Dynamics of a Seasonally Forced Phytoplankton-Zooplankton Model with Impulsive Biological Control

This paper investigates the dynamics of a seasonally forced phytoplankton-zooplankton model with impulsive biological control. It shows that the periodic eradicated solution is unstable. Further, the condition for permanence of the system is established by relations between the model parameters and the intensity of the impulses. The numerical analysis is performed to study the effect of seasonality and impulsive perturbations on plankton dynamics.The numerical results imply that the seasonal forcing can trigger more periodic mode and the impulsive period for control of the size of phytoplankton is more practicable to the system than the impulsive release of zooplankton. These conclusions provide a better understanding of controlling harmful algae blooms.


Introduction
Algae are very diverse and found almost everywhere on the planet.They play an important role in marine, freshwater, and some terrestrial ecosystems.On one hand, phytoplankton are critical to sustain most aquatic food chains and produce half of the world's oxygen in the process of photosynthesis [1].On the other hand, phytoplankton populations can grow explosively and lead to severe oxygen depletion in the relevant waters when harmful algal blooms occur.As a consequence, human activities would be limited and economy would suffer.Therefore, in order to prevent and control harmful algae blooms a better understanding of the mechanisms that trigger the occurrence or explain the absence of a phytoplankton bloom is of considerable significance.Truscott and Brindley [2] were the first to offer a model considering the preypredator system of phytoplankton and zooplankton as a nonlinear excitable system to explain the dynamics of harmful algal blooms.In the framework, the evolution of phytoplankton and zooplankton populations is formulated by where  and  represent population densities of phytoplankton and zooplankton, respectively.The first term in the right side of the first equation in (1) is the logistic growth function.The term  2 /( 2 +  2 ) is called Hollings type III grazing function [3]. is the maximum growth rate of phytoplankton when  is small,  is the environmental carry capacity of phytoplankton,   reflects the maximum specific predation rate,  governs how quickly that maximum is attained as prey densities increase,  denotes the ratio of biomass consumed to biomass of new herbivores produced, and  measures the zooplankton death rate.If we quote the typical parameter values [2,4] for system (1), then the positive equilibrium of ( 1) is asymptotically stable.Assume that the initial phytoplankton concentration is fixed in the stationary value.Then, a phytoplankton bloom that is followed by a delayed zooplankton bloom is triggered by suppressing the initial zooplankton concentration sufficiently far below the stationary value [4].
As a classical predator-prey model, it has been extensively discussed by many researchers [5][6][7][8][9].In fact in spite of noticing the remarkable impact of seasonal forcing on the phytoplankton birth-rate by Truscott and Brindley, there are few models explicitly taken into account.Even though Gao et al. [10] had considered the effect of seasonality and periodicity on the growth rate of phytoplankton in their model, the growth rate taken as a sinusoidal forcing function of time could not reflect sufficiently the explosive growth of phytoplankton.Freund et al. [4] filled the gap, presenting and discussing simulation results of the seasonally forced Truscott-Brindley model with an exponential function depicting the explosive growth.Later, Luo [11] developed the seasonally forced Truscott-Brindley model by including the growth rate and the intrinsic carrying capacity of phytoplankton changing with respect to time and nutrient concentration.For simplicity we only consider the maximum growth rate  in the model (1) as periodically varying function of time due to seasonal variation.And, we adopt the same relation between the phytoplankton growth rate and seasonally varying temperature as Freund et al. [4] have suggested.The effect of changing temperature on growth rate of phytoplankton is where  is the average value of the intrinsic growth rate of phytoplankton,  10 is assumed to be a constant which asserts that a change of the temperature by 10 ∘ C will multiply the rate at mean temperature,  denotes time (days), () =  + Γ sin( + ) represents temperature on time  which is adapted from a fit by using average temperature data,  is the average temperature in one cycle, Γ denotes the amplitude of temperature,  is the angular velocity, and  is the initial phase.For the sake of simplification, the intraspecific competition of phytoplankton is not affected by seasonally varying temperature.The use of impulsive control for ecological systems is proved to be one of the most effective methods and has received much attention from both ecologists and applied mathematicians [12][13][14][15][16].However, almost all the work on the Truscott-Brindley models neglects the impulsive biological control of phytoplankton.Thus, we periodically release zooplankton in laboratories at a constant to reduce the population level of phytoplankton by grazing.Keeping these aspects in view, we establish the following model: where Δ = ( + )−(), Δ = ( + )−(),  is the period of the impulsive effect,  > 0 denotes the concentration change of zooplankton by releasing which is determined by the maximum amount of zooplankton produced by laboratories and  ∈ , and  is the set of all nonnegative integers.In system (3), all parameters are supposed to be positive constants.
This paper is organized as follows.In Section 2, we study the dynamics of system (3) without impulsive effect.In Section 3, we mainly focus on system (3) and obtain the stability of phytoplankton-eradication periodic solution and the condition of permanence of (3).In Section 4, the numerical analysis is performed to investigate the dynamics of (3).Finally, we close with a discussion in Section 5.

Dynamical Properties of (3) without
Impulsive Effect In order to analyze to the dynamical behavior of (3) without impulsive effect, we first consider a periodically logistic equation: where () and () are periodically continuous functions defined on  + with the common period  > 0. According to the results of [11,17], we obtain the following conclusions.
Theorem 5 (see [11]).Assume that then population of zooplankton tends to extinction; more precisely we have for each solution of (6), where  * () is the uniquely periodic solution of ( 5) established by Theorem 2.
Proof.To investigate the local stability of periodic solution (0, ()), we will use the method of small amplitude perturbations.To this purpose, define where () and V() represent the small amplitude perturbations.

Numerical Analysis
In this section, we will study the influence of seasonal forcing and impulsive perturbations.We quote typical parameter values [4]:  = 0.3/day,  = 108 g/L,   = 0.7/day,  = 5.7 g/L,  = 0.05,  = 10 ∘ C, Γ = 6 ∘ C,  = 2/(365 days), /(2) = 0.59, and  10 = 2. Based on Theorems 2 and 4, we see that the sufficiently small value of  leads to be permanent for (6).For enough large value of , in contrast, (6)    a periodic solution that corresponds to zooplankton eradication.Figure 1(b) illustrates our conclusion.By comparison with Figure 1, a seasonal forcing of the phytoplankton growth rate can trigger a bloom mode.This claim is pointed out to be important for considering the seasonal variation in study of the Truscott and Brindley model.First, we analyze the effect of  on the long-run dynamics of system (3).For fixed parameter values  = 0.012/day and  = 12 days, the bifurcation diagrams with respect to  have been plotted in Figure 2(b).It needs to highlight that with the change of the parameter value  system (3) undergoes period-3 attractors and later enters into chaotic behaviors.We know that system (3) without impulsive release of zooplankton presents the harmful algae bloom mode (Figure 1(b)).However, it can be seen in Figure 2(b) that if the impulsive biological control is considered, then the nonbloom mode will occur.That is to say that the occurrence of phytoplankton blooms is successfully prevented.Besides, for the purpose of studying the influence of seasonal forcing of the phytoplankton growth rate on system (3), the bifurcation diagrams of system (3) without considering seasonality are shown in Figure 2(a).As we can see in Figure 2(a), system (3) without seasonal fluctuations in the growth rate of phytoplankton exhibits relatively simple dynamical behaviors: period-1 attractor.It can be observed in Figure 2 that adding seasonal forcing reduces the maximum density of phytoplankton and enhances the zooplankton population size.What is more is that adding seasonal forcing produces chaos.It is noted in Figure 2(b) that although system (3) shows chaotic dynamics, the phytoplankton population does not trigger bloom mode.Second, we will investigate the influence of impulsive period on the population dynamics of system (3).Thus, we set  = 0.012/day and  = 0.5 g/L.It follows that system (3) shown in Figure 3(a) exhibits the only dynamical behavior: period-1 attractor when we do not consider seasonality.It is worth to be stressed that the max density of phytoplankton is monotonically increasing with increase of , and the max density of zooplankton is monotonically decreasing.In Figure 3(a) we need to emphasize that changing of  system (3) without seasonality does not exhibit the phytoplankton bloom.Conversely, if the seasonal factor is included, Figure 3(b) illustrates relatively complex dynamical behavior for system (3).Figures 4 and 5 display the detailed cases.It is worth pointing out that in Figure 4 when the time of releasing presents a 5-day delay, the  periodic solution inflates to +1 until  = 10.If  ≥ 57 days, the phytoplankton population of system (3) exhibits bloom dynamics (Figure 5).By comparison with Figure 4, it concludes that a seasonal forcing augments fluctuation of population.Further analysis shows that  is more practicable to system (3) as a control parameter than .Thus, we suggest that in order to control the size of phytoplankton the impulsive period  should be selected to be feasible.

Discussion
We have studied the dynamics of a seasonally forced phytoplankton-zooplankton model with impulsive biological control.By control of the releasing period of zooplankton, the size of phytoplankton could be reduced.However, the amount of  is determined by laboratories and this kind of production is limited.Therefore, system (3) cannot be used for larger areas of water.We suggest that system (3) should be suitable for smaller areas of water such as urban lakes.In this paper we consider the dynamic behaviors of the system when periodic forcing is solely superimposed on the growth rate of the phytoplankton.In fact, there are a number of ways to apply periodic forcing in an ecological model [19].Thus, in the future, applying periodic forcing to the Truscott and Brindley model with impulsive biological control will be considered.From the above, it is known that releasing zooplankton to control the size of phytoplankton is limited.Therefore, we need study other effective ways to reduce harmful