On the expected number of internal equilibria in random evolutionary games with correlated payoff matrix

The analysis of equilibrium points in random games has been of great interest in evolutionary game theory, with important implications for understanding of complexity in a dynamical system, such as its behavioural, cultural or biological diversity. The analysis so far has focused on random games of independent payoff entries. In this paper, we overcome this restrictive assumption by considering multi-player two-strategy evolutionary games where the payoff matrix entries are correlated random variables. Using techniques from the random polynomial theory we establish a closed formula for the mean numbers of internal (stable) equilibria. We then characterise the asymptotic behaviour of this important quantity for large group sizes and study the effect of the correlation. Our results show that decreasing the correlation among payoffs (namely, of a strategist for different group compositions) leads to larger mean numbers of (stable) equilibrium points, suggesting that the system or population behavioural diversity can be promoted by increasing independence of the payoff entries. Numerical results are provided to support the obtained analytical results.Comment: Revision from the previous version; 27 page

Existence and attractivity results for nonlinear first-order random differential equations

In this paper, the existence and attractivity results are proved for nonlinear first order ordinary random differential equations. An example is indicated to demonstrate a realization of the abstract theory developed in the present paper.Викладено результати про iснування та атракторнiсть розв’язкiв нелiнiйних стохастичних диференцiальних рiвнянь першого порядку. Наведено приклад реалiзацiї абстрактної теорiї