Bounded environmental stochasticity generates secondary Allee thresholds

Bounded environmental stochasticity generates secondary Allee thresholds | 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 Bounded environmental stochasticity generates secondary Allee thresholds View ORCID Profile Sebastian J. Schreiber doi: https://doi.org/10.1101/2025.04.03.647087 Sebastian J. Schreiber * Department of Evolution and Ecology and Center for Population Biology, University of California , Davis, CA 95616, USA Find this author on Google Scholar Find this author on PubMed Search for this author on this site ORCID record for Sebastian J. Schreiber For correspondence: sschreiber{at}ucdavis.edu Abstract Full Text Info/History Metrics Preview PDF Abstract A population exhibits an Allee effect when there is a critical density below which it goes extinct and above which it persists. Classical models with environmental stochasticity predict inevitable extinction, stemming from the assumption that environmental variation is normally distributed with rare but arbitrary large effect sizes. However, environmental fluctuations are bounded and often not normally distributed. To address this reality, I analyze piecewise deterministic Markov models (PDMPs) of populations experiencing Allee effects, where environmental dynamics are governed by a finite-state Markov chain. These models predict that populations can persist through the emergence of two threshold densities. Below the lower threshold, populations deterministically go extinct; above the higher threshold, they deterministically persist. At intermediate densities, populations experience stochastic bistability: with positive, complementary probabilities, they either go extinct or persist. Persistence becomes impossible when the carrying capacity in one environment falls below the Allee threshold in another. Such mismatch occurs only when the environmental state affects percapita growth rates non-monotonically, as when environments supporting higher carrying capacities also produce higher predation levels or greater mate limitation. This work demonstrates that incorporating realistic bounded environmental fluctuations substantially alters predictions about population persistence, with important implications for conservation and management. I ntroduction Populations exhibit an Allee effect when, at low densities, the per-capita growth rate increases with density ( Allee, 1931 ; Stephens et al., 1999 ). Common causes of this effect include larger population sizes that reduce the risk of individual predation and the difficulty of finding mates, or increase cooperative foraging or breeding ( Courchamp et al., 1999 ; Stephens et al., 1999 ; Gascoigne and Lipcius, 2004 ; Courchamp et al., 2008 ; Gascoigne et al., 2009 ; Kramer et al., 2009 ). When an Allee effect is sufficiently strong, it results in a critical density below which a population is driven rapidly to extinction and above which the population grows and persists. This critical density can play an important role in the rescue of threatened populations ( Dennis, 1989 ; Stephens and Sutherland, 1999 ; Berec et al., 2007 ; Courchamp et al., 2008 ), controlling diseases or pest species ( Hilker et al., 2009 ; Tobin et al., 2011 ; Kang and Castillo-Chávez, 2014 ), or managing harvested populations ( Berec et al., 2007 ; Winter et al., 2019 ). Populations that simultaneously experience a strong Allee effect and environmental stochasticity may experience a greater risk of extinction ( Dennis, 1989 , 2002 ; Lande et al., 2003 ). Stochastic fluctuations in environmental conditions can knock a persisting population below the Allee threshold and initiate an exponential decline to extinction. Traditional models of environmental stochasticity have predominantly used diffusion processes or stochastic differential equations (SDEs) ( Dennis, 1989 , 2002 ; Lande et al., 2003 ). These approaches assume that environmental fluctuations are normally distributed where small deviations from the mean occur with the greatest likelihood but where arbitrarily large deviations are possible ( Dennis, 1989 , 2002 ; Liebhold and Bascompte, 2003 ; Lande et al., 2003 ; Dennis et al., 2015 ). An important consequence of this unbounded stochasticity is that populations with a strong Allee effect have no stable non-zero stationary distribution ( Dennis et al., 2015 ) and inevitably go extinct at an exponential rate. Although these models have provided valuable information on extinction probabilities and critical thresholds, they may not adequately capture the nature of environmental variability in many systems. In reality, environmental fluctuations in nature are bounded and may be more often large than small ( d’Onofrio, 2013 ; Fieberg and Ellner, 2001 ; McLaughlin et al., 2002 ; Silva et al., 1991 ; Stacey and Taper, 1992 ). This fundamental difference raises an important question. What is the effect of bounded environmental fluctuations on populations experiencing a strong Allee effect? To answer this question, I analyze ordinary differential equation (ODE) models of populations experiencing Allee effects and bounded fluctuations in their demographic rates. Unlike previous studies that often focus exclusively on demographic stochasticity ( Allen et al., 2005 ; Ackleh et al., 2007 ) or use diffusion approximations, this work models environmental variation using a continuous-time Markov chain with any number of states representing different environmental conditions experienced by the population. The environmental state determines the demographic parameters of the ODEs. Hence, one gets population dynamics driven by a Markovian environment, i.e., a piecewise deterministic Markov process (PDMP) ( Davis, 1984 ). PDMPs allow for modeling environmental regimes with potentially large but bounded fluctuations, making them increasingly valuable for ecological applications (Benaïm and Lobry, 2016; Hening and Strickler, 2019 ; Hening and Nguyen, 2020 ; Hening et al., 2021 ; Benaïm et al., 2023; Monmarché et al., 2024 ). My analysis reveals several key findings. First, even when populations can persist in each of the environmental states individually, a demographic mismatch between the environmental states can result in inevitable extinction. Second, when population persistence is possible, there emerge two distinct Allee thresholds rather than a single critical density. For populations whose density lies below the lower threshold, extinction is inevitable. When the population density lies above the higher threshold, the population persists indefinitely. Most interestingly, for populations between the two thresholds, extinction or persistence occur with complementary, non-zero probabilities–creating a zone of probabilistic outcomes not captured by deterministic models. The analysis further explores how the speed at which the environment switches between states influences these probabilities of extinction, revealing that the relationship between environmental fluctuation rates and extinction risk is more complex than previously understood. To demonstrate ecological applications, these analytical findings are applied to specific scenarios of biological importance: populations experiencing mate limitation and predator satiation. M odels and M ethods The stochastic model of Allee dynamics Consider a population with density N living in a randomly fluctuating environment. The traditional approach to modeling environmental stochasticity in continuous-time is to formulate the model using stochastic differential equations ( Foley, 1994 ; Lande et al., 2003 ; Dennis, 2002 ). However, environmental stochasticity in these models is given by a Brownian motion on log densities. This form of stochasticity has no auto-correlation structure, and fluctuations can be arbitrarily large over arbitrarily small time intervals. In contrast, here environmental fluctuations E ( t ) are modeled as a continuous-time Markov chain with any number ( k ) of states. The environment transitions from state i to state j at a rate λ ij , that is, ℙ [ E ( t + Δ t ) = j | E ( t ) = i ] ≈ λ ij Δ t for small Δ t > 0. This approach to modeling the environment ensures that the fluctuations are bounded, allows for auto-correlations, and provides great flexibility in the long-term distribution of environmental effects. I assume that this Markov chain is irreducible and, consequently, there is a unique stationary distribution π = ( π 1 , … , π k ), where π i represents the long-term probability of the environment being in state i . The per-capita growth rate f E ( N ) of the population is completely dependent on the environmental state E and the population density N . The population dynamics are given by a piece-wise deterministic Markov process (PDMP), which was introduced by Davis (1984) and has been used extensively to model stochastic dynamics in a variety of disciplines: In each environmental state, the population exhibits an Allee effect. Specifically, for the environmental state E = i , there is an Allee threshold A i and a carrying capacity K i such that f i ( A i ) = f i ( K i ) = 0, f i ( N ) < 0 for N K i , and f i ( N ) > 0 for A i < N < K i . Thus, in this environmental state, the population rapidly declines to extinction whenever its initial density is below A i . In contrast, the population asymptotically approaches its carrying capacity K i when its initial density is above A i . The irreducibility of the Markov chain determining the environmental dynamics allows us to define the mean-field model given by: Analytical and Numerical Methods In Appendix Appendix, the dynamics of (1), which describes the population growth in a fluctuating environment, are classified by identifying the initial conditions that lead to long-term extinction or persistence and the probabilities of these outcomes. Extinction for an initial condition ( E (0), N (0)) = ( i, n ) means with non-zero probability. In other words, with non-zero probability, the population density decreases exponentially over time, approaching zero. Alternatively, persistence means there exists a minimal density m > 0 such that N ( t ) ≥ m for sufficiently large t and with non-zero probability. The mathematical arguments mainly rely on classical results about ordinary differential equations and probability. Numerical simulations are implemented using a hybrid of Gillespie (1977 )’s method for exact simulations of Markov chains and an ordinary differential equation (ODE) solver. Specifically, Gillespie’s method is used to determine the times T 1 < T 2 < … at which the environmental state E ( t ) changes and the environmental states E ( T 1 ), E ( T 2 ), … at these times. During each interval [ T i , T i +1 ), the ODE (1) with E = E ( T i ) and the initial condition is solved numerically using the deSolve package in R ( Soetaert et al., 2010 ). R esults A Dynamical Dichotomy The first main finding is a fundamental dichotomy in the dynamical behavior of (1) ( Fig. 1 , Appendix). This dichotomy depends on three key quantities: the minimum of the Allee thresholds A = min i A i , the maximum of the Allee thresholds Ā = max i A i , and the minimum of the carrying capacities K = min i K i . When the maximal Allee threshold Ā is greater than the minimal carrying capacity K (a filled circle is to the left of an unfilled circle in Fig. 1A ), a population at its carrying capacity in one environment state will eventually find itself below the Allee threshold in another environmental state, and decline. While this decline may be followed by increases in population densities due to favorable environmental shifts, the population always asymptotically approaches extinction, i.e., lim t →∞ N ( t ) = 0 with probability one ( Fig. 1B ). When Ā > K , I say there is a demographic mismatch between the environmental states. Download figure Open in new tab Figure 1. A dynamical dichotomy for populations experiencing strong Allee effects in an environment randomly switching between two states ( k = 2). In panels A and C, the per-capita growth rates for both environmental states are plotted with Allee thresholds A i as unfilled circles and carrying capacities K i as filled circles. In B and D, multiple simulations for three initial densities: one below A (light gray), one above A (light gray), and one between A and Ā (black). In A and B, there is demographic mismatch ( K A , i.e., all filled circles lie to the right of the unfilled circles. Furthermore, persistence or extinction occurs with complementary non-zero probabilities for the initial condition lying between A and Ā . Parameters: f i ( N ) = ( N − A i )( K i − N ) with transition rates λ 12 = λ 21 = 1, A 1 = 1, A 2 = 2, and K 2 = 3. For panels A and B, K 1 = 2.6; for panels C and D, K 1 = 1.8. When there is no demographic mismatch ( Ā < K as in Fig. 1C ), the minimum and maximum Allee thresholds determine the long-term fate of the population ( Fig. 1D ). If the initial population density is below the minimum Allee threshold ( N (0) < A ), then the population continuously decreases to extinction at an exponential rate (lower, gray simulation curves in Fig. 1D ). If the initial population density is above the maximum Allee threshold ( N (0) > Ā ), then the population density always remains above this threshold and the population persists, that is, N ( t ) ≥ Ā for all time t ≥0 (upper, gray simulation curves in Fig. 1D ). If the initial population density lies between the minimum and maximum Allee thresholds ( A < N (0) < A ), then the long-term fate of the population is stochastic: with non-zero probability, the population goes asymptotically extinct and with a complementary non-zero probability, it persists (black simulation curves in Fig. 1D ). This stochastic outcome is determined by the specific sequence of environmental states experienced by the population in the early stages of its dynamics, before its density can either fall below A or rise above Ā . Extinction risk depends on the speed of the environmental dynamics For initial densities N (0) between the minimum and maximum Allee thresholds, the non-zero probabilities of extinction versus persistence depend on how quickly the environment switches between different states. Specifically, suppose that the environmental transition rates are of the form sλ ij where s > 0 determines the speed at which the transitions occur. Although the speed s does not affect the fraction π i of time spent in each environmental state i , they have a strong effect on the probabilities of extinction versus persistence ( Fig. 2 , Proposition 1 in Appendix). Specifically, when transition rates are slow (ie s ≈0 but positive), the probability of extinction strongly depends on the initial population density N (0) and the initial environmental state E (0) = i . If N (0) < A i , then the probability of extinction is close to 1, otherwise the probability of extinction is close to 0. Intuitively, if the environmental states are changing slowly, then the Allee threshold of the initial environmental state determines, with high likelihood, the long-term population fate (compare the lighter curves in Fig. 2A versus B). Alternatively, if environmental dynamics is fast ( s ≫0), then the Allee threshold A * of the mean field dynamics of equation (2) determines the fate of the population (darker curves in Fig. 2 ). If the population density is initially below A * , the probability of extinction is high; otherwise, it is low. Although the Allee threshold for the mean field A * always lies between the minimum and maximum Allee thresholds, it is generally not the environmental average ∑ i π i A i of the Allee thresholds for the different environments. This stems from Jensen’s inequality, which implies that for a non-linear function, the average of the function values may differ from the function of the average. In this case, the non-linear relationship between environmental conditions and Allee thresholds causes this discrepancy. Download figure Open in new tab Figure 2. Environmental switching rates alter extinction probabilities. The probability of going asymptotically extinct as a function of the initial population density N (0) and different initial environmental states E (0). Different shaded curves correspond to different transition rates λ 12 = λ 21 . Dashed lines from left to right correspond to minimal Allee threshold A , the Allee threshold A * of the mean model, and the maximal Allee threshold Ā . Parameters: f i ( N ) = ( N − A i )( K i − N ) with transition rates λ 12 = λ 21 as shown, A 1 = 2, A 2 = 4, and K 1 = K 2 = 5. Simple environmental variation never leads to demographic mismatch We say that environmental variation is simple if the environmental effects have a one-dimensional parameterization, say e , and this parameter has a monotonic effect on the per-capita growth rate. More specifically, assume that all per-capita growth rates are of the form f i ( N ) = f ( N, e i ) where either f is non-decreasing in e for all N , or f is non-increasing in e for all N . For example, increasing e always increases birth rates and lowers mortality rates, or increasing e always decreases birth rates and increases mortality rates. The environmental variation of this form ( Lemma 1 in Appendix) never creates a demographic mismatch. Hence, provided there exist positive Allee thresholds in all environments, simple environmental variation cannot lead to unconditional extinction i.e. all initial conditions lead to asymptotic extinction with probaiblity one. Demographic mismatch via complex environmental variation Complex environmental variation, however, can lead to demographic mismatch. To illustrate how, we examine specific models that separately account for mate limitation and predator satiation. In these models, like most demographic models, there are parameter values for which the per-capita growth is negative for all densities and, consequently, would lead to extinction. We call environments with these parameter values inhospitable environments. For these situations, extinction always occurs whenever one of the environments is inhospitable even if other environments have non-zero Allee thresholds A i and carrying capacities K i (Proposition 2 in Appendix). To model mate limitation, let N/ ( h + N ) be the probability that an individual is mated where h is the population density resulting in 50% of individuals being mated ( Dennis, 1989 ; Boukal and Berec, 2002 ). Mated individuals produce offspring at a rate b . All individuals experience density-independent mortality d and density-dependent mortality cN where c measures the strength of intraspecific competition. In environment i , the per-capita growth rate equals This per-capita growth rate has a positive Allee threshold A i and a positive carrying capacity K i whenever where r i = b i − d i is the intrinsic rate of growth in the absence of mate limitation. Whenever this inequality is not satisfied (i.e. density-independent mortality is too high), environments are inhospitable. The environmental variation in any one of the demographic parameters, b, h, d , or c , in equation (3) corresponds to simple environmental variation and consequently does not cause a demographic mismatch. However, variation in multiple parameters can lead to demographic mismatch when some environments have low carrying capacities and other environments have large Allee thresholds. Specifically, this creates a situation where the minimum carrying capacity K falls below the maximum Allee threshold Ā . For example, when one environment has weaker density-dependent mortality (i.e., smaller c i ) and greater mate limitation (i.e., larger h i ) than another environment, then demographic mismatch can occur ( Fig. 3A ). Download figure Open in new tab Figure 3. Demographic consequences of mate-limitation (A) or predator satiation (B) in a fluctuating environment. Regions of parameter space correspond to three types of ecological dynamics: conditional persistence in which there is persistence or extinction depending on initial conditions; demographic mismatch in which the maximal Allee threshold Ā is greater than the minimal carrying capacity K and the population is ultimately driven extinct; and an inhospitable environmental state in which the per-capita growth rate is negative for all densities and the population ultimately goes extinct. To model a prey population experiencing predation from a generalist predator, we use the model of Noy-Meir (1975) . In the absence of predation, the population exhibits Logistic dynamics with intrinsic rate of growth r and carrying capacity κ . The maximal per-capita predation rate is p with a half saturation prey density h . The per-capita growth rate in the environmental state i is In environment i , there is an Allee threshold A i and carrying capacity K i if where H i = h i /κ i . The first inequality ensures a negative per-capita growth rate at low population densities, and the second inequality ensures the existence of A i and K i . When the second inequality is violated -the maximal predation rate is too high), the environment is inhospitable. If the environmental states only affect any one of the demographic rates (i.e. r, κ, h or p ), then the environmental variation is simple and there is no demographic mismatch. However, simultaneous variation in the maximal predation rate p and the prey’s carrying capacity κ can lead to demographic mismatch. When the prey carrying capacity κ is small, the carrying capacity K in the presence of predation is low. Alternatively, if the prey carrying capacity κ is high and the maximal predation p is high (but not too high), then there is a high Allee threshold A . Environments that alternate between these conditions—specifically when one environment produces a low K value and another environment produces a high A value such that K < Ā —create a demographic mismatch that leads to inevitable extinction ( Fig. 3B ). D iscussion In nature, environmental fluctuations are bounded and may be typically large rather than small. However, these realities are not accommodated by stochastic differential equations that are commonly used to model environmental fluctuations in populations with overlapping generations ( May, 1973 ; Dennis, 1989 ; Foley, 1994 ; Lande et al., 2003 ; Nolting and Abbott, 2016 ; Hening and Nguyen, 2018 ). Here, environmental stochasticity is captured by Markovian dynamics between any number of states. These Markovian dynamics are inherently bounded and allow for arbitrarily complex environmental dynamics including a predominance of large rather than small fluctuations. I have shown that, for populations experiencing a strong Allee effect, this additional environmental realism can dramatically impact the long-term population dynamics. Unlike stochastic differential equation models that predict strong Allee effects always drive population extinct ( Dennis et al., 2015 ), models allowing for bounded, stochastic fluctuations generate two distinct scenarios depending on the relationship between environmental states. The first scenario occurs when there is a demographic mismatch between environments: namely, when the carrying capacity in one environment lies below the Allee threshold in another environment. Even when environmental conditions individually allow for conditional persistence of the population, fluctuating between these conditions may lead to unavoidable extinction. Demographic mismatch only occurs when environmental shifts have non-monotonic impacts on the per-capita growth rate—where shifts increase growth rates over some density ranges but decrease them over others. For example, in populations experiencing mate limitation, mismatch can occur when environmental conditions that reduce intraspecific competition simultaneously increase mate limitation. Temperature fluctuations may generate such trade-offs. The metabolic theory of ecology predicts that intraspecific competition increases at higher temperatures (effectively lowering carrying capacity) ( Savage et al., 2004 )—a prediction supported by experimental evidence ( Bernhardt et al., 2018 ). In contrast, a recent metaanalysis suggests that higher temperatures may enhance mating success ( Pilakouta and Baillet, 2022 ), potentially lowering Allee thresholds. When these effects occur together, temperature fluctuations can create the conditions for demographic mismatch. Similarly, the analysis presented here suggests that environments that support higher prey carrying capacities while simultaneously enabling greater maximal predation rates can create a demographic mismatch for prey populations with predator-driven Allee effects. This positive relationship between prey carrying capacities and predation intensity aligns with general scaling laws showing that total predator biomass increases with total prey biomass in diverse terrestrial and aquatic communities ( Hatton et al., 2015 ; Perkins et al., 2022 ). Kaul et al. (2016) showed that Allee thresholds for the bacterivore Cafeteria roenbergensis depend in a complex manner on the availability of carbon resources and predation levels, potentially illustrating these types of demographic trade-offs. The second scenario reveals a novel finding: in the absence of demographic mismatch, populations experiencing environmental fluctuations and Allee effects can persist, an important difference from classical SDE models that predict inevitable extinction ( Dennis, 2002 ; Dennis et al., 2015 ). In these cases, there are two critical Allee threshold densities A < Ā rather than just one. If the initial population density lies below the lower threshold density A , the population always goes extinct. If the initial density exceeds the higher threshold density Ā , the population always persists and remains above this higher threshold. Similar to deterministic models ( Dennis, 1989 ; Boukal and Berec, 2002 ; Schreiber, 2003 ), the fate of populations with initial densities outside the interval [ A , Ā ] is deterministic. However, unlike deterministic models, populations with initial densities within this interval have stochastic fates: they either go extinct or persist with complementary, non-zero probabilities. In the absence of demographic mismatch, the rate at which the environment fluctuates between different states determines the sensitivity of the results to the initial conditions. The dynamics of a system with fast environmental switching can be closely approximated by the mean-field model (2). This deterministic model has a single critical threshold density A * that lies between the two threshold densities of the stochastic model ( A < A * < Ā ). Hence, with fast switching, extinction becomes highly likely (though not certain) when the initial density falls between A and A * , while persistence becomes highly likely when the initial density lies between A * and Ā . In contrast, when environmental switching is slow, the outcomes depend heavily on both the initial population density and the initial environmental state. Specifically, if the initial population density falls below the Allee threshold of the initial environmental state, extinction becomes highly likely but not certain. Conversely, when the initial density exceeds the initial Allee threshold, persistence becomes highly likely. These findings complement and extend previous work on stochastic Allee effects. Liebhold and Bascompte (2003) studied a discrete-time model that incorporated an Allee effect and environmental stochasticity, but unlike our model, theirs lacked negative density-dependence. Their model, parameterized for the North American Gypsy moth pest ( Lymantria dispar ), revealed threshold behavior similar to that observed here. Specifically, when initial densities fell below the critical threshold of their mean-field model, environmental stochasticity increased persistence probability by occasionally pushing populations above the critical density. In contrast, when initial densities exceeded this threshold, stochasticity increased the risk of extinction. Roth and Schreiber (2014) later provided mathematical confirmation of these simulation results. However, important distinctions exist between these studies and the work presented here: these earlier models employed unbounded, log-normal noise similar to SDE approaches, and the probabilistic outcomes they observed stemmed primarily from the absence of negative density-dependence, rather than from the bounded nature of environmental fluctuations. Our findings suggest several promising directions for future research that incorporate additional biological realism. First, spatial structure in deterministic models with Allee effects generates multiple alternative states corresponding to different configurations of patches with low and high occupancy ( Vortkamp et al., 2020 ); extending the analyses presented here to spatially explicit stochastic models could provide insight into how bounded environmental fluctuations affect these spatial patterns. Second, interspecific interactions could be incorporated, particularly since competition can produce complex frequency-dependent dynamics through mechanisms such as shared social information ( Gil et al., 2019 ) or reproductive interference ( Schreiber et al., 2019 ). Third, eco-evolutionary feedbacks may cause Allee thresholds to shift as population traits evolve ( Klausmeier et al., 2020 ), raising questions about how bounded environmental fluctuations might interact with these dynamic thresholds. Most crucially, our demonstration that bounded environmental stochasticity can enable long-term persistence for populations with Allee effects suggests that conservation strategies for threatened populations may need to reconsider traditional extinction risk assessments that rely on unbounded stochastic models. Collectively, this work highlights the importance of incorporating realistic environmental variation when predicting the fate of populations experiencing Allee effects, a critical consideration for both theoretical ecology and conservation biology. Appendix. M athematical P roofs of M ain R esults In this Appendix, we provide precise statements and mathematical proofs of results concerning (1). Let ℝ = ( −∞,∞) denote the real line and ℝ + = [0, ∞) the non-negative reals. For our main result, we need the following assumptions: A1: The per-capita growth rate functions f i : ℝ + → ℝ are Lipschitz. A2: For each i , there exists K i > A i > 0 such that f i ( A i ) = f i ( K i ) = 0, f i ( x ) > 0 for x ∈ ( A i , K i ), and f i ( x ) < 0 for x ∈ [0, A i ) ∪ ( K i , ∞ ). A3: The finite state Markov chain E ( t ) is irreducible with stationary distribution π = ( π 1 , π 2 , … , π k ) Theorem 1. Assume A1 – A3 . Define A = min i A i , Ā = max i A i , K = min i K . If Ā > K , then lim t →∞ N ( t ) = 0 with probability one whenever N (0) > 0. If Ā 0 and ℙ [lim t →∞ N ( t ) = 0|( N (0), E (0)) = ( x, i )] = 1 − a ( x, i ) > 0 whenever x ∈ [ A, Ā ) and 1 ≤ i ≤ k, and lim t →∞ N ( t ) = 0 whenever N (0) ∈ (0, A ). Note: On the event {lim t →∞ N ( t ) = 0}, one can show that with probability one. Proof . Let ϕ i ( t, x ) to denote the flow of in environment i i.e. t ↦ ϕ i ( t, x ) is the solution Of satisfying N (0) = x . The proof makes repeated use of Lévy 0-1 Law. To state this Law in our context, for any integer m ≥ 0, define ℱ m to be the σ -algebra generated by the random variables ℱ 0 , ℱ 1 , … is in an increasing sequence of σ -algebras. Define the σ -algebra ℱ ∞ as ∩ m ≥1 ℱ m . Given any set 𝒜 ∈ ℱ ∞ , let 𝟙 𝒜 : ℱ m →{0, 1} be the indicator function, i.e. equals 1 if the event 𝒜 has occurred and 0 otherwise. Theorem (Lévy 0-1 Law). Let 𝒜 ∈ ℱ ∞ . Then lim m →∞ 𝔼 [𝟙 𝒜 |ℱ m ] = 𝟙 A . Now, assume that Ā > K . As f i ( x ) < 0 for all x ≥ K, N ( t ) eventually enters and remains in the interval . Hence, without loss of generality, assume that N (0) lies in this interval. As f i ( x ) < 0 for all x < A , lim t →∞ N ( t ) = 0 whenever N ( t ) enters the interval [0, A ). Define 𝒜 = { N ( k ) < A for some k ≥ 1} which lies in ℱ ∞ . Given , I will show that there is a positive probability that N ( t ) enters [0, A ). To see why, define i * = arg max i A i and i * = arg min i K i . Choose t * > 0 such that (i) for all and t ≥ t * , and (ii) for all and t ≥ t * . This choice of t * is possible as (i) f i ( x ) 0 such that for all i , 0 ≤ t ≤ t * , and . This choice of t * is possible as ϕ i is continuous and is compact. For any m ≥ 0, let T 1 ( m ) = inf{ t ≥ m : E ( t ) = i * } be the first time after t = m that the environmental state is i * . Let T 2 ( m ) = inf{ t ≥ T 1 ( m ) : E ( t ) ≠ i * } be the first time the environmental state leaves the state i * after T 1 ( m ). On the event { T 2 ( m ) − T 1 ( m ) > t * }, the definition of t * implies that N ( T 2 ( m )) ≤ K + ( Ā − K ) / 4. Define T 3 ( m ) = inf{ t ≥ T 2 ( m ) : E ( t ) = i * } to be the first time the environmental state is i * after T 2 ( m ). On the event { T 2 ( m ) − T 1 ( m ) > t * } ∩ { T 3 ( m ) − T 2 ( m ) < t * }, the definitions of t * , t * imply that N ( T 3 ( m )) ≤ K + ( Ā − K ) / 2 < A . Define T 4 ( m ) = inf{ t ≥ T 3 ( m ) : E ( t ) ≠ i * } to be the first time after t = T 3 ( m ) to leave the state i * . On the event we have N ( T 4 ( m )) < A/ 2. In particular, ℰ ( m ) ⊂ 𝒜. Let As ℰ ( m ) ⊂ 𝒜, 𝔼 [𝟙 𝒜 | ℱ m ] ≥ 𝔼 [𝟙 ℰ ( m ) | ℱ m ] ≥ δ . By the Lévy 0-1 Law, 𝟙 𝒜 = 1 with probability one. Namely, lim t →∞ N ( t ) = 0 with probability one. Now, assume that Ā < K . As f i ( x ) < 0 for all ( x, i ) ∈ (0, A ) × {1, 2, … , k }, lim t →∞ N ( t ) = 0 whenever N (0) 0 for all ( x, i ) ∈ ( Ā, K ) × {1, 2, … , k }, N ( t ) ≥ Ā whenever N (0) > Ā . Assume N (0) ∈ ( A, Ā ). (Re)define à = ( A + Ā ) / 2, i * = arg min i A i and i * = arg max i A i . Choose t 1 > 0 and ε > 0 such that and for all t ≥ t 1 . Define the stopping times T * ( m ) = inf{ t ≥ m : E ( t ) = i * }, T ** = inf{ t ≥ T * ( m ) : E ( t ) ≠ i * }, T * ( m ) = inf{ t ≥ m : E ( t ) = i * }, and T ** ( m ) = inf t T * ( m ) : E ( t ) ≠ i * }. Define the event 𝒜 = N ( t ) ∉ [ A, Ā ] for some t ≥ 0}. Choose δ > 0 such that for all environmental states i and m ≥ 1. It follows that for all m ≥ 1 and For all m ≥ 1, Hence, by the Lévy 0-1 Law, 𝟙 𝒜 = 1 almost surely. In particular, for x ∈ ( A, Ā ) and 1 ≤ i ≤ k , ℙ [lim t →∞ N ( t ) = 0|( N (0), E (0)) = ( x, i )] = 1 − a ( x, i ) where a ( x, i ) := ℙ[ N ( t ) ≥ A for some t ≥ 1|( N (0), E (0)) = ( x, i )]. Finally, I need to show that 0 < a ( x, i ) 0 and ε > 0 such that ϕ i ( t, x ) ∈ ( A + ε, Ā − ε ) for t ∈ [0, t 1 ]. Choose t 2 > 0 such that ; for all t ≥ t 2 and x ∈ [ A + ε, Ā − ε ]. Let T 1 = inf t ≥ 0 : E ( t ) = i * and T 2 = inf { t ≥ T 1 : E ( t ) ≠ i * } . Then a ( x, i ) ≥ℙ [ { T 1 ≤ t 1 , T 2 ≥ t 2 }] > 0. The fact that a ( x, i ) < 1 is proved in a similar way choosing t 2 such that . The next proposition provides limiting values for the probabilities a ( x, i ) in Theorem 1 when switching between environments occurs quickly or slowly. Proposition 1. Assume A1 – A3 hold and Ā 0 and α ij ≥ 0. For each τ > 0, 1 ≤ i ≤ k, and x ∈ [ A, Ā ], define a τ ( x, i ) := ℙ [ N ( t ) ≥ Ā for some t ≥ 1|( N (0), E (0)) = ( x, i )]. Then Furthermore, if there exists a unique A * ∈ [ A, Ā ] such that ∑ i π i f i ( A * ) = 0, then Proof . Let { N τ ( t ), E τ ( t )} be the Markov chain with speed τ > 0. To prove the limit in the case of τ ↓ 0, let i, x > A i , and ε > 0 be given. Choose t i > 0 be such that ϕ i ( t i , x ) ≥ Ā + ( K i − Ā ) / 2. Assume ( N (0), E (0)) =( x, i ) and let T = inf{ t ≥ 0 : E ( t ) ≠ i }. Choose τ > 0 sufficiently small so that ℙ [ T < t i ] = exp(− τ ∑ j ≠ i λ ij t i ) Ā ] ≥1 − ε . As ε > 0 was arbitrary, lim τ ↓0 a τ ( i, x ) = 1 as claimed. The proof that lim τ ↓0 a τ ( i, x ) = 0 for x < A i follows from a similar argument. To prove the limit in the case of τ ↓ 0,let ϕ * ( t, x ) be the flow of the averaged vector field i.e. t ⟼ ϕ * ( t, x ) is the unique solution to this differential equation with initial condition N (0) = x . Let A * , by assumption, be the unique equilibrium in ( A , Ā ). By continuity f * ( N ) < 0 for N 0 for N ∈ ( A * , K ). Let ( x, i ) ∈ ( A * , Ā ) × {1, 2, … , k } be given. Assume that ( N (0), E (0)) = ( x, i ). Choose T > 0 such that . Benaïm and Strickler (2019a, Lemma A.4) implies that for all δ > 0 Choosing implies lim τ →∞ a τ ( x, i ) = 1 as claimed. The proof that lim τ ↓0 a τ ( i, x ) = 0 for x < A * follows from a similar argument. The next lemma implies that “simple” environments can not result in a demographic mismatch. Lemma 1. Let f ( x, y ) be function such that for all y there exist K ( y ) > A ( y ) > 0 such that f ( A ( y ), y ) = f ( K ( y ), y ) = 0 for all y, f ( x, y ) > 0 for A ( y ) < x < K ( y ), and f ( x, y ) y 1 . As f ( x, y 2 ) ≥ f ( x, y 1 ) > 0 for all x ∈ ( A ( y 1 ), K ( y 1 )), A ( y 2 ) ≤ A ( y 1 ) and K ( y 2 ) ≥ K ( y 1 ). Finally, in the cases where there is no Allee effect or there is an inhospitable environment, (Benaïm and Strickler, 2019b, Theorems 3.1 and 3.2) implies lim t →∞ N ( t ) = 0 whenever ∑ i π i f i (0) 0. R eferences ↵ Ackleh , A.S. , Allen , L.J.S. , and Carter , J. ( 2007 ). Establishing a beachhead: a stochastic population model with an Allee effect applied to species invasion . Theoretical Population Biology , 71 : 290 – 300 . OpenUrl CrossRef PubMed ↵ Allee , W.C. ( 1931 ). Animal aggregations, a study in general sociology . University of Chicago Press , Chicago . ↵ Allen , L. , Fagan , J. , Högnäs , G. , and Fagerholm , H. ( 2005 ). Population extinction in discrete-time stochastic population models with an allee effect . Journal of Difference Equations and Applications , 11 : 273 – 293 . OpenUrl CrossRef Benäim , M. and Lobry , C. ( 2016 ). Lotka–Volterra with randomly fluctuating environments or “how switching between beneficial environments can make survival harder” . Annals of Applied Probability , 26 : 3754 – 3785 . OpenUrl Benäim , M. , Lobry , C. , Sari , T. , and Strickler , É. ( 2023 ). Untangling the role of temporal and spatial variations in persistence of populations . Theoretical Population Biology , 154 : 1 – 26 . OpenUrl CrossRef PubMed Benäim , M. and Strickler , E. ( 2019a ). Random switching between vector fields having a common zero . The Annals of Applied Probability , 29 : 326 – 375 . OpenUrl Benäim , M. and Strickler , E. ( 2019b ). Random switching between vector fields having a common zero . Ann. Appl. Probab ., 29 : 326 – 375 . OpenUrl ↵ Berec , L. , Angulo , E. , and Courchamp , F. ( 2007 ). Multiple allee effects and population management . Trends in Ecology and Evolution , 22 : 185 – 191 . OpenUrl CrossRef ↵ Bernhardt , J.R. , Sunday , J.M. , and O’Connor , M.I. ( 2018 ). Metabolic theory and the temperature-size rule explain the temperature dependence of population carrying capacity . The American Naturalist , 192 : 687 – 697 . OpenUrl CrossRef PubMed ↵ Boukal , D.S. and Berec , L. ( 2002 ). Single-species models of the allee effect: Extinction boundaries, sex ratios and mate encounters . Journal of Theoretical Biology , 218 : 375 – 394 . OpenUrl CrossRef PubMed Web of Science ↵ Courchamp , F. , Berec , L. , and Gascoigne , J. ( 2008 ). Allee effects in ecology and conservation , volume 36 . Cambridge University Press . ↵ Courchamp , F. , Clutton-Brock , T. , and Grenfell , B. ( 1999 ). Inverse density dependence and the Allee effect . Trends in Ecology and Evolution , 14 : 405 – 410 . OpenUrl CrossRef ↵ Davis , M.H.A. ( 1984 ). Piecewise-deterministic Markov processes: a general class of nondiffusion sto-chastic models . J. Roy. Statist. Soc. Ser. B , 46 : 353 – 388 . With discussion. OpenUrl CrossRef ↵ Dennis , B. ( 1989 ). Allee effects: Population growth, critical density, and the chance of extinction . Natural Resources Modeling , 3 : 481 – 538 . OpenUrl CrossRef ↵ Dennis , B. ( 2002 ). Allee effects in stochastic populations . Oikos , 96 : 389 – 401 . OpenUrl CrossRef Web of Science ↵ Dennis , B. , Assas , L. , Elaydi , S. , Kwessi , E. , and Livadiotis , G. ( 2015 ). Allee effects and resilience in stochastic populations . Theoretical Ecology , 9 : 1 – 13 . OpenUrl ↵ d’Onofrio , A. ( 2013 ). Bounded noises in physics, biology, and engineering . Springer . ↵ Fieberg , J. and Ellner , S.P. ( 2001 ). Stochastic matrix models for conservation and management: a comparative review of methods . Ecology Letters , 4 : 244 – 266 . OpenUrl CrossRef Web of Science ↵ Foley , P. ( 1994 ). Predicting extinction times from environmental stochasticity and carrying capacity . Conservation Biology , 8 : 124 – 137 . OpenUrl CrossRef ↵ Gascoigne , J. , Berec , L. , Gregory , S. , and Courchamp , F. ( 2009 ). Dangerously few liaisons: a review of mate-finding allee effects . Population Ecology , 51 : 355 – 372 . OpenUrl CrossRef ↵ Gascoigne , J. and Lipcius , R. ( 2004 ). Allee effects driven by predation . Journal of Applied Ecology , 41 : 801 – 810 . OpenUrl CrossRef ↵ Gil , M. , Baskett , M. , and Schreiber , S. ( 2019 ). Social information drives ecological outcomes among competing species . Ecology , 100 : e02835 . OpenUrl CrossRef PubMed ↵ Gillespie , D.T. ( 1977 ). Exact stochastic simulation of coupled chemical reactions . Journal of Physical Chemistry , 81 : 2340 – 2361 . OpenUrl CrossRef PubMed Web of Science ↵ Hatton , I.A. , McCann , K.S. , Fryxell , J.M. , Davies , T.J. , Smerlak , M. , Sinclair , A.R. , and Loreau , M. ( 2015 ). The predator-prey power law: Biomass scaling across terrestrial and aquatic biomes . Science , 349 : aac6284 . OpenUrl Abstract / FREE Full Text ↵ Hening , A. and Nguyen , D. ( 2018 ). Coexistence and extinction for stochastic Kolmogorov systems . The Annals of Applied Probability , 28 : 1893 – 1942 . OpenUrl ↵ Hening , A. and Nguyen , D.H. ( 2020 ). The competitive exclusion principle in stochastic environments . Journal of Mathematical Biology , 80 : 1323 – 1351 . OpenUrl CrossRef PubMed ↵ Hening , A. , Nguyen , D.H. , and Chesson , P. ( 2021 ). A general theory of coexistence and extinction for stochastic ecological communities . Journal of Mathematical Biology , 82 : 56 . OpenUrl CrossRef PubMed ↵ Hening , A. and Strickler , E. ( 2019 ). On a predator-prey system with random switching that never converges to its equilibrium . SIAM Journal on Mathematical Analysis , 51 : 3625 – 3640 . OpenUrl CrossRef ↵ Hilker , F.M. , Langlais , M. , and Malchow , H. ( 2009 ). The allee effect and infectious diseases: Extinction, multistability, and the (dis-)appearance of oscillations . The American Naturalist , 173 : 72 – 88 . OpenUrl CrossRef PubMed ↵ Kang , Y. and Castillo-Chávez , C. ( 2014 ). A simple epidemiological model for populations in the wild with allee effects and disease-modified fitness . Discrete and Continous Dynamical Systems B , 19 : 89 – 130 . OpenUrl CrossRef ↵ Kaul , R.B. , Kramer , A.M. , Dobbs , F.C. , and Drake , J.M. ( 2016 ). Experimental demonstration of an allee effect in microbial populations . Biology letters , 12 : 20160070 . OpenUrl CrossRef PubMed ↵ Klausmeier , C.A. , Osmond , M.M. , Kremer , C.T. , and Litchman , E. ( 2020 ). Ecological limits to evolutionary rescue . Philosophical Transactions of the Royal Society B , 375 : 20190453 . OpenUrl CrossRef PubMed ↵ Kramer , A.M. , Dennis , B. , Liebhold , A.M. , and Drake , J.M. ( 2009 ). The evidence for Allee effects . Population Ecology , 51 : 341 – 354 . OpenUrl CrossRef ↵ Lande , R. , Engen , S. , and Sæther , B. ( 2003 ). Stochastic population dynamics in ecology and conservation: An introduction . Oxford University Press . ↵ Liebhold , A. and Bascompte , J. ( 2003 ). The Allee effect, stochastic dynamics and the eradication of alien species . Ecology Letters , pages 133 – 140 . ↵ May , R.M. ( 1973 ). Stability and complexity in model ecosystems . Princeton University Press , Princeton, N.J . ↵ McLaughlin , J.F. , Hellmann , J.J. , Boggs , C.L. , and Ehrlich , P.R. ( 2002 ). Climate change hastens population extinctions . Proceedings of the National Academy of Sciences , 99 : 6070 – 6074 . OpenUrl Abstract / FREE Full Text ↵ Monmarché , P. , Schreiber , S.J. , and Strickler , É. ( 2024 ). Impacts of tempo and mode of environmental fluctuations on population growth: Slow-and fast-limit approximations of lyapunov exponents for periodic and random environments . arXiv preprint arxiv:2408.11179. ↵ Nolting , B.C. and Abbott , K.C. ( 2016 ). Balls, cups, and quasi-potentials: quantifying stability in stochastic systems . Ecology , 97 : 850 – 864 . OpenUrl CrossRef PubMed ↵ Noy-Meir , I. ( 1975 ). Stability of grazing systems: an application of predator-prey graphs . Journal of Ecology , 63 : 459 – 481 . OpenUrl CrossRef Web of Science ↵ Perkins , D.M. , Hatton , I.A. , Gauzens , B. , Barnes , A.D. , Ott , D. , Rosenbaum , B. , Vinagre , C. , and Brose , U. ( 2022 ). Consistent predator-prey biomass scaling in complex food webs . Nature Communications , 13 : 4990 . OpenUrl CrossRef PubMed ↵ Pilakouta , N. and Baillet , A. ( 2022 ). Effects of temperature on mating behaviour and mating success: A meta-analysis . Journal of Animal Ecology , 91 : 1642 – 1650 . OpenUrl CrossRef PubMed ↵ Roth , G. and Schreiber , S.J. ( 2014 ). Pushed beyond the brink: Allee effects, environmental stochasticity, and extinction . Journal of Biological Dynamics , 8 : 187 – 205 . OpenUrl CrossRef PubMed ↵ Savage , V.M. , Gillooly , J.F. , Brown , J.H. , West , G.B. , and Charnov , E.L. ( 2004 ). Effects of body size and temperature on population growth . The American Naturalist , 163 : 429 – 441 . OpenUrl CrossRef PubMed Web of Science ↵ Schreiber , S.J. ( 2003 ). Allee effects, chaotic transients, and unexpected extinctions . Theoretical Population Biology , 64 : 201 – 209 . OpenUrl CrossRef PubMed ↵ Schreiber , S.J. , Yamamichi , M. , and Strauss , S.Y. ( 2019 ). When rarity has costs: coexistence under positive frequency-dependence and environmental stochasticity . Ecology , 100 : e02664 . OpenUrl CrossRef PubMed ↵ Silva , J.F. , Raventos , J. , Caswell , H. , and Trevisan , M.C. ( 1991 ). Population responses to fire in a tropical savanna grass, andropogon semiberbis: a matrix model approach . The Journal of Ecology , pages 345 – 355 . ↵ Soetaert , K. , Petzoldt , T. , and Setzer , R.W. ( 2010 ). Solving differential equations in R: Package deSolve . Journal of Statistical Software , 33 : 1 – 25 . OpenUrl CrossRef PubMed ↵ Stacey , P.B. and Taper , M. ( 1992 ). Environmental variation and the persistence of small populations . Ecological Applications , 2 : 18 – 29 . OpenUrl CrossRef PubMed Web of Science ↵ Stephens , P.A. and Sutherland , W.J. ( 1999 ). Conseqeuences of the Allee effect for behavior, ecology, and conservation . Trends in Ecology and Evolution , 14 : 401 – 405 . OpenUrl CrossRef ↵ Stephens , P.A. , Sutherland , W.J. , and Freckleton , R.P. ( 1999 ). What is the Allee effect? Oikos , 87 : 185 – 190 . OpenUrl CrossRef Web of Science ↵ Tobin , P.C. , Berec , L. , and Liebhold , A.M. ( 2011 ). Exploiting Allee effects for managing biological invasions . Ecology Letters , 14 : 615 – 624 . OpenUrl CrossRef PubMed ↵ Vortkamp , I. , Schreiber , S.J. , Hastings , A. , and Hilker , F.M. ( 2020 ). Multiple attractors and long transients in spatially structured populations with an allee effect . Bulletin of mathematical biology , 82 : 1 – 21 . OpenUrl CrossRef ↵ Winter , A.M. , Richter , A. , and Eikeset , A.M. ( 2019 ). Implications of Allee effects for fisheries management in a changing climate: evidence from Atlantic cod . Ecological Applications , 30 . View the discussion thread. Back to top Previous Next Posted April 09, 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 Bounded environmental stochasticity generates secondary Allee thresholds 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. Your Personal Message CAPTCHA This question is for testing whether or not you are a human visitor and to prevent automated spam submissions. Share Bounded environmental stochasticity generates secondary Allee thresholds Sebastian J. Schreiber bioRxiv 2025.04.03.647087; doi: https://doi.org/10.1101/2025.04.03.647087 Share This Article: Copy Citation Tools Bounded environmental stochasticity generates secondary Allee thresholds Sebastian J. Schreiber bioRxiv 2025.04.03.647087; doi: https://doi.org/10.1101/2025.04.03.647087 Citation Manager Formats BibTeX Bookends EasyBib EndNote (tagged) EndNote 8 (xml) Medlars Mendeley Papers RefWorks Tagged Ref Manager RIS Zotero Tweet Widget Facebook Like Google Plus One Subject Area Ecology Subject Areas All Articles Animal Behavior and Cognition (7635) Biochemistry (17690) Bioengineering (13892) Bioinformatics (41935) Biophysics (21451) Cancer Biology (18587) Cell Biology (25499) Clinical Trials (138) Developmental Biology (13377) Ecology (19899) Epidemiology (2067) Evolutionary Biology (24318) Genetics (15609) Genomics (22506) Immunology (17736) Microbiology (40394) Molecular Biology (17181) Neuroscience (88601) Paleontology (666) Pathology (2832) Pharmacology and Toxicology (4824) Physiology (7641) Plant Biology (15152) Scientific Communication and Education (2045) Synthetic Biology (4294) Systems Biology (9825) Zoology (2271)