Samares Pal

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.

ROLE OF MIGRATORY BIRDS UNDER ENVIRONMENTAL FLUCTUATION — A MATHEMATICAL STUDY

Chatterjee and Chattopadhyay1 proposed and analyzed a simple one season eco-epidemiological model to understand the role of migratory birds in an eco-epidemiological scenario. In this paper, we have modified their model taking into account standard incidence as horizontal incidence. We have studied the proposed model both in the absence and in the presence of environmental disturbances. From our analysis, we observe that in the absence of the environmental fluctuations, the disease spread by migratory birds may be controlled by either decreasing the contact rate, or increasing the recovery or predation rates. Further, we also observe that if the artificial eutrophication (or intensity of white noise) is kept under a certain threshold value, the control procedure proposed in the deterministic case is also valid in the presence of environmental disturbances.

Stage-structured cannibalism with delay in maturation and harvesting of an adult predator

A three-dimensional stage-structured predator–prey model is proposed and analyzed to study the effect of predation and cannibalism of the organisms at the highest trophic level with non-constant harvesting. Time lag in maturation of the predator is introduced in the system and conditions for local asymptotic stability of steady states are derived. The length of the delay preserving the stability is also estimated. Moreover, it is shown that the system undergoes a supercritical Hopf bifurcation when the maturation time lag crosses a certain critical value. Computer simulations have been carried out to illustrate various analytical results.

Dynamics of a stage-structured system with harvesting and discrete time delays

A two-dimensional single-species stage-structured model is proposed and analyzed to study the effect of intraguild predation with harvesting of the adult species. Time lags in reproduction and maturation of the organism are introduced in the system and conditions for local asymptotic stability of steady states are derived. The length of the delay preserving the stability is also estimated. It is found that there are stability switches for time delays, and Hopf bifurcations when time delays cross through some critical values. The stability and direction of the Hopf bifurcations are determined by applying the normal form method and the center manifold theory. Computer simulations have been carried out to illustrate various analytical results.

The role of space in stage-structured cannibalism with harvesting of an adult predator

The dynamics of a predator-prey reaction-diffusion system with Holling type III functional response, where the predator has two stages, a juvenile stage and an adult stage, is proposed and analyzed to study the effect of predation with non-constant harvesting of an adult predator. Our analysis leads to different thresholds in terms of the model parameters acting as conditions under which the organisms associated with the system cannot thrive even in the absence of predation. Local stability of the system is obtained in the absence of one or more organisms and in the presence of all the organisms. Moreover, it is shown that the system undergoes Hopf bifurcation when the intrinsic growth rate of herbivorous prey crosses certain critical value. Computer simulations have been carried out to illustrate various analytical results.

Hysteresis in coral reefs under macroalgal toxicity and overfishing

Macroalgae and corals compete for the available space in coral reef ecosystems.While herbivorous reef fish play a beneficial role in decreasing the growth of macroalgae, macroalgal toxicity and overfishing of herbivores leads to proliferation of macroalgae. The abundance of macroalgae changes the community structure towards a macroalgae-dominated reef ecosystem. We investigate coral-macroalgal phase shifts by means of a continuous time model in a food chain. Conditions for local asymptotic stability of steady states are derived. It is observed that in the presence of macroalgal toxicity and overfishing, the system exhibits hysteresis through saddle-node bifurcation and transcritical bifurcation. We examine the effects of time lags in the liberation of toxins by macroalgae and the recovery of algal turf in response to grazing of herbivores on macroalgae by performing equilibrium and stability analyses of delay-differential forms of the ODE model. Computer simulations have been carried out to illustrate the different analytical results.

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.

Allelopathy of plasmid-bearing and plasmid-free organisms competing for two complementary resources in a chemostat

We consider a model of competition between plasmid-bearing and plasmid-free organisms for two complementary nutrients in a chemostat. We assume that the plasmid-bearing organism produces an allelopathic agent at the cost of its reproductive abilities which is lethal to plasmid-free organism. Our analysis leads to different thresholds in terms of the model parameters acting as conditions under which the organisms associated with the system cannot thrive even in the absence of competition. Local stability of the system is obtained in the absence of one or both the organisms. Also, global stability of the system is obtained in the presence of both the organisms. Computer simulations have been carried out to illustrate various analytical 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.