at least one positive root.
. .Properties of Hopf Bifurcation.From Theorem 8, we have obtained the conditions for the existence of Hopf bifurcation when  =  0 .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].

By above definitions and conditions, we can compute
where where (74) Based on above analysis, the following values can be given By computing (75), we have the following theorem.
. .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 [   , +∞],  = 1, 2, 3;  = 1, 2, . . . .This includes the values that are not necessarily near the local Hopf bifurcation values.
We rewrite system (4) as u () = (()).Suppose that () is nonconstant periodic solution for system (4); then () = ( 1 ,  2 ,  3 ) is nonconstant periodic solution of the following equations From the Lemma 10, we know that the periodic orbits of system (85) corresponding to () are included in the following area where  and  are the boundedness of the periodic solutions of system (2) in the .
The second compound system is We select vector module for  3 as follows Then, the Lozinski ǔ measure ( [2] ()) of second additive compound matrix corresponding to a vector module above is  ( [2] ()) = max { Thus, if ( 4 ) holds, there exists an  such that () ≤ − < 0 for all  ∈ .
This shows that the conclusion of Lemma 11 follows from Corollary 3.5 in [38].

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 as where  is the constant harvesting cost per unit effort,  1 and  2 are the constant price per unit biomass of the phytoplankton and zooplankton, and V 1 , V 2 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 0 < () <  max ; here,  max is the maximum effect capacity of the harvesting industry.We want to find the optimal harvesting effect  * such that where  is the control set defined by where   =   (),  = 1, 2, 3, are adjoint variables.Transversality conditions are   (  ) = 0,  = 1, 2, 3.
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 [0,   ] satisfying the following three functions.The state equation is   () =   ( () ,   () ,  () ,  ()) , the optimality condition is and the adjoint equation is The condition for the optimal harvesting can be obtained from (105), which is Based on the above analysis, we have the following theorem.
Again, through the above parameter values, the properties of bifurcating periodic solutions at  0 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.
Nextly, we consider a set of parametric values to illustrate optimal control problem as follows:  0 = 1,  = 0.485,  = 0.388,  1 = 0.0555,  1 = 0.35,  = 0.07,  = 0.76,  1 = 0.3,  1 = 0.66,  = 0.03,  = 0.085,  2 = 0.1,  = 5,  = 0.1,  = 0.2,  = 0.2, V 1 = 0.4, V 2 = 0.7,  1 = 1,  2 = 5.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.

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 is where  1 and  2 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].
are the Hopf bifurcation points of system (2)} .

Figure 5 :
Figure 5: Optimal plankton harvesting with respect to different parameters: (a) the constant input of nutrient concentration  0 , (b) the half saturation constant , (c) the rate of toxin liberation .

Figure 6 :
Figure 6: Diagrams for the state variables with control for system (4): (a) the nutrient, (b) the phytoplankton population, (c) the zooplankton population.

Figure 8 :
Figure 8: Variation of the optimal effort  with respect to time with different  in system (2).