On a Class of Generalized Gompertz-Bateman Growth-decay Models
A recently proposed reaction network induces the classical Gompertz equations. The network consist of two reactions, one reaction for a decaying species $S$ and another reaction for a sigmoidally growing species $X$ which is catalyzed by species S. The proposed reaction network provides for a separation of the dynamical evolutions of the two species inherently present in the Gompertz reaction network. More specifically, the reaction equation for the decaying species S is totally independent on (uncoupled from) the growing species X, while the species X uses species S simultaneously both as a catalyst and as a resource (food). Based on the idea of such a separation, in this work we propose a class of growth-decay models formulated in terms of reaction networks that includes the Bateman exponential decay chain for the evolution of the catalist/resource species. An important advantage of the new class of models is the presence of a prolonged lag time of the sigmoidal solutions of the growing species. In this note we show that the Gompertz-type reaction network can be generalized into a class of Gompertzian-type growth-decay models by replacing the first species from the Bateman exponential decay chain by the second species in the chain, that now can play the role of a catalyst/resource for the growing species. A further generalization would be to use the third species (or the k-th species) in the Bateman decay chain as a catalyst/resource for the growing species.
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