$\mathrm{\ell }$- Volterra Quadratic Stochastic Operators: Lyapunov Functions, Trajectories

We consider $\mathrm{\ell }$-Volterra quadratic stochastic operators defined on $(m-1)$-dimensional simplex, where $\mathrm{\ell }\in \{0,1,\mathrm{.}\mathrm{.}\mathrm{.},m\}$. Under some conditions on coefficients of such operators we describe Lyapunov functions and apply them to obtain upper estimates for the set of $\omega$- limit points of trajectories. We describe a set of fixed points of $\mathrm{\ell }$-Volterra operators.