Temperature-dependent herbivore nutritional traits affect population dynamics and persistence

Temperature-dependent herbivore nutritional traits affect population dynamics and persistence | bioRxiv /* */ /* */ <!-- <!-- /*! * yepnope1.5.4 * (c) WTFPL, GPLv2 */ (function(a,b,c){function d(a){return"[object Function]"==o.call(a)}function e(a){return"string"==typeof a}function f(){}function g(a){return!a||"loaded"==a||"complete"==a||"uninitialized"==a}function h(){var a=p.shift();q=1,a?a.t?m(function(){("c"==a.t?B.injectCss:B.injectJs)(a.s,0,a.a,a.x,a.e,1)},0):(a(),h()):q=0}function i(a,c,d,e,f,i,j){function k(b){if(!o&&g(l.readyState)&&(u.r=o=1,!q&&h(),l.onload=l.onreadystatechange=null,b)){"img"!=a&&m(function(){t.removeChild(l)},50);for(var d in y[c])y[c].hasOwnProperty(d)&&y[c][d].onload()}}var j=j||B.errorTimeout,l=b.createElement(a),o=0,r=0,u={t:d,s:c,e:f,a:i,x:j};1===y[c]&&(r=1,y[c]=[]),"object"==a?l.data=c:(l.src=c,l.type=a),l.width=l.height="0",l.onerror=l.onload=l.onreadystatechange=function(){k.call(this,r)},p.splice(e,0,u),"img"!=a&&(r||2===y[c]?(t.insertBefore(l,s?null:n),m(k,j)):y[c].push(l))}function j(a,b,c,d,f){return q=0,b=b||"j",e(a)?i("c"==b?v:u,a,b,this.i++,c,d,f):(p.splice(this.i++,0,a),1==p.length&&h()),this}function k(){var a=B;return a.loader={load:j,i:0},a}var l=b.documentElement,m=a.setTimeout,n=b.getElementsByTagName("script")[0],o={}.toString,p=[],q=0,r="MozAppearance"in l.style,s=r&&!!b.createRange().compareNode,t=s?l:n.parentNode,l=a.opera&&"[object Opera]"==o.call(a.opera),l=!!b.attachEvent&&!l,u=r?"object":l?"script":"img",v=l?"script":u,w=Array.isArray||function(a){return"[object Array]"==o.call(a)},x=[],y={},z={timeout:function(a,b){return b.length&&(a.timeout=b[0]),a}},A,B;B=function(a){function b(a){var a=a.split("!"),b=x.length,c=a.pop(),d=a.length,c={url:c,origUrl:c,prefixes:a},e,f,g;for(f=0;f<d;f++)g=a[f].split("="),(e=z[g.shift()])&&(c=e(c,g));for(f=0;f<b;f++)c=x[f](c);return c}function g(a,e,f,g,h){var i=b(a),j=i.autoCallback;i.url.split(".").pop().split("?").shift(),i.bypass||(e&&(e=d(e)?e:e[a]||e[g]||e[a.split("/").pop().split("?")[0]]),i.instead?i.instead(a,e,f,g,h):(y[i.url]?i.noexec=!0:y[i.url]=1,f.load(i.url,i.forceCSS||!i.forceJS&&"css"==i.url.split(".").pop().split("?").shift()?"c":c,i.noexec,i.attrs,i.timeout),(d(e)||d(j))&&f.load(function(){k(),e&&e(i.origUrl,h,g),j&&j(i.origUrl,h,g),y[i.url]=2})))}function h(a,b){function c(a,c){if(a){if(e(a))c||(j=function(){var a=[].slice.call(arguments);k.apply(this,a),l()}),g(a,j,b,0,h);else if(Object(a)===a)for(n in m=function(){var b=0,c;for(c in a)a.hasOwnProperty(c)&&b++;return b}(),a)a.hasOwnProperty(n)&&(!c&&!--m&&(d(j)?j=function(){var a=[].slice.call(arguments);k.apply(this,a),l()}:j[n]=function(a){return function(){var b=[].slice.call(arguments);a&&a.apply(this,b),l()}}(k[n])),g(a[n],j,b,n,h))}else!c&&l()}var h=!!a.test,i=a.load||a.both,j=a.callback||f,k=j,l=a.complete||f,m,n;c(h?a.yep:a.nope,!!i),i&&c(i)}var i,j,l=this.yepnope.loader;if(e(a))g(a,0,l,0);else if(w(a))for(i=0;i (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0];var j=d.createElement(s);var dl=l!='dataLayer'?'&l='+l:'';j.src='//www.googletagmanager.com/gtm.js?id='+i+dl;j.type='text/javascript';j.async=true;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-M677548'); Skip to main content Home About Submit ALERTS / RSS Search for this keyword Advanced Search New Results Temperature-dependent herbivore nutritional traits affect population dynamics and persistence View ORCID Profile David M. Anderso , Francis Fruin Laid-Low , View ORCID Profile Mary I. O’Connor doi: https://doi.org/10.1101/2025.10.28.685165 David M. Anderso 1 Department of Zoology and Biodiversity Research Centre, University of British Columbia , Vancouver, BC V6T 1Z4, Canada 2 Hawai’i Institute of Marine Biology, University of Hawai’i at Mānoa , Kāne’ohe, HI 96744 Find this author on Google Scholar Find this author on PubMed Search for this author on this site ORCID record for David M. Anderso For correspondence: da4786{at}hawaii.edu Francis Fruin Laid-Low 1 Department of Zoology and Biodiversity Research Centre, University of British Columbia , Vancouver, BC V6T 1Z4, Canada Find this author on Google Scholar Find this author on PubMed Search for this author on this site Mary I. O’Connor 1 Department of Zoology and Biodiversity Research Centre, University of British Columbia , Vancouver, BC V6T 1Z4, Canada 3 Santa Fe Institute , Santa Fe, NM 87501 Find this author on Google Scholar Find this author on PubMed Search for this author on this site ORCID record for Mary I. O’Connor Abstract Full Text Info/History Metrics Preview PDF Abstract The nutritional traits of herbivores affect demographic rates and regulate nutrient and energy fluxes among trophic levels. Herbivore nutritional requirements, and the nutrient contents of herbivore biomass, depend on temperature – at temperature extremes, more nutritious food is required to maximize growth rate and herbivore biomass contains fewer nutrients. Yet, the consequences of these thermal responses for the population dynamics of herbivore-autotroph systems have not been explored. Here, we develop and analyze a stoichiometrically-explicit, temperature-dependent model of herbivore-autotroph systems to answer the question: How does the thermal response of herbivore nutritional traits affect population responses to temperature and nutrient (phosphorus) supply? We find that temperature-dependent herbivore nutritional traits restrict the range of temperatures at which herbivore populations persist, reduce the stability of population dynamics at high phosphorus supplies, and limit the herbivore’s capacity to control autotroph population density. These results reflect temperature-dependent changes in the herbivore’s sensitivity to nutrient-poor autotroph biomass and ability to retain nutrients in biomass (and thereby dilute autotroph nutrient contents). The thermal response of herbivore nutritional traits may therefore be an important factor influencing population and community responses to warming and nutrient enrichment. Introduction Temperature and nutrient supply strongly influence physiology and population dynamics. Temperature constrains organismal metabolism and energetics ( Robinson et al. 1983 , Raven and Geider 1988 , Gillooly et al. 2001 ), and, in doing so, affects ecological processes including growth, mortality, and consumption ( Eppley 1971 , Brown et al. 2004 , Dell et al. 2011 ). Effects of nutrient availability on ecological processes are mediated by organismal stoichiometry, because biomass production and maintenance require a balance of chemical elements ( Sterner and Elser 2002 , Prater et al. 2024 ). Despite substantial work on the individual-level effects of temperature or nutrients on organismal metabolism and stoichiometry, integrating and scaling these effects up to population and community dynamics remains a major challenge ( Allen and Gillooly 2009 , Cross et al. 2015 , Meunier et al. 2024 , O’Connor et al. 2025 ). Efforts to integrate individual energetics and stoichiometry have focused on the nutritional requirements of consumer growth ( Doi et al. 2010 , Cross et al. 2015 ). Consumers require energy stored in carbon-carbon bonds and nutrients in the form of chemical elements (e.g., phosphorus, P) to produce new biomass. Mismatches between consumer requirements and resource stoichiometry can inhibit resource assimilation, energy acquisition and ultimately reduce consumer growth rate ( Urabe and Watanabe 1992 , Frost et al. 2006 , Prater et al. 2024 ). The resource stoichiometry (C:X ratio, where “X” is an element such as P) that distinguishes between carbon-(or energy-) and nutrient-limited growth is often referred to as the Threshold Elemental Ratio (hereafter, TER C:X or simply “nutritional requirements,” ( Urabe and Watanabe 1992 , Frost et al. 2006 )). The TER C:X is an important trait of individual consumers that affects demographic rates and controls the flow of energy and nutrients through consumer biomass. Temperature can affect consumer nutritional traits by altering the assimilation and allocation of carbon and nutrients in biomass ( Cross et al. 2015 ). Laspoumaderes et al. (2022) performed the largest set of standardized experiments to assess the thermal response of nutrient (phosphorus) traits – including five aquatic herbivore species representing diverse taxa (protists, zooplankton, crabs) from both marine and freshwater environments – and consistently found that nutrient (P) demands increase towards thermal extremes, whereas nutrient contents of biomass decline at extreme temperatures. They attributed this inverse relationship between P requirements and contents across temperatures to differences in C and P use efficiencies. The Laspoumaderes et al. (2022) results differ from other reported responses, including increased carbon requirements with warming ( Malzahn et al. 2016 ), increased nutrient (P) requirements with warming ( Persson et al. 2011 ), no change in nutritional requirements with warming ( Doi et al. 2010 , Anderson et al. 2017 ), and an increase in carbon requirements at thermal extremes ( Ruiz et al. 2020 ). These differences have been suggested to reflect the range of temperatures tested, as experiments must test a wide thermal range to capture the complex non-monotonic temperature dependence of nutrient requirements ( Schmitz and Rosenblatt 2017 , Ruiz et al. 2020 , Laspoumaderes et al. 2022 ). Despite these advances in understanding of the thermal response of herbivore nutrient traits, we know little about the implications of these responses for population dynamics. Herbivore nutritional traits have important consequences for herbivore-autotroph population dynamics ( Elser and Urabe 1999 , Urabe et al. 2002 ). Herbivores and autotrophs differ in stoichiometric flexibility: autotroph stoichiometry is plastic and sensitive to environmental conditions ( Droop 1974 ), whereas herbivore stoichiometry is homeostatically regulated ( Andersen and Hessen 1991 , Sterner and Elser 2002 ). These differences affect population dynamics ( Andersen et al. 2004 ). For example, herbivore populations can collapse when autotroph populations are abundant but low in nutritional quality ( Sommer 1992 , Sterner et al. 1998 , Diehl et al. 2022 , Frost et al. 2025 ). Herbivore population growth can be facilitated by increased autotroph nutritional quality that arises from external changes (e.g., nutrient inputs, ( Frost et al. 2025 )) and/or intrinsic feedbacks with herbivore populations (e.g., via nutrient excretion, Sterner 1990 , Urabe et al. 2002 , or a reduction in autotroph density that concentrates nutrients in autotroph biomass, Sommer 1992 , Urabe 1995 , Hall et al. 2007 ). Herbivore nutritional traits affect these dynamics by determining both herbivore sensitivity to poor food quality, and the impact of herbivores on autotroph nutritional contents ( Andersen 1997 , Elser and Urabe 1999 ). The implications of temperature-dependent herbivore nutritional requirements for the response of autotroph-herbivore systems to changing temperatures are unresolved. Analyses of bioenergetic consumer-resource models have shown that the response of community properties (e.g., persistence, stability, interaction strength) to warming depends on the relative thermal sensitivities of key ecological rates, including consumer ingestion rate, consumer metabolic rate, resource growth rate, and resource carrying capacity ( Vasseur and McCann 2005 , Rall et al. 2010 , O’Connor et al. 2011 , Gilbert et al. 2014 , Amarasekare 2015 , Synodinos et al. 2021 ). Although it has been demonstrated that nutrients and stoichiometry constrain thermal responses of resource (autotroph) carrying capacity ( Lemoine 2019 , Bieg and Vasseur 2024 ), autotroph growth rate ( Thomas et al. 2017 , Bestion et al. 2018 , Sunday et al. 2024 , Anderson et al. 2025 ), and consumer-resource dynamics ( Uszko et al. 2017 , Diehl et al. 2022 , Sentis et al. 2022 ), these developments have not been integrated with recent advances in our understanding of how temperature affects consumer nutritional traits ( Ruiz et al. 2020 , Laspoumaderes et al. 2022 ). Here, we evaluate the effects of temperature-dependent herbivore nutritional traits on the response of autotroph-herbivore systems to temperature and nutrient supply. We develop a stoichiometrically-explicit, temperature-dependent model describing the coupled dynamics of autotroph and herbivore populations. Our model builds on work by Sentis et al. (2022) , which integrates general theory describing stoichiometric constraints on consumer-resource interactions ( Andersen 1997 , Loladze et al. 2000 , Muller et al. 2001 , Andersen et al. 2004 , Hall 2009 ) with the model of Uszko et al. (2017) that considers interactive effects of temperature and resources on trophic interactions. Specifically, we modify this modeling framework to allow the herbivore’s phosphorus requirements and phosphorus contents to increase and decrease towards thermal extremes, respectively, following empirical findings by Laspoumaderes et al. (2022) . The model is parameterized to describe the population dynamics of the marine copepod Acartia tonsa feeding on the unicellular autotroph Rhodomonas salina . Our objective is to evaluate how the thermal response of herbivore nutritional requirements and contents impacts the sensitivity of three important community properties – 1) herbivore persistence, 2) system stability and population fluctuations, and 3) the trophic interaction strength ( Gilbert et al. 2014 ) – to temperature and nutrient supply. We find that thermal variation in herbivore nutritional traits alters dynamic feedbacks between autotroph stoichiometry and herbivore population density, and that these changes can impact the structure and dynamics of the community. Model and Methods Model Development We develop a model describing how temperature and phosphorus supply influence interactions between a herbivore population with a fixed Phosphorus:Carbon (P:C) stoichiometry and an autotroph population with a variable P:C stoichiometry. To test the effects of temperature-dependent herbivore nutritional traits on community properties – herbivore persistence, system stability, and interaction strength – we contrast a scenario in which herbivores have constant nutritional requirements and contents across temperatures with a scenario in which nutritional requirements and contents depend on temperature, such that, at thermal extremes, herbivores require more nutritionally dense (i.e., high P:C ratio) autotrophs to maximize conversion efficiency and are themselves less nutritionally dense (i.e., have a lower P:C ratio, ( Laspoumaderes et al. 2022 )). The model consists of two differential equations describing the coupled dynamics of the autotroph ( A, gC m −3 ) and herbivore biomass density ( Z, gC m −3 ), with an algebraic equation that determines the autotroph Phosphorus:Carbon stoichiometry ( Q A , gP gC −1 ; also referred to here simply as phosphorus or nutrient contents) ( Fig. 1 ): where μ ( Q A , T ) describes how the autotroph’s specific rate of carbon production (or growth rate, gC gC −1 d −1 ) depends on its’ phosphorus contents and temperature T (°C), f ( A, T ) describes how the herbivore’s grazing rate ( m 3 d −1 gC −1 ) depends on autotroph population density and temperature, ε ( Q A , T ) captures the effects of autotroph stoichiometry and temperature on herbivore conversion efficiency ( gC gC −1 ), R ( T ) is the temperature-dependent rate of carbon biomass loss to respiration, and m A ( T ) and m Z ( T ) are temperature-dependent rates of biomass loss to mortality for autotrophs and herbivores, respectively ( gC gC −1 d −1 ). This model tracks units of both carbon and phosphorus in biomass, in order to consider the effects of stoichiometry (i.e., phosphorus:carbon ratios, Q A and Q Z ( T )) and metabolism (e.g., respiratory losses in R ( T )) on consumer-resource population dynamics. Download figure Open in new tab Fig. 1. Conceptual diagram of the stocks and flows of biomass in the model. The green and blue circles represent biomass densities of the autotroph A and herbivore Z , respectively, the inset white squares depict phosphorus:carbon stoichiometry, and the arrows show the flows of biomass in the system (note that growth, or the production of new biomass, is represented here by the flow of biomass from the larger circle to the smaller circle). Increases in the phosphorus contents of the autotroph, which may arise from increases in phosphorus availability N tot or decreases in herbivore or autotroph population density (see Eqn. 1c ), leads to an increase in the growth rate of both the autotroph population ( Eqn. 4a ) and herbivore population (via improved conversion efficiency ε , Eqn. 3a). The autotroph phosphorus content ( Eqn. 1c , Q A ) is variable, and depends on how the total phosphorus availability, N tot ( gP m −3 ), is divided into herbivore and autotroph biomass. This model follows from two key assumptions regarding phosphorus cycling: 1) the system is closed for phosphorus, and 2) fluxes of phosphorus from other pools within the system (i.e., P embedded in dead biomass and in inorganic form) back into autotroph biomass happens instantaneously ( Andersen 1997 ). These two assumptions generally do not qualitatively alter model outcomes; Diehl 2007 and Diehl et al. 2022 contrasted this model structure to one explicitly resolving phosphorus cycling through detritus and inorganic pools and found no qualitative difference. Under these assumptions, all available phosphorus is incorporated into either autotroph or herbivore biomass. The phosphorus contents of autotrophs Q A can therefore be calculated by rearranging an equation describing mass-balance constraints on phosphorus ( Eqn. 1c , Andersen 1997 , Loladze et al. 2000 , Andersen et al. 2004 ): the total phosphorus in the system N tot ( gP m −3 ) is the sum of all phosphorus incorporated into herbivore biomass (i.e., Z Q Z ( T ), where Q Z ( T ) is the temperature-dependent herbivore stoichiometry in gP gC −1 ) and autotroph biomass (i.e., A Q A ). We next describe the parameters controlling the dynamics of Eqn. 1 (parameters are also summarized in Table S1 ). The net rate of carbon acquisition by herbivores is the difference between the ingestion rate f ( A, T ) A and the respiration rate R ( T ). These rates vary with temperature T (°C) and/or autotroph population density (expressed as carbon density) as: The clearance rate a ( T ) ( m 3 d −1 gC −1 ), maximum ingestion rate I ( T ) (i.e., the inverse of the handling time, d −1 ), and respiration rate R ( T ) vary unimodally with temperature as described by Sharpe-Schoolfield functions ( Schoolfield et al. 1981 ), with a maximum value at temperature T opta , T optI , or T optR and thermal sensitivity controlled by E aa , E aI , or E aR below the thermal optimum and E da , E dI , or E dR above the thermal optimum (note that Eqns. 2a – d rearrange the Sharpe-Schoolfield function to introduce a thermal optimum parameter). The parameter k = 8.62 ∗ 10 −5 is the Boltzmann’s constant ( eV K −1 ), T K = 270.15 is included to convert temperature T in °C to Kelvin, and T ref = 298.15 K is a reference temperature. These left-skewed, unimodal thermal responses are consistent with evidence based on meta-analyses of the temperature-dependence of the functional response ( Englund et al. 2011 , Uszko et al. 2017 , Uiterwaal and DeLong 2020 ) and describe data from Laspoumaderes et al. (2022) well ( Fig. S1 ). The net acquired carbon ( f ( A, T ) A − R ( T )) is converted into herbivore biomass with efficiency ε ( Q A , T ) ( gC gC −1 ), which is assumed to increase with autotroph phosphorus content Q A as ( Andersen 1997 , Andersen et al. 2004 ): where ε max is the maximum conversion efficiency and Q TER ( T ) is the temperature-dependent herbivore nutritional (phosphorus) requirements ( gP gC −1 ). Under Eqn. 3 , the herbivore is energy (carbon) limited when the autotroph phosphorus content is large relative to herbivore phosphorus requirements ( Q A > Q TER ( T )), but is phosphorus limited when autotroph phosphorus contents Q A are less than Q TER ( T ) ( Fig. 2A ). Download figure Open in new tab Fig. 2. Effect of temperature-dependent herbivore nutritional (phosphorus) requirements (i.e., Q TER ( T )) on herbivore conversion efficiency (i.e., ε ( Q A , T ); A, C) and thermal traits (B, D) in the model. A-B Herbivore traits under the assumption that phosphorus requirements ( Q TER ( T ), solid black line in A ; Eqn. 4a ) are insensitive to temperature (accomplished by setting the thermal breadth S TER to 10 4 ). Conversion efficiency depends only on the prevailing autotroph phosphorus contents ( A ) and is insensitive to temperature when Q A is invariant ( B ). C-D Herbivore traits under the assumption that phosphorus requirements ( Q TER ( T ), solid black line in C ) increase towards warm and cold temperature (i.e., the thermal breadth S TER is reduced to the empirically observed 8.1, Fig. S1D , Table S1 ). Here, conversion efficiency may decline towards thermal extremes when autotroph food quality is invariant ( D ). In panels B and D , our goal is to illustrate how temperature-dependent nutritional requirements impact herbivore thermal traits. As such, we assume Q A and A are fixed at Q A = 0.0125 and A = 0.3 (note, however, that each of these quantities are dynamic in the model, Eqn. 1). Top panels in B and D show how temperature affects the components of herbivore per-capita population growth rate, whereas the bottom panels show the net per-capita population growth rate across temperatures. The herbivore phosphorus requirements Q TER ( T ) and phosphorus contents Q Z ( T ) ( gP gC −1 ) are modeled as U-shaped and unimodal relationships to temperature, respectively: where T opt,TER and T opt,Q give the temperatures minimizing and maximizing these functions and S TER and S Q determine the thermal breadths ( Fig. 2C , Fig. S1 , ( Laspoumaderes et al. 2022 )). Under this model, the herbivores’ demand for phosphorus, Q TER ( T ), increases as temperatures become warmer or colder from T opt,TER , whereas the herbivore phosphorus contents decline as temperatures diverge from T opt,Q . The thermal breadths S TER and S Q are key control parameters for testing the effects of temperature-dependent nutritional traits: by setting S TER and S Q to a large number (e.g., Fig. 2A ), the nutritional requirements and contents are effectively insensitive to temperature and thus consistent with past work (e.g., Sentis et al. 2022 ). When the thermal breadth S TER is relatively small, the herbivore nutritional requirements Q TER ( T ) follow a U-shaped relationship ( Fig. 2C ) that leads to a unimodal relationship of conversion efficiency to temperature when autotroph nutrient contents Q A are invariant (as herbivore phosphorus demands Q TER ( T ) become large relative to Q A as temperatures diverge from T opt,Q ; Fig. 2C-D dashed line). The autotroph population’s specific growth rate increases as its phosphorus content Q A increases above its’ minimum requirements Q min ( gP gC −1 ), up to a maximum temperature-dependent birth rate μ ∞ ( T ) ( gC gC −1 d −1 , ( Droop 1974 )): where the maximum birth rate μ ∞ ( T ) increases with temperature following the Boltzmann-Arrhenius function ( Gillooly et al. 2001 ). The parameter μ 0 is a scaling coefficient ( day −1 ), E μ determines the sensitivity of birth rate to temperature (the apparent activation energy of metabolic reactions sustaining growth, eV −1 ), and k, T K , and T ref are constants defined above. Eqn. 5a – b assumes that temperature impacts the maximum growth rate μ ∞ , but not the minimum nutrient requirements Q min , consistent with past theoretical work ( Sentis et al. 2022 , Sauterey and Ward 2022 , Bieg and Vasseur 2024 ) and empirical findings by Anderson et al. (2025) . Lastly, the mortality rates for the autotroph and herbivore, m A ( T ) and m Z ( T ) respectively, scale with temperature following the Boltzmann-Arrhenius function ( Gillooly et al. 2001 ): where m A 0 and m Z 0 are the scaling coefficients ( day −1 ), and E mA and E mZ determine the thermal sensitivities of these rates. This exponential temperature-dependence of mortality losses follows from the Metabolic Theory of Ecology ( Brown et al. 2004 ) and is consistent with evidence from comparative analyses ( Gillooly et al. 2001 , McCoy and Gillooly 2008 , Munch and Salinas 2009 ), and is therefore commonly employed in temperature-dependent population models ( Uszko et al. 2017 , Sentis et al. 2022 , Vinton and Vasseur 2022 ). Parameterization The model (Eqn. 1-5) is parameterized to represent population dynamics of the marine herbivorous zooplankton Acartia tonsa consuming the unicellular autotroph Rhodomonas salina (phytoplankton). Parameters controlling the thermal dependencies of the herbivore’s maximum ingestion rate ( Eqn. 2c ), respiration rate ( Eqn. 2d ), phosphorus requirements (i.e., TER C:P , Eqn. 4a ), phosphorus contents ( Eqn. 4b ) were all estimated by fitting the respective equations to data for Acartia tonsa from Laspoumaderes et al. (2022) (model fits are shown in Fig. S1A, C, D and F , respectively). The herbivore clearance rate was estimated from a functional response experiment conducted on A. tonsa and Rhodomonas spp . at 18°C by Thor and Wendt (2010) ( Fig. S1B ), and we assumed that the thermal sensitivity of the clearance rate was identical to that of the maximum ingestion rate. The maximum conversion efficiency ε max was estimated by fitting Eqn. 3 to data for conversion efficiency from Laspoumaderes et al. (2022) , with the nutrient requirements Q TER ( T ) and autotroph nutrient contents Q A fixed at values measured experimentally ( Fig. S1E , and Q A = 0.015). The herbivore mortality rate was taken as 0.02 at 20°C (based on the global value from Ward et al. 2012 ), with a thermal sensitivity of E mZ = 0.54 as estimated by Hirst and Kiørboe (2002) for broadcast spawners like Acartia tonsa . The temperature dependence of autotroph birth rate ( Eqn. 5b ) and mortality rate ( Eqn. 6a ) was estimated based on data for the net per-capita population growth rate of Rhodomonas salina across temperatures from Ferreira et al. (2022) (these experiments were conducted in nutrient-replete conditions, and we therefore assumed Q A ≫ Q min so that μ ( Q A , T ) ≈ μ ∞ ( T ); fits are shown in Fig. S1G ). The autotroph Q min was taken to be the average of the estimates of Q min in gP gC −1 from Edwards et al. (2015) . All model fits were obtained using maximum likelihood methods, implemented with the Mathematica function “NonLinearModelFit[]”. Parameter values are provided in Table S1 . Analysis Our objective is to evaluate the impact of temperature-dependent herbivore phosphorus requirements and contents, Q TER ( T ) and Q Z ( T ), respectively, on the response of community properties to temperature and phosphorus availability ( N tot ). To address this goal, we manipulate the presence of thermal variation in Q TER ( T ) and Q Z ( T ) (by altering the thermal breadths, S TER and S Q ; Fig. 2A, C ), and assess community properties across gradients in temperature and phosphorus availability. The two scenarios we are comparing to test the effects of temperature-dependent herbivore nutrient traits – that with constant herbivore nutritional traits, and that with temperature-dependent nutritional traits – are identical at T opt,Q ≈ T opt,TER ≈ 18°C and, as such, the differences between the scenarios are relevant as temperatures diverge from ~18°C. Note that, by fixing traits at Q TER ( T opt,TER ) and Q Z ( T opt,Q ) (the minimum of the U-shaped Q TER function and maximum of the unimodal Q Z function, see Fig. S1D-F ), the constant herbivore nutritional traits scenario models relatively low phosphorus requirements and high nutrient contents across temperatures. This choice provides a useful contrast for understanding the processes underlying effects of temperature-dependent nutritional traits on community properties, and also aligns with existing theory that uses nutritional trait measurements made near the thermal optimum ( Diehl et al. 2022 , Sentis et al. 2022 ). We briefly summarize the community properties considered here and provide additional details in the Supplemental Materials section “Calculating community properties.” We identify three properties of the community and compare how they change over the temperature and nutrient supply gradients: 1) the herbivore persistence boundary, 2) the stability boundary, and 3) the strength of the trophic interaction. We solve for the sets of temperature and phosphorus availability that 1) separate herbivore persistence from extinction (i.e., herbivore persistence boundary; Eqn. S4, Fig. S2A-B ) and 2) separate stable and unstable non-trivial equilibria (i.e., stability boundary; Fig. S2A ). Because unstable equilibria lead to limit cycles with very low autotroph population densities that renders autotrophs (and the herbivores who depend on autotrophs) susceptible to extinction ( Rosenzweig 1971 ), we also find temperatures and phosphorus supplies producing minimum autotroph population densities of 10 −3 , 10 −6 , and 10 −9 (we do not consider herbivore densities here because they are not driven to such low densities, Fig. S2A ). Lastly, we evaluate the trophic interaction strength, defined here as the ratio of autotroph density in the absence of herbivores to the autotroph density in the presence of the herbivore (Eqn. S5) ( Gilbert et al. 2014 ), across temperature and phosphorus supply gradients. Note that the extent to which this measure of trophic interaction strength influences community stability is debated ( Gilbert et al. 2014 , Nilsson and McCann 2016 , Uszko et al. 2017 ), and we report trophic interaction strength here because it is a biologically meaningful measure of the herbivore’s capacity to control autotroph biomass. High trophic interaction strengths occur when herbivores strongly reduce autotroph density relative to the density that autotrophs achieve in the absence of herbivory. Equilibria, persistence boundaries, and stability boundaries were calculated using Mathematica (Version 12.1.1.0) and the EcoEvo package. We solved for equilibria analytically, and filtered out non-trivial equilibria that did not satisfy conservation of phosphorus (see Eqn. S3). We used root finding approaches to assess nutrient supplies required for persistence, destabilization of the equilibrium, and limit cycles producing minimum autotroph population densities of 10 −3 , 10 −6 , and 10 −9 . Results Temperature-dependent herbivore nutritional traits influence community structure and dynamics. When thermal extremes cause herbivores to require more nutritionally dense autotrophs and have less nutritionally dense biomass, we find that, relative to the control scenario in which temperature does not affect nutritional traits: 1) the temperature range permitting herbivore persistence is restricted at all phosphorus concentrations N tot (contrast persistence boundaries in Fig. 3A and 3C ), 2) the phosphorus supply required to destabilize the system decreases (particularly at temperatures 22°C; compare stability boundaries in Fig. 3A and 3C ), and 3) the trophic interaction strength is usually reduced, particularly at temperatures warmer than 20°C (contrast Fig. 3B and 3D ). We next detail the mechanisms by which the temperature-dependence of herbivore nutritional requirements and contents alters the herbivore persistence boundary, stability boundary, and interaction strengths. Download figure Open in new tab Fig. 3. Effects of thermal variation in herbivore nutritional traits on herbivore persistence and system stability (A, C) and trophic interaction strength (B, D; Eqn. S5) across gradients in total phosphorus concentration and temperature. A & C Effect of temperature on the phosphorus availability needed to permit herbivore persistence (solid black lines), to destabilize the system (dashed line), and to produce limit cycles with a minimum autotroph biomass density of 10 −3 , 10 −6 , 10 −9 (colored contours). B & D Trophic interaction strength – the equilibrium autotroph biomass in the presence of herbivores (denoted A Eq 2 ) divided by the equilibrium autotroph biomass in the presence of herbivores ( A Eq 3 ; Eqn. S5) – across temperatures and phosphate supplies. Note that we only present interaction strengths at phosphate concentrations and temperatures at which herbivores persist with autotrophs at a stable equilibrium. Panels A and C assume that herbivore nutritional requirements and contents are effectively independent of temperature (accomplished by setting S Q = S TER = 10 4 ), whereas B and D assume herbivore nutritional requirements increase, and herbivore nutritional contents decrease, as temperatures diverge from ~18°C. Note that we include two panels for each scenario to more clearly display minimum autotroph densities and interaction strengths (persistence and stability boundaries are identical in the two panels). Persistence boundary Increasing herbivore nutritional requirements towards temperature extremes reduces the thermal breadth of the herbivore persistence boundary. This result reflects increases in the autotroph phosphorus contents needed to support a positive per-capita growth rate ( Fig. 4 ). To understand this result, first consider the impact of temperature on the set of autotroph population densities and phosphorus contents that yield a zero net per capita population growth rate under constant ( Fig. 4A-B ) and temperature-dependent ( Fig. 4D-E ) herbivore nutritional requirements (at a constant N tot ). Notice that, at relatively high autotroph population densities (e.g., ~0.2-1 gC m −3 ), the height of the zero net growth isocline (i.e., the autotroph P-content needed to produce a zero net growth rate) increases more rapidly with warming and cooling when herbivore nutritional (phosphorus) requirements increase towards warm and cool temperatures (contrast Fig. 4D to Fig. 4A and Fig. 4B to Fig. 4E ). To persist, the herbivore must have a positive growth rate at the autotroph-only equilibrium population density and phosphorus content (i.e., the colored points in Fig. 4A-B and D-E must lie above the herbivore isocline; see Supplemental Materials section “Calculating community properties” and Fig. S2 ). At the autotroph-only equilibrium, population densities and phosphorus contents are relatively weakly temperature-dependent ( Fig. 4C and Fig. 4F ). It is thus the rapid increase in the height of the herbivore zero-net growth isocline that accompanies temperature-dependent increases in nutritional requirements towards thermal extremes that drives the restriction of herbivore persistence boundaries. Download figure Open in new tab Fig. 4. Increases in herbivore nutritional requirements towards thermal extremes increases the minimum temperature supporting herbivore persistence and decreases the maximum temperature supporting herbivore persistence. A-B and D-E Effects of temperature on the set of autotroph phosphorus contents and autotroph population densities that allow herbivores to achieve a zero per-capita population growth rate (isoclines). These isoclines are L-shaped because herbivores require a large quantity of autotroph biomass to maintain zero net growth when autotrophs are of low nutritional quality (autotroph population densities and P-contents above and below these isoclines yield a positive and negative growth rate, respectively). The points in each figure show the autotroph population density and phosphorus content at the autotroph-only equilibrium at each temperature (note that points in A and D lie on top of one another because these quantities vary weakly with temperature; see C and F ). Importantly, the herbivore population can persist at a given temperature if the autotroph-only equilibrium P-content and density permits a positive per-capita population growth rate ( Fig. S2 ). C and F Effects of temperature on the phosphorus contents of the autotroph and herbivore at the autotroph-only and non-trivial equilibria (top panel), and on the population densities of the autotroph and herbivore at the autotroph-only and non-trivial equilibria (bottom panels). These panels are included to support interpretation of panels A-B and C-D ; the vertical dotted line shows the herbivore persistence boundary for the respective scenario whereas the colored points indicate the autotroph density and P-contents at temperatures considered in A-B and D-E . This analysis assumes N tot = 0.0025. Stability boundary Decreases in herbivore phosphorus contents generally increase the strength of autotroph self-regulation (by decreasing phosphorus retention in herbivore biomass and hence increasing availability of phosphorus for autotrophs), which leads to decreases in the phosphorus enrichment required to destabilize the non-trivial equilibrium ( Fig. 5 ). To understand this result, it is helpful to recall that an equilibrium is resilient to perturbation (e.g., an increase in autotroph density) if the decline in autotroph growth rate with increased autotroph population density (i.e., a stabilizing mechanism that contributes to a reduction of the autotroph population back towards the equilibrium) is strong relative to the decline in autotroph mortality rate with increased autotroph population density that arises from saturation of the herbivore functional response (i.e., a destabilizing mechanism that contributes to growth of the autotroph population away from the equilibrium). Sufficient increases in phosphorus availability N tot always destabilize the equilibrium by concentrating phosphorus in autotroph biomass and thus rendering autotroph growth rate relatively insensitive to autotroph population density near the equilibrium (notice in Fig. 5I that autotroph birth rate is a saturating function of phosphorus contents Q A , such that autotroph self-regulation is weak at equilibria associated with high Q A ; e.g., Fig. 5A-B ). Download figure Open in new tab Fig. 5. Temperature-dependent shifts in herbivore nutritional contents destabilizes population dynamics at warm (and cold) temperatures. A-F Shown are representative isoclines across a gradient in phosphorus availability ( N tot = 0.006, N tot = 0.0045, or N tot = 0.000 for panels A-B, C-D , and E-F , respectively) at a constant temperature ( T =24°C) when herbivores have constant nutritional traits (left column) or temperature-dependent nutritional traits (right column). Importantly, at this temperature, the herbivore in the temperature-dependent nutritional traits scenario has reduced nutritional contents – thus, a given unit of herbivore carbon biomass contains a smaller mass of phosphorus. Herbivores and autotrophs have positive per-capita population growth rates above and below their isoclines, respectively. Unstable and stable non-trivial equilibria are indicated by hollow and solid points, respectively. The gray shading indicates the autotroph phosphorus contents across herbivore and autotroph population density, and white regions indicate densities that violate nutrient conservation (Eqn. S3). The herbivore isocline is unimodal because high herbivore population densities are associated with a large share of total phosphorus that is fixed in herbivore biomass – thereby reducing autotroph P-contents, herbivore conversion efficiency, and ultimately herbivore percapita population growth rate. Notice that instability occurs when the equilibrium lies at densities yielding relatively large autotroph phosphorus contents ( Q A > 0.00), where there are locally weak effects of autotroph population density on autotroph growth rate (see panel I ). I Effect of autotroph phosphorus content on autotroph specific growth rate (Eqn. 5) at 24°C. The points show the equilibrium phosphorus contents and autotroph birth rates at the indicated panels. When herbivores have relatively low phosphorus contents (e.g., at T =24°C under the temperature-dependent nutritional requirements scenario), a given unit of herbivore carbon biomass is tied to a smaller mass of phosphorus, thereby increasing the amount of total phosphorus in autotroph biomass (contrast gray contours in Fig. 5B, D, F to those in Fig. 5A, C , and E). This mechanism reduces the phosphorus enrichment required to weaken the stabilizing influence of negative density dependence sufficiently to destabilize the equilibrium. Thus, decreases in herbivore nutritional contents towards thermal extremes ultimately makes the system more sensitive to the destabilizing influence of nutrient enrichment. Interaction strength Increases in herbivore nutritional requirements towards thermal extremes increases the autotroph population density required to support herbivores, thereby weakening the trophic interaction strength (defined as the reduction in autotroph biomass that occurs in the presence of the herbivore, Eqn. S5, Fig. 6 ). Note that temperature-dependent changes in herbivore nutritional requirements and contents have opposing effects on the interaction strength – increased nutritional requirements weakens the interaction strength, whereas decreased nutritional contents strengthens the interaction (by increasing autotroph nutrient contents and therefore the density at which autotrophs achieve zero growth; contrast autotroph isoclines in Fig. 6D and H ) – but here the effects of temperature-dependent nutritional requirements is stronger. This result can be understood by considering the effect of temperature on the herbivore isocline under constant nutritional requirements ( Fig. 6B – D ) and under temperature-dependent increases in nutritional requirements ( Fig. 6F-H ). Increasing herbivore nutritional requirements increases the autotroph population density needed to permit herbivore persistence (notice the reduction in the height of the herbivore isocline as temperatures increase from 21.5°C in Fig. 6F to 22.5°C in Fig. 6G and 23.5°C in Fig. 6H ). This mechanism ultimately leads to a relatively large equilibrium density of autotrophs when present with herbivores, and therefore weak trophic interaction strengths relative to the scenario in which herbivore nutritional requirements are constant (contrast Fig. 6B-D with Fig. 6F-H ). Download figure Open in new tab Fig. 6. Increases in herbivore nutritional requirements at warm temperatures reduces the trophic interaction strength. A Effect of temperature on trophic interaction strength at a constant phosphorus availability ( N tot = 0.0015 gP m −3 ) when herbivore nutritional requirements and contents are constant (black line) and temperature-dependent (gray, dashed line). This figure is a horizontal slice through the contour plots shown in Fig. 3B & D . The points labeled in A are considered in more detail in subsequent panels. B-H Representative isoclines across a temperature gradient at a constant phosphorus availability when herbivores have constant nutritional traits ( B-D ) and temperature-dependent nutritional traits ( F-H ). In each panel, herbivores and autotrophs have positive per-capita population growth rates above and below their respective isoclines, respectively. The gray shading indicates the autotroph phosphorus contents across herbivore and autotroph population density, and white regions indicate densities that violate nutrient conservation (Eqn. S3). The autotroph-only and non-trivial equilibria are indicated by hollow and solid points, respectively. Recall that the trophic interaction strength is the ratio of the autotroph density at the autotroph-only ( A Eq 2 ) and non-trivial equilibrium ( A Eq 3 ) – thus, the trophic interaction strengths can be visually assessed by examining the horizontal displacement of the non-trivial equilibrium (indicated here by arrows in B-H ). Discussion These analyses demonstrate that the thermal responses of herbivore nutritional traits affect the persistence and dynamics of autotroph-herbivore communities across temperature and nutrient gradients. We found that increases in herbivore nutritional requirements and decreases in herbivore nutrient contents towards thermal extremes led to a restriction of the thermal breadth of the herbivore persistence boundary ( Fig. 3 – 4 ), a decrease in the nutrient enrichment required to destabilize population dynamics ( Fig. 3 , 5 ), and a reduction in the strength of the trophic interaction ( Fig. 3 , 6 ). These results reflect two key consequences of thermal dependencies of herbivore nutrient traits: 1) a decreased retention of the available nutrients in herbivore biomass and corresponding increase in the nutrient density of autotroph biomass, and 2) an increased sensitivity of herbivore conversion efficiency to nutrient-poor autotroph biomass. These analyses provide testable theoretical predictions for the effects of stoichiometric constraints on population responses to warming. For example, this analysis supports the prediction that stoichiometry and population densities vary systematically across temperature gradients: herbivore and autotroph Phosphorus:Carbon ratios respectively exhibit U-shaped and unimodal thermal responses, whereas population densities show unimodal and U-shaped responses to temperature, respectively ( Fig. 4C & F ). Mesocosm experiments examining zooplankton-phytoplankton population dynamics and stoichiometry across environmental gradients have found patterns broadly consistent with these predictions ( Sommer 1992 , Urabe and Sterner 1996 , Hall et al. 2007 , Diehl et al. 2022 , Frost et al. 2025 ). Notably, recent experiments by Diehl et al. (2022) support the prediction that warming decreases phytoplankton P:C stoichiometry and herbivore population density, but increases phytoplankton population density (note that shifts in phytoplankton community composition contributed to the shifts in phosphorus contents observed by Diehl et al. (2022) , whereas our study considered only a single producer with flexible stoichiometry). However, it is unclear whether or how putative temperature-dependent nutritional traits contributed to these population responses because zooplankton nutritional traits were not measured as part of the study ( Diehl et al 2022 ). We assumed that herbivores require more nutritionally dense food to maximize conversion efficiency at relatively cool and warm temperatures (i.e., a unimodal thermal relationship of the TER C:P ), following experimental results of Laspoumaderes et al. (2022) . However, theoretical and experimental findings by Ruiz et al. (2020) indicate that the TER C:P exhibits a U-shaped relationship to temperature, such that herbivores require more energy rich food towards thermal extremes. To reconcile these results, Laspoumaderes et al. (2022) proposed that the TER C:P exhibits a cubic thermal response (i.e., nutritional requirements increase towards cool and warm temperatures, but sufficient warming ultimately leads to increased carbon demands) – a relationship that was observed in one of their study species (the marine crab Carcinus maena ). Our model could readily be extended to consider more complex thermal responses of the TER C:P , and our results indicate that such increases in carbon (energetic) requirements at very warm temperatures are likely to extend the thermal niche of herbivores embedded in simple dynamic trophic communities. We considered autotroph-herbivore interactions in static environments, yet natural environments are characterized by temporal variability in temperature and nutrient supply. To understand how stoichiometric constraints might affect population dynamics in variable environments, it will be important to consider the timescales at which herbivore nutritional requirements and autotroph nutritional contents respond to environmental change. Herbivores may respond to a nutrient-poor diet by investing resources in enzymes or other digestive traits that aid in extracting nutrients from food ( Karasov et al. 2011 ) or by shifting body composition ( Hood and Sterner 2010 ). These physiological responses can mediate performance in environments with temporal variability in food nutritional quality or temperature ( Hood and Sterner 2010 , Koussoroplis et al. 2019 , Van Baelen et al. 2024 ). It is well-established that the nutritional quality of autotrophs such as phytoplankton declines with increasing temperature and decreasing nutrient supply ( Yvon-Durocher et al. 2015 , Tanioka and Matsumoto 2020 , Sauterey and Ward 2022 ), and that the time-scales of these responses have important implications for population dynamics of autotrophs ( Caperon 1968 , Droop 1974 , Bieg and Vasseur 2024 , Anderson et al. 2025 ) and herbivores ( Diehl et al. 2022 ). Future work may consider how flexibility in consumer nutritional requirements interacts with plasticity in autotroph stoichiometry to moderate community structure and dynamics in temporally variable environments. Resource enrichment can arise both from an increase in nutrient availability, as considered in our model, and from an increase in solar energy (light) supply. How might variation in irradiance, and associated changes in autotroph growth and stoichiometry, alter the response of communities to temperature gradients? Autotroph stoichiometry exhibits fundamentally different responses to light enrichment than to nutrient enrichment – increased irradiance leads to decreases in autotroph nutritional contents (i.e., decreased Nutrient:Carbon ratios, as high irradiance enhances carbon fixation), whereas increased nutrient supply increases autotroph nutrient contents ( Urabe and Sterner 1996 , Tanioka and Matsumoto 2020 ). This mechanism leads to qualitative changes in the system’s dynamical response to enrichment: increased irradiance initially destabilizes the system (via the classic paradox of enrichment mechanism ( Rosenzweig 1971 ); also observed here with nutrient enrichment, see Fig. 5 ), but further increases in irradiance and concomitant reductions in autotroph nutritional quality can stabilize population dynamics and ultimately lead to herbivore extinction (referred to as the “paradox of energy enrichment;” ( Loladze et al. 2000 , Diehl 2007 , Elser et al. 2012 )). To our knowledge, effects of temperature on properties of the paradox of energy enrichment (e.g., the irradiance needed for stabilization and for herbivore extinction) have not been investigated. We might expect that the irradiance yielding herbivore extinction exhibits a unimodal relationship to temperature, as temperature extremes impair herbivore productivity (via reduced consumption rates ( Uszko et al. 2017 ), increased metabolic requirements ( Rall et al. 2010 ), and/or reduced conversion efficiency as found here). Our analyses lead to two conclusions to contribute to the state of understanding of the temperature-dependence of herbivore conversion efficiency. First, the model produces a unimodal relationship of conversion efficiency to temperature when autotroph stoichiometry is invariant with respect to temperature ( Fig. 2C-D ). This prediction is consistent with some evidence in fishes (reviewed in Jobling, 1995 ; see Innes-Gold et al. 2025 for an example in herbivorous fish). However, this result contrasts with the results of a cross-species meta-analysis performed by Lang et al. (2017) , which showed that conversion efficiency increases with temperature. Additional research is needed to connect these results of Lang et al. (2017) based on interspecific comparisons to the intraspecific thermal responses of conversion efficiency that emerge from our model. For example, one possibility is that warm-adapted species generally have higher conversion efficiencies than cold-adapted species, even if both cold- and warm-adapted species tend to exhibit unimodal thermal dependencies of conversion efficiency. Second, our analysis shows that, because herbivores are embedded in dynamic food webs, the temperature-dependence of conversion efficiency is influenced by the dynamics of autotroph stoichiometry ( Sentis et al. 2022 ). We found that autotroph Phosphorous:Carbon stoichiometry exhibited a unimodal relationship to temperature in both versions of the model ( Fig. 4C, F ), which strengthens the decline in conversion efficiency at temperature extremes relative to the case of invariant stoichiometry. Continued integration of ecological stoichiometry theory ( Sterner and Elser 2002 , Prater et al. 2024 ) with bioenergetic consumer-resource models ( Yodzis and Innes 1992 , Vasseur and McCann 2005 , Gilbert et al. 2014 ) may offer a theoretical foundation for modeling consumer conversion efficiency across temperatures. Conclusion Nutrients and temperature are key global change factors that influence ecological communities through their effects on organismal stoichiometry and metabolism. In this study, we evaluated the consequences of temperature-dependent changes in herbivore nutritional traits for population dynamics and persistence across gradients in temperature and nutrient supply. We found that increased herbivore nutrient demands and decreased herbivore nutrient contents towards temperature extremes restricted the thermal niche of herbivores and altered the cycling of nutrients between herbivore and autotroph populations, which weakened trophic interaction strengths and destabilized population dynamics. This work advances the understanding of the role of stoichiometry and energetics in moderating the response of population dynamics and persistence to changes in temperature and nutrient availability. Supplemental information Download figure Open in new tab Fig. S1. Fits of model parameters to experimental data. In all panels, the solid black line shows model fits whereas the points or bars show data. A Fit of Eqn. 2c to data for the maximum ingestion rate of Acartia tonsa across a temperature gradient from Laspoumaderes et al. (2022) . B Fit of Eqn. 2a to data for the ingestion rate of Acartia tonsa feeding on Rhodomonas spp . at various densities from Thor and Wendt (2010) . C Fit of Eqn. 2d to data for the respiration rate of Acartia tonsa across temperatures from Laspoumaderes et al. (2022) . Estimates of respiration rate in gO 2 gC −1 day −1 were converted to gC gC −1 day −1 using a respiratory quotient of 0.38 gC gO 2 −1 ( Andersen 1997 ). D Fit of Eqn. 4a to data for the phosphorus requirements of Acartia tonsa across temperatures from Laspoumaderes et al. (2022) . E Fit of Eqn. 3 to data for conversion efficiency of Acartia tonsa from Laspoumaderes et al. (2022) , with the nutrient requirements Q TER ( T ) and autotroph nutrient contents Q A set at values measured experimentally (panel D , and Q A = 0.015). This yields an estimate of the maximum conversion efficiency ε max . F Fit of Eqn. 4b to data for the phosphorus contents of Acartia tonsa across temperatures from ( Laspoumaderes et al. 2022 ). G Fit of autotroph per-capita population growth rate, (1 /A )( dA/dt ), to data for Rhodomonas salina in nutrient-replete conditions and in the absence of grazing mortality from Ferreira et al. (2022) . In this case, Z = 0 and Q Z ≫ Q min (so that μ ( Q A , T ) ≈ μ ∞ ( T )), and thus (1 /A )( dA/dt ) = μ ∞ ( T ) − m Z ( T ). This fit therefore yields estimates of parameters for μ - ( T ) ( Eqn. 4b ) and m .( T ) ( Eqn. 5a ). H Frequency distribution for Q min in units of gP gC −1 from Edwards et al. (2015) . This dataset did not contain an estimate for Rhodomonas salina , so we used the average value (indicated by vertical line). View this table: View inline View popup Table S1. Parameter values and sources. For parameters that vary with temperature, we provide the set of parameters controlling the temperature-dependent function (either a Sharp-Schoolfield, Boltzmann-Arhenius, gaussian, or gaussian function; see the equation provided in the Model Development section of the main text). Calculating community properties To calculate the community properties considered in this study – the herbivore persistence boundary, the stability boundary, and the trophic interaction strength – we must first define the equilibrium states of the system. Equilibria are defined as the set of autotroph and herbivore population densities (i.e., { A Eq , Z Eq }) that satisfy: There are three equilibria: To ensure that the mass of phosphorus is conserved, we only consider equilibria with autotroph and herbivore population densities that satisfy ( Andersen et al. 2004 ): To understand this constraint, consider a simple case in which Z = 0 and autotroph phosphorus content is fixed at its’ minimum possible value Q min ( gP gC −1 ). In this case, sufficient increases in autotroph population density A ( gC m 31 ) leads the total phosphorus embedded in phytoplankton ( Q min A ; gP m −3 ) to exceed the total phosphorus in the system N 424 ( gP m −1 ) – thus violating the law of mass conservation. By restricting dynamics to be within the boundary of Eqn. S3, we avoid this issue ( Andersen et al. 2004 ). We are now in position to define 1) the herbivore persistence boundary, 2) the stability boundary, and 3) the trophic interaction strength. First, we identify the herbivore persistence boundary by solving for the phosphorus availability N tot at which the herbivore maintains a net zero per-capita population growth rate at the autotroph-only equilibrium (Eqn. S2b; Fig. S2A-B ; ( Andersen 1997 , Uszko et al. 2017 )): We then assess the impact of temperature on the phosphorus availability N 424 satisfying Eqn. (S4), yielding the set of phosphorus requirements and temperatures that separate herbivore persistence from extinction. Note that herbivore persistence does not depend on herbivore phosphorus contents Q Z ( T ) because the herbivore population does not influence autotroph nutrient contents Q A at the autotroph-only equilibrium (i.e., Z = 0, canceling Q Z ( T ) from Eqn. 1c ) and Q Z ( T ) does not influence herbivore growth. This approach (Eqn. S4) can be understood conceptually by comparing the autotroph-only and non-trivial equilibria near the persistence boundary: as reductions in phosphorus availability drive herbivores to extinction, the autotroph density at the non-trivial equilibrium approaches the autotroph-only equilibrium ( Fig. S2A-B ). Second, we examined effects of temperature on the total phosphorus concentration N tot at which the system is destabilized, and on the population fluctuations of limit cycle attractors. To do so, we calculated the eigenvalues associated with the Jacobian matrix evaluated at the non-trivial equilibrium (Eqn. S2c), and solved for the N tot at which the real part of the largest eigenvalue equals zero (i.e., the N tot separating a stable equilibrium with negative eigenvalues, and an unstable equilibrium with at least one positive eigenvalue; Fig. S2A , ( Uszko et al. 2017 )). Stability can be understood by inspecting herbivore and autotroph zero-net growth isoclines ( Fig. S2C-D ). An equilibrium is stable if a small increase in autotroph population density leads to negative autotroph growth rates and thus a return to the equilibrium (e.g., ( Uszko et al. 2015 )). This occurs when the autotroph isocline has a negative slope at the intersection with the herbivore isocline (so long as the intersection occurs when the herbivore isocline is vertical), as shown in the example in Fig. S2C (but not Fig. S2D ). Enrichment also leads to limit cycles with very low autotroph population densities ( Fig. S2A ) that leads autotrophs, and thus the herbivores that depend on autotrophs, susceptible to extinction ( Rosenzweig 1971 ). We further track the N tot producing minimum autotroph population densities of 10 −3 , 10 −6 , and 10 −9 across temperatures. Third, we assessed the strength of the trophic interaction, denoted IS , as the ratio of autotroph density in the absence of herbivory (Eqn. S2b) to the autotroph density in the presence of the herbivore (Eqn. S2c) ( Gilbert et al. 2014 ): Biologically, Eqn. S5 quantifies the potential density of the autotroph population that is captured by the herbivore (see Fig. S2A and C-D ), and is a useful metric because it is readily measured experimentally. To understand variability in the IS in the phase-plane, consider the horizontal displacement of the non-trivial equilibrium ( A Eq 3 ) from the autotroph-only equilibrium ( A Eq 2 ) (indicated by arrows in Fig. S2C ). Download figure Open in new tab Fig. S2. Properties of the autotroph-herbivore community considered in this study. A Phosphorus must be available at a sufficient concentration ( N tot ) to support herbivore persistence, and further enrichment will ultimately destabilize the system and create limit cycles that drive autotrophs to low population densities. Shown are the autotroph-only equilibrium population density, and autotroph and herbivore densities at the non-trivial equilibrium. Unstable equilibria are semi-transparent, and in unstable regions we present the minimum and maximum population densities for each population over a limit cycle (opaque lines). The indicated points are explored in more detail in panels B-D. B Effect of autotroph population density and phosphorus content on the herbivore zero net growth isocline. Herbivores need a larger autotroph population density to persist when autotroph biomass is less nutritious. The gray and pink points indicate autotroph population densities and phosphorus contents at the autotroph-only equilibrium near the herbivore persistence boundary (see A ). Note that herbivores can persist at an N tot (and temperature) if the autotroph-only equilibrium permits positive growth (as in the pink point). C-D Representative isoclines (i.e., sets of autotroph and herbivore densities yielding zero per-capita population growth rates) for a stable ( C ) and an unstable non-trivial equilibrium ( D ). In each panel, herbivores and autotrophs have positive per-capita population growth rates at points below their respective isoclines. The gray shading indicates autotroph phosphorus contents as a function of herbivore and autotroph population density ( Eqn. 1c ), whereas white regions are densities that violate nutrient conservation (Eqn. S3). Note that herbivores require a greater quantity of autotroph biomass to achieve a positive population growth rate when herbivores are at a high population density (as increasing herbivore density leads to a higher share of total phosphorus locked in herbivore biomass and thus P-poor autotroph biomass – ultimately reducing herbivore assimilation efficiency), creating a unimodal shape of their isocline in phase space. This figure assumes T = 18°C, which is the temperature at which the constant Q Z And Q TER and temperature-dependent Q Z and Q TER versions of the model are roughly equivalent to one another. Funder Information Declared Simons Foundation, https://ror.org/01cmst727 , 877200 References ↵ Allen , A. P. and Gillooly , J. F. 2009 . 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Effects of warming on predator– prey interactions – a resource-based approach and a theoretical synthesis . - Ecol. Lett . 20 : 513 – 523 . OpenUrl CrossRef PubMed Ward , B. A. , Dutkiewicz , S. , Jahn , O. and Follows , M. J. 2012 . A size-structured food-web model for the global ocean . - Limnol. Oceanogr . 57 : 1877 – 1891 . OpenUrl View the discussion thread. Back to top Previous Next Posted October 30, 2025. Download PDF Email Thank you for your interest in spreading the word about bioRxiv. NOTE: Your email address is requested solely to identify you as the sender of this article. Your Email * Your Name * Send To * Enter multiple addresses on separate lines or separate them with commas. You are going to email the following Temperature-dependent herbivore nutritional traits affect population dynamics and persistence Message Subject (Your Name) has forwarded a page to you from bioRxiv Message Body (Your Name) thought you would like to see this page from the bioRxiv website. 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