We are interested in populations in which the fitness of different genetic types fluctuates in time and space, driven by temporal and spatial fluctuations in the environment. For simplicity, our population is assumed to be composed of just two genetic types. Short bursts of selection acting in opposing directions drive to maintain both types at intermediate frequencies, while the fluctuations due to 'genetic drift' work to eliminate variation in the population.We consider first a population with no spatial structure, modelled by an adaptation of the Lambda (or generalised) Fleming-Viot process, and derive a stochastic differential equation as a scaling limit. This amounts to a limit result for a Lambda-Fleming-Viot process in a rapidly fluctuating random environment. We then extend to a population that is distributed across a spatial continuum, which we model through a modification of the spatial Lambda-Fleming-Viot process with selection. In this setting we show that the scaling limit is a stochastic partial differential equation. As is usual with spatially distributed populations, in dimensions greater than one, the 'genetic drift' disappears in the scaling limit, but here we retain some stochasticity due to the fluctuations in the environment, resulting in a stochastic p.d.e. driven by a noise that is white in time but coloured in space.We discuss the (rather limited) situations under which there is a duality with a system of branching and annihilating particles. We also write down a system of equations that captures the frequency of descendants of particular subsets of the population and use this same idea of 'tracers', which we learned from HALLATSCHEK and NELSON (2008, [23]) and DURRETT and FAN (2016, [13]), in numerical experiments with a closely related model based on the classical Moran model.

We investigate the behaviour of an establishing mutation which is subject to rapidly fluctuating selection under the Lambda-Fleming-Viot model and show that under a suitable scaling it converges to the Feller diffusion in a random environment. We then extend to a population that is distributed across a spatial continuum. In this setting the scaling limit is the SuperBrownian motion in a random environment. The scaling results for the behaviour of the rare allele are achieved via particle representations which belong to the family of 'lookdown constructions'. This generalises the results obtained for the neutral version of the model by Chetwynd-Diggle and Etheridge (2018), which was proved using a duality argument. To our knowledge this is the first instance of the application of the lookdown approach in which other techniques seem unavailable.

We are interested in populations in which the fitness of different genetic types fluctuates in time and space, driven by temporal and spatial fluctuations in the environment. For simplicity, our population is assumed to be composed of just two genetic types. Short bursts of selection acting in opposing directions drive to maintain both types at intermediate frequencies, while the fluctuations due to 'genetic drift' work to eliminate variation in the population.We consider first a population with no spatial structure, modelled by an adaptation of the Lambda (or generalised) Fleming-Viot process, and derive a stochastic differential equation as a scaling limit. This amounts to a limit result for a Lambda-Fleming-Viot process in a rapidly fluctuating random environment. We then extend to a population that is distributed across a spatial continuum, which we model through a modification of the spatial Lambda-Fleming-Viot process with selection. In this setting we show that the scaling limit is a stochastic partial differential equation. As is usual with spatially distributed populations, in dimensions greater than one, the 'genetic drift' disappears in the scaling limit, but here we retain some stochasticity due to the fluctuations in the environment, resulting in a stochastic p.d.e. driven by a noise that is white in time but coloured in space.We discuss the (rather limited) situations under which there is a duality with a system of branching and annihilating particles. We also write down a system of equations that captures the frequency of descendants of particular subsets of the population and use this same idea of 'tracers', which we learned from Hallatschek and Nelson (2008) and Durrett and Fan (2016), in numerical experiments with a closely related model based on the classical Moran model.

No abstract

The risk of bankruptcy of an enterprise is inevitably inscribed in running a business. This risk should be managed effectively in order to avoid negative consequences associated with the bankruptcy of an entity. This article will present the economic and legal causes of bankruptcy, followed by its negative and positive effects. In connection with this topic, the authors will discuss the concept of the microeconomic wage productivity and the management variable in the context of their importance in an enterprise. As an extension of previous studies by one of the authors, an attempt will be made to determine the possibility of applying the microeconomic wage productivity and the management variable as a supplement of the bankruptcy risk analysis of enterprises that are suppliers to the automotive industry in Poland.