Introduction

The most prominent feature of life on Earth is its remarkable species diversity: countless macro- and micro-species fill every corner on land and in the water (Pennisi, 2005; Hoorn et al., 2010; Vargas et al., 2015; Daniel, 2005). In tropical forests, thousands of plant and vertebrate species coexist (Hoorn et al., 2010). Within a gram of soil, the number of microbial species is estimated to be 2,000–18,000 (Daniel, 2005). In the photic zone of the world ocean, there are roughly 150,000 eukaryotic plankton species (Vargas et al., 2015). Explaining this astonishing biodiversity is a major focus in ecology (Pennisi, 2005). A great challenge stems from the well-known competitive exclusion principle (CEP): two species competing for a single type of resources cannot coexist at constant population densities (Gause, 1934; Hardin, 1960), or generically, in the framework of consumerresource models, the number of consumer species cannot exceed that of resources at a steady state (MacArthur and Levins, 1964; Levin, 1970; McGehee and Armstrong, 1977). On the contrary, in the paradox of plankton, a limited variety of resources supports hundreds or more coexisting species of phytoplankton (Hutchinson, 1961). Then, how can plankton and many other organisms somehow liberate the constraint of CEP?

Ever since MacArthur and Levin proposed the classical mathematical proof for CEP in the 1960s (MacArthur and Levins, 1964), various mechanisms have been put forward to overcome the limits set by CEP (Chesson, 2000). Some suggest that the system never approaches a steady state where the CEP applies, due to temporal variations (Hutchinson, 1961; Levins, 1979), spatial heterogeneity (Levin, 1974), or species’ self-organized dynamics (Koch, 1974; Huisman and Weissing, 1999). Others consider factors such as toxins (Lczaran et al., 2002), cross-feeding (Goyal and Maslov., 2018; Goldford et al., 2018; Niehaus et al., 2019), spatial circulation (Martín et al., 2020; Gupta et al., 2021), “kill the winner” (Thingstad, 2000), pack hunting (Wang and Liu, 2020), collective behavior (Dalziel et al., 2021), metabolic trade-offs (Posfai et al., 2017; Weiner et al., 2019), co-evolution (Xue and Goldenfeld, 2017), and other complex interactions among the species (Beddington, 1975; DeAngelis et al., 1975; Arditi and Ginzburg, 1989; Kelsic et al., 2015; Grilli et al., 2017; Ratzke et al., 2020). However, questions remain as to what determines species diversity in nature, especially for quasi-well-mixed systems such as that of plankton (Pennisi, 2005; Sunagawa et al., 2020).

Among the proposed mechanisms, predator interference, specifically the pairwise encounters among consumer individuals, emerges as a potential solution to this issue. Predator interference is commonly described by the classical Beddington-DeAngelis (B-D) phenomenological model (Beddington, 1975; DeAngelis et al., 1975). Through the application of the B-D model, several studies (Cantrell et al., 2004; Hsu et al., 2013) have shown that intraspecific predator interference can break CEP and facilitate species coexistence. However, from a mechanistic perspective, the functional response of the B-D model can be formally derived from a scenario solely involving chasing pairs, representing the consumption process between consumers and resources, without accounting for pairwise encounters among consumer individuals (Wang and Liu, 2020; Huisman and Boer, 1997). Disturbingly, it has been established that the scenario involving only chasing pairs is subject to the constraint of CEP (Wang and Liu, 2020), raising doubt regarding the validity of applying the B-D model to overcome the CEP.

In this work, building upon MacArthur’s consumer-resource model framework (MacArthur, 1969, 1970; Chesson, 1990), and drawing on concepts from chemical reaction kinetics (Ruxton et al., 1992; Huisman and Boer, 1997; Wang and Liu, 2020), we present a mechanistic model of predator interference that extends the B-D phenomenological model (Beddington, 1975; DeAngelis et al., 1975) by providing a detailed consideration of pairwise encounters. The intraspecific interference among consumer individuals effectively constitutes a negative feedback loop, enabling a wide range of consumer species to coexist with only one or a few types of resources. The coexistence state is resistant to stochasticity and can hence be realized in practice. Our model is broadly applicable and can be used to explain biodiversity in many ecosystems. In particular, it naturally explains species coexistence in classical experiments that invalidate CEP (Ayala, 1969; Park, 1954) and quantitatively illustrates the S-shaped pattern of the rank-abundance curves in an extensive spectrum of ecological communities, ranging from the communities of ocean plankton worldwide (Fuhrman et al., 2008; Ser-Giacomi et al., 2018), tropical river fishes from Argentina (Cody and Smallwood, 1996), forest bats of Trinidad (Clarke et al., 2005), rainforest trees (Hubbell, 2001), birds (Terborgh et al., 1990; Martínez et al., 2023), butterflies (Devries et al., 1997) in Amazonia, to those of desert bees (Hubbell, 2001) in Utah and lizards from South Australia (Cody and Smallwood, 1996).

Results

A generic model of pairwise encounters

Here we present a mechanistic model of pairwise encounters (see Fig. 1A), where S_C consumer species {C₁, … } compete for S_R resource species {R₁, … }. The consumers are biotic, while the resources can be either biotic or abiotic. For simplicity, we assume that all species are motile and move at certain speeds, namely, for consumer species C_i (i = 1, …, S_C) and for resource species R_l (l = 1, …,S_R). For abiotic resources, they cannot propel themselves but may passively drift due to environmental factors. Each consumer is free to feed on one or multiple types of resources, while consumers do not directly interact with one another except through pairwise encounters.

Figure 1.

Then, we explicitly consider the population structure of consumers and resources: some wander around freely, undergoing Brownian motions; others encounter one another, forming ephemeral entangled pairs. Specifically, when a consumer individual C_i and a resource R_l come close within a distance of (see Fig. 1A), the consumer can chase the resource and form a chasing pair: (see Fig. 1B), where the superscript “(P)” represents “pair”. The resource can either escape at rate d_il or be caught and consumed by the consumer with rate k_il. Meanwhile, when a C_i individual encounters another consumer C_j (j = 1, …, S_C) within a distance of (see Fig. 1A), they can stare at, fight against, or play with each other, thus forming an interference pair: (see Fig. 1B). This paired state is evanescent, with consumers separating at rate d’_il. For simplicity, we assume that all and are identical, respectively, i.e., ∀i,j,l, and .

In a well-mixed system of size L², the encounter rates among individuals, a_il and a’_ij (see Fig. 1B), can be derived using the mean-field approximation: and (see Materials and Methods, and Appendix-fig. 1). Then, we proceed to analyze scenarios involving different types of pairwise encounters (see Fig. 1B). For the scenario involving only chasing pairs, the population dynamics can be described as follows:

where , g_l is an unspecified function, the superscript “(F)” represents the freely wandering population, D_i denotes the mortality rate of C_i, and w_il is the mass conversion ratio (Wang and Liu, 2020) from resource R_l to consumer C_i. With the integration of intraspecific predator interference, we combine Eq. 1 and the following equation:

where a’_i = a’_ii, d’_i = d’_ii, and represents the intraspecific interference pair (see Fig. 1B). For the scenario involving chasing pairs and interspecific interference, we combine Eq. 1 with the following equation:

where stands for the interspecific interference pair (see Fig. 1B). In the scenario where chasing pairs and both intra- and interspecific interference are relevant, we combine Eqs. 13, and the populations of consumers and resources are given by and , respectively.

Generically, the consumption and interference processes are much quicker compared to the birth and death processes. Thus, in the derivation of the functional response, F(R_l, C_i) ≡ k_ilx_il/C_i, the consumption and interference processes are supposed to be in fast equilibrium. In all scenarios involving different types of pairwise encounters, the functional response in the B-D model is a good approximation only for a special case with d_il ≈ 0 and (see Appendix-fig. 2 and Appendix II for details).

To facilitate further analysis, we assume that the population dynamics of the resources follows the same construction rule as that in MacArthur’s consumer-resource model (MacArthur, 1969, 1970; Chesson, 1990). Then,

In the absence of consumers, biotic resources exhibit logistic growth. Here, η_l and κ_l represent the intrinsic growth rate and the carrying capacity of species R_l. For abiotic resources, ζ_l stands for the external resource supply rate of R_l, and κ_l is the abundance of R_l at a steady state without consumers. For simplicity, we focus our analysis on abiotic resources, although all results generally apply to biotic resources as well. By applying dimensional analysis, we render all parameters dimensionless (see Appendix VI). For convenience, we retain the same notations below, with all parameters considered dimensionless unless otherwise specified.

Intraspecific predator interference facilitates species coexistence and breaks CEP

To clarify the specific mechanisms that can facilitate species coexistence, we systematically investigate scenarios involving different forms of pairwise encounters in a simple case with S_C = 2 and S_R = 1. To simplify the notations, we omit the subscript/superscript “l” since S_R = 1. For clarity, we assign each consumer species of unique competitiveness by setting that the mortality rate D_i is the only parameter that varies with the consumer species.

First, we conduct the analysis within a deterministic framework using ordinary differential equations (ODEs). In the scenario involving only chasing pairs, consumer species cannot coexist at a steady state except for special parameter settings (sets of measure zero) (Wang and Liu, 2020). In practice, if all species coexist, the steady-state equations of the consumer species (Ċ_i = 0, i.e., the zero-growth isolines)yield f_l(R^(F)) = D_i(i = 1,2), where f_i(R^(F)) is defined as f_i(R^(F)) ≡ R^(F)/(R^(F)+ K_i) and K_i ≡ (d_i + k_i)/a_i. These equations form two parallel surfaces in the (C₁, C₂, R) coordinates, making steady coexistence impossible (Wang and Liu, 2020) (see Fig. 1C, F and Appendix-fig. 3A-C).

Meanwhile, in the scenario involving chasing pairs and interspecific interference, if all species coexist, the zero-growth isolines of the three species (see Eqs. S65) correspond to three nonparallel surfaces Ω’_i(R, C₁ ,C₂) = D_i(i = 1, 2), G’(R, C₁, C₂) = 0 (see Fig. 1G and Appendix-fig. 3D; refer to Appendix IV for definitions of Ω’_i and G’), which can intersect at a common point (fixed point). However, this fixed point is unstable (see Fig. 1G and Appendix-fig. 3D, F), and thus one of the consumer species is doomed to extinction (see Fig. 1D).

Next, we turn to the scenario involving chasing pairs and intraspecific interference. Likewise, steady coexistence requires (see Eqs. S38) that three non-parallel surfaces Ω_i (R, C₁, C₂) = D_i(i = 1, 2), G(R, C₁, C₂) = 0 cross ata common point (see Fig. 1H and Appendix-fig. 3G; refer to Appendix III for definitions of Ω_i and G). Indeed, this naturally happens, and encouragingly the fixed point can be stable. Therefore, two consumer species may stably coexist at a steady state with only one type of resources, which obviously breaks CEP (see Fig. 1E and Appendix-fig. 4A). In fact, the coexisting state is globally attractive (see Appendix-fig. 4A), and there exists a non-zero volume of parameter space where the two consumer species stably coexist at constant population densities (see Appendix-fig. 4B-C), demonstrating that the violation of CEP does not depend on special parameter settings. We further consider the scenario involving chasing pairs and both intra- and interspecific interference (see Appendix-fig. 5). Much as expected, the species coexistence behavior is very similar to that without interspecific interference.

Intraspecific interference promotes biodiversity in the presence of stochasticity

Stochasticity is ubiquitous in nature. However, it is prone to jeopardize species coexistence (Xue and Goldenfeld, 2017). Influential mechanisms such as “kill the winner” fail when stochasticity is incorporated (Xue and Goldenfeld, 2017). Consistent with this, we observe that two notable cases of oscillating coexistence (Koch, 1974; Huisman and Weissing, 1999) turn into species extinction when stochasticity is introduced (see Appendix-fig. 6A-B), where we simulate the models with stochastic simulation algorithm (SSA) (Gillespie, 2007) and adopt the same parameters as those in the original references (Koch, 1974; Huisman and Weissing, 1999).

Then, we proceed to investigate the impact of stochasticity on our model using SSA (Gillespie, 2007). In the scenario involving chasing pairs and intraspecific interference, species may coexist indefinitely in the SSA simulations (see Fig. 2A and Appendix-fig. 4D). In fact, the parameter region for species coexistence in this scenario is rather similar between the SSA and ODEs studies (see Appendix-fig. 6C-D). Similarly, in the scenario involving chasing pairs and both inter- and intraspecific interference, all species may coexist indefinitely in company with stochasticity (see Appendixfig. 5D).

Figure 2.

To further mimic a real ecosystem, we resort to individual-based modeling (IBM) (Grimm and Railsback, 2013; Vetsigian, 2017), an essentially stochastic simulation method. In the simple case of S_C = 2 and S_R = 1, we simulate the time evolution of a 2-D square system in a size of L² with periodic boundary conditions (see Materials and Methods for details). In the scenario involving chasing pairs and intraspecific interference, two consumer species coexist for long with only one type of resources in the IBM simulations (see Fig. 2B-C). Together with the SSA simulation studies, it is obvious that intraspecific interference still robustly promotes species coexistence when stochasticity is considered.

Comparison with experimental studies that reject CEP

In practice, two classical studies (Ayala, 1969; Park, 1954) reported that, in their respective laboratory systems, two species of insects coexisted for roughly years or more with only one type of resources. Evidently, these two experiments (Ayala, 1969; Park, 1954) are incompatible with CEP, while factors such as temporal variations, spatial heterogeneity, cross-feeding, etc. are clearly not involved in such systems. As intraspecific fighting is prevalent among insects (Boomsma et al., 2005; Dankert et al., 2009; Chen et al., 2002), we apply the model involving chasing pairs and intraspecific interference to simulate the two systems. Overall, our SSA results show good consistency with those of the experiments (see Fig. 2D-E and Appendix-figs. 7). The fluctuations in experimental time series can be mainly accounted by stochasticity.

A handful of resource species can support a wide range of consumer species regardless of stochasticity

To resolve the puzzle stated in the paradox of the plankton, we analyze the generic case where S_c consumer species compete for S_R resource species (with S_C > S_R) within the scenario involving chasing pairs and intraspecific interference. The population dynamics is described by equations combining Eqs. 1, 2, and 4. As with the cases above, each consumer species is assigned a unique competitiveness through a distinctive D_i(i = 1, …, S_C).

Strikingly, a plethora of consumer species may coexist at a steady state with only one resource species (S_C ≫ S_R, S_R = 1) in the ODEs simulations, and crucially, the facilitated biodiversity can still be maintained in the SSA simulations. The long-term coexistence behavior is exemplified in Fig. 3 and Appendix-fig. 8–10, involving simulations with or without stochasticity. The number of consumer species in long-term coexistence can be up to hundreds or more (see Fig. 3 and Appendixfig. 8). To mimic real ecosystems, we further analyze cases with more than one type of resources, such as systems with S_R = 3 (S_C ≫ S_R). Just like the case of S_R = 1 (S_C ≫ S_R), an extensive range of consumer species may coexist indefinitely regardless of stochasticity (see Fig. 3 and Appendixfigs. 11–14).

Figure 3.

We further analyze the scenario involving chasing pairs and both intra- and interspecific interference, where multiple consumer species compete for one resource species. Similar to the scenario involving chasing pairs and intraspecific interference, all species coexist indefinitely in either ODEs or SSA simulation studies (see Appendix-fig. 5F-H for the cases of S_C = 6, 20 and S_R = 1).

Intuitive understanding: an underlying negative feedback loop

For the case with only one resource species (S_R = 1), if the total population size of the resources is much larger than that of the consumers (i.e., ), the functional response F ≡ k_ix_i/C_i and the steady-state population of each consumer and resource species can be obtained analytically (see Appendix III B-C for details). In fact, the functional response of a consumer species (e.g., C_i) is negatively correlated with its own population size:

where β_i ≡ a’_i/d’_i. The analytical steady-state solutions are highly consistent with the numerical results (see Fig. 1E and Appendix-fig. 3H-I) and can even quantitatively predict the coexistence region of the parameter space (see Appendix-fig. 3I).

Intuitively, the mechanisms of how intraspecific interference facilitates species coexistence can be understood from the underlying negative feedback loop. Specifically, for consumer species of higher competitiveness (e.g., C_i) in an ecological community, as the population size of C_i increases during competition, a larger portion of C_i individuals are then engaged in intraspecific interference pairs which are temporarily absent from hunting (see Eq. S59 and Appendix-fig. 15A-B). Consequently, the fraction of C_i individuals within chasing pairs decreases (see Eq. S59 and Appendixfig. 15A-B) and thus form a self-inhibiting negative feedback loop through the functional response (see Eq. S59 and Appendix-fig. 15C). This negative feedback loop prevents further increases in C_i populations, results in an overall balance among the consumer species, and thus promotes biodiversity (see Appendix III C for details).

The S shape pattern of the rank-abundance curves in a broad range of ecological communities

As mentioned above, a prominent feature of biodiversity is that the species’ rank-abundance curves follow a universal S-shaped pattern in the linear-log plot across a broad spectrum of ecological communities (Fuhrman et al., 2008; Ser-Giacomi et al., 2018; Cody and Smallwood, 1996; Terborgh et al., 1990; Marti’nez et al., 2023; Clarke et al., 2005; Hubbell, 2001; Devries et al., 1997). Previously, this pattern was mostly explained by the neutral theory (Hubbell, 2001), which requires special parameter settings that all consumer species share identical fitness. To resolve this issue, we apply the model involving chasing pairs and intraspecific interference to simulate the ecological communities, where one or three types of resources support a large number of consumer species (S_C ≫ S_R). In each model system, the mortality rates of consumer species follow a Gaussian distribution where the coefficient of variation was taken around 0.3 (Menon et al., 2003) (see Appendix VIII for details). For a broad array of the ecological communities, the rank-abundance curves obtained from the long-term coexisting state of both the ODEs and SSA simulation studies agree quantitatively with those of experiments (see Fig. 3C-D and Appendix-figs. 8–14), sharing roughly equal Shannon entropies and mostly being regarded as identical distributions in the Kolmogorov-Smirnov (K-S) statistical test (with a significance threshold of 0.05). Still, there is a noticeable discrepancy between the experimental data and SSA studies in terms of the species’ absolute abundances (e.g., see Appendix-fig. 8C): those with experimental abundances less than 10 tend to be extinct in the SSA simulations. This is due to the fact that the recorded individuals in an experimental sample are just a tiny portion of that in the real ecological system, whereas the species population size in a natural community is certainly much larger than 10.

Discussion

The conflict between the CEP and biodiversity, exemplified by the paradox of the plankton (Hutchinson, 1961), is a long-standing puzzle in ecology. Although many mechanisms have been proposed to overcome the limit set by CEP (Hutchinson, 1961; Chesson, 2000; Levins, 1979; Levin, 1974; Koch, 1974; Huisman and Weissing, 1999; Lczárán et al., 2002; Goyal and Maslov., 2018; Goldford et al., 2018; Martín et al., 2020; Gupta et al., 2021; Thingstad, 2000; Wang and Liu, 2020; Dalziel et al., 2021; Posfai et al., 2017; Weiner et al., 2019; Xue and Goldenfeld, 2017; Beddington, 1975; DeAngelis et al., 1975; Arditi and Ginzburg, 1989; Kelsic et al., 2015; Grilli et al., 2017; Ratzke et al., 2020), it is still unclear how plankton and many other organisms can flout CEP and maintain biodiversity in quasi-well-mixed natural ecosystems. To address this issue, we investigate a mechanistic model with detailed consideration of pairwise encounters. Using numerical simulations combined with mathematical analysis, we identify that the intraspecific interference among the consumer individuals can promote a wide range of consumer species to coexist indefinitely with only one or a handful of resource species through the underlying negative feedback loop. By applying the above analysis to real ecological systems, our model naturally explains two classical experiments that reject CEP (Ayala, 1969; Park, 1954), and quantitatively illustrates the universal S-shaped pattern of the rank-abundance curves for a broad range of ecological communities (Fuhrman et al., 2008; Ser-Giacomi et al., 2018; Cody and Smallwood, 1996; Terborgh et al., 1990; Martínez et al., 2023; Clarke et al., 2005; Hubbell, 2001; Devries et al., 1997).

In fact, predator interference has been introduced long ago by the classical B-D phenomenological model (Beddington, 1975; DeAngelis et al., 1975). However, the functional response of the B-D model involving intraspecific interference can be formally derived from the scenario involving only chasing pairs without consideration of pairwise encounters between consumer individuals (Wang and Liu, 2020; Huisman and Boer, 1997) (see Eqs. S8 and S24). Yet, it has been demonstrated that the scenario involving only chasing pairs is under the constraint of CEP (Wang and Liu, 2020) (see Appendix-fig. 3A-C). Therefore, it is questionable regarding the validity of applying the B-D model to break CEP (Cantrell et al., 2004; Hsu et al., 2013). From a mechanistic perspective, we resolve these issues and show that the B-D model corresponds to a special case of our mechanistic model yet without the escape rate (see Appendix-fig. 2 and Appendix II for details).

Our model is broadly applicable to explain biodiversity in many ecosystems. In practice, many more factors are potentially involved, and special attention is required to disentangle confounding factors. In microbial systems, complex interactions are commonly involved (Goyal and Maslov., 2018; Goldford et al., 2018; Hu et al., 2022), and species’ preference for food is shaped by the evolutionary course and environmental history (Wang et al., 2019). It is still highly challenging to fully explain how organisms evolve and maintain biodiversity in diverse ecosystems.

Methods and Materials

Derivation of the encounter rates with the mean-field approximation

In the model depicted in Fig. 1A, consumers and resources move randomly in space, which can be regarded as Brownian motions. At moment t, a consumer individual of species C_i(i = 1, …, S_C) moves at speed with velocity (t), while a resource individual of species R_l(l = 1, …, S_R) moves at speed with velocity (t). Here and are two invariants, while the directions of (t) and (t) change constantly. The relative velocity between the two individuals is , with a relative speed of . Then, , where represents the angle between (t) and (t). This system is homogeneous, thus, , where the overline stands for the temporal average. Then, we obtain the average relative speed between the C_i and R_l individuals: . Likewise, the average relative speed between the C_i and C_j individuals is . Evidently, . Meanwhile, the concentrations of species C_i and R_l in a 2-D square system with a length of L are and , while those of the freely wandering C_i and R_l are and .

Then, we use the mean-field approximation to calculate the encounter rates a_il and a’_ij in the well-mixed system. In particular, we estimate a_il by tracking a randomly chosen consumer individual from species C_i and counting its encounter frequency with the freely wandering individuals from resource species R_l (see Appendix-fig. 1). At any moment, the consumer individual may form a chasing pair with a R_l individual within a radius of (see Fig. 1A). Over a time interval of Δt, the number of encounters between the consumer individual and R_l individuals can be estimated by the encounter area and the concentration , which takes the value of (see Appendix-fig. 1). Combined with , for all freely wandering C_i individuals, the number of their encounters with R^(F) during interval Δt is . Meanwhile, in the ODEs, this corresponds to . Comparing both terms above, for chasing pairs, we have . Likewise, for interference pairs, we obtain . In particular, .

Stochastic simulations

To investigate the impact of stochasticity on species coexistence, we use the stochastic simulation algorithm (SSA) (Gillespie, 2007) and individual-based modeling (IBM) (Vetsigian, 2017; Grimm and Railsback, 2013) in simulating the stochastic process. In the SSA studies, we follow the standard Gillespie algorithm and simulation procedures (Gillespie, 2007).

In the IBM studies, we consider a 2D square system with a length of L and periodic boundary conditions. In the case of S_C = 2 and S_R = 1, consumers of species C_i(i = 1,2) move at speed , while the resources move at speed v_R. The unit length is Δl = 1, and all individuals move probabilistically. Specifically, when Δt is small so that Δt ≪ 1, C_i individuals jump a unit length with the probability Δt. In practice, we simulate the temporal evolution of the model system following the procedures below.

Initialization.

We choose the initial position for each individual randomly from a uniform distribution in the square space, which rounds to the nearest integer point in the x–y coordinates.

Moving.

We choose the destination of a movement randomly from four directions (x-positive, x-negative, y-positive, y-negative) following a uniform distribution. The consumers and resources jump a unit length with probabilities Δt and v_RΔt respectively.

Forming pairs.

When a C_i individual and a resource individual get close in space within a distance of r^(C), they form a chasing pair. Meanwhile, when two consumer individuals C_i and C_j stand within a distance of r^(I), they form an interference pair.

Dissociation.

We update the system with a small time step Δt so that d_iΔt, k_iΔt, d’_ijΔt ≪ 1 (i, j = 1,2). In practice, a random number ς is sampled from a uniform distribution between 0 and 1, i.e., U(0,1). If ς < d_iΔt or ς < d’_ijΔt, then the chasing pair or interference pair dissociates into two separated individuals. One occupies the original position, while the other individual moves just out of the encounter radius in a uniformly distributed random angle. For a chasing pair, if d_iΔt < ς < (d_i + k_i)Δt, then, the consumer catches the resource, and the biomass of the resource flows into the consumer populations (updated according to the birth procedure), while the consumer individual occupies the original position. Finally, if ς > (d_i + k_i)Δt or ς > d’_ijΔt, the chasing pair or interference pair maintains the current status.

Birth and death.

For each species, we use two separate counters with decimal precision to record the contributions of the birth and death processes, both of which accumulate in each time step. The incremental integer part of the counter will trigger updates in this run. Specifically, a newborn would join the system following the initialization procedure in a birth action, while an unfortunate target would be randomly chosen from the living population in a death action.

Data and materials availability

All data and codes for this paper have been deposited on GitHub: https://github.com/SchordK/Intraspecific-predator-interference-promotes-biodiversity-in-ecosystems.

Author contributions

X.W. conceived and designed the project; J.K., S.Z., Y.N., F.Z., and X. W. performed research; J.K. and X. W. analyzed data and wrote the paper.

Competing interests

The authors declares no competing interests.

Acknowledgements

We thank Roy Kishony, Eric D. Kelsic and Yang-Yu Liu for helpful discussions. This work was supported by National Natural Science Foundation of China (Grant No.12004443), Guangzhou Municipal Innovation Fund (Grant No.202102020284) and the Hundred Talents Program of Sun Yat-sen University.

Appendix I

The classical proof of Competitive Exclusion Principle (CEP)

In the 1960s, MacArthur (MacArthur and Levins, 1964) and Levin (Levin, 1970) put forward the classical mathematical proof of CEP. We rephrase their idea in the simple case of S_C = 2 and S_R = 1, i.e., two consumer species C₁ and C₂ competing for one resource species R. In practice, this proof can be generalized into higher dimensions with several consumer and resource species. The population dynamics of the system can be described as follows:

Here C_i and R represent the population abundances of consumers and resources, respectively, while the functional forms of f_i(R) and g(R, C₁, C₂) are unspecific. D_i stands for the mortality rate of the species C_i. If all consumer species can coexist at steady state, then f_i(R)/D_i = 1 (i = 1,2). In a 2-D representation, this requires that three lines y = f_i(R)/D_i (i = 1,2)and y = 1 share a common point, which is commonly impossible unless the model parameters satisfy special constraint (sets of Lebesgue measure zero). In a 3-D representation, the two planes corresponding to f_i(R)/D_i = 1 (i = 1,2) are parallel, and hence do not share a common point (see Ref. (Wang and Liu, 2020) for details).

Appendix II

Comparison of the functional response with Beddington-DeAngelis (B-D) model

A B-D model

In 1975, Beddington proposed a mathematical model (Beddington, 1975) to describe the influence of predator interference on the functional response with hand-waving derivations. In the same year, DeAngelis and his colleagues considered a related question and put forward a similar model (DeAngelis et al., 1975). Essentially, both models are phenomenological, and they were called B-D model in the subsequent studies. In practice, the B-D model can be extended into scenarios involving different types of pairwise encounters with Beddington’s modelling method. In this section, we systematically compare the functional response in B-D model with that of our mechanistic model in all the relevant scenarios.

Recalling Beddington’s analysis, the model (Beddington, 1975) consists of one consumer species C and one resource species R(S_C = 1, S_R = 1). In a well-mixed system, an individual consumer meets a resource with rate a, while encounters another consumer with rate a’. There are two other phenomenological parameters in this model, namely, the handling time t_h and the wasting time t_w. Both can be determined by specifying the scenario and using statistical physics modeling analysis. In fact, Beddington analyzed the searching efficiency Ξ_B-D rather than the functional response F_B-D, yet both can be reciprocally derived with Ξ_B-D ≡ F_B-D/R. Here R stands for the population abundance of the resources, and the specific form of Ξ_B-D is (Beddington, 1975):

where C’ = C - 1, and C stands for the population abundance of the consumes. Generally, C ≫ 1, and thus C’ ≈ C.

B Scenario involving only chasing pairs

Here we consider the scenario involving only chasing pair for the simple case with one consumer species C and one resource species R(S_C = 1, S_R = 1). When an individual consumer is chasing a resource, they form a chasing pair:

where the superscript “(F)” stands for populations that are freely wandering, and “(+)” signifies gaining biomass (we count C(^F)(+) as C^(F)). C^(P) ∨ R^(P) represents chasing pair (where “(P)” signifies pair), denoted as x. a, d and k stand for encounter rate, escape rate and capture rate, respectively. Hence, the total number of consumers and resources are C ≡ C^(F) + x and R ≡ R^(F) + x. Then, the population dynamics of the system follows:

Here the functional form of g(R, x, C) is unspecific, while D and w represent the mortality rate of the consumer species and biomass conversion ratio (Wang and Liu, 2020), respectively. Since consumption process is generically much faster than the birth/death process, in deriving the functional response, the consumption process is supposed to be in fast equilibrium (i.e., ẋ = 0). Then, we can solve for x with:

where , and then,

By definition, the functional response and searching efficiency are:

Hence, we obtain the functional response and searching efficiency in this chasing-pair scenario:

Since , using first order approximations in Eq. S7, we obtain . Then the functional response and searching efficiency are:

Evidently, there is no predator interference within the chasing-pair scenario, yet the functional response form is identical to the B-D model involving intraspecific interference (see Eq. S2). Meanwhile, using first order approximations in the denominator of Eq. S5, we have .

Hence,

In the case that R ≫ C, then R ≫ C > x = R - R^(F). By applying R ≈ R^(F) in Eq. S3, we obtain . Then,

To compare these functional responses with that of the B-D model, we determine the parameters t_h and t_w in the B-D model by calculating their ensemble average values in a stochastic framework. Using the properties of waiting time distribution in the Poisson process, we obtain and (in the chasing-pair scenario, a’ = 0). By substituting these calculations into Eq. S2, we have:

In the special case with d = 0 and R ≫ C, the B-D model is consistent with our mechanistic model: Ξ_B-D(R ,C) = Ξ_CP(R ,C)₍₄₎. Outside the special region, however, the discrepancy can be considerably large (see Appendix-fig. 2A-C for the comparison).

C Scenario involving chasing pairs and intraspecific interference

Here we consider the scenario with additional involvement of intraspecific interference in the simple case of S_C = 1 and S_R = 1:

Here C^(P) ∨ C^(P) stands for the intraspecific predator interference pair, denoted as y; a’ and d’ represent the encounter rate and separation rate of the interference pair, respectively. Then, the total population of consumers and resources are C ≡ C^(F) + x + 2y and R ≡ R^(F) + x. Hence the population dynamics of the consumers and resources can be described as follows:

The consumption process and interference process are supposed to be in fast equilibrium (i.e., ẋ = 0,ẏ = 0), then we can solve for x with:

where ϕ₀ = -C R², ϕ₁ = 2C R + K R + R², ϕ₂ = 2βK² - K - C - 2R, with β = a’/d’. The discriminant of Eq. S13 (denoted as Λ) is:

with Ψ = ϕ₁ - (ϕ₂)²/3 and φ = ϕ₀ - ϕ₁ϕ₂/3 + 2(ϕ₂)³/27. When Λ < 0, there are one real solution x₍₁₎ and two complex solutions x₍₂₎, x₍₃₎, which are:

where (i stands for the imaginary unit), , and . On the other hand, when Λ > 0, there are three real solutions x₍₁₎, x₍₂₎, and x₍₃₎, which are:

where ψ’ = (-4ψ/3)^1/2 and φ’ = arccos(-(-ψ/3)^-3/2φ/2)/3. Note that x ∈ [0, min(R, C)], then we obtain the exact feasible solution of x (denoted as x_ext), and hence the functional response and searching efficiency are:

In the case of R ≫ C, then R - R^(F) = x < C ≪ R, and thus R^(F) ≈ R. Still, the consumption process is supposed to be in fast equilibrium (i.e., ẋ = 0, ẏ = 0), and then we obtain:

Consequently,

When or 8βC/(1+R/K)² ≪ 1, using first order approximations in the denominator of Eq. S18, we have:

and then,

In the case that 8βC/(1 + R/K)² ≫ 1, using first order approximations in Eq. S18, we obtain:

and thus,

Meanwhile, the B-D model only fits to the cases with d = 0. By calculating the average values of t_h and t_w in the stochastic framework, we have , . Thus, we obtain the searching efficiency and functional response in the B-D model:

Overall, the searching efficiency (and the functional response) of the B-D model is quite different from either the rigorous form Ξ_intra(R, C)₍₁₎, the quasi rigorous form Ξ_intra(R, C)₍₂₎, or the more simplified forms Ξ_intra(R, C)₍₃₎ and Ξ_intra(R, C)₍₄₎ (Appendix-fig. 2D-F). Still, there is a region where the discrepancies can be small, namely d ≈ 0 and R ≫ C (Appendix-fig. 2D-F). Intuitively, when and d = 0, then . Consequently, if , then . In this case, the difference between and is small.

In fact, the above analysis also applies to cases with more than one types of consumer species (i.e., for cases with S_C > 1).

D Scenario involving chasing pairs and interspecific interference

Next, we consider the scenario involving chasing pairs and interspecific interference in the case of S_C = 2 and S_R = 1:

Here stands for the interspecific interference pair, denoted as z; a’₁₂ and d’₁₂ represent the encounter rate and separation rate ofthe interference pair, respectively. Then, the total population of consumers and resources are and . The population dynamics of the consumers and resources follows:

where the functional form of g(R, x₁, x₂, C₁, C₂) is unspecific, while D_i and w_i represents the mortality rates of the two consumers species and biomass conversion ratios. Still, the consumption/interference process is supposed to be in fast equilibrium, i.e., ẋ_i = 0, ż = 0. In the case that R ≫ C₁ + C₂ > x₁ + x₂, by applying R^(F) ≈ R, we obtain:

Then, the searching efficiencies and functional responses are:

Since , by applying first order approximation to the denominator of Eq. S26, we obtain:

and the searching efficiencies and functional responses are:

Likewise, the B-D model only fits to cases with d = 0. By calculating the average values in a stochastic framework, we obtain , . Then, we obtain the searching efficiencies in the B-D model:

Consequently, the functional responses in the B-D model are:

Evidently, the searching efficiencies in the B-D model are overall different from either the quasi rigorous form Ξ_t(R, C₁, C₂)₁, or the simplified form Ξ_i(R, C₁, C₂)₂ (Appendix-fig. 2G-I). Still, the discrepancy can be small when d ≈ 0 and R ≫ C (Appendix-fig. 2G-I). Intuitively, when we have:

Thus, if , then . In this case, the difference between and is small.

Appendix III

Scenario involving chasing pairs and intraspecific interference

A Two consumers species competing for one resource species

We consider the scenario involving chasing pairs and intraspecific interference in the simple case of S_C = 2 and S_R = 1:

Here, the variables and parameters are just extended from the case of S_C = 1 and S_R = 1 (see Appendix II.). The total number of consumers and resources are and . Then, the population dynamics of the consumers and resources can be described as follows:

The functional form of g(R, x₁, x₂, C₁, C₂) is unspecified. For simplicity, we limit our analysis to abiotic resources, while all results generically apply to biotic resources. Besides, we define K_i ≡ (d_i + k_i)/a_i, α_i ≡ D_i/(w_ik_i), and β_i ≡ a’_i/d’_i(i = 1,2). At steady state, from ẋ_i = 0, γ_i = 0, we have:

Note that and Then,

By substituting Eq. S35a into Eq. S35b, we have:

Then, we can present with C₁, C₂ and R(i = 1,2). By further combining with Eqs. S34, S35a and S36a, we express R^(F), x_i , and y_i using C₁, C₂ and R. In particular, for x_i , we have:

If all species coexist, then the steady-state equations of Ċ_i = 0 (i = 1,2) and Ṙ = 0 are:

where G(R, C₁ ,C₂) ≈ g(R, u₁(R, C₁ ,C₂), u₂(R, C₁, C₂), C₁ ,C₂), and . In practice, Eq. S38 corresponds to three unparallel surfaces, which share a common point (Fig. 1H and Appendix-fig. 3G). Importantly, the fixed point can be stable, and hence two consumer species may coexist at constant population densities.

1 Stability analysis of the fixed-point solution

We use linear stability analysis to study the local stability of the fixed point. Specifically, for an arbitrary fixed point E(x₁, x₂, y₁, y₂, C₁ ,C₂ ,R), only when all the eigenvalues (defined as λ_i ,i = 1, …,7) of the Jacobian matrix at point E own negative real parts would the point be locally stable.

To investigate whether there exists a non-zero measure parameter region for species coexistence, we set D_i (i = 1, 2) to be the only parameter that varies with species C₁ and C₂, and then Δ ≡ (D₁ - D₂)/D₂ reflects the completive difference between the two consumer species. As shown in Appendix-fig. 4B, the region below the blue surface and above the red surface corresponds to stable coexistence. Thus, there exists a non-zero measure parameter region to promote species coexistence, which breaks CEP.

2 Analytical solutions of the species abundances at steady state

At steady state, since ẋ_i = ẏ_i, = Ċ_i = 0 (i = 1, 2), then,

Meanwhile, , and C_i, R > 0 (i = 1,2). Then, we have:

If the resource species owns a much larger population abundance than the consumers (i.e., R ≫ C₁ + C₂), then R ≫ x₁ + x₂, and R^(F) ≈ R. Thus,

By further assuming that the population dynamics of the resources follow identical construction rule as the MacArthur’s consumer-resource model (MacArthur, 1970), we have:

Since Ṙ = 0, then,

where and .

Eqs. S41, S43 are the analytical solutions of species abundances at steady state when R ≫ C₁ + C₂. As shown in Fig. 1E, the analytical solutions agree well with the numerical results (the exact solutions). To conduct a systematic comparison for different model parameters, we assign D_i (i = 1,2) to be the only parameter varying with species C₁ and C₂ (D₁ > D₂), and define Δ ≡ (D₁ - D₂)/D₂ as the competitive difference between the two consumer species. The comparison between analytical solutions and numerical results is shown in Appendix-fig. 3H. Clearly, they are close to each other, exhibiting very good consistency.

Furthermore, we test if the parameter region for species coexistence is predictable using the analytical solutions. Since D_i (i = 1,2) is the only parameter that varies with the two-consumer species, the supremum of the competitive difference tolerated for species coexistence (defined as ) corresponds to the steady-state solutions that satisfy R, C₂ > 0 and C₁ = 0⁺, where 0⁺ stands for the infinitesimal positive number. To calculate the analytical solutions at the upper surface of the coexistence region, where and C₁ = 0⁺, we further combine Eq. S41 and then obtain (note that R > 0):

Meanwhile, α₁ = α₂(Δ + 1). Thus, for the upper surface of the coexistence region:

Combining Eqs. S43–S45, we have:

where When R ≫ C₁, + C₂, the comparison of obtained from analytical solutions with that from numerical results (the exact solutions) are shown in Appendix-fig. 3I, which overall exhibits good consistency.

B S_C consumers species competing for S_R resources species

Here we consider the scenario involving chasing pairs and intraspecific interference for the generic case with S_C types of consumers and S_R types of resources. Then, the population dynamics of the system can be described as follows:

Note that Eq. S47 is identical with Eqs. 1–2, and we use the same variables and parameters as that in the main text. Then, the populations of the consumers and resources are and . For convenience, we define K_il ≡ (d_il + k_il)/a_il,α_il ≡ D_il/(k_ilw_il) and β_i ≡ a’_ii/d’_ii (i=1,…,S_c,l = 1,…,S_R).

1 Analytical solutions of species abundances at steady state

At steady state, from ẋ_il = 0, ẏ_i = 0, and Ċ_i = 0, we have:

Meanwhile , and note that C_i > 0, thus,

Combined with Eq. S49, and then,

We further assume that the specific function of g_i({R_l}, {x_i}, {C_i}) satisfies Eq. 4, i.e.,

By combining Eqs. S48, S49 and S51, we have:

If the population abundance of each resource species is much more than the total population of all consumers (i.e., ), then and . Thus,

with l = 1, …, S_R. Eq. S53 is a set of second-order algebraic differential equations, which is clearly solvable.

In fact, when S_R = 1, S_c ≥ 1, and , we can explicitly present the analytical solution of the steady-state species abundances. To simplify the notations, we omit the “l” in the sub-/super-scripts since S_R = 1.Then, we have:

Here and .

C Intuitive understanding: an underlying negative feedback loop

Intuitively, how can intraspecific predator interference promote biodiversity? Here we solve this question by considering the case that S_C types of consumers compete for one resource species. The population dynamics of the system are described in Eqs. S47 and S51 with S_R = 1. To simplify the notations, we omit the “l” in the subscript since S_R = 1. The consumption process and interference process are supposed to be in fast equilibrium (i.e., ẋ_i = 0, ẏ_i = 0). Then, we have a set of equations to solve for x_i and y_i given the population size of each species:

In the first three sub-equations of Eq. S55, by getting rids of we have,

Then, by regarding R^(F) as a temporary parameter, we solve for x_i and y_i:

If the total population size of the resources is much larger than that of consumers (i.e., ), then and R^(F) ≈ R, and thus we get the analytical expressions of x_i and y_i:

Note that the fraction of C_i individuals engaged in chasing pairs is x_i/C_i, while that for individuals trapped in intraspecific interference pairs is y_i/C_i. With Eq. S58, it is straightforward to obtain these fractions:

where both x_i/C_i and y_i/C_i are bivariate functions of R and C_i. From Eq. S59, it is clear that for a given population size of the resource species, y_i/C_i is a monotonously increasing function of C_i, while x_i/C_i is a monotonously decreasing function of C_i. In Appendix-fig. 15A-B, we see that the analytical results are highly consistence with the exact numerical solutions. By definition, the functional response of C_i species is F ≡ k_ix_i/C_i, and thus,

Evidently, the function response of C_i species is negatively correlated with the population size of itself, which effectively constitutes a self-inhibiting negative feedback loop (Appendix-fig. 15C).

Then, we have a simple intuitive understanding of species coexistence through the mechanism of intraspecific interference. In an ecological community, consumer species that of higher/lower competitiveness tend to increase/decrease their population size in the competition process. Without intraspecific interference, the increasing/decreasing trend would continue until the system obeys CEP. In the scenario involving intraspecific interference, however, for species of higher competitiveness (e.g., C_i), with the increase of C_i’s population size, a larger portion of C_i individuals are then engaged in intraspecific interference pair which are temporarily absent from hunting (Appendix-fig. 15A-B). Consequently, the functional response of C_i drops, which prevents further increase of C_i’s population size, results in an overall balance among the consumer species, and thus promotes species coexistence.

Appendix IV

Scenario involving chasing pairs and interspecific interference

Here we consider the scenario involving chasing pairs and interspecific interference in the case of S_C = 2 and S_R = 1, with all settings follow that depicted in Appendix II.. Then, , and the population dynamics follows (identical with Eq. S25):

Here the functional form of g(R, x₁, x₂, C₁, C₂) is unspecified. For convenience, we define K_i ≡ (d_i + k_i)/a_i, α_i ≡ D_i/(w_ik_i)(i = 1,2), and γ ≡ a’₁₂/d’₁₂. At steady state, from ẋ_i = 0(i = 1, 2) and ż = 0, we have:

Note that and R ≡ R^(F) + x₁ + x₂, then,

Then, we can express and R^(F) with C₁, C₂ and R. Combined with Eq. S62, x_i and z can also be expressed using C₁ ,C₂ and R. In particular, for x_i, we have:

If all species coexist, by defining , then, the steady-state equations of Ċ_i = 0(i = 1,2) and Ṙ = 0 are:

where G’(R, C₁, C₂) ≡ g (R, u’₁(R, C₁, C₂), u’₂(R, C₁, C₂), C₁, C₂).

Here, Eq. S65 corresponds to three unparallel surfaces and share a common point (Fig. 1G and Appendix-fig 3A). However, all the fixed points are unstable (Appendix-fig. 3F), and hence the consumer species cannot stably coexist at steady state (Fig. 1D).

A Analytical results of the fixed-point solution

We proceed to investigate the unstable fixed point where R, C₁, C₂ > 0. From ẋ_i = 0 (i = 1,2), ż = 0, Ċ_i = 0, and note that , we have:

Since C_i > 0, then:

If R ≫ C₁ + C₂, then R ≫ x₁ + x₂ and R^(F) ≈ R, we have:

Still, we assume that the population dynamics of the resource species follows Eq. S42. At the fixed point, Ṙ = 0. We have:

Combined with Eq. S68, we can solve for R :

where and .

Eqs. S68, S70 are the analytical solutions of the fixed point when R ≫ C₁ + C₂. As shown in Appendix-fig. 3E, the analytical predictions agree well with the numerical results (the exact solutions).

Appendix V

Scenario involving chasing pairs and both intra- and interspecific interference

Here we consider the scenario involving chasing pairs and both intra- and inter-specific interference in the simple case of S_C = 2 and S_R = 1:

We adopt the same notations as that depicted in Appendix III. and. Then, and R ≡ R^(F) + x₁ + x₂, and the population dynamics of the system can be described as follows:

Here, the functional form of g(R, x₁, x₂, C₁, C₂) follows Eq. S42. For convenience, we define K_i ≡ (d_i + k_i)/a_i,α_i ≡ D_i/(w_ik_i), β_i ≡ a_i/ d’_i, and γ ≡ a’₁₂/d’₁₂, (i = 1,2). At steady state, from ẋ_i =0, ẏ_i =0, ż = 0, and Ċ_i = 0, (i = 1, 2), we have:

Combined with , and since C_i > 0(i = 1,2), then,

A Analytical solutions of species abundances at steady state

If R ≫ C₁ + C₂, then R ≫ x₁ + x₂ and thus R^(F) ≈ R. Combined with Eq. S73, we obtain:

Using Ṙ = 0 and R > 0, we have:

where , and

Eqs. S74–S75 are the analytical solutions of the species abundances at steady state when R ≫ C₁ + C₂. As shown in Appendix-fig. 5E, the analytical calculations agree well with the numerical results (the exact solutions).

B Stability analysis of the coexisting state

In the scenario involving chasing pairs and both intra- and inter-specific interference, the behavior of species coexistence is similar to that without interspecific interference. Evidently, the influence of interspecific interference would be negligible if d’₁₂ is extremely large, and vice versa for intraspecific interference if both d’¹ and d’₂ are tremendous. In the deterministic framework, the two-consumer species may coexist at constant population densities (Appendix-fig. 5B), and the fixed points are globally attracting (Appendix-fig. 5C). Furthermore, there is a non-zero measure of parameter set where both consumer species can coexist at steady state with only one type of resources (Appendix-fig. 5A). In the stochastic framework, just as the scenario involving chasing pairs and intraspecific interference, the coexistence state can be maintained along with stochasticity (Appendix-fig. 5D).

Appendix VI

Dimensional analysis for the scenario involving chasing pairs and both intra- and inter-specific interference

The population dynamics of the system involving chasing pairs and both intra- and inter-specific interference are shown in Eqs. 1–4:

with l = 1, …, S_R; i, j = 1, …, S_C, and i ≠ j. Here and represent the population abundances of the consumers and resources in the system. In fact, there are already several dimensionless variables and parameter in Eq. S76, namely x_il, y_i, z_ij, , , C_i, R_l, w_iI, k_l. To make all terms dimensionless, we define , where and is a reducible dimensionless parameter which is freely to take any positive values. Besides, we define dimensionless parameters , , , , , , , and . By substituting all the dimensionless terms into Eq. S76, we have:

For convenience, we omit the notation “~” and use dimensionless variables and parameters in the simulation studies unless otherwise specified.

Appendix VII

Approximations applied in the pairwise encounter model

For consumers within a paired state, either in a chasing pair or an interference pair, the consumer may die following the mortality rate. Thus, in the scenario involving chasing pairs and both intra- and inter-specific interference, the population dynamics of the system should be described as follows:

However, since predation or interference processes are generally much faster than birth and death processes, i.e., D_i << k_il, d_il, d’_i, d’_ij, the influence of mortality rate in a paired state is negligible. Therefore, we have used the following approximations throughout our manuscript: (k_il + d_il + D_i) ≈ (k_il,+d_il), (d’_i+ D_i) ≈ d’_i,(d’_ij + D_i + D_j) ≈ d’_ij. Hence, the approximated population dynamics is described as follows:

which is identical to those shown in the main text.

Appendix VIII

Simulation details of the main text figures

In Fig. 1C, F: a_i = 0.1, d_i = 0.5, w_i = 0.1, k_i = 0.1 (i = 1,2); D₁ = 0.002,D₂ = 0.001, κ = 5, ζ = 0.05. In Fig. 1D, G: a_i = 0.02, a’. = 0.021, d_i = 0.5, d’_ij = 0.01, w_i = 0.08, k_i = 0.03, i,j = 1,2, i ≠ j,D₂ = 0.001, D 1 = 0.0011 κ = 20, ζ = 0.01. In Fig. 1E, H: a_i = 0.5, a’_i = 0.625, d_i = 0.5, d’_i = 0.02, w_i = 0.2, k_i = 0.4 (i = 1,2), D₁ = 0.0286, D₂ = 0.022, κ =10, ζ = 0.5. Fig. 1C, F were calculated or simulated from Eqs. 1, 4. Fig. 1D, G were calculated or simulated from Eqs. 1, 3, 4. Fig. 1E, H were calculated or simulated from Eqs. 1, 2, 4. The analytical solutions in Fig. 1E were calculated from Eqs. S41 and S43.

In Fig. 2A: a_i = 0.02, a_i^’ = 0.025, d_i = 0.7, d_i^’ = 0.7, w_i = 0.4, k_i = 0.05 (i = 1,2); D₁ = 0.0160, D₂ = 0.0171, κ = 2000, ζ = 5.5. In Fig. 2B-C: L = 100, r^(C) = 5,r^(I) = 5, , v_R = 0.1, a_i = 0.2010, a’_i = 0.2828, d’_i = 0.8, d_i = 0.7, w_i = 0.33, k_i = 0.2 (i = 1,2); D₁ = 0.0605, D₂ = 0.0600, κ = 1000, ζ = 100. In Fig. 2D: a_i = 0.3, a’_i = 0.33, w_i = 0.018, k_i = 4.8, d’_i = 5, d_i = 5.5 (i = 1,2); D₂ = 0.010, ζ = 35, κ = 10000, D₁ = 0.011. In Fig. 2E: w_i = 0.02, k_i = 4.5, d’_i = 4, d_i = 4.5 (i = 1,2);D₂ = 0.010, ζ = 35, κ = 10000, a_i = 0.2, a’_i = 0.24 (i = 1,2); D 1 = 0.0120. In Fig. 2D-E: We set τ = 0.4 Day (see). This results in an expected lifespan of Drosophila serrata in the model settings of τ/D₂ = 40 days and that of Drosophila pseudoobscura τ/D₁ = 36.4 days, which roughly agrees with experimental data showing that the average lifespan of D. serrata is 34 days for males and 54 days for females (Narayan et al., 2022), and the average lifespan of D. pseudoobscura is around 40 days for females (Gowaty et al., 2010). The time averages () and standard deviations (δc_i) of the species’ relative/absolute abundances for the experimental data or SSA results are as follows: , , , , , , , where the superscript “(R)” represents relative abundances. A comparison of Shannon entropies in the time series between experimental data and SSA results is presented in Appendix-fig. 7C-D. Fig. 2A-E were simulated from Eqs. 1, 2, 4. See Appendix-fig. 7E, G for the long-term time series of all species in Fig. 2D-E, respectively.

Model settings in Fig. 3A-B, D (plankton): a _il = 0.1, a _i^’_i = 0.125, d_il = 0.5, d ^’_i = 0.2, w _il = 0.3, k _il = 0.2, κ₁ = 8 X 10⁴, κ2 = 5 X 10⁴, κ3 = 3 X 10⁴, ζ₁ = 280, ζ₂ = 200, ζ₃ = 150, D_i = 0.03 X N(1,0.25) (i = 1, …, S_c,l = 1, …, S_R), S_c = 140 and S_R = 3. Model settings in Fig. 3C (bird): a_i = 0.1, a’_i = 0.125, d_i = 0.5, d’_i = 0.5, w_i = 0.3, k_i = 0.2, D_i = 0.02xN(1,0.28) (i = 1, …, S_c);ζ = 110, κ = 10⁵, S_c = 250 and S_R = 1. Model settings in Fig. 3C (fish): a_i = 0.1, a’_i = 0. 14, d_i = 0.5, d’_i = 0.5, w_i = 0.2, k_i = 0.1, D_i = 0.015 x N(1,0.32) (i = 1, …,45); ζ = 550, κ = 10⁶, S_c = 45 and S_R = 1. Model settings in Fig. 3C (butterfly): a_i = 0.1, a’_i = 0.125, d_i = 0.5, d’_i = 0.3, w_i = 0.3, k_i = 0.2, D_i = 0.034 x N (1,0.35) (i = 1, … ,S_c);ζ = 300, κ = 10⁵, S_c = 150 and S_R = 1. Model settings in Fig. 3D (bat): a_i = 0. 1, a’_i = 0.125, d_i = 0.5, d’_i = 0.5, w_i = 0.2, k_i = 0.1, D_i = 0.013 x N (1,0. 34) (i = 1, …, S_c); ζ = 250, κ = 10⁶, S_c = 40 and S_R = 1. Model settings in Fig. 3D (lizard): a_i = 0.1, a’_i = 0. 125, d_i = 0.5, d’_i = 0.5, w_i = 0.2, k_i = 0.1, D_i = 0.014 x N(1,0.34) (i = 1, …, S_c);ζ = 250, κ = 10⁶, S_c = 55 and S_R = 1. In Fig. 3A-D, the mortality rate D _i (i = 1, … , S_C) is the only parameter that varies with the consumer species, which was randomly sampled from a Gaussian distribution N(μ,σ), where μ and σ are the mean and standard deviation of the distribution. The coefficient of variation of the mortality rates (i.e., σ/μ) was chosen to be around 0.3, or more precisely, the best-fit in the range of 0.15–0.43. This range was estimated from experimental results (Menon et al., 2003) using the two-sigma rule. These settings for the mortality rates also apply to those in Appendix-figs. 8–14. Fig. 3A-D were simulated from Eqs. 1, 2, 4. See Appendix-figs. 14C, 10K, C, D, H, I, J, Fig. 3A, Fig. 3B for the time series of Fig. 3C , , , , , , , and , respectively. The Shannon entropies of the experimental data and simulation results for each ecological community are: , , , ; , , . Here the Shannon entropy , where P. is the probability that a consumer individual belongs to species C_i.

Appendix Tables

Appendix-table 1.

Appendix-table 2.

Appendix Figures

Appendix-figure 1.

Appendix-figure 2.

Appendix-figure 3.

Appendix-figure 4.

Appendix-figure 5.

Appendix-figure 6.

Appendix-figure 7.

Appendix-figure 8.

Appendix-figure 9.

Appendix-figure 10.

Appendix-figure 11.

Appendix-figure 12.

Appendix-figure 13.

Appendix-figure 14.

Appendix-figure 15.