DYNAMICS OF A NUTRIENT-PLANKTON MODEL WITH DELAY AND TOXICITY

In this paper, a dynamical system exploring the nutrient-plankton interaction has been studied to determine the impact of toxin liberation delay. It is well known that the toxins liberated by phytoplankton species are harmful to the growth and the life cycle of zooplankton species. Moreover, the process of toxin liberation is not immediate, but it follows some time delay. We have observed many significant features of the given model system like boundedness, positivity, Hopf-bifurcation and its direction, etc. From the analysis of this model, it is observed that the toxin liberation delay can include complexity in the system as time delay passes through its critical value. All analytical results are verified through numerical simulations, and some significant findings are interpreted from the ecological point of view.


INTRODUCTION
Ecology is the branch of science, which studies the pattern of the interrelationship of different organisms and their interaction with the environment. The primary concern of ecological research is the deep study of those parameters or factors which influence the interactivity of organisms and their relationship with the environment to get results that help to the endurable progress of the ecosystem. The study of the plankton ecosystem is significant for ecological balance because 71 percent of our biosphere is covered with water. Plankton species (phytoplankton, zooplankton) are the crucial link in the pelagic food chain, generate about 80 percent oxygen of earth's atmosphere and biological wealth for whole water bodies.
The appearance and disappearance in the concentration of the phytoplankton population are known as bloom. The blooms produced by toxin-producing phytoplankton (TPP) are known as Harmful algal blooms (HABs). The HABs are the crucial ecological problem, as it can cause serious harm to see organisms, environments, and economies. Moreover, the toxicant produced by TPP is also very dangerous as this toxicant may regulate the growth of its predator population. Thus, it is fascinating to study the effects of toxicants on biological organisms.
Researchers in [1,2] have proposed the plankton nutrient interaction models to study the relation and importance of nutrients in the growth and life cycle of plankton species. In [3,4], Raun et al. have analyzed the complexity of the nutrient-plankton system when the factor of toxicity is also present in the marine ecosystem. [25,26,27,28] have developed some mathematical models involving nutrients and plankton population to study the stability and instability of these systems. Different plankton dynamics have been presented to investigate the effects of toxin and nutrients on the marine ecosystem [5,6,7,8]. The toxins produced by the phytoplankton species help in reducing the predation effect of the zooplankton, which results in the termination of planktonic blooms [9,10]. It is also notable that zooplankton species try to avoid those regions where the concentration of TPP is dense.
Nothing is instantaneous in the real world, every biological system is delayed by some time lag, which can convert the stable dynamical systems to unstable ones [11,12,13,21,22,23,32,33]. Therefore, the study of time delay in dynamical systems has become a crucial field of research.
Many mathematical models have been established to discuss the impacts of different time delays (predation, maturation, gestation, and toxin liberation delay, etc.) on the plankton dynamics [11,12,13,14,15,29,30]. Sharma et al. [21,22,23] have analyzed some ecological dynamics showing plankton interaction with multiple delays and observed that these dynamics become unstable, and a Hopf-bifurcation occurs at the interior steady state as the time lag crosses its critical value. Ecologists in [16] have studied a delayed model of phytoplankton and zooplankton interaction to investigate the impact of toxin liberation delay in the formation of HABs. [17,18] have extended the research of [16] and determine the global stability of the dynamical system involving the zooplankton population and TPP. It is observed that sufficient research has been made by scientists to understand the effect of toxin liberation delay (TLD) on the plankton dynamics and its consequences on the environment [21,22,23]. But, the impact of toxin liberation delay and instantaneous recycling of dead mass of both phytoplankton and zooplankton on the nutrient-plankton system is rarely observed by us. The main motive of this analytical study is to examine the impact of the (TLD) on the nutrient-plankton dynamics. So, we have proposed a delayed nutrient-plankton dynamical system involving liquefied nutrient (N(t)), phytoplankton (x(t)), and zooplankton (y(t)), respectively. We organize this study as follows. The ecological model, its assumptions, boundedness, positivity, and persistence are discussed in Section 2, followed by the stability analysis of the plankton dynamics in Section 3.
In Section 4, the direction and stability of the bifurcating solutions are obtained. All analytical results of our study are verified through numerical simulations in Section 5. The manuscript ends with certain outcomes in the concluding Section 6.

THE ECOLOGICAL MODEL
We have proposed our ecological dynamical system by assuming certain assumptions as follow; Let N(t), x(t), and y(t) denote the biomass of liquefied nutrient, phytoplankton, and zooplankton population, respectively, at time t. The phytoplankton depends upon nutrients for their growth with Holling-I type interaction. Some phytoplankton species like Cylindrospermopsis, protoperidinium, and Karenia produce toxins such as Anatoxin-a, Azaspiracid, Brevetoxin, and Microcystins, etc. In these toxins, some are hepatotoxins and others are neurotoxins, which cause mortality of the predator zooplankton. The process of toxin liberation is not immediate.
Instead, it follows some time delay and the ecological importance of TLD lies in the fact that it is the time required by TPP to become mature enough to avoid the harvesting impact of predator zooplankton. The dead mass of phytoplankton and zooplankton species is converted into nutrients. Based upon these assumptions, our ecological model is represented by the following set of simultaneous equations, where the biological interpretation of all the parameters is as follows: N 0 is the natural availability of nutrients in the water, a is the washout rate, and b denotes its uptake rate by phytoplankton. The parameters k 1 and k 2 represent the recycling rate of the dead mass of phytoplankton and zooplankton species, respectively. The death rate of phytoplankton is denoted by b 1 and the natural mortality rate of zooplankton species is represented by α 2 . The parameter α 1 is a maximum conversion rate of nutrients for the growth of phytoplankton and β 1 is the maximal conversion rate of phytoplankton for the growth of zooplankton. The parameter β represents the zooplankton's maximal ingestion rate, ρ is the rate of toxin secreted by TPP, γ is the half-saturation constant, and τ is the time delay. (1)

Positivity and Boundedness
The following theorems show that the system dynamics (1) is biological valid.
As t → ∞, we have V (t) ≤ N 0 ψ , Therefore, all solutions of (1) are bounded for 0 ≤ V (t) ≤ N 0 ψ . Hence, all the results of the dynamical system (1) are lying in the octant, [24] gives us lim inf t→∞ V (t) ≥ N 0 Θ = M 1 (say) Hence, we get the required result.

DYNAMICAL BEHAVIOR OF THE MODEL SYSTEM ABOUT DIFFERENT EQUILIB-RIUM POINTS
The system (1) has three steady states namely; The axial equilibria E 1 = ( N 0 a , 0, 0), which always exists, a zooplankton free equilibrium When τ = 0, the equation (2) can be written as, The interior equilibrium E * is LAS if H 1 holds true (due to Routh-Hurwitz criterion). Now, considering time delay τ as bifurcation parameter, we shall observe its effects on the behavior of the given plankton dynamics (1) around feasible steady state E * . Next, the equation (2) can be written in a second order exponential polynomial in λ as, where I 1 (λ , τ) = λ 3 + Aλ 2 + Bλ +C and I 2 (λ , τ) = (D + Eλ ).

DIRECTION OF PERIODIC TRAJECTORIES
Presently, the stability, direction and period of bifurcated periodic trajectories will be determined, using theorems given in [19,20].

NUMERICAL SIMULATION
In the present section, we will verify our analytical results through numerical examples.
Firstly, we take a set of parametric values axis which is clearly locally asymptotically stable as condition N 0 < ab 1 α 1 i.e (3 < 3.080) stated in proposition 3.1. hold good. Next, when we take the values of the parameters as, We get the equilibrium point E 2 (0.5505, 1.8306, 0) and it can be verified from proposition 3.1.
Thus, the transversality condition is evidently satisfied, which ensures the occurrence of Hopf bifurcation at τ 0 = 6.4. Therefore, positive equilibrium E * remains stable for 0 ≤ τ ≤ 6.4 and a Hopf-bifurcation appears when τ passes through its thresh hold value τ 0 = 6.4. Fig.2., and Fig.3. show the existence of stability, Hopf-bifurcation and a limit cycle around E * , respectively. A bifurcation diagram w.r.t. time delay of different populations i.e. nutrient, phytoplankton, and zooplankton is also shown in Fig.3(c). Thus, our numerical results show that when the toxication delay crosses its threshold value, the given system starts to oscillate periodically and exhibits a Hopf-bifurcation. Biologically, these periodic oscillations can be interpreted as the occurrence of plankton bloom. So, it is observed that the delay in the process of toxin liberation can destabilize the system with the existence of planktonic bloom.

CONCLUSION AND DISCUSSION
In this manuscript, we study the impact of toxin libration delay on a 3-D nutrient-plankton dynamics. It is assumed that the process of toxin libration by the phytoplankton population is not immediate. Rather, it is followed by some discrete-time variation, which is known as toxin liberation delay. Initially, we have discussed the stability of the non-delayed system dynamics under certain conditions around E 1 and E 2 (preposition 3.1.). It is shown that the steady state E * is locally asymptotically stable if (H 1 ) holds good. In the presence of a toxin liberation delay, it is determined that the dynamical system remains stable for τ ∈ [0, 6.4] (see Fig.2.). It enters into a Hopf-bifurcation when toxin libration delay τ passes through its threshold value τ 0 = 6.4 (see Fig.3.) with the existence of periodic trajectories around E * . Lemma 4.2. proves that the Hopf-bifurcation is supercritical. Thus, it is investigated that the predation delay beyond the thresh hold value of τ can include excitability in the dynamical system with the existence of planktonic bloom.

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.