Kinetics of the populations

This paper presents a phytoplankton population model in natural waters, constructed on the basis of the principle of conservation of mass. This is an elementary physical law which is satisfied by macroscopic natural systems. The use of this principle is dictated primarily by the lack of any more specific physical laws which can be applied to these biological systems. An alternate conservation law, that of conservation of energy, can also be used. However, the details of how mass is transferred from species to species are better understood than the corresponding energy transformations. The mass interactions are related, among other factors, to the kinetics of the populations, and it is this that the bulk of the paper is devoted to exploring. 

These equations describe only the kinetics of the populations in a single volume element Vj. However, in a natural water body there exists significant mass transport as well. The mass transport mechanisms can be conveniently represented by the matrix A with elements Oij. If for particular segments i and / the matrix element ay is nonzero, then the volume segments Vi and Vj interact, and mass is transported between the two segments. Letting P, Z, and N be the vectors of elements Pj, Zj, and Nj and letting SP, SN, Sz be the vectors of elements SPj, SZj,... 

We now turn our attention to steady states of reaction-transport systems. We focus first on steady states that arise in RD models on finite domains. Such models are important from an ecological point of view, since they describe population dynamics in island habitats. The main problem consists in determining the critical patch size, i.e., the smallest patch that can minimally sustain a population. As expected intuitively, the critical patch size depends on a number of factors, such as the population dynamics in the patch, on the nature of the boundaries, the patch geometry, and the reproduction kinetics of the population. The first critical patch model was studied by Kierstead and Slobodkin [228] and Skellam [414] and is now called the KISS problem. A significant amount of work has focused on systems with partially hostile boundaries, where individuals can cross the boundary at some times but not at others, or systems where individuals readily cross the boundary but the region outside the patch is partially hostile, or a combination of the above. In this chapter we deal with completely hostile boundaries and calculate the critical patch size for different geometries, reproduction processes, and dynamics. 

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