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If α denotes the class of all QTAG-modules M such that M/H β (M) is totally projective for every ordinal β < α, then these modules are called α-modules. Here we study the relation between the structure of fully invariant submodules of certain QTAG-modules and the structure of containing modules. It is found that if F is a fully invariant submodule of the totally projective QTAG-module M, then both F and M/F are totally projective. We show that if for some sequence β = (β k ) k<ω , both M β and M/M β are totally projective, then M itself is necessarily totally projective.

In this paper, a nonlinear mathematical model to study the effect of a toxicant on a biological population is proposed and analyzed. We have taken the case in which some members of a biological population get severely affected by the toxicant and show abnormal symptoms, like deformity, fecundity, necrosis, etc. It has been assumed that the toxicant is produced by the population itself. This model can be applied to the human population which creates pollution and affects itself. The analysis of the model suggests the need of a regulatory agency to control the emission of toxicant from manmade projects. The stability analysis of the equilibria of the proposed model and existence of Hopf-bifurcation are determined. We have also determined the direction and stability of bifurcating periodic solutions to clearly understand the effect of emission rate of the toxicant on the biological species. Finally, numerical simulation has been given to illustrate the mathematical results.

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We show the inheritance of summable property for certain fully invariant submodules by the QT AG-modules and vice versa. Important generalizations and extensions of classical results in this direction are also established.