An Effective Resource-Competition Model for Species Coexistence Deepak Gupta1;2,∗ Stefano Garlaschi1,∗ Samir Suweis1, Sandro Azaele1, and Amos Maritan1 1Dipartimento di Fisica e Astronomia \Galileo Galilei", Universit`adegli Studi di Padova, via Marzolo 8, 35131 Padova, Italy and 2Department of Physics, Simon Fraser University, Burnaby, British Columbia V5A1S6, Canada (Dated: April 6, 2021) Local coexistence of species in large ecosystems is traditionally explained within the broad frame- work of niche theory. However, its rationale hardly justifies rich biodiversity observed in nearly homogeneous environments. Here we consider a consumer-resource model in which effective spatial effects, induced by a coarse-graining procedure, exhibit stabilization of intra-species competition. We find that such interactions are crucial to maintain biodiversity. Herein, we provide conditions for several species to live in an environment with a very few resources. In fact, the model displays two different phases depending on whether the number of surviving species is larger or smaller than the number of resources. We obtain conditions whereby a species can successfully colonize a pool of coexisting species. Finally, we analytically compute the distribution of the population sizes of coex- isting species. Numerical simulations as well as empirical distributions of population sizes support our analytical findings. Our planet hosts an enormous number of species [1], in agreement with the empirical data [35{40]. However, which thrive within a variety of environmental condi- it lacks a convincing mechanism of coexistence, which is tions. The coexistence of this enormous biological diver- usually maintained only by an external source of indi- sity is traditionally explained in terms of local adaptation viduals. Within this context, competition for resources [2,3], environmental heterogeneity [4,5], species' abilities plays a relatively minor role with respect to the niche to aptly respond to the distribution of resources [6,7], setting, and the total number of species sustained by a and other abiotic factors which broadly define a niche region cannot be inferred by the availability of resources [8]. When species are geographically separated, they may in the habitat. Finally, because species are competitively survive because they match a specific environmental con- equivalent, and therefore, they lack any specific identity, dition and inter-specific competition is not detrimental. it is difficult to explain whether an individual is successful However, several microbial species seem to coexist de- in invading an ecosystem made of coexisting species. spite they occupy very similar niches in close-by regions In this Letter, we propose an alternative ap- [9{11]. This scenario is known as the paradox of plankton proach based on a generalization of the aforementioned [12]. Now, on timescales that are larger than one gener- MacArthur's consumer-resource model. This new formu- ation but smaller than speciation timescales, the fittest lation explains why a large number of species can co- species should outcompete all the others. Then, why do exist even in the presence of a limited number of re- we still observe coexistence? Consistent with this ratio- sources, thus violating CEP. Secondly, it predicts how nale, theoretical work based on MacArthur's consumer- many species will survive depending on the amount of resource model [13, 14] confirms the validity of the so resource present in the habitat. Thirdly, we find the con- called competitive exclusion principle (CEP): the number ditions under which an invading species outcompetes a of coexisting species competing for the same resources is pool of coexisting species. Finally, we analytically obtain bounded by the number of resources themselves [15{21]. the species abundance distribution (SAD), i.e., the prob- Such a stark contrast between the predictions of CEP ability distribution of the population sizes of the species, and coexistence of many species has been resolved by in- and show that it justifies the empirical SAD calculated arXiv:2104.01256v1 [q-bio.PE] 2 Apr 2021 corporating metabolic trade-off [22], adaptive metabolic from the plankton data presented in Ref. [41]. strategies [23, 24], enrichment of resources [25] and multi- ple stable states [26]. Finally, consumer-resource models The key feature of this new framework is the effect of are well suited to investigate how interactions reverber- a stabilizing intra-species interaction term, which affects ate on coexistence, but it is usually hard to calculate the dynamics on top of the traditional inter-species in- community patterns which we more easily observe. teraction terms which account for the indirect resource consumption. This stabilizing factor emerges naturally A complementary framework, which has been able in all ecosystems when spatial effects are not negligible. to explain several biodiversity patterns at macroscopic Indeed, by introducing a coarse-graining of the spatial scales, is the neutral theory of biogeography [27{34]. In- degrees of freedom, we show that a density-dependent stead of looking at what specific traits facilitate species' inhibition term emerges, which stabilizes the dynamics survival, this approach highlights the general features through a carrying capacity term (see full derivation in which tend to make species more similar to each other. the supplementary material [42]). For example in tree This framework has the merit to predict several patterns communities, such density-dependent term may model 2 ∗ ∗ ∗ > Janzen-Connell effect (JCE) [43{45], that describes the (c1; c2; : : : ; cR) . In Eq. (3) we have imposed the con- inhospitability for the seedlings in the proximity of par- dition that all species, present at the initial time, have ~ ∗ ent trees due to host-specific pathogens. This results survived at a large-time, i.e., (N )σ = nσ > 0 8 σ. Later, in penalization of their growth and it inhibits the local we will also discuss and analyze the case when a sub-set crowding of individuals belonging to the same species of them go extinct. [46{48]. Now, substituting Eq. (3) in Eq. (4), we find To begin with, we consider an ecological community T −1 T −1 composed of M different species competing for R re- GQ E QGU~ − GQ E B~ = µ(L~ − ~χ); (5) sources. These species are characterized by their max- which gives R coupled equations that can be solved for imum consumption rates, α , at which a species σ up- σi r (c∗) as a function of other parameters. Further, the takes the i-th resource and converts it into its biomass at i i condition for all species to survive can be written using high concentration of the resource, c . In what follows, i Eq. (3): we also refer to the α's as metabolic strategies. Since per- capita growth rates are proportional to the resource con- (QG U~)σ > B~σ 8 σ; (6) centrations when they are low, the overall dependence of resource concentration is typically captured by multiply- and it gives the coexistence region in the R-dimensional ∗ ∗ ∗ ing the maximum growth rate by the Monod function, space whose axes are r1(c1); r2(c2); : : : ; rR(cR). Thus, ri(ci) = ci=(ki + ci), where ki is a resource dependent a necessary condition for all initial species to coexist is constant. Further, resources degrade in time with a rate that the solution of Eq. (5) lies in the coexistence region; µi. Similarly, populations decay with intrinsic mortality otherwise, some of them get extinct. This is our first rate βσ. Therefore, the system evolves with an extended main result. consumer-resource model [13, 14]: To illuminate the above result, we first consider a case when several species are competing for one resource. A R X discussion for higher number of resources is relegated to n_ = n α r (c ) − β − n ; (1) σ σ σi i i σ σ σ the supplemental material [42]. In what follows, unless i=1 specified, we consider E to be a diagonal matrix [see M X Eq. (1)]. In this case, removing the immaterial index i, c_ = µ (Λ − c ) − r (c ) n α ; (2) i i i i i i σ σi Eq. (3) becomes n∗ = [α r(c∗) − β ]/ for all σ. Since σ=1 σ σ σ σ ∗ ∗ nσ > 0, we find r(c ) > βσ/ασ. Moreover, we can write 2 ∗ ∗ where we have included an additional contribution σnσ r(c ) > r(¯c) ≡ maxσfβσ/ασg, where r(c ) is the solution in Eq. (1) that corresponds to intra-species interactions. of Eq. (5): Ar2(c∗) − Br(c∗) − µ(Λ − c∗) = 0, in which −1 P 2 P Herein, σ may be treated as proportional to the carry- the coefficients A = σ ασ/σ and B = σ ασβσ/σ ing capacity for the species σ. In Eq. (2), the quantity carry the characteristic features of the species. Note that ∗ µiΛi is the rate of supplying abiotic resources. Notice r(c ) ≤ 1, therefore, the metabolic strategies, in order to that biotic resources are typically modeled by substitut- guarantee a coexistence of all species, should be greater ing µi with µici. Then, µiΛi and Λi, respectively, corre- than the death rates: ασ > βσ. Thus, for fixed parame- spond to the growth rate and the carrying capacity of the ters that characterize the species, i.e., fασ; βσ; σg, coex- i-th resource. In what follows, for simplicity, we report istence of all species is achieved when tuning the resource the results for the case of abiotic resources with degra- supply rate by varying Λ at a fixed µ in such a way that ∗ dation rates of all resources to be the same: µ = µi. The the condition r(c ) > r(¯c) is satisfied. Such a critical other cases do not display qualitatively different results. value of Λ is given by Λ¯ = r(¯c)[Ar(¯c) − B]/µ +c ¯. In As the time progresses, we expect the system to reach Fig.1, we consider an example of ecosystem having 4 a stationary state.