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In this paper, classes of mathematical models of phytoplankton–zooplankton systems with toxic phytoplankton are proposed and analysed.

We investigate various situations, by changing the uptake function, the mortality of the zooplankton or the toxin liberation process, and we compare the corresponding dynamics.

We show notably that a stable oscillatory regime (after a Hopf bifurcation) can only be observed if the uptake function is nonlinear.

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.

The present paper studies the naturally observed phenomenon of population fluctuation in the context of a minimal plankton model. The analysis of the basic model reveals asymptotic stable behavior that is unable to explain any kind of population outburst. We introduce the nonuniform spatial distribution of plankton by considering physical diffusion of the species concerned. The resulting reaction-diffusion equation model is first analyzed with constant diffusivity and then with variable diffusivity for the zooplankton. The model with variable diffusivity is analyzed by using Floquet theory. In both cases, the model is seen to exhibit stable behavior. Finally, we study another characteristic feature of any ecological population, namely, environmental fluctuations. We achieve this by perturbing the growth rate of phytoplankton and death rate of zooplankton by using colored noise. The resulting model is analyzed by evaluating the spectral density functions. It is observed that the stochastic model can generate a large fluctuation in population concentration for high amplitude driving forces.

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.

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.