Empirical evidence for large-scale abrupt changes in ecosystems such as lakes and vegetation of semi-arid regions is growing. Such changes, called regime shifts, can lead to degradation of ecological services. We study simple ecological models that show a catastrophic transition as a control parameter is varied and propose a novel early warning signal that exploits two ubiquitous features of ecological systems: nonlinearity and large external fluctuations. Either reduced resilience or increased external fluctuations can tip ecosystems to an alternative stable state. It is shown that changes in asymmetry in the distribution of time series data, quantified by changing skewness, is a model-independent and reliable early warning signal for both routes to regime shifts. Furthermore, using model simulations that mimic field measurements and a simple analysis of real data from abrupt climate change in the Sahara, we study the feasibility of skewness calculations using data available from routine monitoring.

No abstract

No abstract

No abstract

The two-impurity Kondo problem is studied by use of perturbative scaling techniques. The physics is determined by the interplay between the Ruderman-Kittel-KasuyaYosida (RKKY) interaction between the two impurity spins and the Kondo effect. In particular, for a strong ferromagnetic RKKY interaction the susceptibility exhibits three structures as the temperature is lowered, corresponding to the ferromagnetic locking together of the two impurity spins followed by a two-stage freezing out of their local moments by the conduction electrons due to the Kondo effect.PACS numbers: 75.20.Hr, 72.15.Qm In this Letter we discuss the possible temperature-dependent susceptibilities for two Kondo impurities separated by a distance R imbedded in a metal of noninteracting conduction electrons. The various behaviors depend on the relative magnitude (and sign) of the Ruder man-Kittel-KasuyaYosida (RKKY) interaction compared with the Kondo temperature of an isolated impurity.The essential ingredient of this work is the ability to trace the evolution of an effective Hamiltonian that describes successively lower energy (and hence temperature) scales. For a given temperature T, the effective Hamiltonian is obtained by integrating out the conduction electrons and holes of energy bigger than a cutoff D -10T. Using a thermodynamic scaling method, 1 ' 2 we are able to describe the variation of the effective Hamiltonian with decreasing cutoff. In this Letter we exhibit only the most important interactions in the effective Hamiltonian. Further, we choose the parameters of the problem so that the rele-, H 2LM (D) =ff el (D) -78, -3 2 -J s (s, + s 2 ) -(S, +3 2 ) -

Transcriptional pulsing has been observed in both prokaryotes and eukaryotes and plays a crucial role in cell-to-cell variability of protein and mRNA numbers. An important issue is how the time constants associated with episodes of transcriptional bursting and mRNA and protein degradation rates lead to different cellular mRNA and protein distributions, starting from the transient regime leading to the steady state. We address this by deriving and then investigating the exact time-dependent solution of the master equation for a transcriptional pulsing model of mRNA distributions. We find a plethora of results. We show that, among others, bimodal and long-tailed ͑power-law͒ distributions occur in the steady state as the rate constants are varied over biologically significant time scales. Since steady state may not be reached experimentally we present results for the time evolution of the distributions. Because cellular behavior is determined by proteins, we also investigate the effect of the different mRNA distributions on the corresponding protein distributions using numerical simulations.

Ecosystems can undergo large-scale changes in their states, known as catastrophic regime shifts, leading to substantial losses to services they provide to humans. These shifts occur rapidly and are difficult to predict. Several early warning signals of such transitions have recently been developed using simple models. These studies typically ignore spatial interactions, and the signal provided by these indicators may be ambiguous. We employ a simple model of collapse of vegetation in one and two spatial dimensions and show, using analytic and numerical studies, that increases in spatial variance and changes in spatial skewness occur as one approaches the threshold of vegetation collapse. We identify a novel feature, an increasing spatial variance in conjunction with a peaking of spatial skewness, as an unambiguous indicator of an impending regime shift. Once a signal has been detected, we show that a quick management action reducing the grazing activity is needed to prevent the collapse of vegetated state. Our results show that the difficulties in obtain- Electronic supplementary materialThe online version of this article (ing the accurate estimates of indicators arising due to lack of long temporal data can be alleviated when high-resolution spatially extended data are available. These results are shown to hold true independent of various details of model or different spatial dispersal kernels such as Gaussian or heavily fat tailed. This study suggests that spatial data and monitoring multiple indicators of regime shifts can play a key role in making reliable predictions on ecosystem stability and resilience.

No abstract