We consider a two-competitor/one-prey model in which both competitors exhibit a general functional response and one of the competitors exhibits a density-dependent mortality rate. It is shown that the two competitors can coexist upon the single prey.As an example, we consider a two-competitor/one-prey model with a Holling II functional response. Our results demonstrate that density-dependent mortality in one of the competitors can prevent competitive exclusion. Moreover, by constructing a Liapunov function, the system has a globally stable positive equilibrium.

The microquasar GRS 1915+105, exhibits a large variety of characteristic states, according to its luminosity, spectral state, and variability. The most interesting one is the so-called ρ-state, whose light curve shows recurrent bursts. This paper presents a model based on Fitzhugh-Nagumo equations containing two variables: x, linked to the source photon emission rate detected by the MECS, and y related to the mean photon energy. We aim at providing a simple mathematical framework composed by non-linear differential equations useful to predict the observed light curve and the energy lags for the ρ-state and possibly other classes of the source. We studied the equilibrium state and the stability conditions of this system that includes one external parameter, J, that can be considered a function of the disk accretion rate. Our work is based on observations performed with the MECS on board BeppoSAX when the source was in ρ and ν mode, respectively. The evolution of the mean count rate and photon energy were derived from a study of the trajectories in the count rate-photon energy plane. Assuming J constant, we found a solution that reproduces the x profile of the ρ class bursts and, unexpectedly, we found that y exhibited a time modulation similar to that of the mean energy. Moreover, assuming a slowly modulated J the solutions for x quite similar to those observed in the ν class light curves is reproduced. According these results, the outer mass accretion rate is probably responsible for the state transitions, but within the ρ-class it is constant. This finding makes stronger the heuristic meaning of the non-linear model and suggests a simple relation between the variable x and y. However, how a system of dynamical equations can be derived from the complex mathematical apparatus of accretion disks remains to be furtherly explored. © 2014 Springer Science+Business Media Dordrecht

Lyapunov functions are given to prove the global asymptotic stability of a large class of predator-prey models, including the ones in which the intrinsic growth rate of the prey follows the Ricker-law or the Odell generalization of the logistic law, and the functional predator response is of Holling type.

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This paper introduces a biparametric family of Lyapunov functions for a nonlinear mathematical model based on the FitzHugh-Nagumo equations able to reproduce some main features of the X-ray bursting behaviour exhibited by the microquasar GRS 1915+105. These functions are useful to investigate the properties of equilibrium points and allow us to demonstrate a theorem on the global stability. The transition between bursting and stable behaviour is also analyzed.