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We consider a system of several nonlinear equations with a distributed delay and obtain absolute asymptotic stability conditions, independent of the delay. The ideas of the proofs are based on the notion of a strong attractor. The results…
In complex systems, the interplay between nonlinear and stochastic dynamics, e.g., J. Monod's necessity and chance, gives rise to an evolutionary process in Darwinian sense, in terms of discrete jumps among attractors, with punctuated…
Populations evolving under the joint influence of recombination and resampling (traditionally known as genetic drift) are investigated. First, we summarise and adapt a deterministic approach, as valid for infinite populations, which assumes…
A general framework of evolutionary dynamics under heterogeneous populations is presented. The framework allows continuously many types of heterogeneous agents, heterogeneity both in payoff functions and in revision protocols and the entire…
An array system of coupled maps is proposed as a model for economy evolution. The local dynamics of each map or agent is controlled by two parameters. One of them represents the growth capacity of the agent and the other one is a control…
We appeal to results from combinatorial random matrix theory to deduce that various random graph $\mathrm{C}^*$-algebras are asymptotically almost surely Kirchberg algebras with trivial $K_1$. This in particular implies that, with high…
Populations interact non-linearly and are influenced by environmental fluctuations. In order to have realistic mathematical models, one needs to take into account that the environmental fluctuations are inherently stochastic. Often,…
Populations of replicating entities frequently experience sudden or cyclical changes in environment. We explore the implications of this phenomenon via a environmental switching parameter in several common evolutionary dynamics models…
This contribution is concerned with mathematical models for the dynamics of the genetic composition of populations evolving under recombination. Recombination is the genetic mechanism by which two parent individuals create the mixed type of…
When studying the dynamics of trait distribution of populations in a heterogeneous environment, classical models from quantitative genetics choose to look at its system of moments, specifically the first two ones. Additionally, in order to…
We study the asymptotic stability properties of nonlinear switched systems under the assumption of the existence of a common weak Lyapunov function. We consider the class of nonchaotic inputs, which generalize the different notions of…
We study the behavior of an infinite system of ordinary differential equations modeling the dynamics of a metapopulation, a set of (discrete) populations subject to local catastrophes and connected via migration under a mean field rule; the…
We study asymptotic behaviour of stochastic approximation procedures with three main characteristics: truncations with random moving bounds, a matrix valued random step-size sequence, and a dynamically changing random regression function.…
In this communication, complex systems with a near trivial dynamics are addressed. First, under the hypothesis of equiprobability in the asymptotic equilibrium, it is shown that the (hyper) planar geometry of an $N$-dimensional multi-agent…
We consider non-i.i.d. random holomorphic dynamical systems whose choice of maps depends on Markovian rules. We show that generically, such a system is mean stable or chaotic with full Julia set. If a system is mean stable, then the…
In this paper we give a new sufficient condition for asymptotic periodicity of Frobenius-Perron operator corresponding to two--dimensional maps. The result of the asymptotic periodicity for strictly expanding systems, that is, all…
We derive asymptotic properties for a stochastic dynamic network model in a stochastic dynamic population. In the model, nodes give birth to new nodes until they die, each node being equipped with a social index given at birth. During the…
Large populations of globally-coupled identical maps subjected to independent additive noise are shown to undergo qualitative changes as the features of the stochastic process are varied. We show that for strong coupling, the collective…
Quantum walks obey unitary dynamics: they form closed quantum systems. The system becomes open if the walk suffers from imperfections represented as missing links on the underlying basic graph structure, described by dynamical percolation.…
In this paper, under an abstract setting we establish the spreading properties and the existence, non-existence and global attractivity of spatially heterogeneous steady states for a large class of monotone evolution systems without the…