Related papers: Completely Integrable Replicator Dynamics Associat…
In this paper we address a class of replicator dynamics, referred as polymatrix replicators, that contains well known classes of evolutionary game dynamics, such as the symmetric and asymmetric (or bimatrix) replicator equations, and some…
We present a wide class of differential systems in any dimension that are either integrable or complete integrable. In particular, our result enlarges a known family of planar integrable systems. We give an extensive list of examples that…
It is known that there is a one-to-one mapping between oriented directed graphs and zero-sum replicator dynamics (Lotka-Volterra equations) and that furthermore these dynamics are Hamiltonian in an appropriately defined nonlinear Poisson…
We propose a simple model of network co-evolution in a game-dynamical system of interacting agents that play repeated games with their neighbors, and adapt their behaviors and network links based on the outcome of those games. The…
The replicator equation is one of the fundamental tools to study evolutionary dynamics in well-mixed populations. This paper contributes to the literature on evolutionary graph theory, providing a version of the replicator equation for a…
We propose a minimal model of the dynamics of diversity -- replicator equations with extinction, invasion and mutation. We numerically study the behavior of this simple model and show that it displays completely different behavior from the…
The set of linear, differential operators preserving the vector space of couples of polynomials of degrees n and n-2 in one real variable leads to an abstract associative graded algebra A(2). The irreducible, finite dimensional…
We consider a class of complex manifolds constructed as multiplicative quiver varieties associated with a cyclic quiver extended by an arbitrary number of arrows starting at a new vertex. Such varieties admit a Poisson structure, which is…
A partial differential equation is derived, describing the replicator dynamics with mutations of games with a continuous strategy space. This equation is then applied to continuous versions of symmetric 2x2 games, such as the Prisoners…
This chapter is an overview of foundational results in the mathematical theory of replicator systems. Its primary aim is to provide a unified framework for the mathematical formalisation of evolutionary processes in the spirit of…
We prove that evolution families on complex complete hyperbolic manifolds are in one to one correspondence with certain semicomplete non-autonomous holomorphic vector fields, providing the solution to a very general Loewner type…
We introduce an algorithm to find possible constants of motion for a given replicator equation. The algorithm is inspired by Dirac geometry and a Hamiltonian description for the replicator equations with such constants of motion, up to a…
A parameter-dependent class of Hamiltonian (generalized) Lotka-Volterra systems is considered. We prove that this class contains Liouville integrable as well as superintegrable cases according to particular choices of the parameters. We…
We explore various combinatorial problems mostly borrowed from physics, that share the property of being continuously or discretely integrable, a feature that guarantees the existence of conservation laws that often make the problems…
We consider the general properties of the replicator dynamical system from the standpoint of its evolution and stability. Vector field analysis as well as spectral properties of such system has been studied. Lyaponuv function for…
We introduce a class of o.d.e.'s that generalizes to polymatrix games the replicator equations on symmetric and asymmetric games. We also introduce a new class of Poisson structures on the phase space of these systems, and characterize the…
Starting with a group of reinforcement-learning agents we derive coupled replicator equations that describe the dynamics of collective learning in multiagent systems. We show that, although agents model their environment in a…
To any tree on $n$ vertices we associate an $n$-dimensional Lotka-Volterra system with $3n-2$ parameters and, for generic values of the parameters, prove it is superintegrable, i.e. it admits $n-1$ functionally independent integrals. We…
An analytic theory of species abundance patterns (SAPs) in biological networks is presented. The theory is based on multispecies replicator dynamics equivalent to the Lotka-Volterra equation, with diverse interspecies interactions. Various…
We report a class of symmetry-intergable third-order evolution equations in 1+1 dimensions under the condition that the equations admit a second-order recursion operator that contains an adjoint symmetry (integrating factor) of order six.…