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Abstract The ecological literature often features phenomenological dynamic models lacking robust validation against observational data. Reverse engineering ecological models from data is an alternative approach, where time series data are utilized to infer or fit a stochastic differential equation. This process, known as system reconstruction, presents significant challenges especially when data resolution is low. This paper addresses the estimation of the (often) non‐linear deterministic and stochastic parts of Langevin models from sparsely sampled time series.We introduce a maximum likelihood estimation (MLE) inference method, termed Euler reconstruction, tailored for time series data with high resolution. However, the Euler approach is not reliable for low‐resolution data. To fill the gap for sparsely sampled data, we present an MLE inference method pioneered by Aït‐Sahalia that we term Hermite reconstruction. We employ splines to detect inherent nonlinearities in the unknown data‐generating system with high accuracy and acceptable computational burden.We applied both methods to a range of simulated, ecological, and climate datasets, with different data resolutions. We provide a practical measure (‘relaxation time’) to distinguish between different data resolutions. For simulated data, we show that the Euler reconstruction can accurately reveal the underlying system when data resolution is high, while Hermite reconstruction can recover the system even with low resolution. We recommend using Hermite reconstruction for real data even when the resolution is relatively high. Only when the resolution of real data is exceptionally high might Euler reconstruction suffice.We provide a MATLAB package and a tutorial to assist researchers in applying the method to their own data. 

Summer temperature chain data were collected from three lakes (Peter, Paul, and Tuesday) using NexSens temperature thermistors. Data are available for five years: 2013–2015 and 2018–2019. Peter and Paul Lakes have complete records for all five years, while Tuesday Lake has data for three years. During 2013–2015, thermistors recorded temperatures at depths of 0.5, 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0, and 5.0 meters. Beginning in 2018, thermistors were added at 4.5 and 6.0 meters. 

This is data from a SUNA V2 nitrate sensor and a YSI EXO 2 sonde instrumented with water temperature, dissolved oxygen, pH, chlorophyll, phycocyanin, conductivity, turbidity, and fDOM sensors. The sensors are located at the lake end of the pier serving the Center for Limnology on the UW-Madison campus. The YSI sonde is fixed on the pier with sensors nominally at 0.5 meters depth, while the SUNA is suspended below the pier at one meter depth in the open water season. In the winter season, the SUNA is placed in a cage on the lake bottom close to shore with the sensor 18cm off the bottom. The depth of the sensors will vary with lake level over the season. The water depth at the lake end of the pier is normally about 3 meters. YSI sonde data are sampled once per minute. Hourly and daily averages are provided as separate CSV files. The SUNA sample rate varies. In the winter (under the ice) it relies on single battery charge, so the wiper is deactivated and the sensor samples every 1-2 hours. Daily averages of SUNA are also provided as a separate CSV. The YSI sonde is deployed only during the ice-free season coinciding with the placement of the pier. Sensors are cleaned and maintained roughly every two weeks. Number of sites: 1. Location lat/long: 43.07758, -89.40297 

High-frequency continuous data for temperature, dissolved oxygen, pH, chlorophyll-a, and phycocyanin in Paul Peter lakes from mid-May to early September for the years 2018 and 2019. Inorganic nitrogen and phosphorus were added to Peter in 2019, while Paul Lake was an unfertilized reference. 

Abstract Resilience, measured by the distribution of passage times between alternate states, indicates persistence of a state in stochastic dynamic systems such as blooms of cyanobacteria in lakes. We used high‐frequency datasets to compare the resilience of low and high states of phycocyanin, a pigment indicator of cyanobacteria, in Lake Mendota, Wisconsin, USA, for three growing seasons that ranged sevenfold in external phosphorus (P) load. Each year we observed 139–265 passage times across the unstable threshold that separated the low‐ from high‐phycocyanin states. Each sample of passage times is highly skewed with low median, larger mean, much larger SD, and wide tails extending to long lifetimes of a state. About 25% of events, whether low or high phycocyanin, lasted a day or more. Among these 3 years of contrasting external P load, there were no discernible differences in the resilience of either ecosystem state. We attribute this lack of contrast to the sustained recycling of P from sediments and the high stochasticity of phycocyanin in this lake. 

This data set is a derived data set based on fish catch data. Data are collected annually to enable us to track the fish assemblages of eleven primary lakes (Allequash, Big Muskellunge, Crystal, Sparkling, Trout, bog lakes 27-02 [Crystal Bog] and 12-15 [Trout Bog], Mendota, Monona, Wingra and Fish). Sampling on Lakes Monona, Wingra, and Fish started in 1995; sampling on other lakes started in 1981. Sampling is done at six littoral zone sites per lake with seine, minnow or crayfish traps, and fyke nets; a boat-mounted electrofishing system samples three littoral transects. Vertically hung gill nets are used to obtain two pelagic samples per lake from the deepest point. A trammel net samples across the thermocline at two sites per lake. In the bog lakes only fyke nets and minnow traps are deployed. Parameters measured include species-level identification and lengths for all fish caught, and weight and scale samples from a subset. Derived data sets include species richness, catch per unit effort, and size distribution by species, lake, and year. Species richness for a lake is the number of fish species caught in that lake during the annual fish sampling. Hybrids captured are only included in the richness value if neither of the two hybridized species are caught in the lake that year. Fish identified only to genus or higher taxonomic level are not included if any fish identified to species within that genus or higher taxonomic level are caught. E.g., Unidentified Chub would be only included in the richness value if no other chub is caught in that lake that year. Sampling Frequency: annually. Number of sites: 11 Notes: Beach seining was discontinued after 2019. 2020 data does not exist due to insufficient sampling. In 2021, sampling in Fish Lake was suspended due to significant lake level changes. Data is missing for the two bogs in 2022. Please consult NTL's website for information on experimental lake manipulations and the DNR's website for management activities 

Data are collected annually to enable us to track the fish assemblages of eleven primary lakes (Allequash, Big Muskellunge, Crystal, Sparkling, Trout, bog lakes 27-02 [Crystal Bog] and 12-15 [Trout Bog], Mendota, Monona, Wingra and Fish). Sampling on Lakes Monona, Wingra, and Fish started in 1995; sampling on other lakes started in 1981. Sampling is done at six littoral zone sites per lake with seine, minnow or crayfish traps, and fyke nets; a boat-mounted electrofishing system samples four littoral transects. Vertically hung gill nets are used to obtain two pelagic samples per lake from the deepest point. A trammel net samples across the thermocline at two sites per lake. In the bog lakes only fyke nets and minnow traps are deployed. Parameters measured include species-level identification and lengths for all fish caught, and weight and scale samples from a subset. Dominant species vary from lake to lake. Perch, rockbass, and bluegill are common, with walleye, large and smallmouth bass, northern pike and muskellunge as major piscivores. Cisco have been present in the pelagic waters of four lakes, and an exotic species, rainbow smelt, is present in two. The bog lakes contain mudminnows. Beach seining was discontinued after the 2019 season. The only sampling done in 2020 were a single gill-netting replicate in Sparkling, Crystal, and Trout lakes. Sampling in Fish Lake was missed in 2021 due to significant lake level changes. Data from the two bogs is missing in 2022. Sampling Frequency: annually Number of sites: 11. 

This data set is a derived data set based on fish catch data. Data are collected annually to enable us to track the fish assemblages of eleven primary lakes (Allequash, Big Muskellunge, Crystal, Sparkling, Trout, bog lakes 27-02 [Crystal Bog] and 12-15 [Trout Bog], Mendota, Monona, Wingra and Fish). Sampling on Lakes Monona, Wingra, and Fish started in 1995; sampling on other lakes started in 1981. Sampling is done at six littoral zone sites per lake with seine, minnow or crayfish traps, and fyke nets; a boat-mounted electrofishing system samples three littoral transects. Vertically hung gill nets are used to obtain two pelagic samples per lake from the deepest point. A trammel net samples across the thermocline at two sites per lake. In the bog lakes only fyke nets and minnow traps are deployed. Parameters measured include species-level identification and lengths for all fish caught, and weight and scale samples from a subset. Derived data sets include species richness, catch per unit effort, and size distribution by species, lake, and year. Protocol used to generate data: Day seines were only used in 1981 and have been eliminated from this data set to make sampling effort across years comparable. Number caught for each species is summed over repetitions of a gear within a lake and over depth. For information on fish stocking by the Wisconsin Department of Natural Resources in LTER lakes in Dane and Vilas counties, see https://dnr.wi.gov/fisheriesmanagement/Public/Summary/Index. Beach seining was discontinued after 2019. The only sampling done in 2020 were a single gill-netting sample in Sparkling, Crystal, and Trout lakes. Sampling in Fish Lake was missed in 2021 due to significant lake level changes. Data from the two bogs is missing in 2022. Sampling Frequency: annually. Number of sites: 11 

This data set is a derived data set based on fish catch and length data. Data are collected annually to enable us to track the fish assemblages of eleven primary lakes (Allequash, Big Muskellunge, Crystal, Sparkling, Trout, bog lakes 27-02 [Crystal Bog] and 12-15 [Trout Bog], Mendota, Monona, Wingra and Fish). Sampling on Lakes Monona, Wingra, and Fish started in 1995; sampling on other lakes started in 1981. Sampling is done at six littoral zone sites per lake with seine, minnow or crayfish traps, and fyke nets; a boat-mounted electrofishing system samples three littoral transects. Vertically hung gill nets are used to obtain two pelagic samples per lake from the deepest point. A trammel net samples across the thermocline at two sites per lake. In the bog lakes only fyke nets and minnow traps are deployed. Parameters measured include species-level identification and lengths for all fish caught, and scale samples and weight from a subset. Derived data sets include species richness, catch per unit effort, and size distribution by species, lake, and year. Dominant species vary from lake to lake. Perch, rockbass, and bluegill are common, with walleye, large and small mouth basses, northern pike and muskellunge as major piscivores. Cisco have been present in the pelagic waters of four lakes, and the exotic species, rainbow smelt, is present in two. The bog lakes contain mudminnows. Protocol used to generate data: The number of fish caught in each five mm length interval (0