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When Mariuá, a 1.5-year-old female jaguar, set foot in our breeding centre in Argentina in December 2018, we did not know that she would make history. Two years later, she walked out with two cubs: the first jaguars to roam the 1.4 million hectares of the Iberá wetlands of northeastern Argentina for at least 70 years. Mariuá and her cubs have started to reverse a process that some had thought irreversible.Within decades, one million species out of a total of some eight million could go extinct globally1. Hunting, habitat loss and ecosystem degradation are propelling this unprecedented biodiversity crisis. Current extinction rates are 100 to 1,000 times higher than in the past several million years.Argentina is no exception. Over the past 150 years, 5 bird and 4 mammal species have gone extinct. Today, about 17% of the country’s 3,000 vertebrate species are imperilled2, and 13 out of the 18 extant species of large mammal, from anteaters to tapirs, are experiencing catastrophic declines, in terms of both number and geographical range (see http://cma.sarem.org.ar).In 1998, we started a rewilding programme in Argentina to try to reverse this appalling loss. Our non-profit foundation, Fundación Rewilding Argentina, was spun out from the US non-profit organization Tompkins Conservation. We create protected areas where we can reintroduce native species, re-establish their interactions, restore ecosystem functionality and build valuable ecotourism based on wildlife viewing.Both rewilding and ecotourism can be controversial. We think that our work is an instructive example of how active restoration of crucial species, when done responsibly, can benefit both ecosystems and local people. It should be in the toolkit for meeting the 2030 biodiversity targets that will be discussed at the Convention on Biological Diversity’s Conference of the Parties in Kunming, China, next month.Three stepsThe popularity of rewilding projects is growing. These include: wolves brought back to Yellowstone National Park in Wyoming, beavers to England, bison and musk ox to northern Russia, leopards to Mozambique and Tasmanian devils to mainland Australia. The International Union for Conservation of Nature reports that, since 2008, at least 418 reintroduction projects have been started3. Most of these projects occur in protected areas and involve one or a few species. Our work in Argentina is broader.As a first step, we acquire private lands with philanthropic funds, reintroduce many species and form government-protected areas that are donated to federal and provincial governments. So far, we have purchased and donated about 400,000 hectares, with an estimated market value of US$91 million. This has created and enlarged six national parks, one national reserve and two provincial parks. Another 100,000 hectares are being donated. Together, these lands comprise a little over 10% of the total terrestrial area currently managed by the National Parks Administration of Argentina.The second step is to restore ecosystems, mainly by reintroducing species at an unprecedented scale. We spend more than $3 million each year on rewilding activities in three regions: the Iberá wetlands in the northeast, the dry Chaco forests in the north and the Patagonian steppe and coast in the south. Most often, we work with species deemed to have large impacts at the ecosystem level, such as large predators and herbivores.

Jaguars now roam Argentina’s Iberá wetlands for the first time in more than 70 years.Credit: Matías Rebak

Thus far, we have successfully reintroduced pampas deer, giant anteaters and collared peccaries (a pig-like, hoofed animal). We have also started founding populations of jaguars, coypus (large aquatic rodents), Wolffsohn’s viscachas (rodents that resemble a large chinchilla), red-and-green macaws and bare-faced curassows (birds related to chickens and pheasants). We are currently working on the reintroduction of 14 species.As they become abundant, reintroduced species re-weave the fabric of ecological relationships. For example, jaguars (Panthera onca) and macaws (Ara chloropterus) are reviving a crucial interaction: predation. Jaguars have begun to prey on eight species, including native rodents and feral hogs, which could limit those populations and thus benefit vegetation growth. The macaws are consuming 49 plant species, which could enhance seed dispersal, although this remains to be tested.
Include the true value of nature when rebuilding economies after coronavirus
Third, we invest heavily in infrastructure, capacity building and publicity to create an economy based on ecotourism. The species we work with are often highly charismatic, which benefits local communities, creating an economic incentive to conserve native wildlife and habitats. We organize workshops and courses so that locals can train as nature guides, cooks, craftspeople and more. In Iberá, where our work is most advanced, tourist visits increased by 87% between 2015 and 2021, according to official data from the Iberá wetland management agency. There were more than 50,000 visitors last year, despite the COVID-19 pandemic.All of these steps are important: simply setting aside protected areas is not enough. Globally, most modern ecosystems are ecologically damaged4, even in long-standing protected areas5. In Argentina, for example, functional populations of jaguars are missing from 19 of 22 national parks where historical distribution data suggest this key apex predator should occur.Jaguars and capybarasOur flagship project is the rewilding of the Iberá wetland. There, we are working on the restoration of nine species, including jaguars, which were eradicated from this area more than 70 years ago. We have now established a founding population of eight individuals: one adult male and three adult females, two of which (including Mariuá) were each released with two cubs aged four months. Our goal is to release a total of 20 individuals by 2027.Of all the species we work with, giant otters (Pteronura brasiliensis) and macaws have been the most difficult. Both species are extinct in the wild in Argentina. Bureaucratic hurdles have made sourcing wild individuals from neighbouring countries impossible.We obtained two pairs of giant otters from European zoos, and are holding them in pens in the core of Iberá. After several attempts, one pair bred successfully and the female gave birth to three cubs, producing the first litter born in the country for more than 30 years. We plan to release this family to the wild next year.

This female giant river otter, together with a male and their three cubs, will be released to the wild in Argentina next year to create a founding population.Credit: Matías Rebak

We source macaws, which have been extinct in the wild in Argentina for 100 years, from zoos, wildlife shelters and breeding centres. Because of their captive origin, we must give them the opportunity to practise flying in an aviary. We provide them with native foods, so that they learn what to eat, and we use a remote-controlled stuffed fox to teach them to avoid predators. This training isn’t always successful. Out of the 87 macaws that we have worked with, 48 were healthy and skilled enough to release. Two founding populations now thrive in the wild; one of them began reproducing in 2020.Efforts elsewhere have demonstrated the powerful effects of restoring species. In the northeast Pacific Ocean, reintroduced sea otters (Enhydra lutris) have voraciously eaten sea urchins, which in turn has allowed the return of lush kelp forests6. In Yellowstone Park, some researchers argue that reintroduced wolves have discouraged herbivores from foraging along stream edges, which might have increased tree growth and stabilized stream banks7. In Mozambique’s Gorongosa Park, the return of wildebeest and other large herbivores has curtailed Mimosa pigra, an undesirable invasive shrub8.
Biodiversity needs every tool in the box: use OECMs
Our rewilding work in Argentina could also have profound impacts. Close monitoring of the female jaguars and their cubs in the Iberá wetland has shown that they are largely feeding on the most abundant native prey: capybaras (Hydrochoerus hydrochaeris). Reducing the number of capybaras is expected to allow more vegetation to thrive, providing habitat for arthropods and small vertebrates, and possibly increasing carbon sequestration9. It could also help to reduce the transmission of sarcoptic mange, a density-dependent disease plaguing the capybara population. Jaguars also prey on foxes, which might benefit threatened bird species. We are working with several academic institutions to test how the return of the jaguar is reshaping the ecosystem.Challenges and caveatsAs our rewilding work gained momentum, critics ramped up from different fronts. At first, some were fearful of our policy of acquiring private lands with funds provided largely by foreign philanthropists. Those concerns faded when we began donating the land to federal and provincial governments.Then, ranchers argued that we were taking agricultural land out of production and reintroducing or boosting populations of animals that would conflict with their livestock. For example, in Patagonia, we established several protected areas where pumas (Puma concolor) and guanacos (Lama guanicoe, a relative of the llama) thrive. For almost a century, ranchers have trapped, shot and poisoned these animals, blaming them for killing sheep and competing for forage, respectively. We are conducting research to quantify the impact of pumas and guanacos on livestock, and offering alternative job opportunities based on wildlife viewing.

Red-and-green macaws went extinct in Argentina in the late 1800s. Rewilding efforts that began in 2016 have now established two founding populations in the Iberá wetlands.Credit: Matías Rebak

Federal and state managers, and often academics, argue that some founding populations of reintroduced species are too small and genetically related to create a viable, long-term population. This is true in some cases. But careful releases of unrelated animals can sidestep this issue. Worries about the spread of diseases when translocating individuals is also often invoked as a reason to halt rewilding activities. We implement thorough health checks and rigorous quarantines to decrease the risk of introducing unwanted diseases in the regions where we work.Concerns are sometimes raised about whether reintroduced species will recreate historical conditions, or instead create something new. Rewilding, however, seeks to regenerate and maintain ecological processes and biodiversity, rather than reaching some specific, historical equilibrium10. We think it is preferable to assume the uncertainties in trying to restore ecosystems, rather than accepting their degraded state.
Protect the last of the wild
Another worry is the possible impacts that tourism can have on climate, biodiversity and society — for instance, on water use, aviation emissions, road building and so on. Our strategy is to limit visitor numbers and avoid crowding by constructing multiple access gates on existing dirt roads.There are many policies that hinder rather than help rewilding. In Argentina, the laws that regulate transportation of wildlife species are built on the assumption that such activities always represent a threat to conservation. Wild animals can typically be imported to the country only through an airport in Buenos Aires. Because of this, an animal that could be driven in a truck from Brazil in a few hours must instead fly more than 1,500 kilometres and then be driven all the way back to its release area. Receiving wild animals at another international port, or moving them around within the country, requires special permits that often take months to obtain. Regulations could be altered to ease rewilding efforts while still policing the illegal wildlife trade.Next stepsNature-based tourism has been growing globally at rates of more than 4% per year, particularly in low- and middle-income countries11. Charismatic fauna, including large predators, are becoming increasingly important. In the Brazilian Pantanal, the world’s largest wetland, wildlife viewing — mostly of jaguars — generated an annual revenue of $6.8 million in 2015. This is three times the revenue obtained from traditional cattle ranching in that region12.With about 97% of the planet’s land surface ravaged by humans4, nature is facing its last stand. Urgent measures are needed not only to halt but also to reverse ecosystem and biodiversity loss. The active reintroduction of key species is one powerful way to heal some degraded ecosystems.This daunting task should not fall solely to non-profit organizations that have limited funds and staff, like us. The United Nations launched its Decade on Ecosystem Restoration in June 2021, calling for massive restoration efforts worldwide to heal nature and the climate. To achieve meaningful results at a global scale, rewilding needs the support of many stakeholders and effective international cooperation. Crucially, it requires the active involvement of governments to facilitate, fund and lead restoration efforts.

Nature 603, 225-227 (2022)
doi: https://doi.org/10.1038/d41586-022-00631-4

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The authors declare no competing interests.

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Field experiments: collective structuresWe found that self-assembled Eciton hamatum bridges adaptively adjust in response to shifts in the terrain on which they are built. Detailed methods are included in Methods: Field experiments. Briefly, we moved foraging trails onto an apparatus where we could introduce a terrain gap. We repeatedly changed the size of this gap by first incrementally increasing it to 30 mm, by 1 mm every 30 s, and then incrementally contracting it at the same rate (See Fig. 1, Methods: Field experiments, and Supplementary Movie 1). As the size of the gap was expanded (the period before the dotted line in Fig. 2a, b) both the volume and number of ants increased to mean maximum values of 1080 mm3 (standard error, s.e. 84) and 18.9 ants (s.e. 1.6), respectively. Ants typically began forming a bridge when the gap was ~5 mm. As the gap size was decreased (period after dotted line), volume and the number of ants decreased back to zero as ants left the bridge. These broad dynamics across the ten complete trials were similar (Fig. 2a, b, inset panels and Supplementary Figs. 2, 3). Additionally, bridge volume (Fig. 2a) strongly correlated with the number of ants in the bridge (Fig. 2b), indicating that the density of ants per unit volume in these structures is relatively consistent (Pearson correlation coefficients range from 0.88 to 0.98 across the ten trials, see also Supplementary Fig. 4). Bridges broke and quickly reformed in eight of the ten trials; breaks occurred in both experimental phases, and these broken periods were excluded from analyses. Overall, these results show that bridges adjust dynamically to changing terrain geometry, as stretching the bridges caused them to become larger, with more ants, and contracting bridges caused them to become smaller, with fewer ants.Fig. 1: Experimental procedure and data extraction summary.Experiments were conducted on robust E. hamatum foraging trails, which were moved onto the experimental apparatus while it was closed. a Experimental procedure: The size of the gap was increased by 1 mm every 30 s until the gap reached 30 mm (expansion phase), then decreased at the same rate till no gap remained (the contraction phase). b Field setup: Experiments were recorded from both the side and the top, examples of bridges during each phase of the same trial are shown. c Data extraction: Example images and silhouettes from the maximum size bridge (30 mm) of the same trial as the images of 20 mm bridges shown in panel a. The envelopes of the bridges were extracted at a temporal resolution of 1 s; for each focal second, image frames were averaged over 10 s to remove ants walking on the bridge from the extracted envelopes. Envelopes were automatically extracted using hue-saturation-value (HSV) thresholding, with thresholds checked independently for each trial due to lighting differences. Locations of fixed points on the platform were used to re-scale and combine data from the side and top views into a single coordinate system in which 100 pixels = 1 cm. Estimates of bridge volume, mean cross-sectional area, and relative height of the center of mass were recorded from the extracted envelopes as shown. See Methods: Data extraction and Supplementary Note 1 for additional details of the data extraction process, including additional bridge metrics.Full size imageFig. 2: Changes in collective structures in experiments.a, b Volume and group size of self-assembled bridges: a Estimated volume of collective bridge structures over time for one focal trial (main figure) and three other examples (inset). The dotted vertical line indicates the time when the experiment shifted from the expansion phase (increasing gap size) to the contraction phase (decreasing gap size). Gray shading indicates that the bridge was broken or recovering from a break; result metrics may be inaccurate during these periods and they were, therefore, excluded from analyses. b The number of ants in the bridge structure over time for the same focal trial (main figure) and three other examples (inset). c–f Hysteresis: Trials consistently show hysteresis, with bridge status at a particular gap size differing during the expansion and contraction phases, for volume (c), number of ants (d), mean cross-sectional area (e), and tautness, or the height of the center of mass of the bridge from the side view (f; lower values indicate bridge is hanging lower). c–f Panels show result metrics over gap size for the same focal trial as in panels a and b, as well as for three other examples (inset). Points show individual measurements, taken every second, lines are smoothed LOESS (local regression) for the expansion (orange points, dashed orange line) and contraction (green points, solid green line) phases. The area between the smoothed lines (shaded gray) shows the extent of hysteresis. c, e, f) Points are jittered to improve clarity. a–f See Supplementary Figs. 2, 3, 5–8 for all complete trials.Full size imageHowever, these changes were not symmetric—adjustments in the contraction phase were not the inverse of adjustments in the expansion phase. We found consistent hysteresis in several metrics; for a given gap size, bridges were larger and made up of more individuals during the contraction of the gap than the expansion (Fig. 2c, d; t-test for volume: mean extent of hysteresis = 0.43, 95% CI = 0.29 to 0.58, t = 6.7, df = 9, p  More

Model descriptionThe model involves a discrete time, discrete valued stochastic Markov process. Model variables and parameters are given in Tables 1 and 2 respectively. Both time and the number of individuals of each type are constrained to be integer valued. Death and reproductive mutation are stochastic processes derived from sampling from binomial distributions given by the relevant probabilities. All ensemble results give the 100-replicate average for the parameter choices in question.Growth of individuals from species ({S}_{1}) and ({S}_{2}) is proportional to the bio-available level of environmental substances ({R}_{1}) and ({R}_{2}) respectively. At time (t) (where time is in units of biological generations) the change in the number ({N}_{q,j}) of individuals of genotype (j) (non-producer, producer, plastic) within species (q) (({S}_{1}) or ({S}_{2})) can be written as a function of the state of the variables at the previous time-step:$${N}_{q,j}left(t+1right)=left(left({N}_{q,j}left(tright)-{rho }_{q,j}left(tright)right)cdot {G}_{q,j}left(tright)-{{{{{{rm{{Upsilon }}}}}}}}_{q,j}left(tright)-{delta }_{q,j}left(tright)+{{{{{{rm{{Upsilon }}}}}}}}_{q,xne j}left(tright)right)cdot left(1-frac{{S}_{q}left(tright)}{K}right)$$
(1)
The leftmost bracket on the right-hand side represents the number of individuals escaping starvation (death due to insufficient environmental substance) at the previous time-step and ({G}_{q,j}left(tright)) is the per capita reproductive growth rate. ({{{{{{rm{{Upsilon }}}}}}}}_{q,j}left(tright)) gives the number of mutant offspring individuals produced during reproduction from parent individuals of genotype (j). ({{{{{{rm{{Upsilon }}}}}}}}_{q,xne j}left(tright)) represents the number of (j) genotype individuals derived from mutation in parent individuals of other genotypes. ({delta }_{q,j}left(tright)) is the number of individuals of genotype (j) lost to random death, and the rightmost bracket relates the total number ({S}_{q}left(tright)) of individuals of species (q) to carrying capacity (K), which represents limitation of growth by any factor other than the relevant environmental substance, e.g. space. (The steady state population size in all simulations shown is below (K) and limited by the environmental substance influx. The carrying capacity is included in the model for computational reasons and as a “crash preventer” but has no qualitative effect on the results).The total number of individuals ({S}_{q}left(tright)) in species (q) is the sum of the number of individuals of each genotype (producer, non-producer and plastic, as discussed in the main text):$${S}_{q}left(tright)=mathop{sum }limits_{j=1}^{{j}_{{total}}}{N}_{q,j}left(tright)={N}_{q,{prod}}left(tright)+{N}_{q,{non}-{prod}}left(tright)+{N}_{q,{plast}}left(tright)$$
(2)
The genotype-specific reproductive growth rate ({G}_{q,j}left(tright)) (again for genotype (j) within species (q), time (t)), gives the number of offspring individuals produced per parent individual, per time-step. Growth rate is an increasing function of the bio-available level of environmental substance ({R}_{q,{BIOAVAIL}{ABLE}}) (the subscript (q) being identical because species ({S}_{1}) and ({S}_{2}) assimilate substances ({R}_{1}) and ({R}_{2}) respectively). Growth rate also includes a substance-to-biomass conversion efficiency parameter ({f}_{{conv}}) and a genotype-specific per capita term ({G}_{q,j,{PR}}) (number of offspring per parent, per unit environmental substance assimilated, per unit time). In the absence of growth-limitation by environmental substance levels, growth rate is capped at a genotype-specific maximum ({G}_{q,{jMAX}}):$${G}_{q,j}left(tright)={MIN}[{G}_{q,j,{PR}}cdot {R}_{q,{BIOAVAILABLE}}(t)cdot {f}_{{conv}},{G}_{q,{jMAX}}]$$
(3)
$${G}_{q,{non}-{prod},{PR}}={G}_{0}$$
(4)
$${G}_{q,{non}-{prod},{MAX}}={G}_{0}cdot {R}_{{assimMAX}}$$
(5)
({G}_{0}) is the baseline number of offspring, per parent, per unit substance assimilated. ({R}_{{assimMAX}}) is a universal maximum potential number of units of environmental substance that can be assimilated by a single individual per time-step (i.e. representing basic physiological constraints on growth). The producer genotype incurs a per capita reproductive growth rate cost ({kappa }_{{prod}}) relative to the non-producer:$${G}_{q,{prod},{PR}}={G}_{0}cdot (1-{kappa }_{{prod}})$$
(6)
$${G}_{q,{prod},{MAX}}={G}_{0}cdot (1-{kappa }_{{prod}})cdot {R}_{{assimMAX}}$$
(7)
This growth rate formulation is therefore a highly simplified linearization of the Michaelis-Menten kinetics normally used in models of resource and nutrient assimilation.The plastic genotype switches phenotype depending upon the level of environmental substance relative to a fixed threshold ({{R}_{q,{BIOAVAILABLE}}}_{{crit}}), in effect becoming a second non-producer genotype below this threshold and a second producer genotype above it:$${IF}[{R}_{q,{BIOAVAILABLE}}(t)ge {{R}_{q,{BIOAVAILABLE}}}_{{crit}}],{G}_{q,{plast}}left(tright)={G}_{q,{prod}}left(tright)$$$${ELSEIF}[{R}_{q,{BIOAVAILA}{BLE}}left(tright) , < , {{R}_{q,{BIOAVAILABLE}}}_{{crit}}],{G}_{q,{plast}}left(tright)={G}_{q,{non}-{prod}}left(tright)$$ (8) There is no spatial structure whatsoever, thus access to environmental substance is uniform across individuals. The bioavailable quantity of each environmental substance is simply the total amount ({R}_{q,{NET}}(t)) divided by the total number of individuals assimilating it:$${R}_{q,{BIOAVAILABLE}}left(tright)=frac{{R}_{q,{NET}}(t)}{{S}_{q}left(tright)}$$ (9) We allow the per capita reproductive growth rate to fall below ({G}_{q,j}left(tright)=1), which, if interpreted deterministically at the individual level would correspond to an individual failing to sustain its biomass to the next time-step and thus dying. However, a population-level average ({G}_{q,j}left(tright) , < , 1) is interpretable in terms of a thinning factor that maps between discretized individuals and continuously distributed environmental substance. Thus, a thinning factor of (left(1-{G}_{q,j}left(tright)right)) is used to calculate the total number of individuals dying of starvation ({rho }_{q,j}) (again genotype (j), species (q)). This represents pre-reproduction deaths, corresponding to the difference between the actual population size and the population size that the environmental substance pool is capable of supporting. ({rho }_{q,j}left(tright)) is constrained to be an integer and is zero for ({G}_{q,j}left(tright) , > , 1):$${rho }_{q,j}left(tright)={N}_{q,j}left(tright)cdot {MAX}left[0,left(1-{G}_{q,j}left(tright)right)right]$$
(10)
A subset of offspring are a different genotype from their parent via mutation. For parent genotype (j), the number ({{{{{{rm{{Upsilon }}}}}}}}_{q,j}left(tright)) of mutant offspring with genotype (ne j) is calculated using baseline mutation probability per reproductive event ({mu }_{0}), with the total number of new individuals produced by the parent individuals surviving starvation ({G}_{q,j}left(tright)cdot left({N}_{q,j}left(tright)-{rho }_{q,j}left(tright)right)). The total number of mutant offspring ({{{{{{rm{{Upsilon }}}}}}}}_{q,j}left(tright)) is thus a binomially distributed random variable with success probability ({mu }_{0}) and number of trials ({G}_{q,j}left(tright)cdot left({N}_{q,j}left(tright)-{rho }_{q,j}left(tright)right)). The expected value (Eleft[{{{{{{rm{{Upsilon }}}}}}}}_{q,j}left(tright)right]) is the product of these two numbers:$$ {{{{{{rm{{Upsilon }}}}}}}}_{q,j}left(tright) sim Bleft({G}_{q,j}left(tright)cdot left({N}_{q,j}left(tright)-{rho }_{q,j}left(tright)right),{mu }_{0}right), \ Eleft[{{{{{{rm{{Upsilon }}}}}}}}_{q,j}left(tright)right]={G}_{q,j}left(tright)cdot left({N}_{q,j}left(tright)-{rho }_{q,j}left(tright)right)cdot {mu }_{0}$$
(11)
Mutation to genotype (j) from the other genotypes is calculated in exactly the same way using the number and reproductive growth rates of the relevant (other) genotypes. Any particular mutant offspring is randomly allocated to one of the other genotypes with equal probability ({p}_{kto j}=frac{1}{{j}_{{total}}-1}=0.5) (where ({j}_{{total}}=3) is the total number of genotypes per species). The expected number of offspring with genotype (j) produced by mutation within parent offspring of other genotypes (kne j) is therefore:$$E[{{{{{{rm{{Upsilon }}}}}}}}_{q,x , ne , j}left(tright)]={left(mathop{sum }limits_{k=1}^{{k}_{{to}{tal}}}{{{{{{rm{{Upsilon }}}}}}}}_{q,k}left(tright)right)}_{kne j}cdot frac{1}{{j}_{{total}}-1}$$
(12)
Independently of reproduction and assimilation of environmental substance, any given individual has a probability ({delta }_{0}) at each time point of death due to stochastic factors. The genotype/species specific number of such deaths is again a random sample from a binomial distribution, with success probability ({delta }_{0}):$${delta }_{q,j}left(tright) sim Bleft({N}_{q,j}left(tright),{delta }_{0}right),Eleft[{delta }_{q,j}left(tright)right]={N}_{q,j}left(tright)cdot {delta }_{0}$$
(13)
The net quantity of growth-limiting environmental substance at each time-step is given by the difference between total biotic assimilation ({A}_{{R}_{q}}) and the production ({P}_{{R}_{q}}) and abiotic input ({varphi }_{{R}_{q}}) fluxes:$${R}_{q,{NET}}(t+1)={varphi }_{{R}_{q}}(t)+{P}_{{R}_{q}}(t)-{A}_{{R}_{q}}(t)$$
(14)
The abiotic net influx is the sum of two fluxes. First, an input term that is the product of a baseline scaling factor ({{varphi }_{0}}_{{R}_{q}}) and a model forcing (frac{partial {t}_{{geo}}}{partial {t}_{{bio}}}) representing the mapping between abiotic-geological and biotic-evolutionary timescales. In practice (frac{partial {t}_{{geo}}}{partial {t}_{{bio}}}(t)) was set to either (1) or (0) or (in fluctuation runs) a time-dependent switching between the two. (More sophisticated implementations of (frac{partial {t}_{{geo}}}{partial {t}_{{bio}}}(t)), e.g. sinusoidal oscillations and stochastic time dependence, were attempted but made little qualitative difference to the results). Second, an abiotic removal term that scales linearly with the quantity of environmental substance:$${varphi }_{{R}_{q}}(t)={{varphi }_{0}}_{{R}_{q}}cdot frac{partial {t}_{{geo}}}{partial {t}_{{bio}}}left(tright)-frac{{R}_{q,{NET}(t)}}{{R}_{q,{NET}0}}$$
(15)
where ({R}_{q,{NET}0}) is a normalization factor representing the sensitivity of the abiotic efflux to the influx. In the absence of any biota and for (frac{partial {t}_{{geo}}}{partial {t}_{{bio}}}=1) the steady state environmental substance level is immediately given by (15) as ({R}_{q,{NET}(t)}={R}_{q,{NET}0}cdot {{varphi }_{0}}_{{R}_{q}}), thus the numerical value of ({R}_{q,{NET}0}) corresponds to the abiotic steady state residence time.Total biotic assimilation ({A}_{R}) of each environmental substance is given by:$${A}_{{R}_{q}}left(tright)=mathop{sum }limits_{k=1}^{{j}_{{total}}}frac{{G}_{q,k}left(tright)cdot left({N}_{q,k}left(tright)-{rho }_{q,k}left(tright)right)}{{G}_{q,{kPR}}}$$
(16)
The numerator gives the total number of individuals produced as a result of biological assimilation of environmental substance ({R}_{q}) and the denominator is the genotype specific number of individuals produced per unit substance assimilated, dividing through by which therefore converts to total units of substance assimilated by the population as a whole.Net biotic ({P}_{{R}_{q}{NET}}) production of substance ({R}_{q}) by the producer genotype in the other species (p) is calculated equivalently, via the product of the per capita production rate ({P}_{q,{prod}}) and the total number of reproducing individuals:$${P}_{q,{prod}}left(tright)=frac{{MIN}[{G}_{q,{prod}}left(tright)cdot {f}_{{convprod}},{G}_{q,{prodMAX}}]}{{G}_{p,{PRODUCER;PR}}}$$
(17)
$${P}_{{R}_{q}{NET}}left(tright)={P}_{q,{prod}}left(tright)cdot left({N}_{p,{PRODUCER}}left(tright)-{rho }_{p,P{RODUCER}}left(tright)right)$$
(18)
Where ({f}_{{conv},{PROD}}) is the per capita efficiency by which producers convert the environmental substance that they assimilate into the by-product they produce (note that the equivalent conversion efficiency for assimilation ({f}_{{conv}}) already appears in the growth functions of each genotype, therefore does not appear in Eq. (17)).The residence time ({T}_{{R}_{q}})of each environmental substance is given by the net quantity of this substance divided by the influx, to give units of the average number of biological generations a unit of environmental substance spends in the relevant pool before being removed. In those simulations in which the abiotic influx ({varphi }_{{R}_{q}}(t)) was set to zero (i.e. during the shut-off intervals) production ({P}_{{R}_{q}{NE}T}left(tright)) was used as an alternative denominator:$${IF}left[{varphi }_{{R}_{q}}left(tright) , > , 0,{T}_{{R}_{q}}left(tright)=frac{{R}_{q,{NET}}left(tright)}{{varphi }_{{R}_{q}}left(tright)}right]!,{IF}left[left(left({varphi }_{{R}_{q}}left(tright)=0right){& }left({P}_{{R}_{q}{NET}}left(tright) , > ,0right)right)!,{T}_{{R}_{q}}left(tright)=frac{{R}_{q,{NET}}left(tright)}{{P}_{{R}_{q}{NET}}left(tright)}right]{ELSE}[{T}_{{R}_{q}}left(tright)=0]$$
(19)
The cycling ratio ({{CR}}_{{R}_{q}}) of each substance is given by the ratio between net biotic assimilation of that substance ({A}_{{R}_{q}}left(tright)) and the abiotic influx of that substance ({varphi }_{{R}_{q}}left(tright)). As with the residence time, when the abiotic influx was zero, the input from biological production was used as an alternative denominator:$${IF}left[{varphi }_{{R}_{q}}left(tright) , > , 0,{{CR}}_{{R}_{q}}left(tright)=frac{{A}_{{R}_{q}}left(tright)}{{varphi }_{{R}_{q}}left(tright)}right]{IF}left[left(left({varphi }_{{R}_{q}}left(tright)=0right){{& }}left({P}_{{R}_{q}{NET}}left(tright) , > ,0right)right),{{CR}}_{{R}_{q}}left(tright)=frac{{A}_{{R}_{q}}left(tright)}{{P}_{{R}_{q}{NET}}left(tright)}right]{ELSE}[{{CR}}_{{R}_{q}}left(tright)=0]$$
(20)
Deterministic approximation to steady stateAssume that at steady state substance assimilation will reach a maximal state such that the level of environmental substance is limiting to population size. Assume that such a state is below the level ({{R}_{q,{BIOAVAILABLE}}}_{{crit}}) at which the plastic genotype effectively becomes a second non-producer genotype and can thus be subsumed into non-producer frequency, such that (2) becomes ({S}_{q}left(tright)=mathop{sum }nolimits_{j=1}^{{j}_{{total}}}{N}_{q,j}left(tright)={N}_{q,{prod}}left(tright)+{N}_{q,{non}-{prod}}left(tright)). Assume that there are non-zero starvations at each time-step for all genotypes, which implies growth rate ({G}_{q,j}left(tright) , < , 1,ll {G}_{q,{jMAX}},forall j,q), which gives by (3)({G}_{q,j}left(tright)={G}_{q,j,{PR}}cdot {R}_{q,{BIOAVAILABLE}}left(tright)cdot {f}_{{conv}}). Substituting this into (10), then the first bracketed term in (1), then labeling the post-starvation number of individuals as ({({N}_{q,j}left(tright))}_{{NET}}):$${({N}_{q,j}left(tright))}_{{NET}}=left({N}_{q,j}left(tright)-{rho }_{q,j}left(tright)right)={N}_{q,j}left(tright)cdot left(1-left(1-{G}_{q,j}left(tright)right)right)={N}_{q,j}left(tright)cdot {G}_{q,j}left(tright)$$ (21) Approximate (14) deterministically by a fixed fractional parameter corresponding to the baseline random death rate:$${delta }_{q,j}left(tright)approx {N}_{q,j}left(tright)cdot {delta }_{0}$$ (22) Doing the same for mutation:$${{{{{{rm{{Upsilon }}}}}}}}_{q,j}left(tright)={G}_{q,j}left(tright)cdot {left({N}_{q,j}left(tright)right)}_{{NET}}cdot {mu }_{0}={N}_{q,j}left(tright)cdot {{G}_{q,j}left(tright)}^{2}cdot {mu }_{0}$$ (23) The term in (12) for mutation to (j) from other genotypes simplifies to$${{{{{{rm{{Upsilon }}}}}}}}_{q,x ,ne , j}left(tright)={sum }_{k,=,1}^{{j}_{{total}}}frac{{G}_{q,k}left(tright)cdot {left({N}_{q,k}left(tright)right)}_{{NET}}cdot {mu }_{0}}{{j}_{{total}}-1}={N}_{q,k}left(tright)cdot {{G}_{q,k}left(tright)}^{2}cdot {mu }_{0}$$Substituting Eqs. (21–23) into (1):$${N}_{q,j}left(tright)cdot {{G}_{q,j}left(tright)}^{2}cdot (1-{mu }_{0})-{N}_{q,j}left(tright)cdot {delta }_{0}+{N}_{q,k}left(tright)cdot {{G}_{q,k}left(tright)}^{2}cdot {mu }_{0}=0$$ (24) Noting that by Eqs. (3)–(5) combined with the above assumptions, the growth rate of the producer can be written as:$${G}_{q,{prod}}left(tright)={G}_{q,{cheat}}left(tright)cdot (1-{kappa }_{{prod},q})$$ (25) Writing (24) explicitly for each genotype:$${N}_{q,{non}-{prod}}left(tright)cdot {{G}_{q,{non}-{prod}}left(tright)}^{2}cdot (1-{mu }_{0})-{N}_{q,{non}-{prod}}left(tright)cdot {delta }_{0}+{N}_{q,{prod}}left(tright)cdot {{G}_{q,{prod}}left(tright)}^{2}cdot {mu }_{0}=0$$$${N}_{q,{prod}}left(tright)cdot {{G}_{q,{prod}}left(tright)}^{2}cdot (1-{mu }_{0})-{N}_{q,{prod}}left(tright)cdot {delta }_{0}+{N}_{q,{non}-{prod}}left(tright)cdot {{G}_{q,{non}-{prod}}left(tright)}^{2}cdot {mu }_{0}=0$$Adding:$${N}_{q,{non}-{prod}}left(tright)cdot {{G}_{q,{non}-{prod}}left(tright)}^{2}cdot left(1-{mu }_{0}right)-{N}_{q,{non}-{prod}}left(tright)cdot {delta }_{0}$$$$+{N}_{q,{prod}}left(tright)cdot {left({G}_{q,{non}-{prod}}left(tright)cdot left(1-{kappa }_{{prod},q}right)right)}^{2}cdot {mu }_{0}+{N}_{q,{prod}}left(tright)cdot {left({G}_{q,{non}-{prod}}left(tright)cdot left(1-{kappa }_{{prod},q}right)right)}^{2}cdot left(1-{mu }_{0}right)$$$$-{N}_{q,{prod}}left(tright)cdot {delta }_{0}+{N}_{q,{non}-{prod}}left(tright)cdot {{G}_{q,{non}-{prod}}left(tright)}^{2}cdot {mu }_{0}=0$$Because the mutation terms cancel:$$left({N}_{q,{non}-{prod}}left(tright)+{N}_{q,{prod}}left(tright)cdot {left(1-{kappa }_{{prod},q}right)}^{2}right)cdot left({{G}_{q,{non}-{prod}}left(tright)}^{2}-{delta }_{0}right)=0$$ (26) Substituting in for the growth rate terms (3–7) gives:$$left({N}_{q,{non}-{prod}}(t)+{N}_{q,{prod}}(t)cdot {left(1-{kappa }_{{prod},q}right)}^{2}right)cdot left({left({G}_{0}cdot {R}_{q,{BIOAVAILABLE}}(t)cdot {f}_{{conv}}right)}^{2}-{delta }_{0}right)=0$$ (27) By (16), (4), (6) and the above, total steady state assimilation of the growth limiting environmental substance by species (q) is:$${A}_{{R}_{q}}left(tright)=mathop{sum }limits_{k=1}^{{j}_{{total}}}frac{{G}_{q,k}left(tright)cdot left({N}_{q,k}left(tright)-{rho }_{q,k}left(tright)right)}{{G}_{q,{kPR}}}$$$$kern2.4pc=frac{{N}_{q,{non}-{prod}}left(tright)cdot {left({G}_{q,{non}-{prod}}left(tright)right)}^{2}}{{G}_{0}}+frac{left({N}_{q,{prod}}left(tright)cdot {left(1-{kappa }_{{prod},q}right)}^{2}right)cdot {left({G}_{q,{non}-{prod}}left(tright)right)}^{2}}{{G}_{0}cdot left(1-{kappa }_{{prod}}right)}$$$$=left({N}_{q,{non}-{prod}}left(tright)+{N}_{q,{prod}}left(tright)cdot left(1-{kappa }_{{prod},q}right)right)cdot {{G}_{0}cdot ({R}_{q,{BIOAVAILABLE}}left(tright)cdot {f}_{{conv}})}^{2}$$ (28) By (16–18), production of this substance by the producer allele in the other species (p , ne , q), assuming the various arguments above simultaneously apply to this species, is:$${P}_{{R}_{q}}left(tright)= frac{{G}_{p,{prod}}left(tright)cdot left({N}_{p,{prod}}left(tright)-{rho }_{p,{PRODUCER}}left(tright)right)}{{G}_{p,{prodPR}}}cdot {f}_{{conv},{PROD}}\ = {N}_{p,{prod}}left(tright)cdot left(1-{kappa }_{{prod},p}right) cdot {G}_{0}cdot {({R}_{p,{BIOAVAILABLE}}left(tright)cdot {f}_{{conv}})}^{2}cdot {f}_{{conv},{PROD}}$$ (29) Balance between input and output fluxes of each environmental substance requires ({varphi }_{{R}_{q}}left(tright)+{P}_{{R}_{q}}left(tright)={A}_{{R}_{q}}left(tright)), meaning that by substituting in ({A}_{{R}_{q}}left(tright)) from (28) it is possible to solve for bioavailable substance level, then substitute in the production flux of this substance derived from the producer allele in the other species (p,ne, q):$${R}_{q,{BIO}{AVAILABLE}}left(tright) =frac{1}{{f}!_{{conv}}}sqrt{frac{{varphi }_{{R}_{q}}left(tright)+{P}_{{R}_{q}}left(tright)}{left({N}_{q,{non}-{prod}}left(tright)+{N}_{q,{prod}}left(tright)cdot left(1-{kappa }_{{prod},q}right)right)cdot {G}_{0}}}\ =frac{1}{{f}!_{{conv}}}sqrt{frac{{varphi }_{{R}_{q}}left(tright)+{N}_{p,{prod}}left(tright)cdot left(1-{kappa }_{{prod},p}right)cdot {G}_{0}cdot {({R}_{p,{BIOAVAILABLE}}left(tright)cdot {f}_{{conv}})}^{2}cdot {f}_{{conv},{PROD}}}{left({N}_{q,{non}-{prod}}left(tright)+{N}_{q,{prod}}left(tright)cdot left(1-{kappa }_{{prod},q}right)right)cdot {G}_{0}}}$$ (30) Substituting this into (27) gives a symmetrical condition for steady state genotype frequencies and substance levels across the system:$$ left({N}_{q,{non}-{prod}}left(tright)+{N}_{q,{prod}}left(tright)cdot {left(1-{kappa }_{{prod},q}right)}^{2}right)cdot\ left(frac{{varphi }_{{R}_{q}}left(tright)+{N}_{p,{prod}}left(tright)cdot left(1-{kappa }_{{prod},p}right)cdot {G}_{0}cdot {({R}_{p,{BIOAVAILABLE}}left(tright)cdot {f}_{{conv}})}^{2}cdot {f}_{{conv},{PROD}}}{left({N}_{q,{non}-{prod}}left(tright)+{N}_{q,{prod}}left(tright)cdot left(1-{kappa }_{{prod},q}right)right)cdot {G}_{0}}-{delta }_{0}right)=0$$ (31) This solution illustrates the intuitive ideas that growth and reproduction balance random death at steady state and that the associated producer frequency is lower than that of the non-producer by a factor of the cost. (This factor is of second order because the growth rate is used both directly and (by (10)) in the calculation of starvations). Because our model is a discrete stochastic process, (31) can be viewed as an approximation to a steady state condition, subject to the above assumptions combined with the continuous generation of producers by mutation at a sufficient rate to preclude their extinction. The key point is that over long timescales in the finite populations with which we deal, organism-level selection unavoidably favors the non-producer, with no possibility for multi-level fecundity selection. The producer’s stable presence is thus attributable to the combination of mutation and cycle-level selection. More