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Resonance-Induced Multimodal Body-Size Distributions in Ecosystems

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Resonance-Induced Multimodal Body-Size Distributions in Ecosystems Resonance-induced multimodal body-size distributions in ecosystems Adam Lamperta,1,2 and Tsvi Tlustya,b aDepartment of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100, Israel; and bSimons Center for Systems Biology, Institute for Advanced Study, Princeton, NJ 08540 Edited by Andrea Rinaldo, Ecole Polytechnique Federale de Lausanne, Lausanne, Switzerland, and approved November 12, 2012 (received for review July 31, 2012) The size of an organism reflects its metabolic rate, growth rate, classic metapopulation framework, we assume each patch is mortality, and other important characteristics; therefore, the distri- either empty or occupied by a single species (15–17). In our bution of body size is a major determinant of ecosystem structure model, each species is specified by its maximal growth rate q, and function. Body-size distributions often are multimodal, with which is negatively correlated with its body size. The higher several peaks of abundant sizes, and previous studies suggest that growth rate of smaller individuals results in more migrants sent this is the outcome of niche separation: species from distinct peaks to other patches. Larger individuals, on the other hand, are ad- avoid competition by consuming different resources, which results vantageous in interference and exclude the smaller individuals in selection of different sizes in each niche. However, this cannot during within-patch competitions. explain many ecosystems with several peaks competing over the Specifically, we considered the following processes (Fig. 1): same niche. Here, we suggest an alternative, generic mechanism First, occupied patches are emptied at a rate m. This may occur underlying multimodal size distributions, by showing that the size- because of mortality or because of path destruction and creation dependent tradeoff between reproduction and resource utilization in which occupied patches vanish and new patches enter the entails an inherent resonance that may induce multiple peaks, all system. Second, a patch occupied by a q-species spreads its off- competing over the same niche. Our theory is well fitted to empirical spring as migrants at a rate q. Each migrant arrives at a randomly data in various ecosystems, in which both model and measurements picked patch. If the destination patch is empty, the migrant takes show a multimodal, periodically peaked distribution at larger sizes, over. If it is occupied by a q′-species, the invasion probability is followed by a smooth tail at smaller sizes. Moreover, we show given by the sigmoidal function (Fig. 1C) a universal pattern of size distributions, manifested in the collapse h i of data from ecosystems of different scales: phytoplankton in a lake, À Á −1 ; ′ = + sðq − q′Þ ; [1] metazoans in a stream, and arthropods in forests. The demon- gs q q 1 e strated resonance mechanism is generic, suggesting that multi- modal distributions of numerous ecological characters emerge from where s > 0 measures the selection strength. This function may the interplay between local competition and global migration. be seen as a phenomenological rule reflecting stochastic com- petition in which a species with an inferior fitness (higher q) still adaptive dynamics | evolutionary ecology | universal scaling | species may take over, but also may be derived from microscopic dy- packing | species assembly namics of competition between the resident and migrant sub- populations. In the deterministic limit, s → ∞, the dynamics are ultimodal body-size distributions, in which a population reduced to the “colonization-competition” model of Hastings exhibits several peaks of abundant sizes, characterize local M (16) and Tilman (17): the sigmoid (gs) becomes a step-function ecosystems and have puzzled ecologists for many years (1–4). To and the larger species (lower q) always takes over. resolve this puzzle, most previous studies concentrated on one To examine the possibility of multimodal distributions, we important aspect of body size, namely resource partitioning or solve the long-term evolution of the entire size spectra. The niche separation: larger individuals consume certain resources hallmark of multimodal distributions is the emergence of dis- more efficiently than smaller ones, whereas smaller individuals crete peaks within the continuous spectrum of possible q-values, have access to some resources that are unavailable to larger indi- 0 ≤ q < ∞. When the number of patches is large, the normalized viduals. As a result, multimodal size distributions may originate distribution of patch occupancy (relative abundance), ρðqÞ, from community-wide character displacement, in which natural evolves according to the dynamics: selection pushes species to assort into packs, each of which fits to 2 3 a particular niche (1, 4, 5). Z∞ dρðqÞ À ÁÀ Á À Á d2ρðqÞ Another aspect of size, however, relates to a growth-vs.- = ρð Þ4 − − ρ ′ + ′ ′; ′5+ μ ; [2] q q m q q q gs q q dq 2 efficiency tradeoff (6, 7). Smaller individuals generally invest less dt dq 0 effort in their development and have a higher maximal growth – rate when the resource is abundant (8 12), whereas larger indi- where the term q corresponds to colonization without competi- fi viduals have more ef cient metabolism and are more successful in tion, the term m to mortality, the negative integral term to direct competitive interferences (Fig. 1A) (6, 13, 14). As a result, smaller individuals are more likely to be the first to populate newly formed patches, and the larger individuals follow only later. Author contributions: A.L. and T.T. designed research; A.L. performed research; and A.L. In this study, we show how this tradeoff may promote multimodal and T.T. wrote the paper. body-size distributions, and we identify several characteristics of The authors declare no conflict of interest. the underlying dynamics. This article is a PNAS Direct Submission. 1To whom correspondence should be addressed. E-mail: [email protected]. Stochastic Subdivided Population Model 2 – fi Present address: Department of Environmental Science and Policy, University of Califor- To examine how the growth ef ciency tradeoff leads to multi- nia, Davis, Davis, CA 95616. ECOLOGY modal body-size distributions, we considered a population that is This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. subdivided into several habitable resource patches. As in Levins’s 1073/pnas.1211761110/-/DCSupplemental. www.pnas.org/cgi/doi/10.1073/pnas.1211761110 PNAS | January 2, 2013 | vol. 110 | no. 1 | 205–209 Downloaded by guest on October 1, 2021 AB ytilatroM Migration noititepmoC size Competitiveness Max. growth−rate, q C D q m o −1 1 s qq+ o q + q 0.5 qq+ qg(q,q’) 0 qq’ 0 + q’g(q’,q) + q’q’ Fig. 1. Stochastic colonization-competition model. (A) Body size is associated with a tradeoff between maximal growth rate, q, and competitiveness abilities such as direct interference. (B) Schematic illustration of the dynamic processes. Mortality (or path destruction and creation): occupied patches are emptied at aratem. Migration: each species spreads migrants from its home patch to random patches at a rate proportional to its maximal growth rate q. Competition:if the destination patch is empty, the migrant invades; if the patch is already occupied by a q′-species, the q-migrant takes over with a probability gðq; q′Þ and is being eliminated otherwise. (C) The invasion probability gðq; q′Þ has a sigmoidal shape with a width ∼1=s, which corresponds to stochasticity. Species with higher q have a smaller chance of taking over occupied patches. (D) Competition is much faster than migration, which implies that each patch is either empty (∅) or occupied by a single species, entitled by its maximal growth rate (q). The resulting dynamics are that of a well-mixed population of patches subject to the illustrated rate equations for all q and q′: exchange (up), colonization of empty patches (middle), and invasion into occupied patches (down). The cor- responding mean-field equation is Eq. 2. inhibition of colonization by competition, both due to the in- distribute smoothly. (Evolution pushes species to differ, i.e., to be- vasion of competing migrants and due to the inability to invade come further away from their neighboring species along the q-axis.) into already occupied patches, and the last term corresponds to In contrast, when the competition becomes stochastic, as the mutations that slightly tune q at a rate μ (SI Methods). selection strength s decreases below a certain threshold, peaks emerge and the steady-state distribution becomes multimodal Emergence of Multimodal Size Distributions (Fig. 2 B–D). For s just below the transition, peaks are most In the long term, the distribution ρðqÞ approaches a unique steady prominent near the singularity at q = m.Ass decreases further, the state that we examine for multimodality. The case in which peaks become significant even further away from the singularity. competition over patches is deterministic (s → ∞), i.e., a q- This implies species packing: evolution pushes species toward the migrant invades a populated q′ patch if and only if q < q′ (16, 17), peaks even when they already are populated by other species. exhibits a monotonous steady-statepffiffiffiffi solution (Fig. 2A). For μ = 0, Peaks are located at approximately equal distances that are of this solution is ρ*ðqÞ = ð m=2Þq−3=2 if q ≥ m; and ρ*ðqÞ = 0 oth- the order of the q range over which competition is stochastic, erwise (18): species with q < m vanish whereas species with q > m δq ∼ 1=s. ACB D 1.5 s = ∞ s = 8 s = 4 s = 2 1 0.5 Abundance, 0 m 2 3 4 5 6 7 m 2 3 4 5 6 7 m 2 3 4 5 6 7 m 2 3 4 5 6 7 Max. growth−rate, Fig. 2. Emergence of multimodal body-size distributions. Steady-state distributions of the model (Eqs.
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