Evolutionary Games

International audienceEvolutionary games constitute the most recent major mathematical tool for understanding, modelling and predicting evolution in biology and other fields. They complement other well establlished tools such as branching processes and the Lotka-Volterra [6] equations (e.g. for the predator - prey dynamics or for epidemics evolution). Evolutionary Games also brings novel features to game theory. First, it focuses on the dynam- ics of competition rather than restricting attention to the equilibrium. In particular, it tries to explain how an equilibrium emerges. Second, it brings new de nitions of stability, that are more adapted to the context of large populations. Finally, in contrast to standard game theory, players are not assumed to be \rational" or \knowledgeable" as to anticipate the other players' choices. The objective of this article, is to present founda- tions as well as recent advances in evolutionary games, highlight the novel concepts that they introduce with respect to game theory as formulated by John Nash, and describe through several examples their huge potential as tools for modeling interactions in complex systems

Fluctuation scaling in complex systems: Taylor's law and beyond

Complex systems consist of many interacting elements which participate in some dynamical process. The activity of various elements is often different and the fluctuation in the activity of an element grows monotonically with the average activity. This relationship is often of the form "$fluctuations \approx const.\times average^\alpha$", where the exponent $\alpha$ is predominantly in the range $[1/2, 1]$. This power law has been observed in a very wide range of disciplines, ranging from population dynamics through the Internet to the stock market and it is often treated under the names \emph{Taylor's law} or \emph{fluctuation scaling}. This review attempts to show how general the above scaling relationship is by surveying the literature, as well as by reporting some new empirical data and model calculations. We also show some basic principles that can underlie the generality of the phenomenon. This is followed by a mean-field framework based on sums of random variables. In this context the emergence of fluctuation scaling is equivalent to some corresponding limit theorems. In certain physical systems fluctuation scaling can be related to finite size scaling.Comment: 33 pages, 20 figures, 2 tables, submitted to Advances in Physic

Potential Landscape and Probabilistic Flux of a Predator Prey Network

Predator-prey system, as an essential element of ecological dynamics, has been recently studied experimentally with synthetic biology. We developed a global probabilistic landscape and flux framework to explore a synthetic predator-prey network constructed with two Escherichia coli populations. We developed a self consistent mean field method to solve multidimensional problem and uncovered the potential landscape with Mexican hat ring valley shape for predator-prey oscillations. The landscape attracts the system down to the closed oscillation ring. The probability flux drives the coherent oscillations on the ring. Both the landscape and flux are essential for the stable and coherent oscillations. The landscape topography characterized by the barrier height from the top of Mexican hat to the closed ring valley provides a quantitative measure of global stability of system. The entropy production rate for the energy dissipation is less for smaller environmental fluctuations or perturbations. The global sensitivity analysis based on the landscape topography gives specific predictions for the effects of parameters on the stability and function of the system. This may provide some clues for the global stability, robustness, function and synthetic network design

Measuring and modelling the response of Klebsiella pneumoniae KPC prey to Bdellovibrio bacteriovorus predation, in human serum and defined buffer

In worldwide conditions of increasingly antibiotic-resistant hospital infections, it is important to research alternative therapies. Bdellovibrio bacteriovorus bacteria naturally prey on Gram-negative pathogens, including antibiotic-resistant strains and so B. bacteriovorus have been proposed as "living antibiotics" to combat antimicrobially-resistant pathogens. Predator-prey interactions are complex and can be altered by environmental components. To be effective B. bacteriovorus predation needs to work in human body fluids such as serum where predation dynamics may differ to that studied in laboratory media. Here we combine mathematical modelling and lab experimentation to investigate the predation of an important carbapenem-resistant human pathogen, Klebsiella pneumoniae, by B. bacteriovorus in human serum versus buffer. We show experimentally that B. bacteriovorus is able to reduce prey numbers in each environment, on different timescales. Our mathematical model captures the underlying dynamics of the experimentation, including an initial predation-delay at the predator-prey-serum interface. Our research shows differences between predation in buffer and serum and highlights both the potential and limitations of B. bacteriovorus acting therapeutically against K. pneumoniae in serum, informing future research into the medicinal behaviours and dosing of this living antibacterial

Global Analysis of Dynamical Decision-Making Models through Local Computation around the Hidden Saddle

Bistable dynamical switches are frequently encountered in mathematical modeling of biological systems because binary decisions are at the core of many cellular processes. Bistable switches present two stable steady-states, each of them corresponding to a distinct decision. In response to a transient signal, the system can flip back and forth between these two stable steady-states, switching between both decisions. Understanding which parameters and states affect this switch between stable states may shed light on the mechanisms underlying the decision-making process. Yet, answering such a question involves analyzing the global dynamical (i.e., transient) behavior of a nonlinear, possibly high dimensional model. In this paper, we show how a local analysis at a particular equilibrium point of bistable systems is highly relevant to understand the global properties of the switching system. The local analysis is performed at the saddle point, an often disregarded equilibrium point of bistable models but which is shown to be a key ruler of the decision-making process. Results are illustrated on three previously published models of biological switches: two models of apoptosis, the programmed cell death and one model of long-term potentiation, a phenomenon underlying synaptic plasticity

Stability and monotonicity of Lotka–Volterra type operators

In the present paper,we investigate stability of trajectories ofLotka–Volterra (LV) type operators defined in finite dimensional simplex.We prove that any LV type operator is a surjection of the simplex. It is introduced a newclass of LV-type operators, called MLV type ones, and we show that trajectories of the introduced operators converge. Moreover, we show that such kind of operators have totally different behavior than f-monotone LV type operators