Replicator dynamics for the game theoretic selection models based on state.

The paper presents an attempt to integrate the classical evolutionary game theory based on replicator dynamics and the state-based approach of Houston and McNamara. In the new approach, individuals have different heritable strategies; however, individuals carrying the same strategy can differ in terms of state, role or the situation in which they act. Thus, the classical replicator dynamics is completed by the additional subsystem of differential equations describing the dynamics of transitions between different states. In effect, the interactions described by game structure, in addition to the demographic payoffs (constituted by births and deaths), can lead to the change in state of the competing individuals. Special cases of reversible and irreversible incremental stage-structured models, where the state changes can describeenergy accumulation, developmental steps or aging, are derived for discrete and continuous versions. The new approach is illustrated using the example of the Owner-Intruder game with explicit dynamics of the role changes. The new model presents a generalization of the demographic version of the Hawk-Dove game,with the difference being that the opponents in the game are drawn from two separate subpopulations consisting of Owners and Intruders. Here, the Intruders check random nest sites and play the Hawk-Dove game with the Owner if they are occupied. Meanwhile, the Owners produce newborns that become Intruders, since they must find a free nest site to reproduce. An interesting feedback mechanism is produced via the fluxes of individuals between the different subpopulations. In addition, the population growth suppression mechanism resulting from the fixation Bourgeois strategy is analyzed.

On convergence and asymptotic behaviour of semigroups of operators.

The classical and modern theorems on convergence, approximation and asymptotic stability of semigroups of operators are presented, and their applications to recent biological models are discussed. This article is part of the theme issue 'Semigroup applications everywhere'.

Correction to: From nest site lottery to host lottery: continuous model of growth suppression driven by the availability of nest sites for newborns or hosts for parasites and its impact on the selection of life history strategies.

Unfortunately, part of the article title was updated as subtitle which in turn resulted with complete title not appearing on website and in the bibliographic data. The complete version of title is updated here.

From nest site lottery to host lottery: continuous model of growth suppression driven by the availability of nest sites for newborns or hosts for parasites and its impact on the selection of life history strategies.

The idea that selection works in different ways during free population growth and at the equilibrium population size has been present in theoretical biology for a long time. It was first expressed as an r and K selection concept and later clarified in the debate on fitness measures in life history theory. The latest discussion related to this topic is focused on the nest site lottery mechanism and the resulting new population growth model. In this mechanistic biphasic model, the suppression of growth is induced by a shortage of free nest sites for newborns. Before it occurs, the population can grow exponentially. In this paper, the continuous version of the model and its selective properties are analysed. We show a continuous smooth transition between different fitness measures operating during the exponential growth and suppressed growth phase and at the equilibrium population size. Then, the model is extended to the case of a population of parasites, where a constant number of nest sites is replaced by the dynamics of a population of their hosts, in the role of the limiting supply. Parasite strategies are selected under exponential and suppressed growth phases of the population of hosts. Transitions between different fitness measures and conditions for extinction of hosts by parasites are analysed. An interesting result is the possibility of a continuum of fitness measures of parasites for the unsuppressed exponential growth of the host population.

Nest site lottery revisited: towards a mechanistic model of population growth suppressed by the availability of nest sites.

In the "nest site lottery" mechanism, newborns form a pool of candidates and randomly drawn candidates replace the dead adults in their nest sites. However, the selection process has only been analyzed under the assumption of an equilibrium population size. In this study, we extend this model to cases where the population size is not at an equilibrium, which yields a simplified (but fully mechanistic) biphasic population growth model, where the suppression of growth is driven only by the availability of free nest sites for newborns. This new model is free of the inconsistency found in the classical single phase models (such as the logistic model), where the number of recruited newborns can exceed the number of free nest sites. We analyzed the stability of the stationary density surfaces and the selection mechanisms for individual strategies described by different vital rates, which are implied by the new model.

On a stochastic gene expression with pre-mRNA, mRNA and protein contribution.

In this paper we develop a model of stochastic gene expression, which is an extension of the model investigated in the paper [T. Lipniacki, P. Paszek, A. Marciniak-Czochra, A.R. Brasier, M. Kimmel, Transcriptional stochasticity in gene expression, J. Theor. Biol. 238 (2006) 348-367]. In our model, stochastic effects still originate from random fluctuations in gene activity status, but we precede mRNA production by the formation of pre-mRNA, which enriches classical transcription phase. We obtain a stochastically regulated system of ordinary differential equations (ODEs) describing evolution of pre-mRNA, mRNA and protein levels. We perform mathematical analysis of a long-time behavior of this stochastic process, identified as a piece-wise deterministic Markov process (PDMP). We check exact results using numerical simulations for the distributions of all three types of particles. Moreover, we investigate the deterministic (adiabatic) limit state of the process, when depending on parameters it can exhibit two specific types of behavior: bistability and the existence of the limit cycle. The latter one is not present when only two kinds of gene expression products are considered.

Model of phenotypic evolution in hermaphroditic populations.

We consider an individual based model of phenotypic evolution in hermaphroditic populations which includes random and assortative mating of individuals. By increasing the number of individuals to infinity we obtain a nonlinear transport equation, which describes the evolution of phenotypic distribution. The main result of the paper is a theorem on asymptotic stability of trait distribution. This theorem is applied to models with the offspring trait distribution given by additive and multiplicative random perturbations of the parental mean trait.

Chaoticity of the blood cell production system.

We present a structured model of stem cells given by a partial differential equation. This equation generates a semiflow acting on the set of densities. We show that this semiflow possesses an invariant exact measure positive on open sets. From this it follows that the system is chaotic, i.e., it has dense trajectories and each trajectory is unstable.

Influence of stochastic perturbation on prey-predator systems.

We analyse the influence of various stochastic perturbations on prey-predator systems. The prey-predator model is described by stochastic versions of a deterministic Lotka-Volterra system. We study long-time behaviour of both trajectories and distributions of the solutions. We indicate the differences between the deterministic and stochastic models.

A model for the evolution of paralog families in genomes.

We introduce and analyse a simple probabilistic model of genome evolution. It is based on three fundamental evolutionary events: gene loss, duplication and accumulated change. This is motivated by previous works which consisted in fitting the available genomic data into, what is called paralog distributions. This formalism is described by a system of infinite number of linear equations. We show that this system generates a semigroup of linear operators on the space l (1). We prove that size distribution of paralogous gene families in a genome converges to the equilibrium as time goes to infinity. Moreover we show that when probabilities of gene removal and duplication are close to each other, then the resulting distribution is close to logarithmic distribution. Some empirical results for yeast genomes are presented.

Phytoplankton dynamics.

We present a model of the phytoplankton dynamics. The distribution of the size of the phytoplankton aggregates is described by a non-linear transport equation that contains terms responsible for the growth of phytoplankton aggregates, their fragmentation and coagulation. We study asymptotic behaviour of moments of the solutions and we explain why phytoplankton tends to create large aggregates.