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. 2023 May 10;13:7613. doi: 10.1038/s41598-023-34408-0

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© The Author(s) 2023

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

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Relationships between the mean total length of (a) age-1 and (b) age-2 walleye and the estimated total number of older (age 3 +) walleye in Lake Erie during 1974–2015. Years progress across a color gradient from yellow to blue, with orange and purple representing intermediate decades. Solid lines indicate the best pre-breakpoint relationships, whereas dashed lines represent post-breakpoint relationships (modeled using piecewise linear regression). Gray lines show relationships with a downweighted influence of potentially high leverage points (age-1: 1974, 1983, 1984, and 2007; age-2: 1974, 1983, 1984, and 2008). Relationships were initially negative (age-1: slope = − 51.2, R²_lik = 0.37, P = 0.041; age-2: slope = − 42.0, R²_lik = 0.36, P = 0.046), but switched to positive after the breakpoint (age-1: slope = 52.2, R²_lik = 0.13, P = 0.028; age-2: slope = 46.4, R²_lik = 0.12, P = 0.042). Note the log-scale on the x-axis.