On a nonlinear age-structured model of semelparous species
Ryszard Rudnicki Radosław Wieczorek
Discrete & Continuous Dynamical Systems - B 2014, 19(8): 2641-2656 doi: 10.3934/dcdsb.2014.19.2641
We study a nonlinear age-structured model of a population such that individuals may give birth only at a given age. Properties of measure-valued periodic solutions of this system are investigated. We show that in some cases the age profile of the population tends to a Dirac measure, which means that the population asymptotically consists of individuals at the same age. This phenomenon is observed in nature in some insects populations.
keywords: measure-valued solutions demographic cycle Age-structure semelparous species convergence to singular distributions.
An ergodic theory approach to chaos
Ryszard Rudnicki
Discrete & Continuous Dynamical Systems - A 2015, 35(2): 757-770 doi: 10.3934/dcds.2015.35.757
This paper is devoted to the ergodic-theoretical approach to chaos, which is based on the existence of invariant mixing measures supported on the whole space. As an example of application of the general theory we prove that there exists an invariant mixing measure with respect to the differentiation operator on the space of entire functions. From this theorem it follows the existence of universal entire functions and other chaotic properties of this transformation.
keywords: chaos Invariant measure differentiation operator. universal functions
Does assortative mating lead to a polymorphic population? A toy model justification
Ryszard Rudnicki Radoslaw Wieczorek
Discrete & Continuous Dynamical Systems - B 2018, 23(1): 459-472 doi: 10.3934/dcdsb.2018031

We consider a model of phenotypic evolution in populations with assortative mating of individuals. The model is given by a nonlinear operator acting on the space of probability measures and describes the relation between parental and offspring trait distributions. We study long-time behavior of trait distribution and show that it converges to a combination of Dirac measures. This result means that assortative mating can lead to a polymorphic population and sympatric speciation.

keywords: Assortative mating phenotypic evolution polymorphic population nonlinear operator of trait inheritance convergence to multimodal distribution
Stability of stochastic semigroups and applications to Stein's neuronal model
Katarzyna PichÓr Ryszard Rudnicki
Discrete & Continuous Dynamical Systems - B 2018, 23(1): 377-385 doi: 10.3934/dcdsb.2018026

A new theorem on asymptotic stability of stochastic semigroups is given. This theorem is applied to a stochastic semigroup corresponding to Stein's neuronal model. Asymptotic properties of models with and without the refractory period are compared.

keywords: Stochastic semigroup asymptotic stability Stein's model Markov process neuron activity

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