Philippa Kaur

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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

Continental, regional, and local spatiotemporal patterns of carbon lossFor Africa’s primary tropical humid forest, carbon losses due to forest disturbances reached 42.2 ± 5.1 MtC yr−1 (mean ± standard deviation, where MtC yr−1 is one million metric tons of carbon loss per year) in 2019 and 53.4 ± 6.5 MtC yr−1 in 2020. Just 9 countries out of the 23 analyzed accounted for 95.0% of total gross losses in 2019 and 94.3% in 2020. These countries contain about 95.7% of all primary tropical humid forests of Africa, with the DRC accounting for 52.8%, Gabon 11.8%, the Republic of the Congo 11.0%, and Cameroon 9.8%. Of these, DRC and Cameroon were responsible for 49.3% and 19.1% of losses in 2019 and 44.7% and 20.6% in 2020. DRC and Cameroon had an annual increase of 15.0% and 36.5% respectively, between 2019 and 2020. From countries with at least 1 MtC emitted in the two years analyzed, Madagascar had the highest annual increase in carbon loss (+153.9%), while Equatorial Guinea is the only country with a decrease in carbon loss (−20.1%). Extending the carbon loss analysis for both past and future will help to better understand these variations and whether the COVID-19 global pandemic had any influence on the general increase between 2019 and 202019. While the absolute numbers for carbon loss estimates should be treated carefully and a sample-based approach should be preferred for an unbiased estimate of absolute numbers20, we focused our analysis on the trends of carbon loss at the continental, country, and local scale (Fig. 1 and Supplementary Fig. 1).Fig. 1: Carbon loss across Africa’s rainforests.We analyzed 23 countries containing primary moist forest. The aboveground carbon stock (green palette) underlies the carbon loss estimations (red palette). Several hotspots can be seen across these regions. The uncertainties of the carbon loss estimations are expressed as standard deviations and shown in Supplementary Fig. 1.Full size imageThe high temporal detail of the analysis revealed various monthly patterns of carbon losses for countries, highly related to local rainfall patterns18 (Fig. 2). Countries like Cameroon, Liberia, Nigeria, Central African Republic (CAR), and Madagascar showed a clear dry-wet seasonal variation in carbon loss per year, while the Republic of the Congo and the DRC, due to their latitudinal extent, exhibited two dry-wet season variations per year with varying intensities (Fig. 2). The seasonal variation can be explained by higher accessibility to forests during the dry months when activities related to smallholder agriculture and logging are more feasible than in the wet season when many roads become inaccessible.Fig. 2: Temporal patterns of carbon loss for the top 10 countries.We show monthly statistics for 2019 and 2020 and the associated uncertainty (black lines). We separate between high (red bars) and low (yellow bars) confidence alerts, the latter showing up for the last 3 months of 2020.Full size imageOne of the highest differences between the months with the most and the least carbon losses was found for Madagascar (72 times more carbon loss in March compared to November 2019). In CAR, the three consecutive months with the highest cumulative carbon loss (January to March 2020) contributed to 75.7% of the total annual loss (between February and April 2020), in Nigeria 73.9% (January to March 2020), Liberia 73.1% (February to April 2020), Madagascar 70.7% (September to November 2020), and Cameroon 62.2% (January to March 2020). Lower percentages were found for countries with mixed seasonality and patterns, like DRC 36.7% (January to March 2020), and the Republic of the Congo 32.8% (January to March 2020) (Fig. 2). For the latter two countries, we expect better-defined peaks of carbon loss at local scales, where climatic conditions are not mixed. The annual cumulative carbon loss (%) per country (Fig. 3) showed that Liberia, Nigeria, CAR, and Cameroon reached between 70-90% of their annual carbon loss in April, while Madagascar reached 60% in October. The DRC, Gabon, Republic of the Congo, Equatorial Guinea, and Ghana have a more gradual monthly increase of cumulative carbon loss with less contrasting seasonality effects. Monthly patterns of carbon losses between the two years analyzed resulted in a correlation coefficient of 0.94 for the CAR, 0.92 for the DRC, 0.91 for Madagascar, 0.90 for Gabon, and 0.83 for Cameroon (Supplementary Fig. 2). For the Republic of the Congo, the two years correlated 0.51. Knowing the peak months of carbon loss for each country and that these patterns are repeatable from one year to another can contribute to better target and prioritize enforcement activities, as well as predicting future patterns and early reporting of annual forest carbon losses.Fig. 3: Annual cumulative carbon loss (%) for both years analyzed, 2019 and 2020.Africa’s total cumulative carbon loss is shown with a black line. The 10 topmost emitting countries out of 23 countries analyzed are shown and represented by distinct colored lines.Full size imageSeveral hotspots of carbon losses can be seen in Fig. 1. The high spatial and temporal details of our analysis are shown in Fig. 4, where several local examples with different drivers of forest disturbances are shown, like logging roads, selective logging, mining, oil palm plantations, urban expansion, and small-holder agriculture. This kind of information, coupled with auxiliary datasets (e.g., legal concessions, protected areas) can identify the legality of forest disturbance21.Fig. 4: Local examples of approx. 10 × 10 km in extent showing different spatiotemporal patterns and drivers of carbon loss.The first column shows the carbon loss, the second column the associated uncertainty, the third column the day-of-the-year when the loss occurred, and the last column shows the monthly distribution of carbon loss and associated uncertainty for each local example. The center coordinates of each location are shown in the third column as latitude and longitude. Exact locations are shown in Supplementary Fig. 3. a Logging roads and selective logging in the Central African Republic, b mining of gold and titanium in the Republic of the Congo, c development of an oil palm plantation in Cameroon, d forest disturbance related to building a new capital city in Equatorial Guinea, and e small-scale agriculture expansion at the edge of the forest in the DRC.Full size imageImplications of rapid monitoring of local carbon lossNear-real time alerts combined with biomass maps result in spatially explicit forest carbon loss, unlike global tabular statistics of national data22,23. We provide new insights into the spatiotemporal dynamics of carbon loss with consistent assessment of accuracy that could enable transparency and completeness for countries reporting on their REDD + progress to the UNFCCC24. We provide monthly carbon loss estimates that could play a key role in local, national, and international forest initiatives for global carbon policy goals25. Such a system can be implemented with minimal costs and is based on open-source datasets and Google Earth Engine cloud computing platform26, thus enabling cost-effective national monitoring of forest carbon loss7. Providing rapid reporting on the location, time, and amount of carbon lost across Africa’s primary humid forest will help undertake immediate action to protect and conserve carbon-rich threatened forests. Furthermore, countries will be able to predict and estimate their annual carbon loss before a reporting period ends, thus having the opportunity to adjust their practices to meet their country-specific commitments for climate change mitigation initiatives.Limitations and future improvementsWe used the RADD alerts (Radar for Detecting Deforestation)18 with a minimum mapping unit (MMU) of 0.2 ha as accuracy estimates were available for this MMU. Events smaller than 0.2 ha would add to the total carbon loss but are by nature associated with higher uncertainties18. The implications of the RADD alerts using a global humid tropical forest product as a forest baseline for 201816,27,28 are twofold. First, the global nature of this product might result in inconsistencies at the local level18. Second, because the forest cover loss information used to generate the forest baseline is based on optical Landsat data, persistent cloud cover in the second half of 2018 in some areas led to missed reporting of forest disturbances, thus being detected at the beginning of 2019 by the RADD alerts. This possible overestimation of carbon loss at the start of 2019 is not an issue for a near-real-time alerting system since later months are not affected. Furthermore, the alerts do not distinguish between human-induced disturbances and natural forest disturbances18. When a new forest disturbance alert is detected, it will be confirmed or rejected within 90 days by subsequent Sentinel-1 images18. That is why our carbon loss reporting separates between high and low confidence alerts for the last three months of 2020, which is common for most forest disturbance alerting products18,29. We separated all the alerts into core and boundary pixels. Core alerts represent complete tree cover removal and we assumed complete carbon loss within a pixel. For boundary alerts, we assumed a 50% carbon loss since these mainly represent forest disturbances with partial tree cover removal. Detecting and quantifying the level of degradation remains challenging and future developments will minimize this uncertainty by providing variable percentages of degraded forest30. The timeliness and spatial details of future forest disturbance alerting products will improve with the availability of open access long-wavelength radar data from near-future satellite missions (e.g., NISAR L-band SAR in 202331), by using a combination of optical and radar forest disturbance alert products, and integration with high-resolution satellite products.We relied on an aboveground biomass baseline map from 201832, prior to RADD alerts starting from 2019. Biomass estimation for the tropical moist forests is based on ALOS-2 PALSAR-2 L-band satellite and its usage needs to account for the local biases, especially underestimating AGB values higher than 250 Mg ha−1 (ref. 32). Although we reduced this underestimation by adjusting the AGB map based on ground field data, more research is needed on providing up-to-date high-resolution aboveground carbon estimates33 that could further increase the accuracy of local carbon loss estimation. Radar-based estimation of forest carbon stocks is challenging over mountainous terrain and is less accurate in complex canopies3 and future integration of radar and optical satellite data will provide more robust estimates33. Nevertheless, new spaceborne missions (e.g., GEDI34, BIOMASS35) will provide an unprecedented amount of forest structure samples that will improve the algorithms and thus the final accuracy of aboveground biomass estimates.We focused on exploring and analyzing local carbon losses and showing high temporal and spatial patterns of carbon losses. We showed the country statistics to emphasize the temporal dynamics of carbon losses and compare the temporal profiles across our study region. Our approach was not to provide stratified area estimations36 associated with forest disturbances but we used this concept in the sense that we had a stratified sample of higher quality reference data18 to estimate the omission and commission errors and consider those in our uncertainty estimation on the pixel level. The analysis showed that omission and commission errors are small and rather balanced, and thus do not result in a major area bias for the forest disturbances. The uncertainties of the aboveground biomass product32 were adjusted for known regional biases using regional forest biomass plot data sources. With this approach, the original aboveground biomass map bias was partly corrected using a model-based approach deemed to be an alternative to a sample-based approach whenever country data are unavailable37. Our uncertainty analysis and error reduction showed that we expect only minor bias in the forest disturbance and the biomass data and the remaining uncertainties are propagated in our pixel-based uncertainty layer. More

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doi: https://doi.org/10.1038/d41586-022-00592-8

ReferencesMonk, J. D. et al. Ecol. Lett. https://doi.org/10.1111/ele.13983 (2022).PubMed 
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