Growth, Death, and Resource Competition in Sessile Organisms

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APPLIED ECOLOGY MATHEMATICS sometimes substantial deviations from predictions (39, 42). Here, we present a minimal model to account for these missing factors. We start with allometric scaling theory of forest growth in section 1 and connect deviations from metabolic scaling theory to organism density, resource variability, and competitive interactions in section 2. We explore the implications of competition through space in section 3 and time in section 4, concluding with section 5. Although we explicitly develop our framework using the language of forests, referring, for instance, to individuals as trees and dimensions as stem radii, our formulation is straightforwardly generalizable to other sessile organisms (62). As an example, we extend our model beyond allometric assumptions to consider the emergence of spatial order (SI Appendix, section D). 1. A Size-Class Model for Population Growth Fig. 1. Regular to random spatial distributions and transient to slow tem- The fractal structure of a forest exists both at the level of poral evolution in sessile organisms. (Upper) Trees in Alaskan rainforest the physical branching of individual trees and at the level of (circles indicate basal stem diameter of > 2.5 cm increased by a factor of self-similar packing of differently sized individuals. This fractal five) (5), view of the Panamian rainforest canopy, semiregularly packed ter- structure reflects energetic constraints that have shaped the long- mite mounds reprinted from ref. 3 (empty circles are inactive mounds), and term evolutionary dynamics of forest life (33, 50, 63). Connecting hexagonally packed fairy circles reprinted from ref. 1. (Lower) Newly built energy expenditure with the physical limits of how vasculature ant nests (4), termite mounds with size shown by circles (2), and peren- nial agricultural crops. Dynamics range from transience dominated, in the distributes energy leads to an allometric scaling theory of growth. case of crops razed at the end of the season or newly built ant nests that For the rate of basal stem radius growth, r˙, we have (40, 48) die within days as indicated by open circles, to long-lasting structures such as fairy circles, which can live individually for decades or forests at demo- 3 1−b b graphic equilibrium lasting millennia. Scale is unavailable for fairy circles, r˙(r) ≈ cm ar¯ . [1] but they range from 2 to 12 m in diameter, meaning that the shown plot 8 covers some hundreds of meters on a side (1). Panamanian rainforest image credit: Christian Ziegler (photographer). Termite mounds image reprinted For sufficiently long times, r ∼ t 1/(1−b) for time t. Eq. 1 from ref. 3, which is licensed under CC BY 4.0. Fairy circles image reprinted expresses the general principle of biomass production in terms of from ref. 1, which is licensed under CC BY 4.0. Solenopsis invicta nests a constant determined from biological energetics a¯, how radius image reprinted by permission of ref. 4: Springer Nature, Oecologia, copy- 3/8 right 1995. M. michaelseni mounds image reprinted by permission of ref. scales with tree mass m, r = cm m , and a growth exponent b = 2: Springer Nature, Insectes Sociaux, copyright 2010. Crops image credit: 1/3. Other sessile organisms fill available space determined by Pxhere. analogous mechanisms of growth, death, and competition, suggesting that metabolic principles reflecting vascular or other constraints (with differing allometric exponents) bridge well-studied the origins of scaling laws in organism growth across a large forests and sessile organisms more generally (8). range of body sizes derived from energetic constraints (40, 48, Building on the metabolic picture of growing individuals, we 49). Given constraints on average resource consumption per unit consider size classes labeled by radial dimension rk with index area, individual growth follows power law, allometric scaling k of population number nk (t), which is a function of time t. relations connecting accumulation of biomass m, or the organ- Using forests as our example, these size classes group together ism’s physical dimensions such as the stem radius r with age. trees of various species, roles, and microenvironments, and so In the context of forests where individuals are fixed in loca- we describe properties averaged over such variety. The smallest tion, metabolic scaling can be connected with population-level size class k = 0 is filled with saplings of stem radius r0 that have statistics such as spatial density, biomass production, and stand grown from seedlings with rate g0 (42). As new saplings appear energetics determined by the balance of individual growth and in the system, older ones grow into the next class k = 1, reflected mortality (34, 43). Such predictions have been verified for indi- in the rate of change of stem radius r˙k , where the discrete classes vidual organisms (41, 50) and have highlighted ecosystem-level encompass stems of radius within the interval [rk , rk + ∆r). Fur- regularities such as total population density and predator–prey thermore, trees die with a size-dependent inherent mortality rate relations (51, 52). The presence of universal patterns suggests µk , which we consider independent of competition-based mor- that unifying principles act across systems (53, 54) such as from tality. Accounting for these individual properties of metabolic energetic constraints (34, 43, 55, 56). Such observations form growth and death, we obtain a dynamical equation for change the basis of using power law, allometric relationships to describe in population number per unit time for saplings: the rates of processes, but this is not essential. Any number of assumptions or mathematical relationships could replace these in our overall framework. ∂t n0(t) = g0 − n0(t)[˙r0/∆r + µ0]. [2] One surprising prediction of metabolic scaling theory is that it is not necessary to explicitly include local competitive inter- For larger trees, the change in the population is determined by actions to explain steady-state population distributions (33, 34), the rate at which smaller plants in size class k − 1 grow into the even if local competition is one of the major factors that drives size class k, long-term evolutionary dynamics to optimize fundamental energetic constraints (57). However, competition coupled with other ∂ n = n r˙ /∆r − n [˙r /∆r + µ ], [3] timescales can introduce complex dynamics (58, 59) such as in t k k−1 k−1 k k k response to exogenous perturbations (60, 61), which goes beyond steady-state assumptions. Other than mechanistic additions to describing a sequence of ever larger tree sizes that are populated metabolic scaling theory (44), competition, perturbation, and by an incoming flux of younger and smaller trees and depopu- other dynamics present potential explanations for significant and lated as trees grow to a larger size or die. Thus, Eqs. 2 and 3,

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With 64). 39, (34, mortality for n form functional a function growth mass size-class starting a the as mortality equa- of natural These and hypothesis. formulation growth A). only continuum section including Appendix, model, the (SI determine (32) referring state when tions steady theory equilibrium the demographic to as known also n )ntins ae,adsnih olcini re through trees in collection sunlight and water, (65–67); nutrients, patterning 3) spatial regulate and and that growth by zones mat foraging microbial depletion and local diffusive biofilm 2) 1) (18); include termites con- Examples and (62). fundamentally ants area is local organisms to sessile nected in collection Resource Resources Fluctuating for Competition 2. ref. metabolic for in accounting only 1 model a scaling. [figure in interactions, included competitive trees not as are such large which as processes in from (such arise state observed could by 34]) steady distributions fixed the at size filling be determine for space may only from which features deviations coefficients, determined these exponent, scaling is Since but the constraints. rates of energetic that mortality ratio and fact the any growth when by the typical virtually forests, reflects on as in rately This well filling scaling. as con- space number theory predictions population scaling both other metabolic encompassing with states sistent range steady wide a possible obtain we of allometry, tree individual of and assumptions mortality which at driving timescales exponent (32). relative mortality the act the in growth both also scaling of but combination of growth a forms as the population the mortality fixes law, and power model when a size-class by whereas determined simple is growth curve, metabolic When growth birth. quickly, the dominates, individuals growth by overtakes When solely timescales second. relative mined the the in recover and would death we term and first growth the of in Eq. growth in metabolic exponent number population The Eqs. from mortality model metabolic size-class of simple form the the in fixes directly stationarity Thus, condition boundary sapling with rwh et,adrsuc optto nssieorganisms sessile in competition resource and death, Growth, al. et Lee exponent: number population the determines reality in is growth r radius tree continuous theory, of function and a observation both in classes or environment either competitors. to local reference without growth tree pendent growth for forms mortality functional and particular the specifying without (r uhthat such aigpeitosfo loercsaigter htrelate that theory scaling allometric from predictions Taking rmti iia oe fte rwhudrtescaling the under growth tree of model minimal this From lhuhpplto stpclybne nodsrt size discrete into binned typically is population Although ) ∝ r −α ∂ ∂ t r n n tstationarity at and t k n (r ≡ µ (r 0 k r , , ecietesmls osbefr o inde- for form possible simplest the describe , 0 t t α = ) + = ) ≈ µ(r k dm g −∂ b ∆r 0 α A ¯ 1 = = ) − /dt = r = eobtain we , [n n n ouainnme sdeter- is number population and /3, b Ar ¯ 3 8 (r (r ihte aisi Eq. in radius tree with + ac ¯ 0 ∂ α , b , r t 3¯ −1 t α m 1−b t otessriebeyond survive trees no ∞, → n eaigteindex the Relating . )˙ ac )[˙ r 2 = 8 (r , r (r A ¯ m 1−b [α (r , )] osntdpn sepa- depend not does , t 0 − 0, = ) )/∆ − . b 7 n 4 ]. (r r niae h oeof role the indicates and + , t µ(r )µ(r 5 nawythat way a in 0 eobtain we 1, )], ) k oradius to 1 and [5] [6] [7] [4] r ˙ k h ugtt cl ihpyia ieso osm exponent some to dimension expect physical we with capture, scale η resource to local budget For the feasible. is growth which is availability to need. resource of relative metabolic fluctuations basal how environmental example and consider overlap an by we modulated compet- (36), a (4), symmetric As of ones interactions consequences later. itive taller the discuss of to we formulation that general not interaction but available competitive light neighbors asymmetric trees reduce between shorter particular overlap competi- in to area canopies where by Overlapping 68), determined (19). (49, largely volumes is canopy tion or root overlapping muto hrn htte ihrsuc area resource area, with in overlap i the tree given j that neighbor sharing of the amount induce is that it requirements (49), metabolic size mortality. basal tree below maximum fluctuations determine may availability limited tightly distribution are the tions dominate scale-free events extreme a consider we that tails, fluctuations law of mod- power be precipitation distribution can with low fluctuations accurately of S2 )—resource durations eled Fig. as , Appendix cases—such (SI some (69) in that ing variable mean random the a fluc- about distribution randomly resource tuates averaged long-time quantity patterns, diurnal area, time-fluctuating tree resource a that a resource is of area amount unit (49). per trees for observations Eq. ftm,w osdrtema-edefc rmsc competition C such section from Appendix, effect mean-field (SI the consider we time, of with strengthens area. that overlapping competition resource with needs metabolic when resource tions from mortality when scarcity, deprivation rate high with or occurs When overlap stress symbiotic. large for is with and relationship as competing, the individ- for or if cost reusable as for a overall pay however, availability uals game; resource zero-sum reduces competition a resources, available When neighbor. competing tion neighbors the pc n neatwal,bti sue htitrcin are interactions rate attrition available that competitive the assumes Then, dimensions. fill it two just in but than that weakly, stronger area interact large and time a space over given trees any many sidering at Eq. flux in approximation resource fraction incoming saps neighbors total with competition resource Thus, † * When uptake, water of context the In 1 vrgn vrmn pta ragmnsoe ogperiod long a over arrangements spatial many over Averaging l raim aesm aa eorebudget resource basal some have organisms All utn hs oehr noigrsuc aedpnso the on depends rate resource incoming together, these Putting n osatparameter constant and 8 f ξ ¯ ν ∈ niae eoreetato efficiency extraction resource a indicates 1 = ≤ 0 1] [0, ,w utas xa pe ii otedsrbto oesr nt mean. finite a ensure to distribution the to limit upper an fix also must we 2, ∆Q . † s h exponent The frsucsspoe f ie vra iheach with overlap given off siphoned resources of (t ∆ hij = ) hs Eq. Thus, 1. Q ¯ i a s i (t ερ(t ξ nee fte ,adacntn frac- constant a and i, tree of j indexed ≡ = ) ersnigscarcity, representing a s 1 9 a details): has > ν )a (r η uhta re r estv oresource to sensitive are trees that such ερ(t 1 n eaieyrbs osc fluctua- such to robust relatively and sa cuaedsrpinwe con- when description accurate an is i i ≈ )   lhuhtm-vrgdresource time-averaged Although 2. ∝ (49). 1.8 ∗ h 1 β f )a ν (ξ r 1 − ftettlaon fresource of amount total the If 2 1 = i or , i 2α = ) rmrl umrzswhether summarizes primarily https://doi.org/10.1073/pnas.2020424118 [1 f atrsteblneo basal of balance the captures ol oetal banfrom obtain potentially could r X opttr qal split equally competitors /2, hiji is , − atrdb iiinwith division by captured ρ, ¯ ξ Q ∆ 0 1−ν f 0 ∆ ]− i a ρ (r i j (t ξ a : f Q = ) −ν j )a <   0 1 (r where , − ρ(t i < ν < β eore are resources 1/2, eodperiodic Beyond . i 1 Q ). r ¯ = ) 0 ρ(t η u over sum a ε, (r 1 ∆Q nprdby inspired , ερ(t 2 a ρ/ξ i PNAS hnthe then ), ). i ξ rfluctua- or Q oswith does 0 < (t )a 0 f ensures such 0, Not- ). f | i > above The . from f9 of 3 1/2, [8] [9]

APPLIED ECOLOGY MATHEMATICS matters when incoming resources are insufficient to cover basal R ∞ metabolic rate. With probability p(ξ > ξbasal) = h(ξ) dξ, ξbasal such insufficiency occurs, where ∆Q¯ = 0 defines a minimum sustainable level of scarcity ξbasal. The resulting probability of fatal fluctuations is

κ p(ξ > ξbasal) = Br , A −κ [10] B = rmax , κ = (ν − 1)(η1 − 2αr ).

The probability is normalized by constant B set by recognizing that there is only a single largest tree with radius rmax by defini- tion, which then relates the phenomenological parameters ε and f . Note that we expect κ > 0. When resource area grows slower than metabolic need, as is the case for η1 > 2αr , larger trees have less margin for low resources because they sit close to the boundary of basal metabolic need. Yet, if it were possible (although BCD unrealistic) for resource area to grow faster, η1 < 2αr and κ < 0, then growth is unconstrained, and larger trees have more buffer to withstand environmental fluctuations.‡ Furthermore, the form of exponent κ shows that growth in metabolic cost is mediated by Fig. 2. (A) Population number n(r) with varying strength of area com- the fluctuations in resource availability described by exponent ν petition (α = 2, ν = 5/2, L = 200, averaged over time, and K = 15 random such that when ν > 2, its distribution is narrow, and we expect forests). Mean-field approximation (solid lines) mirrors the shape of the 2D there to be sharp difference in the impact of competition for forest simulation (circles) for varying basal metabolic coefficient β1.(B–D) large trees below and above a cutoff. For ν → 1, all trees, small Simulated forest plots; automaton model details are in SI Appendix, section or large, will pay substantial costs for competition. F. Brown circles represent root competition area centered about gray dots. Combining metabolic growth and mortality from Eqs. 4 and 5 and competition from Eq. 10, we obtain the generalized size-class model extensible by, for example, modifying resource-sharing fraction f to reflect cooperative or noncooperative interactions or an allo- κ metric dependency. Such modifications do not change underlying ∂t n(r, t) = −∂r [n(r, t)r ˙(r)] − n(r, t)[µ(r) + Bs r ]. [11] metabolic scaling but do change the probability of fatal fluctuations, p(ξ > ξ ). Thus, the derived scaling form summarizes a By solving Eq. 11 for steady state, we find basal variety of competitive effects in the exponent κ, suggesting how physical scaling might lead to universal, or similar, demographic r −α F κ+1−b n(r) =n ˜0 exp − r , [12] scaling across different biological systems. r0 κ + 1 − b 3. From Random to Regular Spatial Patterns b−1 with normalization constant n˜0 and F ≡ 8Bscm /3¯a. This shows Different organisms, and even the same organism in another that an exponentially decaying tail truncates the simple scaling environment, may show systematic variation in spacing (70). form, imposing a cutoff on a scale of [F /(κ + 1 − b)]−1/(κ+1−b). Such variation reflects individual growth dynamics and the Eq. 11 is the general formulation incorporating both metabolic strength of resource area-based competition due to the local scaling theory and the impact of area-mediated resource use properties of competitor species or the environment (7, 12, from pairwise competitive interactions, yielding the product 71). Returning to Fig. 1, we again point out randomness in of a scaling law with a decaying tail that wiggles as resource spatial surveys of an Alaskan rainforest along with Macroter- fluctuations are varied. mes michaelseni mounds and ant nests. In contrast, the We show numerical simulations of an explicit two-dimensional spacing between Macrotermes falciger mounds is more regu- (2D) simulation of trees in a large plot in Fig. 2A in comparison lar. Other than intertaxonomic variation, there is also evi- with the mean-field approximation from Eq. 12. The mean-field dence of random and systematic variation between different approximation does not exactly capture the tail of the distribu- plots in nearby regions.§ Thus, natural spatial distributions tion, but it does surprisingly well. Furthermore, it captures the of sessile organisms may be attributable to the assorted qualitative intuition that for large trees, r 1, metabolic con- effects of individual allometries and local competition in our straints dominate, introducing an upper cutoff on the maximum model. possible tree size in the system whose radius varies with the We survey such variety in Fig. 3 along a schematic region of growth coefficient. The suddenness of this cutoff is controlled spatial patterns generated by our model. We vary the rates of by the fluctuation exponent ν such that we find strong curvature growth, death, and competition in Eq. 11. The planes jutting out away from purely scale-free metabolic scaling in the stationary from the back corner in Fig. 3A all correspond to theories where distribution for smaller ν. one of the terms is negligible. When competitive attrition is neg- Importantly, the mean-field argument clearly links resource ligible, or s → 0, population scaling is pinned to the plane where area with resource fluctuations. This means that deviations from there is a perfect scaling law determined by mortality and growth. metabolic scaling theory may result from different combinations For example, the idealized West, Enquist, and Brown (WEB) of resource-area growth and fluctuations in a way that make model for forest distributions contains only growth and death effects hard to disentangle (31). Beyond the particular form of and corresponds to the point where α = 2 (33, 34). The other competitive interactions we consider, this framework is naturally limits of no natural mortality or no growth lead to qualitatively

‡ § In the marginal case η1 = 2αr , the coefﬁcient of competition modiﬁes the scaling Figure 4 in ref. 70 shows termites in several soil types, and SI Appendix, Fig. S8 shows exponent α. This is a mathematical possibility but unrealistic. variation among plots in Alaskan rainforest.

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Fig. in shown and )/r ersnsahlsi a fmauigtesrnt fcom- of strength the measuring of way holistic a represents and niae oi,hxgnlypce hs.Function phase. packed hexagonally solid, indicates i, Loca- (A) models. of realm through trajectories Characterizing r cutoff D C eitosfo admesidct mrec forder of emergence indicate randomness from Deviations ) KL ¯ a ∆r ≡ sdie ozr.Around zero. to driven is [p [F = ||q 3c /(κ orsod osrnteigcmeiieinter- competitive strengthening to corresponds omlzdniho est tdistance at density neighbor Normalized (D) 1/20. ]= m 2/3 14 + A ¯ Z a ¯ 2/3)] ssal hl opttv trto rate attrition competitive while small, is r r eursdtriigabnsz o inte- for size bin a determining requires A 0.3 = /8 0 ¯ ∞ with 0 = p −1/(κ (r p(r min f n rv rwhrt ozr,we zero, to rate growth drive and (0) min √ +2/3) log ) πσ n ayn h et rate, death the varying and https://doi.org/10.1073/pnas.2020424118 rmrno points random from ) = n(r n spotdaantdistance against plotted is and 1 rcn h ryclne nFig. in cylinder gray the tracing , ie density given Eq. E. section Appendix, SI erae scmeiiecosts competitive as decreases ) s 2 = ,w n lqi”phase, “liquid” a find we 1, p q (r (r min min ) ) σ o comparison, For . dr s hsregion This . q(r min PNAS min A ¯ . f = (r ). m 2/3 counts ) ,while 0, s B | s varies. shows ¯ a/8 corre- f9 of 5 [14] A, ¯ = r ,

APPLIED ECOLOGY MATHEMATICS In this limit and for moderate competitive attrition m, the system condenses into a disordered packing or liquid-like phase (SI Appendix, Fig. S4). Nevertheless, long-range order fails to appear because nearest-neighbor statistics are dominated by disorder A introduced by turnover from randomly placed seedlings and con- tinuously changing tree size. In other words, metabolic growth and death act on sufficiently fast timescales that regular patterns in spacing take too long to stabilize. In organisms with different rules for metabolic scaling, we may expect to find stronger tendencies for self-organization. Such an example manifests in the semiregular packing of the fairy circles shown in Fig. 1. Such spacing entails a relatively narrow and peaked distribution of mound areas at some maximum size, a phenomenon incompatible with scale-free growth. Instead, this distribution implies that mounds that approach the maximum size are stable and that strong competitive interactions inhibit the formation of new smaller mounds. We can approxi- B mate such dynamics by driving growth and natural mortality to zero and vastly enhancing competitive mortality. This ensures that mounds are fixed at a typical size, with rigid boundaries delineated by strong competitive interactions and close to hexagonal spacing that minimizes survival of randomly placed colony seeds.¶ As we draw in black dashed lines across Fig. 4 B and C, the simultaneous limits of slow growth (a¯ → 0), slow death (A¯ → 0), and lethal competition (s → ∞) return large values of the KL divergence relative to random (SI Appendix, Fig. S5). As a more direct check, we show that the density of neighbors oscillates (Fig. 4D), analogous to fairy circle data from ref. 18 for which several mechanisms have been proposed. This is not the case for the disordered packing regime, where local exclusion is important but does not lead to long-range order. Fig. 5. Oscillations in population number n(r, t) from asymmetric compe- Thus, hexagonal packing is confined to a tight region of param- tition such as canopy cover. (A) Population number distribution n(r, t) at eter space in our metabolic growth framework where growth different times. Inset shows data from two tropical forests from ref. 32, is bounded (SI Appendix, section D). This region corresponds where markers correspond to data and lines correspond to their model. to a wide separation of timescales: Growth must be sufficiently Dashed black line shows predicted slope at steady state from Eq. 4. (B) slow to avoid introducing spatial disorder on the timescales with Population number oscillations for trees of different sizes. SI Appendix, Fig. which relatively fast competitive death stabilizes regular spatial S6 shows examples of oscillations in 2D automaton simulation. We use the 0 0 patterning (8, 73). Heaviside theta function for 1 − Λ(r − r) = Θ(r − r − ∆rcrit). 4. Transient Dynamics and Population Shock Waves Demographic stability among a population of sessile organisms destroy incipient colonies adjacent to their borders than face a is not guaranteed since ecosystems are buffeted by a wide range threat (8). Susceptibility to exogenous disturbances like wind also of endogenous and exogenous perturbations (20, 60, 74). For depends on size, although sometimes to the benefit of smaller example, local competition is negligible in a young plot until individuals (77, 78). As with symmetric competition, we formu- individuals reach a size where they impinge upon neighbors, a late asymmetric competition in terms of its effects on population phenomenon known as self-thinning (75). This dynamic is espe- growth ∂t n(r, t). A mean-field framework means that the rate of cially prominent in agriculture, where spacing is regular, plants decrease in population number is proportional to typical overlap are genetically identical, and competition onset is almost uniform between trees of radius r with all sizes larger than it up to rmax: (6, 7). Natural stands also vary with plot age, but they are more Z rmax stochastic, and height differences can be prominent (34, 39). 0 0 0 0 Remarkably in previous measurements, we find oscillations in n(r, t) acan(r) n(r , t) acan(r )[1 − Λ(r − r)] dr . [15] population number with radius as depicted in Fig. 5 A, Inset taken r

from ref. 39, suggesting the presence of long-lived transience 0 not captured by the steady state. Inspired by this observation, We assume that the competitive effect Λ(r − r) is some we consider how competitive asymmetry, specifically forces that sigmoid-like function that decays from Λ(0) = 1 to the limit increase mortality of smaller organisms, could excite population Λ(∞) = 0 (figure 4 in ref. 49) at which point the tallest trees waves. completely obscure all light incident on ground area spanned 2αcan Asymmetric competition takes various forms such as how by the canopy acan(r) = ccanr , where αcan = 2 (33). A sig- moidal form indicates some characteristic length scale for Λ canopy shading reduces light incident on shorter plants lying 0 underneath or around the larger ones with little cost to the lat- such that when the distance r − r reaches some critical value ter (35, 43, 76). Large termite colonies are much more likely to ∆rcrit, a substantial portion of light is obscured from above. This is distinct from symmetric interactions that scale with radius r and lack a typical length scale distinguishing competitors from noncompetitors. Thus, we consider asymmetric ¶In the “zero-temperature” limit where competitive mortality always selects out the competition that is area delimited and favors larger organ- weaker of two competitors, it is clear that tight packing is stable to disorder because any isms. This assumption is analogous to canopy shading and newcomer must overlap with more than one other individual, whereas every individual forming the lattice only intersects with the newcomer. Hexagonal packing is the densest more generally captures the competitive advantage of larger of packings and thus, most stable to infiltration. organisms (36, 79).

6 of 9 | PNAS Lee et al. https://doi.org/10.1073/pnas.2020424118 Growth, death, and resource competition in sessile organisms Downloaded by guest on September 26, 2021 Downloaded by guest on September 26, 2021 rwh et,adrsuc optto nssieorganisms sessile in competition resource and death, Growth, al. et Lee data. with aligned suggestively and consequences upon dynamical remarked hardly whose competi- are cause, 71), that (7, endogenous literature remarkable one the in present is widely discussed it although dynamics, origins, tion exogenous of share values likely on with limits incompatible cal generally value scaling its for form different a exponent to leads competition resource metric cet u eed ntewyta aoyae rw with grows area canopy that way the on radius, depends but ficients Eq. with contrast In smerccmeiinfie h ouainnme exponent G) section number Appendix, population (SI to the fixes as goes competition that wide asymmetric number a population the exists When in there region size. that scaling weak sustainable sufficiently are largest limitations the where resource point delimit the limitations near dominates resource an that introduces symmetric tail of constraints decaying case exponentially resource the for in accounting As maximum waves. competition, in population cutoff dampen limiting and a size impose constraints competi- Resource symmetric from tion. differences important find we tuations, may scaling. deviations metabolic overlaying such dynamics that to as due shows arise yields presented model filling are explanation—our space data qual- poor scaling—indeed, metabolic demographic Although against 58). in evidence (25, curves affect- dynamics similar or ecological itatively by classic influenced other perhaps maturity ing cycles of oscillatory points generate different competi- at can all, organisms between After dynamics (74). widespread tive perturbations from from exogenous originate generated repeated or internally and parameters are competitive oscillations of Mudumalai, such tuning for whether ques- case is unanswered the An tion not oscillations. sim- is wide our intriguingly this from shows although widths which example 5A, the other an Fig. Planada, in with across La ulation consistent visible from seem are example undulations the oscillations of For Similar forests. 32. tropical ref. from 5 Fig. ples forests. tropical on data across or stochastic- endogenous infer following to effects 73). used (60, competitive perturbation be and exogenous growth could in waves speed ity and shock shape in population variation Such of growth. tree in because stochasticity disperse of also waves population growth, superlinear from r Fig. in nonlinear waveforms of of case 5B movement the rightward In the H ). growth, section metabolic , Appendix competition of (SI feature sizes generic between a model be may mean-field dif- oscillations the that of size suggests analysis of stability function Linear a sharp. competitive as is ference competition when and prominent generates strong be are delay interactions may and The that dips, suddenly. oscillations cover increases size-class canopy population Eventually, tree growth. small up with propagates trees slowly number larger population to in dip pop- 5A This Fig. the times. in Likewise, short trees shock. small population of a sharp number or correspondingly ulation trees a causes young and in sudden die-off is matter to petition ic h aiu rectf ssapi h ouainnumber disadvantage. population the competitive in sharp sudden n is a cutoff tree at maximum is the are ones Since shortest the trees the between smaller and difference reach trees point, the tallest trees the point tall of which when height at strongly density matters critical but some growth forest initial (t (r hncnieigaymti optto ihrsuc fluc- resource with competition asymmetric considering When curves number population in oscillations find we Remarkably, aoycmeiini elgbeduring negligible is competition canopy 5B, Fig. in show we As ) ceeae ihae aisgossprierywt time with superlinearly grows Radius age: with accelerates , ∝ t h perneo ufiinl ag re o ih com- light for trees large sufficiently of appearance the ), t 2α 3/2 α can hnmtblcgot exponent growth metabolic when hnta fcnnclmtblcsaigtheory, scaling metabolic canonical of that than n eaoi rwhexponent growth metabolic and , Eq. 7, α α 4 = can 16 α lhuhpplto oscillations population Although . can sfe fmtblcgot coef- growth metabolic of free is 2 + − A, b α Inset ipasasde i at dip sudden a displays . 2 = b ipastoexam- two displays eas fphysi- of because 1 = b te than Other /3. ∆ hs asym- Thus, . n r crit (r ) tthis At . ∼ r [16] −α , rpisa ela pta atrigmybte pcf the 18, specify (4, better environments across may 83). acting patterning 82, forces demo- 60, spatial competitive in as of scaling range frame- well predicted integrated as mea- from an graphics to deviations Instead, approaches considering of limited. work statistical are form that competition the suring suggesting diver- comparing strength, KL by disor- the itive show with (Eq. introduces distribution we gence size distance As nearest-neighbor organism (2). the varying spacing weaker when dered with measures regularity correlation-based forests spatial in of of limitations fluctuations—our accurate suggests environmental more work smaller be and large competition may to local it limit that the namely taking by exact made a 81). (80, be to features, systems way can realistic a more that as many mapping serve incorporating allometries models about organize assumptions perturbed, uni- constantly idealized and quantitatively our to finite, (Eq. noisy, that are maps scaling systems metabolic real diversity exponents Although and such fluctuations by environmental how summarized link highlights features or also versal it allome- response diverse of Yet, stress distribution a tries. resource incorporating by in as interac- environment- such sharing capture variation competitive to extended organism-specific of be or to formulation basis a Our establishes tions cutoffs. (Eq. exponent and number mani- scaling is population competition a the die, Then, level. in in grow, every fest at competitors toll same larger the a compete relatively exacts and because however, way competition, scale-free Asymmetric competition: 2). cutoffs symmetric (Fig. competition-mediated for and number scaling population individual-dominated of the differ- two regimes for indicating ent (41), dominates sublinearly scale it mortality and must growth that such competition (Eq. largest organisms radius the area-delimited largest with of because superlinearly statistics is scale population the This in organisms. effects prominent succinctly competitive most competition, that resource-area are relations. of theory exponent context in the mean-field effects In competitive tractable and a metabolic relates with model altered in our evolves the competition sense, how chang- this of 60). of (18, In also only environments spa- not but 1). indicator the conditions an (Fig. as ing in (8) serve may reflected organisms distribution spatial be of (e.g., will distribution interactions tial perturbations competitive these changes symbiosis), or construction) (e.g., fluctuations niche resource inter- dampens When environment by the resources. with driven action fluctuating principles, forces and competitive basic for requirements these framework sessile metabolic general From of a obtain competition. model we area-based minimal the- and a scaling allometric ory propose of aspects we incorporating growth picture, Inspired organism 62). forest (8, the competition analogous and by by sessile death, determined other growth, ways Yet, of of variety (33). mechanisms a space in canopy space fill available organisms requirement the the fill consid- of into trees from translated set that self-similar emerges constraints in This the energetic scaling, (34). on range, of law trees so eration some power smaller and of in over set reflected size consider, scaled structure, smaller can a a as particu- we trees of a that large of trees to way trees into seem a the off” trees such that “branch of such groups size law even lar fractal that pervasive of then this network remarkable obey branching is a photo- to It of canopy roots. only products self-similar not its shuttles from fractal that synthesis vasculature beautiful down microscopic but a twigs, the to is branches to to tree trunk constituents, a visible its of along structure physical The Discussion 5. te hnidctn iiain fmtblcsaigtheory— scaling metabolic of limitations indicating than Other in forces competitive tuning by variation such explore We ,ti esr hne ekywt compet- with weakly changes measure this 13), .I oprsn niiulmetabolic individual comparison, In 11). https://doi.org/10.1073/pnas.2020424118 ,afcigboth affecting 4), PNAS | f9 of 7 12).

APPLIED ECOLOGY MATHEMATICS Beyond competitive forces, we find strong additional con- fixed by principles like space filling as is assumed for metabolic straints necessary to stabilize spatial order in models with scaling of forests (33), and so, they relay a universe of possible metabolic growth (Fig. 3). Whereas metabolic scaling tends to allometries and environments. Nevertheless, power law scaling inject spatial disorder by constantly changing organism size and relations are an approximation of a population, and individu- by opening free space upon organism death, regular tiling such as als will deviate from them (42). Accounting for such variation seen for fairy circles and some termite mounds requires the elim- may be important in high-resolution models (93, 94). More fun- ination of unbridled growth along with slow natural mortality and damentally, organism growth or death may not neatly abide by overwhelming competitive attrition in a background of sparse allometric scaling relations with respect to age. We show one newcomers. This emergence of order starting with individuals extreme in the example of hexagonal packing, where growth is and their metabolic dynamics presents a complementary per- truncated. Although technically, this is still a form of allomet- spective to field-theoretic derivations of vegetation patterns that ric scaling (if a trivial one), the analogy is only mathematically start instead with densities (17, 84–86). In a similar vein, some exact in the limit where individuals live for a long time, and field-theoretic models, such as those introduced by Mart´ınez- thus, the growth period is short. Individual organisms that do not Garc´ıa et al. (87, 88), have incorporated competition with local obey such behavior, perhaps displaying persistent oscillations in (logistic) limits on growth. That hexagonal packing occurs only in size, would present a new set of dynamics extending beyond the a corner of the much broader model space encompassed by our canonical picture of metabolic growth. We emphasize that such theory (Fig. 3) reflects the extraordinary nature of such regular extensions would be natural and interesting to incorporate into patterns. the framework we present here. Complementary to the connection between spatial patterns The most striking ecological patterns occur when local interac- and asymmetric competition (79, 89), we explore transient tions generate large-scale regularities, propagating information dynamics in the context of a size-based competitive hierarchy coherently over large scales and long times (95, 96). Fairy cir- (58). Asymmetric competition can couple different timescales cles and termite mounds are a breathtaking example. Although to one another and lead to oscillatory modes in size classes forests, fairy circles, and termite mounds all seem to obey forces and thus, population number (Fig. 5)—although touching on driving the cycle of birth, growth, and death at the level of the the topic of self-thinning, our model extends beyond the typi- individual, population-level structure varies widely. Even among cal focus on monoculture stands (6, 7, 71, 75). When there is a forests, some are characterized by randomly spaced trees, such threshold at which such effects become important, such as with as the examples we show here, but others, such as the pinyon– canopy light competition, we expect to find similar population juniper ecosystem of the US Southwest, are more spatially shock waves. Remarkably, oscillatory modes manifest in multi- regular. To connect the wide range of spatial patterns shaped by ple datasets of tropical forest demography (39). Such die-offs competitive forces in sessile organisms, we build on theoretical also may be observable in other systems or directly measurable foundations of metabolic scaling. The resulting realm of models if future data collection permits highly resolved temporal data may frame analogies between organisms across species, environ- on organism death. Furthermore, the lifetimes of these transient ments, and times in the language of competitive forces acting on phenomena, indicated by width evolution, may allow us to dis- top of individual properties constrained by metabolic principles. tinguish internal competitive forces by leveraging demographic Previously published data were used for this work from perturbation (20, 90). Oscillatory modes and instabilities are a Tschinkel (1), Grohmann et al. (2), Muvengwi et al. (3), Adams widely studied feature of biological populations [for example, and Tschinkel (4), and Schneider et al. (5). with the classic logistic equation (91, 92)], and metabolic growth in sessile organisms presents an unexplored mechanism by which Data Availability. Code for reproducing the results shown they can arise. in this work is available at https://github.com/eltrompetero/ Growth, death, and competition are essential characteristics of simple sessile. life. Yet, for particular ecosystems, assumed allometric forms for these forces will need to be refined. More specifically, we assume ACKNOWLEDGMENTS. E.D.L. and G.B.W. were supported by NSF Grant PHY1838420. C.P.K. and G.B.W. thank Toby Shannan and Charities Aid Foun- that demographic diversity is summarized in terms of individual dation of Canada (CAF) for supporting this work. We acknowledge useful age, which then determines individual properties like size and discussions with Sungho Choi about forest data and thank Deborah Gordon growth rate. Notably, the relations we derive do not have to be for bringing up self-thinning.

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