Anal Chatterjee

Bottom up and top down effect on toxin producing phytoplankton and its consequence on the formation of plankton bloom

Abstract We consider a plankton–nutrient interaction model consisting of phytoplankton, zooplankton and dissolved limiting nutrient with general nutrient uptake functions and instantaneous nutrient recycling. In this model, it is assumed that phytoplankton releases toxic chemical for self defense against their predators. The model system is studied analytically and the threshold values for the existence and stability of various steady states are worked out. It is observed that if the maximal zooplankton conversion rate crosses a certain critical value, the system enters into Hopf bifurcation. Finally it is observed that to control the planktonic bloom and to maintain stability around the coexistence equilibrium we have to control the nutrient input rate specially caused by artificial eutrophication. In case if it is not possible to control the nutrient input rate, one could use toxic phytoplankton to prevent the recurrence bloom.

EFFECT OF DILUTION RATE ON THE PREDICTABILITY OF A REALISTIC ECOSYSTEM MODEL WITH INSTANTANEOUS NUTRIENT RECYCLING

An analysis is made on a three dimensional mathematical model for the interaction of nutrient, phytoplankton and their predator zooplankton population in an open marine system. For a realistic representation of the open marine plankton ecosystem, we have incorporated various natural phenomena such as dissolved limiting nutrient with general nutrient uptake function, nutrient recycling, interspecies competition and grazing at a higher level. For the model with constant nutrient input and different constant washout rates, conditions for boundedness of the solutions, existence and stability of non negative equilibria, as well as persistence are given. The model system is studied analytically and the threshold values for the existence and stability of various steady states are worked out. It is observed that if the dilution rate of nutrient crosses certain critical value, the system enters into Hopf-bifurcation. Finally, it is observed that planktonic bloom can be controlled and stability around the equilibrium of coexistence can be obtained if the dilution rate of phytoplankton population is increased. Computer simulations have been carried out to illustrate different analytical results.

Plankton nutrient interaction model with effect of toxin in presence of modified traditional Holling Type II functional response

A three-dimensional plankton-nutrient interaction model is proposed and analysed which mediated by a toxin-determined functional response. The new functional response is a modification of the traditional Holling Type II functional response by explicitly including a reduction in the consumption of phytoplankton by the zooplankton due to chemical defenses. Our analysis leads to different thresholds in terms of model parameters to find out different steady-states behaviour. It is found that constant nutrient input and dilution rate of nutrient influence the plankton ecosystem model and maintain stability around the coexistent equilibrium. Our observations indicate that if the constant nutrient input crosses a certain critical value, the system enters into Hopf bifurcation. In addition, we have studied the direction of Hopf bifurcation by applying the normal form method. The maximal amount of toxin of phytoplankton species plays a crucial role to change the steady-state behaviour. Computer simulations illustrate the results.

COEXISTENCE OF PLANKTON MODEL WITH ESSENTIAL MULTIPLE NUTRIENT IN CHEMOSTAT

We consider a phytoplankton–zooplankton interaction model which depends on two complementary nutrients. For a realistic representation in chemostat plankton ecosystem, we have incorporated various natural phenomena such as dissolved limiting nutrients with nutrient uptake functions and yield constants. For the model with two different constant nutrient inputs with same constant washout rate, existence and stability of non-negative equilibria as well as persistence are given. We analyze the behavior of solution of model in order to answer the biological question and seek to determine the limiting behavior of the surviving organisms and the nutrients. It is observed that the constant nutrient inputs of two complementary nutrients play an important role to change steady state behavior of the system. Further it is observed that if the dilution rate of chemostat crosses certain critical value, the system enters into Hopf-bifurcation. Finally, we have derived the explicit algorithm which determines the direction of Hopf-bifurcation. Computer simulations have been carried out to illustrate different analytical results.

A Plankton-Nutrient Model with Holling Type III Response Function

A plankton model including the latest mathematical features introduced in a very recent specialistic contribution showing the emergence of the Holling type III response function is here formulated and developed in its deterministic and stochastic counterparts. The effects of additional food source and harvesting rate of zooplankton are analyzed. The results indicate that if the intensity of environmental fluctuation is kept under a certain threshold value, the control procedure proposed in the deterministic case is also valid in the presence of environmental disturbances.

Role of constant nutrient input and mortality rate of plankton population in marine plankton ecosystem

In this paper, we have proposed a plankton-nutrient interaction model consisting of phytoplankton, zooplankton and dissolved limiting nutrient with type II functional response and instantaneous nutrient recycling. The model system is studied analytically and the threshold conditions for the existence and stability of various steady states are worked out. It is shown that the positive equilibrium loses its stability if the constant nutrient input crosses a certain critical value and the system enters into Hopf bifurcation that induces oscillations of the populations. Our observations indicate that mortality rate of plankton population plays an important role to change steady state to oscillatory behaviour of the system. We derive the bifurcation scenarios when both the parameters constant nutrient input and mortality rate of plankton vary together. It is interesting to observe that the additional food source for zooplankton population plays a vital role to change the steady state behaviour. The results are illustrated by numerical simulations.

Interspecies competition between prey and two different predators with Holling IV functional response in diffusive system

This paper deals with a prey-middle predator-top predator ecosystem model with Holling type IV predator response in the unreserved zone. The model system is studied analytically and the threshold values for the existence and stability of various steady states are worked out. The global stability analysis is carried out. It is observed that if the intrinsic growth rate of prey population crosses a certain critical value, the system enters into Hopf bifurcation. The existence of bionomic equilibrium of the system has been discussed. Further, we study a path of optimal harvesting policy by introducing the Pontryagins maximum principle. Moreover we have found out a condition for diffusive instability of a locally stable equilibrium. Finally, some numerical simulations are performed to justify analytical findings.

Dynamics of the interaction of plankton and planktivorous fish with delay

This paper is devoted to the study of a plankton–fish ecosystem model. The model represents the interaction between phytoplankton, zooplankton, and fish with Holling II functional response consisting of carrying capacity and constant intrinsic growth rate of phytoplankton. It is observed that if the carrying capacity of phytoplankton population crosses a certain critical value, the system enters into Hopf bifurcation. We have introduced discrete time delay due to gestation in the functional response term involved with the growth equation of planktivorous fish. We have studied the effect of time delay on the stability behavior. In addition, we have obtained an estimate for the length of time delay to preserve the stability of the model system. Existence of Hopf bifurcating small amplitude periodic solutions is derived by considering time delay as a bifurcation parameter. It is observed that constant intrinsic growth rate of phytoplankton and mortality rate of planktivorous fish play an important role in changing one steady state to another steady state and oscillatory behavior of the system. Computer simulations illustrate the results.