A parametric model of competition in ecology. II (Q2760710)

The authors deal with the system of nonlinear ordinary differential equations NEWLINE\[NEWLINE\dot x_1 = x_1(1 - x_1 - bx_2 - ax_3),\quad \dot x_2 = x_2(1 - ax_1 - x_2 - bx_3),\quad \dot x_3 = x_3(1 - bx_1 - cx_2 - x_3),NEWLINE\]NEWLINE which arises in ecology for mathematical modeling of competition of three species. Here, \(a\), \(b\), \(c\) are positive constants such that \(a\neq c\); the case \(a=c\) was studied by \textit{R. M. May} and \textit{W. J. Leonard} [SIAM J. Appl. Math. 29, 243-253 (1975; Zbl 0314.92008)].NEWLINENEWLINENEWLINEThe aim of the article is to prove the asymptotic stability (or instability) for the equilibrium state to the system. The authors analyze various cases of behavior of the coefficients of the model and, as a result, obtain theorems which guarantee the asymptotic stability (or instability) for the equilibrium state. NEWLINENEWLINENEWLINEFor part I see ibid. 7, No. 1, 3-10 (2000; Zbl 0956.92033).