Hopf bifurcation and other dynamical behaviors for a fourth order differential equation in models of infectious disease
The authors consider the model of host-parasite population dynamics (1) \(\dot x_ 1 = r_ 1 x_ 1 - k_{11} x_ 1 y_ 1 - k_{12} x_ 1 y_ 2 + e_ 1 y_ 1\), \(\dot y_ 1 = k_{11} x_ 1 y_ 1 + k_{12} x_ 1 y_ 2 - d_ 1 y_ 1\), \(\dot x_ 2 = r_ 2 x_ 2 - k_{22} x_ 2 y_ 2 - k_{21} x_ 2 y_ 1 + e_ 2 y_ 2\), \(\dot y_ 2 = k_{22} x_ 2 y_ 2 + k_{21} x_ 2 y_ 1 - d_ 2 y_ 2\), proposed by \textit{Holt} and \textit{Pickel} [American Naturalist 126, No. 2, 196-211 (1985)]. In this paper using Hopf bifurcation theory they reveal that the model (1) has periodic solutions and prove the reliability of the numerical solution obtained by \textit{W.-M. Liu} [Lect. Notes Biomath. 81, 102-108 (1989; Zbl 0682.92011)].
scientific article; zbMATH DE number 760234