Hopf bifurcation and bistability of a nutrient–phytoplankton–zooplankton model

In this paper we consider a nutrient–phytoplankton–zooplankton model in aquatic environment and study its global dynamics. The existence and stability of equilibria are analyzed. It is shown that the system is permanent as long as the coexisting equilibrium exists. The discontinuous Hopf and classical Hopf bifurcations of the model are analytically verified. It is shown that phytoplankton bloom may occur even if the input rate of nutrient is low. Numerical simulations reveal the existence of saddle-node bifurcation of nonhyperbolic periodic orbit and subcritical discontinuous Hopf bifurcation, which presents a bistable phenomenon (a stable equilibrium and a stable limit cycle).     

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

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.     

MODELING NUTRIENT (DISSOLVED INORGANIC NITROGEN) AND PLANKTON DYNAMICS AT SAGAR ISLAND OF HOOGHLY–MATLA ESTUARINE SYSTEM,WEST BENGAL,INDIA

Abstract Degradation of litter from mangrove forests adjacent to the creeks at Sagar Island of the Hooghly–Matla estuarine ecosystem is one of the principal sources of nutrient to the estuary. Nutrients augment the growth of phytoplankton, which in turn stimulates the production of zooplankton. Zooplankton serves as major food source for fish population of this estuarine system. Here, a dynamic model with three state variables (nutrient, phytoplankton, and zooplankton) is proposed using nitrogen (mgN/l) as currency. Input of dissolved inorganic nitrogen as nutrient, water temperature, surface solar irradiance, and salinity of upstream and downstream of the estuary, collected from the field, are incorporated as graph time functions in the model. Calibration and validation are performed by using collected data of two consecutive years. Model results indicate that the growth of zooplankton and phytoplankton are enhanced by increase in nutrient input in the system. Zooplankton biomass is affected by decrease in the salinity of the estuary. Sensitivity analysis results at ±10% indicate that maximum growth rate of phytoplankton (P_max) is the most sensitive parameter to the nutrient pool although growth rate of zooplankton (g_z) and half saturation constant for phytoplankton grazing by zooplankton (K_z) are most sensitive parameters to phytoplankton and zooplankton compartments, respectively. The model depicts the present status of plankton dynamics, which serve as major food resource for herbivorous and carnivorous fish species of the estuary. Effect of deforestation is tested in the model. Therefore, from management perspective, this model can be used to predict the impact of mangroves on nutrient and plankton dynamics, which will give complete information of both shell and fin fish productions in the estuary.     

Stability and bifurcation analysis of a prey–predator model with age based predation

It is observed that in large animals only adult predators take part in direct predation while suckling feed on milk of adult predators and juveniles are dependent on the dead prey stock killed by the adult predators. Some parts of the dead prey population is consumed by adult predators and remaining parts are consumed by juveniles and the remaining portion decays naturally. In light of this, a mathematical model is proposed to study the stability and bifurcation behaviour of a prey–predator system with age based predation. All the feasible equilibria of the system are obtained and the conditions for the existence of the interior equilibrium are determined. The local stability analysis of all the feasible equilibria is carried out and the possibility of Hopf-bifurcation of the interior equilibrium is studied. Finally, numerical simulation is conducted to support the analytical results.     

Harvesting of a phytoplankton–zooplankton model

Considering that some phytoplankton and zooplankton are harvested for food, a phytoplankton–zooplankton model with harvesting is proposed and investigated. First, stability conditions of equilibria and existence conditions of a Hopf-bifurcation are established. Our results indicate that over exploitation would result in the extinction of the population and an appropriate harvesting strategy should ensure the sustainability of the population which is in line with reality. Furthermore, the existence of bionomic equilibria and the optimal harvesting policy are discussed. The present value of revenues is maximized by using Pontryagin’s maximum principle subject to the state equations and the control constraints. We discussed the case of optimal equilibrium solution. It is found that the shadow prices remain constant over time in optimal equilibrium when they satisfy the transversality condition. It is established that the zero discounting leads to the maximization of economic revenue and that an infinite discount rate leads to complete dissipation of economic rent. Finally, some numerical simulations are given to illustrate our results.     

Stability and bifurcation in a two harmful phytoplankton–zooplankton system

In this paper, a mathematical model consisting of two harmful phytoplankton and zooplankton with discrete time delays is considered. We prove that a sequence of Hopf bifurcations occur at the interior equilibrium as the delay increases. Meanwhile, the phenomenon of stability switches is found under certain conditions. The direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions are determined by using the theory of normal form and center manifold. Numerical simulations are given to support the theoretical results.     

A delay model for viral infection in toxin producing phytoplankton and zooplankton system

An eco-epidemiological delay model is proposed and analysed for virally infected, toxin producing phytoplankton (TPP) and zooplankton system. It is shown that time delay can destabilize the otherwise stable non-zero equilibrium state. The coexistence of all species is possible through periodic solutions due to Hopf bifurcation. In the absence of infection the delay model may have a complex dynamical behavior which can be controlled by infection. Numerical simulation suggests that the proposed model displays a wide range of dynamical behaviors. Different parameters are identified that are responsible for chaos.     

Plankton growth dynamic driven by plankton body size in deterministic and stochastic environments

In this paper, we establish a new phytoplankton–zooplankton model by considering the effects of plankton body size and stochastic environmental fluctuations. Mathematical theory work mainly gives the existence of boundary and positive equilibria and shows their local as well as global stability in the deterministic model. Additionally, we explore the dynamics of V-geometric ergodicity, stochastic ultimate boundedness, stochastic permanence, persistence in the mean, stochastic extinction, and the existence of a unique ergodic stationary distribution in the corresponding stochastic version. Numerical simulation work mainly reveals that plankton body size can generate great influences on the interactions between phytoplankton and zooplankton, which in turn proves the effectiveness of mathematical theory analysis. It is worth emphasizing that for the small value of phytoplankton cell size, the increase of zooplankton body size can not change the phytoplankton density or zooplankton density; for the middle value of phytoplankton cell size, the increase of zooplankton body size can decrease zooplankton density or phytoplankton density; for the large value of phytoplankton cell size, the increase of zooplankton body size can increase zooplankton density but decrease phytoplankton density. Besides, it should be noted that the increase of zooplankton body size cannot affect the effect of random environmental disturbance, while the increase of phytoplankton cell size can weaken its effect. There results may enrich the dynamics of phytoplankton-zooplankton models.     

Modeling and analysis of a marine plankton system with nutrient recycling and diffusion

This article describes a nutrient‐phytoplankton‐zooplankton system with nutrient recycling in the presence of toxicity. We have studied the dynamical behavior of the system with delayed nutrient recycling in the first part of the article. Uniform persistent of the system is examined. In the second part of the article, we have incorporated diffusion of the plankton population to the system and dynamical behavior of the system is analyzed with instantaneous nutrient recycling. The condition of the diffusion driven instability is obtained. The conditions for the occurrence of Hopf and Turing bifurcation critical line in a spatial domain are derived. Variation of the system with small periodicity of diffusive coefficient has been studied. Stability condition of the plankton system subject to the periodic diffusion coefficient of the zooplankton is derived. It is observed that nutrient‐phytoplankton‐zooplankton interactions are very complex and situation specific. Moreover, we have obtained different exciting results, ranging from stable situation to cyclic oscillatory behavior may occur under different favorable conditions, which may give some insights for predictive management. © 2014 Wiley Periodicals, Inc. Complexity 21: 229–241, 2015     

Mathematical modelling on harmful algal blooms in the presence of toxic substances

Effect of toxin producing plankton and its control is an intriguing problem in marine plankton ecology. In this paper we have proposed a three-component model consisting of a non-toxic phytoplankton (NTP), toxin producing phytoplankton (TPP) and zooplankton (Z), where the growth of zooplankton species reduce due to toxic chemicals released by phytoplankton species. It is observed that the three components persist if the predation rate of zooplankton population on toxic phytoplankton is bounded in certain regions. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dynamical behavior of systems of two phytoplankton and one zooplankton populations with toxin producing phytoplankton

Three‐compartment mathematical models of non‐toxic phytoplankton (NTP), toxin producing phytoplankton (TPP), and zooplankton are proposed to explore the role of TPP in algal blooms. The mutual interference between predator zooplankton and avoidance of TPP by zooplankton are incorporated into the model. The NTP and TPP engage in exploit competition and the toxin produced by TPP has no effect on NTP. Using the concept of uniform persistence, we establish coexistence of NTP, TPP, and zooplankton in certain parameter regimes. We study the effects of mutual interference and avoidance by zooplankton upon the population interactions. In addition to the toxin producing mechanism, it is concluded that mutual interference of zooplankton is an important factor for diminishing harmful blooms. Copyright © 2017 John Wiley & Sons, Ltd.     

Simulation of a nonlinear stochastic ecology model

A nonlinear ecological system model is formulated as a stochastic process. The formulation traces a chemical nutrient as it undergoes random exchanges between the phytoplankton, the zooplankton, and the euphotic zone of an aquatic ecosystem. The Chapman-Kolmogorov equations are derived, and simulation techniques are used to study both single realizations and the statistical parameters which govern the system.     

Spatiotemporal pattern formation in a diffusive predator-prey system: an analytical approach

In this paper, we propose and analyse a mathematical model to study the mathematical aspect of reaction diffusion pattern formation mechanism in a predator-prey system. An attempt is made to provide an analytical explanation for understanding plankton patchiness in a minimal model of aquatic ecosystem consisting of phytoplankton, zooplankton, fish and nutrient. The reaction diffusion model system exhibits spatiotemporal chaos causing plankton patchiness in marine system. Our analytical findings, supported by the results of numerical experiments, suggest that an unstable diffusive system can be made stable by increasing diffusivity constant to a sufficiently large value. It is also observed that the solution of the system converges to its equilibrium faster in the case of two-dimensional diffusion in comparison to the one-dimensional diffusion. The ideas contained in the present paper may provide a better understanding of the pattern formation in marine ecosystem.     

Dynamical analysis of toxin producing Phytoplankton–Zooplankton interactions

In the present paper we consider a toxin producing phytoplankton–zooplankton model in which the toxin liberation by phytoplankton species follows a discrete time variation. Firstly we consider the elementary dynamical properties of the toxic-phytoplankton–zooplankton interacting model system in absence of time delay. Then we establish the existence of local Hopf-bifurcation as the time delay crosses a threshold value and also prove the existence of stability switching phenomena. Explicit results are derived for stability and direction of the bifurcating periodic orbit by using normal form theory and center manifold arguments. Global existence of periodic orbits is also established by using a global Hopf-bifurcation theorem. Finally, the basic outcomes are mentioned along with numerical results to provide some support to the analytical findings.     

Impact of noise in a phytoplankton-zooplankton system

In this paper, we investigate the dynamics of a delayed toxic phytoplankton-two zooplankton system incorporating the effects of Levy noise and white noise. The value of this study lies in two aspects: Mathematically, we first prove the existence of a unique global positive solution of the system, and then we investigate the sufficient conditions that guarantee the stochastic extinction and persistence in the mean of each population. Ecologically, via numerical simulations, we find that the effect of white noise or Levy noise on the stochastic extinction and persistence of phytoplankton and zooplankton are similar, but the synergistic effects of the two noises on the stochastic extinction and persistence of these plankton are stronger than that of single noise. In addition, an increase in the toxin liberation rate or the intraspecific competition rate of zooplankton was found to be capable to increase the biomass of the phytoplankton but decrease the biomass of zooplankton. These results may help us to better understand the phytoplankton-zooplankton dynamics in the fluctuating environments.     

A modified Leslie–Gower predator–prey model with prey infection

The disease effect on ecological systems is an important issue from mathematical and experimental point of view. In this paper, we formulate and analyze a predator–prey model for the susceptible population, infected population and their predator population with modified Leslie–Gower (or Holling–Tanner) functional response. Mathematical analysis of the model equations with regard to invariance of nonnegativity and boundedness of solutions, local and global stability of the biological feasible equilibria and permanence of the system are presented. When the rate of infection crosses a critical value, we determine that the strictly positive interior equilibrium undergoes Hopf bifurcation. From our numerical simulations, we observe that the predation rate also plays an important role on the dynamic behavior of our system.     

An epidemic model of a vector-borne disease with direct transmission and time delay

This paper considers an epidemic model of a vector-borne disease which has direct mode of transmission in addition to the vector-mediated transmission. The incidence term is assumed to be of the bilinear mass-action form. We include both a baseline ODE version of the model, and, a differential-delay model with a discrete time delay. The ODE model shows that the dynamics is completely determined by the basic reproduction number R₀. If R₀?1, the disease-free equilibrium is globally stable and the disease dies out. If R₀>1, a unique endemic equilibrium exists and is locally asymptotically stable in the interior of the feasible region. The delay in the differential-delay model accounts for the incubation time the vectors need to become infectious. We study the effect of that delay on the stability of the equilibria. We show that the introduction of a time delay in the host-to-vector transmission term can destabilize the system and periodic solutions can arise through Hopf bifurcation.     

Modeling and analysis of a prey–predator system with disease in the prey

A three dimensional ecoepidemiological model consisting of susceptible prey, infected prey and predator is proposed and analysed in the present work. The parameter delay is introduced in the model system for considering the time taken by a susceptible prey to become infected. Mathematically we analyze the dynamics of the system such as, boundedness of the solutions, existence of non-negative equilibria, local and global stability of interior equilibrium point. Next we choose delay as a bifurcation parameter to examine the existence of the Hopf bifurcation of the system around its interior equilibrium. Moreover we use the normal form method and center manifold theorem to investigate the direction of the Hopf bifurcation and stability of the bifurcating limit cycle. Some numerical simulations are carried out to support the analytical results.     

A simple mathematical model for seasonal planktonic blooms

We show how the inclusion of the defense strategy by different species can alter the prediction of simple models. One of the defense strategy by the phytoplankton population against their grazer is the release of toxic chemicals. In turn the zooplankton population reduces there predation rate over toxin producing phytoplankton (TPP) to protect themselves from those toxic chemicals. Thus, when the level of toxicity is high, the grazing pressure is low and when the level of toxicity is low or when the toxin is absent, the grazing pressure is high. Here we have considered a TPP–zooplankton system where the rate of toxin liberation and the predation rate vary with zooplankton abundance. We observe that our proposed model has the potential to show different dynamical behaviour that are similar to that seen in real‐world situations. Further, we consider three different functional forms for the distribution of the toxins and compare them using latin hypercube sampling technique and found that the functional forms seem to have no effect in determining the final outcome of the system. Copyright © 2009 John Wiley & Sons, Ltd.     

Dynamics of an intraguild predation food web model with strong Allee effect in the basal prey

Since intraguild predation (IGP) is a ubiquitous and important community module in nature and Allee effect has strong impact on population dynamics, in this paper we propose a three-species IGP food web model consisted of the IG predator, IG prey and basal prey, in which the basal prey follows a logistic growth with strong Allee effect. We investigate the local and global dynamics of the model with emphasis on the impact of strong Allee effect. First, positivity and boundedness of solutions are studied. Then existence and stability of the boundary and interior equilibria are presented and the Hopf bifurcation curve at an interior equilibrium is given. The existence of a Hopf bifurcation curve indicates that if competition between the IG prey and IG predator for the basal resource lies below the curve then the interior equilibrium remains stable, while if it lies above the curve then the interior equilibrium loses its stability. In order to explore the impact of Allee effect, the parameter space is classified into sixteen different regions and, in each region, the number of interior equilibria is determined and the corresponding bifurcation diagrams on the Allee threshold are given. The extinction parameter regions of at least one species and the necessary coexistence parameter regions of all three species are provided. In addition, we explore possible dynamical patterns, i.e., the existence of multiple attractors. By theoretical analysis and numerical simulations, we show that the model can have one (i.e. extinction of all species), two (i.e. bi-stability) or three (i.e. tri-stability) attractors. It is also found by simulations that when there exists a unique stable interior equilibrium, the model may generate multiple attracting periodic orbits and the coexistence of all three species is enhanced as the competition between the IG prey and IG predator for the basal resource is close to the Hopf bifurcation curve from below. Our results indicate that the intraguild predation food web model exhibits rich and complex dynamic behaviors and strong Allee effect in the basal prey increases the extinction risk of not only the basal prey but also the IG prey or/and IG predator.