Mass Energy and Flow in closed ecosystems

The general equations of biomass and energy transfer for an n-species, closed ecosystem are written. It is demonstrated how in "ecological time" the parameters describing the dynamics of biomass transfer are related to the parameters of energy transfer, such as respiration, fixation, and energy content. This relationship is determinate for the straight-chain ecosystem, and a simple example is worked out. The results show how the density dependent terms in population dynamics arise naturally, and how the stable system exhibits a hierarchy in energy per unit biomass. A procedure is proposed for extending the theory to include webbed systems, and the particular difficulties involved in the extension are brought before the scientific community for discussion.

Fixed Points of Closed and Compact Composite Sequences of Operators and Projectors in a Class of Banach Spaces

Some results on fixed points related to the contractive compositions of bounded operators in a class of complete metric spaces which can be also considered as Banach's spaces are discussed through the paper. The class of composite operators under study can include, in particular, sequences of projection operators under, in general, oblique projective operators. In this paper we are concerned with composite operators which include sequences of pairs of contractive operators involving, in general, oblique projection operators. The results are generalized to sequences of, in general, nonconstant bounded closed operators which can have bounded, closed, and compact limit operators, such that the relevant composite sequences are also compact operators. It is proven that in both cases, Banach contraction principle guarantees the existence of unique fixed points under contractive conditions.