Some simple but challenging Markov processes

In this note, we present few examples of Piecewise Deterministic Markov Processes and their long time behavior. They share two important features: they are related to concrete models (in biology, networks, chemistry,. . .) and they are mathematically rich. Their math-ematical study relies on coupling method, spectral decomposition, PDE technics, functional inequalities. We also relate these simple examples to recent and open problems

Investment in Irrigation Systems under Weather Uncertainty

Irrigated agriculture will play a crucial role to meet future food demand, but a sustainable water resource management in agriculture is crucial as well. Therefore, the European Water Framework Directive promotes several measures, e.g., the adoption of adequate water pricing mechanisms or the promotion of water-saving irrigation techniques. Since production conditions such as weather and climate development are uncertain, farmers might be reluctant to invest in a water-saving but capital intensive irrigation system. We apply a stochastic dynamic programming approach to analyze a farmer’s optimal investment strategy for either a water–saving drip irrigation system or sprinkler irrigation system under weather uncertainty and assess the probability of adopting either irrigation system until the year 2040. We design two policy scenarios: (i) irrigation water pricing and (ii) equipment subsidies for drip irrigation, and investigate how they affect the farmer’s optimal investment strategy. Our case study analysis is performed for the region Marchfeld, a typical semi-arid agricultural production region in Austria. We use data from the bio-physical process simulation model EPIC (Environmental Policy Integrated Climate) which accounts for site and management related characteristics as well as weather parameters from a statistical climate change model. We find that investment in drip irrigation is unlikely unless subsidies for equipment cost are granted. Even water prices do not increase the probability to adopt a drip irrigation system, but rather decrease the probability to invest into either irrigation system.Irrigation investment, stochastic dynamic programming approach, water policies, weather uncertainty, EPIC, Farm Management, Risk and Uncertainty,

Switchable Genetic Oscillator Operating in Quasi-Stable Mode

Ring topologies of repressing genes have qualitatively different long-term dynamics if the number of genes is odd (they oscillate) or even (they exhibit bistability). However, these attractors may not fully explain the observed behavior in transient and stochastic environments such as the cell. We show here that even repressilators possess quasi-stable, travelling-wave periodic solutions that are reachable, long-lived and robust to parameter changes. These solutions underlie the sustained oscillations observed in even rings in the stochastic regime, even if these circuits are expected to behave as switches. The existence of such solutions can also be exploited for control purposes: operation of the system around the quasi-stable orbit allows us to turn on and off the oscillations reliably and on demand. We illustrate these ideas with a simple protocol based on optical interference that can induce oscillations robustly both in the stochastic and deterministic regimes.Comment: 24 pages, 5 main figure