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.