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Yet most established demographic methods rely on linear assumptions that rank the importance of vital rates but are insensitive to whether the returns of the population growth rate are diminishing or accelerating. Here, we introduce the second\u2010order elasticities of mean individual fitness in the population, the S\u2010elasticity. The S\u2010elasticity is a proportional, element\u2010wise measure that can quantify the curvature of mean individual fitness with respect to single matrix population model (MPM) elements (self S\u2010elasticity) and to pairs of elements (mixed S\u2010elasticity). S\u2010elasticities (1) classify concavity versus convexity of local fitness surfaces, (2) provide a relative\u2010curvature diagnostic by scaling second\u2010order to first\u2010order effects, (3) sum to zero across elements\u2014facilitating comparisons across populations and species\u2014and (4) connect fitness curvature to biologically feasible trade\u2010offs via the elasticity ratio\n                    <jats:italic>K<\/jats:italic>\n                    . We provide a simple inequality based on the S\u2010elasticities and the\n                    <jats:italic>K<\/jats:italic>\n                    \u2010ratio that detects local fitness peaks along constraint directions corresponding to each pair of matrix elements. We illustrate the framework with an average 4\u2009\u00d7\u20094 MPM for\n                    <jats:italic>Didelphis aurita<\/jats:italic>\n                    (Marsupialia, Didelphidae) and compare our results to classical second\u2010order derivatives of population growth rate applied to the same dataset. First\u2010order elasticities identify survival of weaned and subadult survival as dominant linear levers of fitness; their negative self S\u2010elasticities reveal diminishing returns for increases and amplified losses for decreases relative to linear expectations. Mixed S\u2010elasticities expose a costly trade\u2010off between survival of weaned and aged females' fertility: The curvature\u2010only surface is concave near the baseline and the trade\u2010off direction is a local peak, whereas several other element pairs lie along flatter, more substitutable directions. Together, S\u2010elasticities provide a compact diagnostic for when linear rankings suffice, when nonlinearity dominates, and where feasible substitutions are most constrained. Because they are proportional, additive (sum\u2010to\u2010zero), and rooted in life cycle pathways, S\u2010elasticities complement classic elasticity and make second\u2010order demographic information directly comparable across systems\u2014yielding actionable guidance for evolutionary inference and for prioritizing management actions under realistic trade\u2010offs.\n                  <\/jats:p>","DOI":"10.1002\/ecy.70479","type":"journal-article","created":{"date-parts":[[2026,9,12]],"date-time":"2026-09-12T02:22:26Z","timestamp":1789179746000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Second\u2010order elasticities for ecology and evolution: Unraveling nonlinear fitness responses to perturbations"],"prefix":"10.1002","volume":"107","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9996-5897","authenticated-orcid":false,"given":"Maja","family":"Kajin","sequence":"first","affiliation":[{"name":"Department of Biology, Biotechnical Faculty University of Ljubljana  Ljubljana Slovenia"},{"name":"Department of Biology University of Oxford  Oxford UK"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Shripad","family":"Tuljapurkar","sequence":"additional","affiliation":[{"name":"Department of Biology Stanford University  Stanford California USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Wenyun","family":"Zuo","sequence":"additional","affiliation":[{"name":"Department of Biology Stanford University  Stanford California USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1563-4917","authenticated-orcid":false,"given":"Harman","family":"Jaggi","sequence":"additional","affiliation":[{"name":"Department of Ecology and Evolutionary Biology Princeton University  Princeton New Jersey USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2984-1810","authenticated-orcid":false,"given":"Samuel","family":"Gascoigne","sequence":"additional","affiliation":[{"name":"Department of Biology University of Oxford  Oxford UK"},{"name":"School of Biological Sciences, University of Aberdeen  Aberdeen UK"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Roberto","family":"Salguero\u2010G\u00f3mez","sequence":"additional","affiliation":[{"name":"Department of Biology University of Oxford  Oxford UK"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"311","published-online":{"date-parts":[[2026,9,11]]},"reference":[{"key":"e_1_2_8_2_1","doi-asserted-by":"publisher","DOI":"10.1111\/gcb.16454"},{"key":"e_1_2_8_3_1","doi-asserted-by":"publisher","DOI":"10.1029\/2005JD006290"},{"key":"e_1_2_8_4_1","doi-asserted-by":"publisher","DOI":"10.1111\/ele.13488"},{"key":"e_1_2_8_5_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.tpb.2013.07.002"},{"key":"e_1_2_8_6_1","doi-asserted-by":"publisher","DOI":"10.1111\/ele.12563"},{"key":"e_1_2_8_7_1","doi-asserted-by":"publisher","DOI":"10.1086\/285843"},{"key":"e_1_2_8_8_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.tree.2005.11.018"},{"key":"e_1_2_8_9_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0169-5347(00)89117-X"},{"key":"e_1_2_8_10_1","doi-asserted-by":"publisher","DOI":"10.1111\/j.0014-3820.2005.tb01760.x"},{"key":"e_1_2_8_11_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.tpb.2008.03.004"},{"key":"e_1_2_8_12_1","doi-asserted-by":"publisher","DOI":"10.1111\/j.1365-2656.2009.01569.x"},{"key":"e_1_2_8_13_1","doi-asserted-by":"publisher","DOI":"10.1016\/0040-5809(78)90025-4"},{"key":"e_1_2_8_14_1","doi-asserted-by":"publisher","DOI":"10.2307\/2265507"},{"key":"e_1_2_8_15_1","volume-title":"Matrix Population Models: Construction, Analysis, and Interpretation","author":"Caswell H.","year":"2001"},{"key":"e_1_2_8_16_1","doi-asserted-by":"publisher","DOI":"10.1016\/0044-8486(84)90016-4"},{"key":"e_1_2_8_17_1","doi-asserted-by":"publisher","DOI":"10.1890\/15-0693.1"},{"key":"e_1_2_8_18_1","doi-asserted-by":"publisher","DOI":"10.1086\/303261"},{"key":"e_1_2_8_19_1","doi-asserted-by":"publisher","DOI":"10.2307\/1939225"},{"key":"e_1_2_8_20_1","doi-asserted-by":"publisher","DOI":"10.1890\/0012-9658(2000)081[0607:EAROMA]2.0.CO;2"},{"key":"e_1_2_8_21_1","doi-asserted-by":"publisher","DOI":"10.2307\/1938700"},{"key":"e_1_2_8_22_1","doi-asserted-by":"publisher","DOI":"10.1016\/0025-5564(69)90009-1"},{"key":"e_1_2_8_23_1","doi-asserted-by":"publisher","DOI":"10.1086\/430642"},{"key":"e_1_2_8_24_1","doi-asserted-by":"publisher","DOI":"10.1111\/j.0030-1299.2004.13119.x"},{"key":"e_1_2_8_25_1","volume-title":"Plant and Animal Populations: Methods in Demography","author":"Ebert T.","year":"1999"},{"key":"e_1_2_8_26_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF00365565"},{"key":"e_1_2_8_27_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmaa.2006.05.032"},{"key":"e_1_2_8_28_1","volume-title":"The Genetical Theory of Natural Selection. 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