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The dynamic behavior of tri-trophic food chains consisting of resources, prey, predator and top-predator is dealt with. We compare a formulation whereby the prey growth is logistic, with a mass balance formulation. In the case of the mass balance formulation both the linear and the hyperbolic functional response are discussed. The consequences of the different formulations on the dynamics of a microbial food chain in chemostat situation are described. Bifurcation diagrams for the nonlinear dynamic systems are given. When the prey grows logistically there is no coexistence of the three species for biologically realistic parameter values for a microbial food chain. The same holds for the mass balance equations with a linear functional response for the prey. For a hyperbolic functional response, however, there is a stable equilibrium for the whole food chain in a rather large region of the parameter space. Furthermore, this model shows more complex dynamic behaviors; besides point attractors, limit cycles and chaotic attractors.
Production of energy is a foundation of life. The metabolic rate of organisms (amount of energy produced per unit time) generally increases slower than organisms’ mass, which has important implications for life organization. This phenomenon, when considered across different taxa, is called interspecific allometric scaling. Its origin has puzzled scientists for many decades, and still is considered unknown. In this paper, we posit that natural selection, as determined by evolutionary pressures, leads to distribution of resources, and accordingly energy, within a food chain, which is optimal from the perspective of stability of the food chain, when each species has sufficient amount of resources for continuous reproduction, but not too much to jeopardize existence of other species. Metabolic allometric scaling (MAS) is then a quantitative representation of this optimal distribution. Taking locomotion and the primary mechanism for distribution of energy, we developed a biomechanical model to find energy expenditures, considering limb length, skeleton mass and speed. Using the interspecific allometric exponents for these three measures and substituting them into the locomotion-derived model for energy expenditure, we calculated allometric exponents for mammals, reptiles, fish, and birds, and compared these values with allometric exponents derived from experimental observations. The calculated allometric exponents were nearly identical to experimentally observed exponents for mammals, and very close for fish, reptiles and the basal metabolic rate (BMR) of birds. The main result of the study is that the MAS is a function of a mechanism of optimal energy distribution between the species of a food chain. This optimized sharing of common resources provides stability of a food chain for a given habitat and is guided by evolutionary pressures and natural selection.