The dynamics of strictly non-Volterra quadratic stochastic operators on the 2-simplex. (English. Russian original) Zbl 1194.47077
Sb. Math. 200, No. 9, 1339-1351 (2009); translation from Mat. Sb. 200, No. 9, 81-94 (2009).
The mapping \(V : \mathbb{R}^3 \to \mathbb{R}^3 : (x,y,z) \mapsto (x',y'z')\) defined by \(x' = \alpha y^2 + c z^2 + 2 y z\), \(y' = a x^2 + d z^2 + 2 x z\), \(z' = b x^2 + \beta y^2 + 2 x y\) maps the simplex \(\Sigma = \{ (x,y,z) : x, y, z, \geq 0, \;x+ y + z = 1 \}\) to itself provided that \(\alpha, \beta, a, b, c, d \geq 0\) and \(\alpha + \beta = a + b = c +d = 1\). The authors prove that any such mapping of \(\Sigma\) to itself has a unique fixed point, which may or may not be hyperbolic (i.e., such that the eigenvalues of the linear part lie off the unit circle in \(\mathbb{C}\)), and if hyperbolic cannot be a sink. For certain subclasses of mappings of this form, distinguished by conditions on the parameters \(\alpha, \beta, a, b, c\), and \(d\), the authors are able to describe the \(\omega\)-limit set of any point in \(\Sigma\). They state several conjectures based on theoretical considerations and numerical calculations.
Reviewer: Douglas S. Shafer (Charlotte)
MSC:
47H40 | Random nonlinear operators |
47H10 | Fixed-point theorems |
37C20 | Generic properties, structural stability of dynamical systems |
37C25 | Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics |