Plankton Models and Its Attractors in a Local Approximation

Plankton Models and Its Attractors in a Local Approximation

DOI: 10.4018/978-1-5225-9651-6.ch003

Abstract

The previously accepted models of plankton consisting of two interacting populations—phytoplankton and zooplankton—are considered in a local approximation. The analysis of models is carried out with the help of a qualitative study of systems of differential equations as a whole (i.e., in the entire phase space of systems, not limited to a neighborhood of equilibrium positions). Analytical conditions for the occurrence of a Hopf bifurcation are obtained for each model using the Lyapunov stability theory. A comparison of various models is given, and their shortcomings associated with the incompleteness of research are indicated. It has been established that in some cases the loss of stability of the equilibrium position does not lead to the formation of a limit cycle (Hopf bifurcation) but to the formation of a limit continuum with a chaotic behavior of the trajectories in a large part of the phase space. It is shown that the parameters significantly influencing the dynamics of the development of plankton are the natural mortality of populations as an environmental characteristic of the environment.
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Plankton Models

Plankton is a collection of floating organisms belonging to many types and living in the oceans, seas, freshwater bodies and major rivers. The functional classification of plankton organisms is based on their place in the food chain, their size and distribution in nature (Erhard, & Seguin, 1978; Kiselev, 1982). By size, picoplankton is distinguished (2 µm), micro - plankton (0.2 - 2 mm), macro - plankton (more than 20 mm). Phytoplankton are microscopic plants that largely determine the development of all marine communities and the life within them. Due to the fact that the growth of phytoplankton is due to photosynthesis, the global phytoplankton reserves produce half the amount of oxygen that humanity needs to maintain its existence, and absorb half the amount of carbon that could trigger global warming. Along with oxygen and carbon, there are other elements and substances, which are regenerated by phytoplankton. These include, above all, phosphorus and nitrogen - containing compounds. Thus, phytoplankton, in general, is one of the main factors for the development of climate on Earth. Zooplankton are planktonic animals. In the marine zooplankton can be found both herbivores and predators. At the same time, herbivores feed on phytoplankton and are food for zooplankton predators. In the complex, phytoplankton and zooplankton are the basis of all trophic chains and networks in the ocean. In turn, the reproduction of species forming plankton is determined by many factors, such as ambient temperature, sunlight intensity, availability of nutrients, etc. An increase in the concentrations of certain substances (mainly nitrogen and phosphorus), which are a food product for algae, leads to a significant increase in the concentration of algae, i.e. to eutrophication, and, as a result, to the accumulation of organic matter in water, water pollution, the spread of infections and the gradual withering away of all life. For this reason, eutrophication should be classified as an environmental risk caused by an excessive increase in the concentration of nutrients as pathogenic elements (Slepyan, 2002). The increase in nutrients in water can be the result of natural processes or targeted actions. In the latter case, these actions can be considered as acts of environmental terrorism (Slepyan, 2003).

The beginning of the mathematical modeling of the processes underlying the formation of plankton was laid in the work of R. Fleming (Fleming, 1939). He proposed to describe the change in the number of phytoplankton by a differential equation

978-1-5225-9651-6.ch003.m01
(1)

Here 978-1-5225-9651-6.ch003.m02 is the concentration of phytoplankton in water; F is speed of the photosynthesis; R is speed of the breath; G is phytoplankton excretion rate by zooplankton.

In the future, this simplest model became more complicated in two directions. First, the kinetics of phytoplankton development was refined, taking into account its interaction with zooplankton. Secondly, the influence of the mobility of the waters of the seas and oceans and the mobility of individuals of planktonic populations on the distribution of phytoplankton and zooplankton over the space of these waters was considered. Numerous studies of the distribution of plankton for a long time discovered the heterogeneity (mosaic) of the distribution of plankton in both horizontal and vertical (i.e. in depth) directions in lakes, seas and oceans (Beklemishev, 1969; Kiselev, 1982; Erhard, & Sezhen, 1978; Zenkevich, 1951; Kamshilov, Zelikman, & Rouhiyainen, 1958; Morozova - Vodyanitskaya, 1948; Vinogradov, 1968: Sirenko, & Gavrilenko, 1978).

Key Terms in this Chapter

Neuston: The layer in the water are attached to the surface of the water inhabited by living organisms.

Zooplankton: ? collection of animals that inhabit the stratum of sea and fresh water carried by currents.

Trophic Function: The rate of phytoplankton eating by zooplankton.

Phytoplankton: A collection of plants inhabiting a series of sea and fresh waters passively carried by currents.

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