It is well known that, for quadratic stochastic operators defined on a finite-dimensional simplex, the \(\omega \) -limit set of any initial point is nonempty. In the infinite-dimensional setting, the simplex fails to be compact in the strong topology of the \(\ell ^1\) space, and this non-compactness presents substantial challenges for the analysis of the dynamics of infinite-dimensional quadratic stochastic operators. Moreover, even in the finite-dimensional case, the dynamics of quadratic stochastic operators remains far from being completely understood, underscoring the complexity of the infinite-dimensional problem. In the present work, we focus on a specific subclass of quadratic stochastic operators, namely the b-bistochastic operators. For this class, we establish a canonical representation. It is noteworthy that the class of b-bistochastic operators encompasses both Volterra and non-Volterra operators. Nonetheless, we demonstrate that their dynamical behavior exhibits a closer resemblance to that of Volterra operators.
