Analytical solutions of one class of Volterra quadratic stochastic operators of a two-sex population with continuous time
This paper investigates analytical solutions of a continuous analogue of a class of Volterra quadratic stochastic operators of a two-sex population using the homotopy analysis method. The positive invariance of the state space of the system is established. By means of Lyapunov functions, inclusions for the $ω$-limit sets are obtained, together with exponential decay estimates for individual components of the trajectories under the corresponding conditions on the coefficients of the system. Sufficient conditions are obtained for the coincidence of the $ω$-limit sets of the discrete and continuous models. For the diagonal invariant subset, sufficient conditions are established for the periodicity of all interior non-equilibrium trajectories. By means of a majorant series, a sufficient condition for the absolute and uniform convergence of the homotopy series on $[0,+\infty)$ is obtained under the Hurwitz stability of the reduced Jacobian of the system. In addition, explicit analytical solutions of the Cauchy problem are found for special relations between the heredity coefficients. Computational experiments show a rapid decay of the terms of the homotopy series and good approximation accuracy already when using the first few terms of the expansion.