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The production of public goods is a necessary condition for the survival of the species, but it comes at the expense of individual growth. In a prototype bacterial colony, we model the role of quorum sensing as a resource redistribution mechanism. Two types of bacterial colonies are analyzed, one made up of a single strain and one made up of two different strains. Based on a recent series of experimental data present in the literature, we analyze two types of strains with different extinction times: strains that consume available resources very quickly, therefore becoming extinct quickly, and strains that consume resources slowly and die due to aging. We show that the proposed quorum sensing model describes the main experimental result that coexistence may favor the survival of both strains. Furthermore, the production of public goods is maximized when both types of individuals have the maximum proliferation output. Finally, we highlight the role played by so-called dormant cells in the duration of survival time. These cells are of particular interest because their ability to counteract different types of stress (e.g., the use of antibiotics) still constitutes a challenge. [ABSTRACT FROM AUTHOR]

Stefan–Boltzmann law, stating the fourth power temperature dependence of the radiation emission by a black-body, was empirically formulated by Stefan in 1874 by fitting existing experiments and theoretically validated by Boltzmann in 1884 on the basis of a classical physical model involving thermodynamics principles and the radiation pressure predicted by Maxwell equations. At first sight the electromagnetic (EM) gas assumed by Boltzmann and following Rayleigh (1900) identifiable as an ensemble of N classical normal-modes, looks like an extension of the classical model of the massive ideal-gas. Accordingly, for this EM gas the internal total energy, U, was assumed to be function of volume V and temperature T as U = U (V , T) , and the equation of state was given by U = 3 P V , with P the radiation pressure. In addition, Boltzmann implicitly assumed that, for given values of V and T, U and the number of modes N would take finite values. However, from one hand these assumptions are not justified by Maxwell equations and classical statistics since, in vacuum (i.e., far from the EM sources), the values of N and U diverge, the so-called ultraviolet catastrophe introduced by Ehrenfest in 1911. From another hand, Boltzmann derivation of Stefan law is found to be macroscopically compatible with its derivation from quantum statistics announced by Planck in 1901. In this paper, we present a justification of this puzzling classical/quantum compatibility by noticing that the implicit assumptions made by Boltzmann is fully justified by Planck quantum statistics. Furthermore, we shed new light on the interpretation of recent classical simulations of a black body carried out by Wang, Casati, and Benenti in 2022 who found an analogous puzzling consistency between Stefan–Boltzmann law and their simulations to induce speculations on classical physics and black body radiation that are claimed to require a critical reconsideration of the role of classical physics for the understanding of quantum mechanics. [ABSTRACT FROM AUTHOR]