Effect of toxic substance on delayed competitive allelopathic phytoplankton system with varying parameters through stability and bifurcation analysis

We have studied the combined effect of toxicant and fluctuation of the biological parameters on the dynamical behaviors of a delayed two-species competitive system with imprecise biological parameters. Due to the global increase of harmful phytoplankton blooms, the study of dynamic interactions between two competing phytoplankton species in the presence of toxic substances is an active field of research now days. The ordinary mathematical formulation of models for two competing phytoplankton species, when one or both the species liberate toxic substances, is unable to capture the oscillatory and highly variable growth of phytoplankton populations. The deterministic model never predicts the sudden localized behavior of certain species. These obstacles of mathematical modeling can be overcomed if we include interval variability of biological parameters in our modeling approach. In this investigation, we construct imprecise models of allelopathic interactions between two competing phytoplankton species as a parametric differential equation model. We incorporate the effect of toxicant on the species in two different cases known as toxic inhibition and toxic stimulatory system. We have discussed the existence of various equilibrium points and stability of the system at these equilibrium points. In case of toxic stimulatory system, the delay model exhibits a stable limit cycle oscillation. Analytical findings are supported through exhaustive numerical simulations.     

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.     

Stochastic modeling of phytoplankton allelopathy

The study of dynamic interactions between two competing phytoplankton species in the presence of toxic substances is an active field of research due to the global increase of harmful phytoplankton blooms. Ordinary differential equation models for two competing phytoplankton species, when one or both the species liberate toxic substances, are unable to capture the oscillatory and highly variable growth of phytoplankton populations. The deterministic formulation never predicts the sudden localized extinction of certain species. These obstacles of mathematical modeling can be overcome if we include stochastic variability in our modeling approach. In this investigation, we construct stochastic models of allelopathic interactions between two competing phytoplankton species as a continuous time Markov chain model as well as an Itô stochastic differential equation model. Approximate extinction probabilities for both species are obtained analytically for the continuous time Markov chain model. Analytical estimates are validated with the help of numerical simulations.     

An assessment of two models for the subgrid scale tensor in the rational LES model

In this paper, we develop computational methods for a three-dimensional model of competition for light between phytoplankton species. The competing phytoplankton populations are exposed to both horizontal and vertical mixing. The vertical light-dependence of phytoplankton photosynthesis implies that the three-dimensional model is formulated in terms of integro-partial differential equations that require an efficient numerical solution technique.Due to the stiffness of the discretized system we select an implicit integration method. However, the resulting implicit relations are extremely expensive to solve, caused by the strong coupling of the components. This coupling originates from the three spatial dimensions, the interaction of the various species and the integral term. To reduce the amount of work in the linear algebra part, we use an Approximate Matrix Factorization technique.The performance of the complete algorithm is demonstrated on the basis of two test examples. It turns out that unconditional stability (i.e., A-stability) is a very useful property for this application.     

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.     

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.     

A space-time state-space model of phytoplankton allelopathy

An integro-differential equation system with nonlocal effects of interspecific allelopathic interaction has been studied to investigate the formation of spatio-temporal structures in toxin producing phytoplankton population. The model is inherently more realistic than the usual kind of reaction-diffusion model. Bifurcation from uniform steady-state solution has been examined. Evolution of steady-state spatially periodic structure and periodic standing waves have been studied. The model helps to investigate the blooms, pulses and succession in different patches of phytoplankton population. Numerical simulations for a hypothetical set of parameter values and experimental observations have been presented to substantiate the analytical findings.     

Dynamical analysis of a delay model of phytoplankton-zooplankton interaction

The interaction of toxic-phytoplankton-zooplankton systems and their dynamical behavior will be considered in this paper based upon nonlinear ordinary differential equation model system. We induced a discrete time delay to the both of the consume response function and distribution of toxic substance term to describe the delay in the conversion of nutrient consumed to species and the fact that the time required for the phytoplankton species to mature before they can produce toxic substances. We generalized the model in [1] and explicit results are derived for globally asymptotically stability of the boundary equilibrium. Using numerical simulation method, we determine there is a parameter range for the delay parameter τ where more complicated dynamics occurs, and this appears to be a new result. Significant outcomes of our numerical findings and their interpretations from ecological point of view are provided in this paper.     

Modeling on an ecological food chain with recycling

We propose two nutrient-phytoplankton models with instantaneous and time delayed recycling, investigate the dynamics and examine the responses to model complexities. Instead of the familiar specific uptake rate and growth rate functions, we assume only that the nutrient uptake and phytoplankton growth rate functions are positive, increasing and bounded above. We use geometrical and analytical methods to find conditions for the existence of none, one, or at most two positive steady states and analyze the stability properties of each of these equilibria. With the variation of parameters, the system may lose its stability and bifurcation may occur. We study the occurrence of Hopf bifurcation and the possibility of stability switching. Numerical simulations illustrate the analytical results and provide further insight into the dynamics of the models, biological interpretations are given.     

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)     

Effect of stochastic perturbation on a two species competitive model

In this paper first we study the stability and bifurcation of a two species competitive model with a delay effect. Next we extend the deterministic model system to a stochastic delay differential system by incorporating multiplicative white noise terms in growth equations of both species. We consider the stochastic stability of a co-existing equilibrium point in terms of mean square stability by constructing a suitable Lyapunov functional. We perform a numerical simulation to validate our analytical findings.     

Stochastic persistence and stability analysis of a modified Holling–Tanner model

The article aims to study the basic dynamical features of a modified Holling–Tanner prey–predator model with ratio‐dependent functional response. We have proved the global existence of the solution for the deterministic model. The parametric restriction for persistence of both species is also obtained along with the proof of local asymptotic stability of the interior equilibrium point(s). Conditions for local bifurcations of interior equilibrium points are provided. The global dynamic behavior is examined thoroughly with supportive numerical simulation results. Next, we have formulated the stochastic model by perturbing the intrinsic growth rates of prey and predator populations with white noise terms. The existence uniqueness of solutions for stochastic model is established. Further, we have derived the parametric restrictions required for the persistence of the stochastic model. Finally, we have discussed the stochastic stability results in terms of the first and second order moments. Numerical simulation results are provided to support the analytical findings. Copyright © 2012 John Wiley & Sons, Ltd.     

Role of toxin producing phytoplankton on a plankton ecosystem

In this paper, a mathematical model is proposed to study the role of toxin producing phytoplankton on a phytoplankton–zooplankton system with nutrient cycling. The model includes three state variables, viz., nutrient concentration, phytoplankton biomass and zooplankton biomass. It is assumed in the model that phytoplankton biomass is producing toxicant harmful for the zooplankton biomass. 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 are carried out and the possibility of Hopf-bifurcation of the interior equilibrium is studied. The threshold value in terms of constant input rate of nutrient is determined both analytically and numerically.     

Existence and nonexistence of positive steady states in multi-species phytoplankton dynamics

In this paper we study a nonlocal reaction–diffusion–advection system modeling the growth of multiple competitive phytoplankton species in a vertical water column with incomplete mixing. We find that when the diffusion of the system is large, there is no positive steady states, and when the diffusion is not large, there exists at least one positive steady states under certain conditions. The main tools we use are the fixed point index theory, a refined comparison theorem and fine properties of the principal eigenvalues.     

Analytical approximations for a population growth model with fractional order

In this paper, we apply the homotopy analysis method (HAM) to solve the fractional Volterra’s model for population growth of a species in a closed system. This technique is extended to give solutions for nonlinear fractional integro–differential equations. The whole HAM solution procedure for nonlinear fractional differential equations is established. Further, the accurate analytical approximations are obtained for the first time, which are valid and convergent for all time t. This indicates the validity and great potential of the homotopy analysis method for solving nonlinear fractional integro–differential equations.     

Competitive exclusion in a nonlocal reaction–diffusion–advection model of phytoplankton populations

We continue our study on the global dynamics of a nonlocal reaction–diffusion–advection system modeling the population dynamics of two competing phytoplankton species in a eutrophic environment, where both populations depend solely on light for their metabolism. In our previous work, we proved that system (1.1) is a strongly monotone dynamical system with respect to a non-standard cone related to the cumulative distribution functions, and further determined the global dynamics when the species have either identical diffusion rate or identical advection rate. In this paper, we study the trade-off of diffusion and advection and their joint influence on the outcome of competition. Two critical curves for the local stability of two semi-trivial equilibria are analyzed, and some new competitive exclusion results are obtained. Our main tools, besides the theory of monotone dynamical system, include some new monotonicity results for the principal eigenvalues of elliptic operators in one-dimensional domains.     

Stochastic predator–prey model with Allee effect on prey

We study a predator–prey model with the Allee effect on prey and whose dynamics is described by a system of stochastic differential equations assuming that environmental randomness is represented by noise terms affecting each population. More specifically, we consider a term that expresses the variability of the growth rate of both species due to external, unpredictable events. We assume that the intensities of these perturbations are proportional to the population size of each species. With this approach, we prove that the solutions of the system have sample pathwise uniqueness and bounded moments. Moreover, using an Euler–Maruyama-type numerical method we obtain approximated solutions of the system with different intensities for the random noise and parameters of the model. In the presence of a weak Allee effect, we show that long-term survival of both populations can occur. On the other hand, when a strong Allee effect is considered, we show that the random perturbations may induce the non-trivial attracting-type invariant objects to disappear, leading to the extinction of both species. Furthermore, we also find the Maximum Likelihood estimators for the parameters involved in the model.     

The role of viral infection in phytoplankton dynamics with the inclusion of incubation class

The role of viral infection in phytoplankton dynamics without and with incubation population class is studied. It is observed that phytoplankton species in the absence of incubated class are unstable around an endemic equilibrium but the presence of delay in the form of incubated class has made it conditionally stable around an endemic equilibrium. We also observe that the dynamical system is very sensitive to the transfer rate from susceptible to incubated class and when it crosses a certain threshold the phytoplankton population start oscillating around the endemic equilibrium, shown both analytically and numerically.     

Local and global stability analysis of a two prey one predator model with help

In this paper we propose and study a three dimensional continuous time dynamical system modelling a three team consists of two preys and one predator with the assumption that during predation the members of both teams of preys help each other and the rate of predation of both teams are different. In this work we establish the local asymptotic stability of various equilibrium points to understand the dynamics of the model system. Different conditions for the coexistence of equilibrium solutions are discussed. Persistence, permanence of the system and global stability of the positive interior equilibrium solution are discussed by constructing suitable Lyapunov functional. At the end, numerical simulations are performed to substantiate our analytical findings.     

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.