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In this study, we consider a nonlocal almost periodic reaction–diffusion–advection model to study the global dynamics of a single phytoplankton population under the assumption that nutrients are abundant and their metabolism is only affected by light intensity. First, we prove that the single phytoplankton species model is strongly monotone with respect to the order induced by cone $X$. Second, we characterize the upper Lyapunov exponent ${\lambda }^{\ast }$ for a class of almost periodic reaction–diffusion–advection equations, and provide a numerical method to compute it. On this basis, we prove that ${\lambda }^{\ast }$ is the threshold parameter for studying the global dynamic behavior of the population model. Our results show that if ${\lambda }^{\ast }<0$, phytoplankton species will become extinct, and if ${\lambda }^{\ast }>0$, phytoplankton species will be uniformly persistent. Finally, we verified the above results using numerical simulations.

This paper describes a cholera disease transmission model in the human population through the consumption of zooplankton as food by humans. Here the plankton population is classified into two subpopulations such as phytoplankton and zooplankton. Also, human population is classified into two subpopulations such as susceptible human and infected human. The proposed system reflects the impacts of using time delay in the cholera disease transmission. Different possible equilibrium points of our proposed system have been determined. Here local and global stabilities of our proposed system have been analyzed. The existence of Hopf bifurcation has been studied at the interior equilibrium point. The normal form method and center manifold theorem have been used to test the nature of Hopf bifurcation. It is observed that the interior equilibrium is locally asymptotically stable when the time delay in disease transmission term is large, while the change of stability of positive equilibrium will cause a bifurcating periodic solution at the time delay $\tau$ to be at less than its critical value. Finally, some numerical simulation results have been presented for the better understanding of our proposed system.

We develop a six-compartment model consisting of phosphorus, detritus, phytoplankton, zooplankton, planktonivorous fish and pisciphagous fish. In this model, we study the implications that the body sizes of phytoplankton and zooplankton have on the system dynamics. We use ascendency as a goal function or indicator of system performance. Ascendency quantifies growth and development of an ecosystem as a product of total system throughflow and the mutual information inherent in the pattern of internal system flows. Different physiological rate parameters of phytoplankton and zooplankton are assessed by means of allometric relationships applied to their body sizes. We let the phytoplankton body size range from 10 μm³ to 10⁷ μm³ and the zooplankton body size range from 10 μm³ to 10⁴ μm³ in volume. We also investigate the effects of phosphorus input conditions, corresponding to oligotrophic, mesotrophic and eutrophic systems on system dynamics. Ascendency (to be maximized over phytoplankton and zooplankton sizes) was computed after the system had reached a steady state. Since it always was a seasonal cycle, and the ascendency followed this behavior, we averaged the ascendency over 365 successive days (duration of one year) in the oscillatory phase. Under all types of nutrient conditions, the smallest phytoplankton size yielded the maximal values of the ascendency, while the corresponding zooplankton size varied. Under oligotrophic conditions, a phytoplankton size of 10 μm³ combined with a zooplankton size of 10^1.25 μm³ to give the maximum value of the ascendency. Under mesotrophic and eutrophic conditions, maxima were obtained for zooplankton sizes 10^2.26 μm³ and 10^3.20 μm³, respectively.

Toxic or allelopathic compounds liberated by toxin-producing phytoplankton (TPP) acts as a strong mediator in plankton dynamics. On an analysis of a set of phytoplankton biomass data that have been collected by our group in the northwest part of the Bay of Bengal, and by analysis of a three-component mathematical model under a constant as well as a stochastic environment, we explore the role of toxin-allelopathy in determining the dynamic behavior of the competing phytoplankton species. The overall results, based on analytical and numerical wings, demonstrate that toxin-allelopathy due to the TPP promotes a stable co-existence of those competitive phytoplankton that would otherwise exhibit competitive exclusion of the weak species. Our study suggests that TPP might be a potential candidate for maintaining the co-existence and diversity of competing phytoplankton species.

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.

We consider a chemostat model of phytoplankton competing for nitrogen taking into account effects of both intra- and interspecific crowding and the light limitation. We consider crowding as an additive density-dependent mortality rate. Crowding effects may be classified into intra- and interspecific crowding depending on whether the additional mortality is caused by the same or alternate species.

We analyze the existence and local and global stability of single species and coexistence equilibria using the linearization and stability method of Lyapunov. We present a numerical example illustrating the fact that the crowding effects may lead to the bistable coexistence of two phytoplankton species. We demonstrate that the crowding effects and the light limitation affect the outcome of exploitative competition for a single resource and promote coexistence. We also show that while the crowding has a stabilizing effect on phytoplankton community, the light limitation may destabilize the system and produce sustained oscillations.

We construct models of continuous-time Markov chain (CTMC) and Itô stochastic differential equations of population interactions based on a deterministic system of two phytoplankton and one zooplankton populations. The mechanisms of mutual interference among the predator zooplankton and the avoidance of toxin-producing phytoplankton (TPP) by zooplankton are incorporated. Sudden population extinctions occur in the stochastic models that cannot be captured in the deterministic systems. In addition, the effect of periodic toxin production by TPP is lessened when the birth and death of the populations are modeled randomly.

The production of oxygen through phytoplankton photosynthesis is a crucial phenomenon in the dynamics of marine ecosystems. A generic oxygen-phytoplankton interaction model is considered to comprehend its underlying mechanism. This paper investigates the discrete-time dynamics of oxygen and phytoplankton in aquatic ecosystems, incorporating factors that cause phytoplankton mortality due to external influences. We explore the conditions for the local stability of steady states concerning the oxygen content in dissolved water and phytoplankton density. The analysis reveals that the model undergoes a co-dimension one bifurcation, encompassing flip and Neimark–Sacker bifurcations, utilizing the center manifold theorem and bifurcation theory. To manage the chaos resulting from the Neimark–Sacker bifurcation, we apply the OGY feedback control method and a hybrid control methodology. Finally, we present numerical simulations to validate the theoretical discussion.

Although extensive research on annual cycles of phytoplankton communities in the open sea has been conducted, there have been less continuous measurements on short term variations in semi-enclosed bays. To estimate the conditions necessary for red tide occurrences in an area affected by eutrophication, we carried out continuous field measurements in the inner part of Tokyo Bay at three stations where red tides have often been observed in Spring. The blooms of phytoplankton occur under high solar radiation conditions. Mixed layer thickness and the vertical distribution of PAR are also significant in accounting for the levels of phytoplankton blooms. Under optimum conditions of mixed-layer thickness and the euphotic zone, phytoplankton increased rapidly even under average solar radiation. At this time, north-wind induced outflow and vertical mixing result in diluting phytoplankton and terminating blooms. These bloom conditions will not continue due to self shading of phytoplankton, even if there isn't a strong wind. Therefore, these physical conditions are significant in controlling the levels of blooms in an area affected by eutrophication. Following the phytoplankton blooms, dissolved oxygen and phosphate concentrations show greater temporal variability through decomposition processes of the phytoplankton.

In this paper, a fractional order model of the phytoplankton–toxic phytoplankton–zooplankton system with Caputo fractional derivative is investigated via three computational methods, namely, residual power series method (RPSM), homotopy perturbation Sumudu transform method (HPSTM) and the homotopy analysis Sumudu transform method (HASTM). This model is constituted by three components: phytoplankton, toxic phytoplankton and zooplankton. Phytoplankton species are self-feeding members of the plankton community and play a very significant role in ecosystems. A wide range of sea creatures get food through phytoplankton. This paper focuses on the implementation of the three above-mentioned computational methods for a nonlinear time-fractional phytoplankton–toxic phytoplankton–zooplankton (PTPZ) model with a perception to study the dynamics of a model. This study shows that the solutions obtained by employing the suggested computational methods are in good agreement with each other. The computational procedures reveal that the HASTM solution generates a more general solution as compared to RPSM and HPSTM and incorporates their results as a special case. The numerical results presented in the form of graphs authenticate the accuracy of computational schemes. Hence, the implemented methods are very appropriate and relevant to handle nonlinear fractional models. In addition, the effect of variation of fractional order of a time derivative and time $t$ on populations of phytoplankton, toxic–phytoplankton and zooplankton has also been studied through graphical presentations. Moreover, the uniqueness and convergence analyses of HASTM solution have also been discussed in view of the Banach fixed-point theory.

A stochastic model describing the planktonic interaction is presented. The environmental fluctuations are incorporated by perturbing the growth rate of phytoplankton and death rate of zooplankton with coloured noises. Spectral density functions are studied and statistical linearization technique is used to show the unstability and periodicity of the model system indicating the cyclic nature of bloom.

This paper deals with a nutrient-phytoplankton-zooplankton ecosystem model consisting of dissolved limiting nutrient with nutrient uptake functions. We use a Holling type-II harvest function to model density dependent plankton population. It is assumed that phytoplankton release toxic chemical for self defense against their predators. The model system is studied analytically and the threshold conditions for the existence and stability of various steady states are worked out. It is observed that if the rate of toxin produced by phytoplankton population crosses a certain critical value, the system enters into Hopf bifurcation. We have derived the direction of Hopf-bifurcation. Our observations indicate that constant nutrient input and the maximal zooplankton conversion rate influence the nutrient-plankton ecosystem model and maintain stability around the coexistence equilibrium in the presence of toxic chemical release by phytoplankton for self defense. It is observed that harvesting rates of the plankton population play a vital role in changing the stability criteria. Computer simulations have been carried out to illustrate different analytical results.

We propose a general model with n parallel food chains through the stage structured maturation time delay, which can cover most of the prey-predator models in the literature. We discuss some basic dynamical properties of the system with single or multiple patches and with general or some particular functional responses, including the existence of equilibrium points and their local and global stabilities. Numerical simulations are given to compliment the theoretical analysis.