Vector Autoregressive (VAR) models have been used for a long time now to study pro…t-squeeze cycles, most of the time using problematic Hodrick-Prescott (HP) …ltered time series. In a recent paper, Hamilton (2018) has provided a simple alternative that overcomes the main drawbacks of the HP procedure. In order to evaluate the empirical relevance of the pro…t-squeeze mechanism, we compare both methodologies using quarterly data for the United States from 1948-67 to 2016. Furthermore, we present an extension of Goodwin's (1967) growth-cycle model that includes employment rates, income distribution, and capacity utilisation as endogenous variables. We show analytically that the system always admits a family of periodic solutions. The model is estimated econometrically using the Autoregressive Distributed Lag (ARDL) approach. Through numerical simulations and making use of our estimations, we con…rm that ‡uctuations are persistent and bounded.
The TH1/TH2 paradigm has been largely used in the interpretation of several diseases, particularly in leishmaniasis. As far as we know there is no mathematical description of this model related to leishmaniasis. We have extended and modified a previous published set of equations1in order to adapt it to leishmanial disease particularities. The main modifications were: (1) the analysis of logistic and exponential parasite growth curves, (2) the assumption of the TH2 arm of the immune response having a positive action on parasite growth. The set of three simultaneous differential equations describing the TH1 arm, TH2 arm and parasite growth were analyzed for conditions of existence and stability of the solutions.Stable solutions valid for the logistic and exponential parasite growth models, with its possible clinical correlations, were obtained in the following situations: (1) parasite and TH2 extinction [TH1 cure], (2) parasite extinction and TH1/TH2 co-existence [TH1/TH2 cure], (3) TH1 and parasite co-existence, TH2 extinction [stable TH1 infection], and (4) TH1, TH2 and parasite co-existence [stable TH1/TH2 infection]. TH2 and parasite co-existence associated to TH1 extinction [stable TH2 infection] was obtained only with the logistic growth model. The model also provides an alternative hypothesis for TH1 bias in resistant mice and emphazises the importance of natural immunity for the existence of chronic states.
We study a system of ODE's modelling the interaction of one predator and one prey dx/dt = xg(x) - yp(x), dy/dt = gamma y[- delta - nu y - alpha y2 + h(x)]. This system defines a two-species community which incorporates competition among prey in the absence of any predators as well as a density-dependent predator specific death rate. This system is investigated under ecologically natural regularity conditions and assumptions on g, p and h to ensure the existence and uniqueness of limit cycles. The proof uses the standard Hopf-Andronov bifurcation theory and the technique of Liénard's equation.
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