we grew the budding yeast saccharomyces cerevisiae in 200 l batch culture on bd falcon 96-well microtest plates at 30.7 c ( 0.2c , standard deviation ) using synthetic media ( yeast nitrogen bases + nitrogen , complete supplement mixture ) supplemented with 2% sucrose .
cultures were maintained in a well - mixed condition by growing on a shaker at 825 r.p.m .
serial dilutions were performed daily ( 23 hours of growth ) with variable dilution factors .
population densities were recorded each day before the serial dilution by measuring optical density at 600 nm using a thermo scientific varioskan flash multimode reader .
the calibration between optical density and cell density was based on the previous characterization of this system . in the group of connected populations , for each dilution factor there were 4 replicate arrays each consisting of 10 patches .
populations were connected by controlled dispersal between nearest neighbors ( dispersal rate d=0.5 , which is defined as the fraction of population going out of a patch ) . reflecting boundary conditions
were adopted , meaning that a population on the edge would have 75% of its cells remaining in the patch during the dispersal step . in the group of isolated populations ,
the experiment was performed in a similar spatial setting except that there was no dispersal ( d=0 ) ; for each dilution factor there were 4 arrays each consisting of 5 patches , thus a total of 20 replicate populations isolated from each other .
the dilution factors for the data presented in fig . 2 are 500 , 1000 , 1200 , 1400 and 1600 . in the experiment to measure recovery length ,
populations were connected by nearest - neighbor dispersal ( d=0.5 , reflecting boundary conditions ) .
the dilution factor for the bad region ( 1 patch ) was 2500 ; the dilution factor for the good region ( 5 patches ) was varied as the environmental driver .
the dilution factors for the data presented in fig . 4 are 500 , 750 , 1000 , 1133 , 1200 , 1266 , 1350 and 1400 .
statistical indicators for the connected populations were calculated among 10 populations in one array on each day and averaged over a span of at least 5 days , after the populations stabilized .
the mean value of 4 replicate arrays and sem ( n=4 ) are shown in figure 2 .
for the isolated populations , statistical indicators were calculated on each day among 20 populations over 5 days .
we used bootstrap to compute ses of the indicators by resampling 1000 times the ensemble of replicate populations ( for the coefficient of variation and the temporal correlation ) or arrays ( for the spatial correlation ) .
the coefficient of variation ( cv ) was calculated as the sample standard deviation ( supplementary fig .
3b ) divided by the sample mean . because the local populations in our experiment were grown in a homogeneous environment , in principle they could all be treated as replicates . assuming the system is ergodic , the cv calculated over an ensemble of replicates can be interpreted either as spatial cv of many populations at one time point or temporal cv of a single population over many time points .
the temporal correlation , defined as the lag-1 autocorrelation , was estimated by the pearson s correlation coefficient between the population densities at subsequent days . to correct for negative bias in small samples
n is a fixed number for different dilution factors , so using the modified estimators would not affect the trend of indicators .
the spatial correlation , defined as the two - point correlation between all neighboring pairs , was estimated by the moran s coefficient .
the expectation of moran s coefficient is 1n1 in the absence of spatial correlation ; we used a modified estimator with an additional term 1n1 so that the expectation is 0 . in this case , the sample size n is the number of patches in an array : n=10 for connected populations ; n=5 for isolated populations . for detailed formula of the statistical indicators , see supplementary note 4 . in the analysis we ensured environmental homogeneity by removing a linear gradient of population density observed in connected populations
the small gradient is presumably caused by some heterogeneity in experimental conditions ( temperature , dilution errors , etc . ) across the plate .
removing gradient - type spatial heterogeneity before statistical analysis is similar to the detrending procedure commonly used in time - series analysis ; it prevents spurious signals such as positive spatial correlation ( supplementary fig .
after the recovery profile stabilized , we tracked the population density profiles of at least 6 replicates over several days .
the half - point recovery length l was estimated by performing a shape - preserving interpolation ( matlab function pchip , piecewise cubic hermite interpolating polynomial ) to the recovery profile and then locating the position of half recovery at which n(x = lhalf)=12n(x=5 ) .
the population density of the bad region ( dilution factor 2500 ) in our experiment was close to 0 ( fig . 4 and supplementary fig .
7 ) . in the more general scenario with a sharp boundary between two regions of different quality ( supplementary fig . 6 and 8) ,
the position of half recovery can be defined as the midpoint between the equilibrium population density of the region of interest and the population density at the boundary .
the exponential recovery length was estimated by fitting an exponential function with three parameters c1 exp(x / l ) + c2 to the recovery profile n(x ) .
the data points used for exponential fitting are from position 1 to 5 ( except for dilution factor 500 , the data for fitting are from position 0 to 5 ) .
we note that our definition of exponential recovery length is phenomenological , because : 1 ) the deviation is expected to be exponential only close enough to the equilibrium ; 2 ) at higher dilution factors the profile can deviate from an exponential form ( supplementary fig .
4c ) may be due to the limited data points used in fitting or experimental errors . for both the half - point recovery length and the exponential recovery length , we used bootstrap to compute ses for the indicators by resampling the ensemble of steady - state profiles 100 times and fitting the average recovery profile .
we grew the budding yeast saccharomyces cerevisiae in 200 l batch culture on bd falcon 96-well microtest plates at 30.7 c ( 0.2c , standard deviation ) using synthetic media ( yeast nitrogen bases + nitrogen , complete supplement mixture ) supplemented with 2% sucrose .
cultures were maintained in a well - mixed condition by growing on a shaker at 825 r.p.m .
serial dilutions were performed daily ( 23 hours of growth ) with variable dilution factors .
population densities were recorded each day before the serial dilution by measuring optical density at 600 nm using a thermo scientific varioskan flash multimode reader .
the calibration between optical density and cell density was based on the previous characterization of this system . in the group of connected populations , for each dilution factor there were 4 replicate arrays each consisting of 10 patches .
populations were connected by controlled dispersal between nearest neighbors ( dispersal rate d=0.5 , which is defined as the fraction of population going out of a patch ) . reflecting boundary conditions
were adopted , meaning that a population on the edge would have 75% of its cells remaining in the patch during the dispersal step . in the group of isolated populations ,
the experiment was performed in a similar spatial setting except that there was no dispersal ( d=0 ) ; for each dilution factor there were 4 arrays each consisting of 5 patches , thus a total of 20 replicate populations isolated from each other .
the dilution factors for the data presented in fig . 2 are 500 , 1000 , 1200 , 1400 and 1600 . in the experiment to measure recovery length ,
populations were connected by nearest - neighbor dispersal ( d=0.5 , reflecting boundary conditions ) .
the dilution factor for the bad region ( 1 patch ) was 2500 ; the dilution factor for the good region ( 5 patches ) was varied as the environmental driver .
4 are 500 , 750 , 1000 , 1133 , 1200 , 1266 , 1350 and 1400 .
statistical indicators for the connected populations were calculated among 10 populations in one array on each day and averaged over a span of at least 5 days , after the populations stabilized .
the mean value of 4 replicate arrays and sem ( n=4 ) are shown in figure 2 .
for the isolated populations , statistical indicators were calculated on each day among 20 populations over 5 days .
we used bootstrap to compute ses of the indicators by resampling 1000 times the ensemble of replicate populations ( for the coefficient of variation and the temporal correlation ) or arrays ( for the spatial correlation ) .
the coefficient of variation ( cv ) was calculated as the sample standard deviation ( supplementary fig .
3b ) divided by the sample mean . because the local populations in our experiment were grown in a homogeneous environment , in principle they could all be treated as replicates .
assuming the system is ergodic , the cv calculated over an ensemble of replicates can be interpreted either as spatial cv of many populations at one time point or temporal cv of a single population over many time points .
the temporal correlation , defined as the lag-1 autocorrelation , was estimated by the pearson s correlation coefficient between the population densities at subsequent days . to correct for negative bias in small samples
n is a fixed number for different dilution factors , so using the modified estimators would not affect the trend of indicators .
the spatial correlation , defined as the two - point correlation between all neighboring pairs , was estimated by the moran s coefficient .
the expectation of moran s coefficient is 1n1 in the absence of spatial correlation ; we used a modified estimator with an additional term 1n1 so that the expectation is 0 . in this case , the sample size n is the number of patches in an array : n=10 for connected populations ; n=5 for isolated populations . for detailed formula of the statistical indicators , see supplementary note 4 .
in the analysis we ensured environmental homogeneity by removing a linear gradient of population density observed in connected populations .
the small gradient is presumably caused by some heterogeneity in experimental conditions ( temperature , dilution errors , etc . ) across the plate .
removing gradient - type spatial heterogeneity before statistical analysis is similar to the detrending procedure commonly used in time - series analysis ; it prevents spurious signals such as positive spatial correlation ( supplementary fig .
after the recovery profile stabilized , we tracked the population density profiles of at least 6 replicates over several days .
the half - point recovery length l was estimated by performing a shape - preserving interpolation ( matlab function pchip , piecewise cubic hermite interpolating polynomial ) to the recovery profile and then locating the position of half recovery at which n(x = lhalf)=12n(x=5 ) .
the population density of the bad region ( dilution factor 2500 ) in our experiment was close to 0 ( fig . 4 and supplementary fig .
7 ) . in the more general scenario with a sharp boundary between two regions of different quality ( supplementary fig . 6 and 8)
, the position of half recovery can be defined as the midpoint between the equilibrium population density of the region of interest and the population density at the boundary .
the exponential recovery length was estimated by fitting an exponential function with three parameters c1 exp(x / l ) + c2 to the recovery profile n(x ) .
the data points used for exponential fitting are from position 1 to 5 ( except for dilution factor 500 , the data for fitting are from position 0 to 5 ) .
we note that our definition of exponential recovery length is phenomenological , because : 1 ) the deviation is expected to be exponential only close enough to the equilibrium ; 2 ) at higher dilution factors the profile can deviate from an exponential form ( supplementary fig .
4c ) may be due to the limited data points used in fitting or experimental errors . for both the half - point recovery length and the exponential recovery length , we used bootstrap to compute ses for the indicators by resampling the ensemble of steady - state profiles 100 times and fitting the average recovery profile
 slower recovery from perturbations near a tipping point and its indirect signatures in fluctuation patterns have been suggested to foreshadow catastrophes in a wide variety of systems1,2 . 
 recent studies of populations in the field and in the laboratory have used time - series data to confirm some of the theoretically predicted early warning indicators , such as an increase in recovery time or in the size and timescale of fluctuations36 . 
 however , the predictive power of temporal warning signals is limited by the demand for long - term observations . 
 large - scale spatial data are more accessible , but the performance of warning signals in spatially extended systems710 needs to be examined empirically3,1113 . here 
 we use spatially extended yeast populations , an experimental system displaying a fold bifurcation6 , to evaluate early warning signals based on spatio - temporal fluctuations and to identify a novel warning indicator in space . 
 we found that two leading indicators based on fluctuations increased before collapse of connected populations ; however , the magnitude of increase was smaller than that observed in isolated populations , possibly because local variation is reduced by dispersal . 
 furthermore , we propose a generic indicator based on deterministic spatial patterns ,  recovery length  . as the spatial counterpart of recovery time14 
 , recovery length is defined as the distance for connected populations to recover from perturbations in space ( e.g. a region of poor quality ) . in our experiments , 
 recovery length increased substantially before population collapse , suggesting that the spatial scale of recovery can provide a superior warning signal before tipping points in spatially extended systems . 
Methods
Experimental protocols
Calculation of statistical indicators
Recovery length
Supplementary Material