Overcome Competitive Exclusion in Ecosystems.

Explaining biodiversity in nature is a fundamental problem in ecology. An outstanding challenge is embodied in the so-called Competitive Exclusion Principle: two species competing for one limiting resource cannot coexist at constant population densities, or more generally, the number of consumer species in steady coexistence cannot exceed that of resources. The fact that competitive exclusion is rarely observed in natural ecosystems has not been fully understood. Here we show that, by forming chasing pairs and chasing triplets among the consumers and resources in the consumption process, the Competitive Exclusion Principle can be naturally violated. The modeling framework developed here is broadly applicable and can be used to explain the biodiversity of many consumer-resource ecosystems and hence deepens our understanding of biodiversity in nature.

Introduction

In Darwin's theory of evolution, survival of the fittest, i.e., the less competitive species die out, implicating the notion of competition exclusion (Darwin, 1859). In 1928, Volterra illustrated mathematically that when two species compete for a single resource, one must die out unless the hunting to death rate ratio is exactly the same for the two competing species (Volterra, 1928). Those results were absorbed in the famous Competition Exclusion Principle (CEP) (Hardin, 1960, Gause, 1934, Armstrong and McGehee, 1980), also named as Gause's law (Gause, 1934): two species competing for one type of resource cannot coexist at steady state. In the 1960s, MacArthur and Levins extended this principle to ecosystems with arbitrary number of resource species (MacArthur and Levins, 1964, Levin, 1970, McGehee and Armstrong, 1977). Consider types of consumer species competing for types of resources. Each consumer can feed on one or multiple types of resources. Consumers do not directly interact with each other via other mechanisms except competing for the resources. According to the CEP (MacArthur and Levins, 1964, Levin, 1970, McGehee and Armstrong, 1977), at steady state the number of coexisting species of consumers cannot exceed that of resources, i.e., (see also Figure S1).

The classical proof (MacArthur and Levins, 1964, Levin, 1970, McGehee and Armstrong, 1977) of the CEP is demonstrated in Figure 1. Consider the simplest case: and , i.e., two consumer species and compete for one type of resource (Figure 1A). The generic population dynamics of this consumer-resource system can be described as follows:

Figure 1

Classical proof of the Competitive Exclusion Principle

(A) The scenario of two consumer species () and one resource species (). The green arrows denote the biomass flow among the consumption relationships.

(B) At steady state, if the two consumer species coexist, then according to Equation 1, (i = 1, 2). This requires that the following three lines (i = 1, 2) and intersect at a single point, which normally cannot happen.

(C) Representative trajectories of the two consumer species, which cannot coexist at steady state when . Here (i = 1, 2); See Table S1 for simulation details.

(D) The scenario of three consumer species () and two resource species (). Predation or other interactions are forbidden among consumers but allowed (denoted by gray arrows) among resources.

(E) If the three consumer species coexist at steady state, then according to Equation 2, (i = 1, 2, 3). Generically, three curves would not intersect at exactly the same point, hence the three consumer species cannot coexist at steady state.

(F) Representative trajectories of the three consumer species, which cannot all coexist at steady state (see Figure S2A for the case that two of the three consumer species coexist). Here (i = 1, 2, 3); (j = 1, 2).

See Table S1 for simulation details. See also Figures S1 and S2.

Here and are unspecified functions, stands for mortality rate of the consumer . At steady state, if the two consumer species coexist, we have , i = 1, 2. This requires that the two curves and should cross the line at the same point, which is typically impossible (Figure 1B), unless the model parameters satisfy certain constraint (with Lebesgue measure zero, see Figure S2B). Hence, generically the two consumer species cannot coexist at steady state (Figure 1C). In the case of , , the general population dynamics of the system can be written as

Here and are unspecified functions. Similar proof strategy used in the case of and can be applied here (see Figures 1D–1F and S2A), or more complicated cases with any positive and (MacArthur and Levins, 1964).

Interestingly, an astonishing level of biodiversity has been witnessed in most natural ecosystems. In aquatic biology, Hutchinson first proposed the paradox of the plankton: a limited number of resource types supports an unexpectedly large number of plankton species (Hutchinson, 1961). In tropical rainforests, one gram of soil contains a spectacular 2,000 to 18,000 distinct microbial genomes (Daniel, 2005). The vast diversity of microbial species plays an important role in the biogeochemical nutrient cycling of our planet. Yet, how could this biodiversity naturally emerge and sustain? Explaining biodiversity is a great challenge in ecology. In the past five decades, many mechanisms have been proposed to overcome the limitation on biodiversity set by CEP. Some argued that ecosystem never approaches steady state due to temporal (Hutchinson, 1961, Levins, 1979, Descamps-Julien and Gonzalez, 2005) or spatial factors (Levin, 1974, Richerson et al., 1970) or species self-organized dynamics (Koch, 1974, Huisman and Weissing, 1999, Benincà et al., 2008). Some considered special cases when the system parameters satisfy certain constraints (Volterra, 1928). The rest considered aspects such as cross-feeding (Turner et al., 1996, Goyal and Maslov, 2018, Goldford et al., 2018), toxin (Czárán et al., 2002), rock-paper-scissors relation (Kerr et al., 2002, Kelsic et al., 2015, Grilli et al., 2017), predator interference (Skalski and Gilliam, 2001, Beddington, 1975, Crowley and Martin, 1989, DeAngelis et al., 1975, Kuang et al., 2003), complex interactions (Kelsic et al., 2015, Bairey et al., 2016, Grilli et al., 2017), metabolic trade-off (Posfai et al., 2017), or co-evolution (Xue and Goldenfeld, 2017) (see Supplemental Information Sec.II.A for details).

Many of the mechanisms mentioned above are broadly relevant to promote biodiversity in nature. In the context of CEP itself, here we present a mechanism that considers the details of the consumption process. Specifically, consumer and resource species can form a chasing pair when an individual consumer is chasing an individual resource, whereas they form a chasing triplet when two individuals of consumer chase an individual resource simultaneously. We find that forming chasing pairs and chasing triplets among the consumers and resources can naturally break the CEP and hence facilitate biodiversity.

Results

Consumption Process with Chasing Pairs

We realize that none of the previous studies explicitly considered the detailed consumption process. Although the timescale of the consumption process is generally much faster than that of the birth and death processes, it can have remarkable impact on the population dynamics (see Supplemental Information Sec.III.B for details). Hence, in our modeling framework, we explicitly consider the consumption process between the consumers and resources (see Figure S3A). The consumers are biotic (i.e., living organisms), whereas the resources can be either biotic or abiotic (i.e., supplied nutrients that are not alive). First, we consider the case that both consumer species and resource species are biotic, and for simplicity we assume both species are motile. Then we can explicitly consider the population structure of consumers and resources: some are wandering around freely; some are chasing each other. When a consumer encounters a resource with rate , they form a chasing pair, denoted as , where the superscript “P” stands for “pair.” The resource can either “escape” with rate or be caught and consumed by the consumer with rate . For abiotic resources, they cannot actively escape from the consumers, yet they may passively “escape” owing to environmental factors. In this case, the “escape” rate corresponds to that the consumer fails to capture the resource in a chasing pair, which is analogous to a non-effective collision in chemical reactions. Such a consumption kinetics commonly takes the Michaelis-Menten form:

, with , which corresponds to the Holling's type-II functional response (Holling, 1959) in ecology and is widely adopted in consumer-resource models (Koch, 1974, Momeni et al., 2017). This form, in fact, agrees with the growth rate function in the classical proof (MacArthur and Levins, 1964, Levin, 1970, McGehee and Armstrong, 1977), where and is a biomass conversion ratio constant (see Supplemental Information Sec.III.B for details). Nevertheless, the Michaelis-Menten kinetics is a good approximation only if the resource population is much larger than the consumer population, i.e., .When this condition is not satisfied, the growth rate function follows (Liu et al., 2015) rather than (see Supplemental Information Sec.III.A for details). The -dependency in the growth rate function invalidates the classical proof (MacArthur and Levins, 1964, Levin, 1970, McGehee and Armstrong, 1977), implying a potential mechanism to break the CEP. Actually, existing mechanisms involving predator interference (Skalski and Gilliam, 2001, Beddington, 1975, Crowley and Martin, 1989, DeAngelis et al., 1975) or ratio-dependent predation (Arditi and Ginzburg, 1989, Abrams and Ginzburg, 2000) also have -dependency in the growth rate function or functional response.

Forming Chasing Pairs Still Cannot Break the CEP

Interestingly, we find that the presence of chasing pair and the -dependent growth rate functions are still not enough to break the CEP. For example, in case and (Figure 2A), the population dynamics of the system can be described as follows:with i = 1, 2. Here consumers and resources that are freely wandering around are denoted as and , respectively, where the superscript “F” stands for “freely wandering.” The variable represents the chasing pair, is the encounter rate between a consumer and a resource to form a chasing pair , is the “escape” rate of a resource out of a chasing pair , and is the capture rate of consumer in a chasing pair . If the two consumers can coexist, we prove that the steady-state equations yield , with and (see Supplemental Information Sec.IV for details), which corresponds to parallel planes in the coordinate system (Figure S3C), rendering coexistence impossible (Figures 2C, 2E, and S3, see Supplemental Information Sec.IV-V for details).

Figure 2

Modeling the Consumption Process between Consumers and Resources Explicitly May Naturally Break the CEP

For simplicity, we consider the case of two consumer species () and one biotic resource species (, see Figures S5C and S5D for the case of abiotic resource species).

(A) Formation of a chasing pair between a consumer and a resource.

(B) Formation of a chasing triplet among two consumers of the same species and a resource. We denote the scenario combining chasing pair (A) and triplet (B) as P-T Model.

(C and E) Time courses of the species abundances involving only chasing pair. (C) Consumer species cannot coexist at steady state. (E) Only one type of consumer species exists for long, the oscillating dynamics resembles that of the classical predator-prey models (May, 1972).

(D and F) Time courses of the species abundances in P-T Model with the presence of chasing pairs and chasing triplet. Both consumer species can coexist at steady state. The dotted lines in (F) are the steady-state analytical solutions (labeled with superscript “Analytical”) calculated in Equations S30–S32. (C) and (E) were simulated from Equation 3, where . (D) and (F) were simulated from Equations 4 and 5.

See Table S1 for simulation details. See also Figures S3–S10.

Consumption Process with Chasing Triplets

Pack hunting is prevalent across different organisms in the wild (Creel and Creel, 1995, Muro et al., 2011, Geisen et al., 2015, Merron, 1993, Stander, 1992, Boesch, 1994, Bshary et al., 2006, Vail et al., 2013, Berleman and Kirby, 2009, Seccareccia et al., 2015). Intraspecific pack hunting is very general and commonly occurs, whereas interspecific pack hunting has also been reported for a handful of species (Bshary et al., 2006, Vail et al., 2013). This means that two or more consumer individuals can chase the same resource individual simultaneously. To take this into account, we revisit the consumption process and naturally extend the idea of chasing pair to chasing triplet, i.e., two consumers (within the same or from different species) can chase the same resource (Figures 2B and S4). For intraspecific pack hunting, in case and , a consumer can join an existing chasing pair to form a chasing triplet (Figure 2B, in combination with Figure 2A, denoted as P-T Model), where the superscript “T” stands for “triplet.” The population of consumers and resources consists of freely wandering individuals (, ) and those involved in a chasing pair () or triplet (). Mathematically, they are given by (i = 1, 2) and , respectively. The population dynamics of the system can be described as follows:with i = 1, 2. Here is the encounter rate between a consumer and an existing chasing pair to form a chasing triplet ; and are the escape rates of a consumer out of a chasing triplet (Figure 2B). Consumer species can capture resource either from a chasing pair with rate or from a triplet with rate .

Forming Both Chasing Pairs and Chasing Triplets Can Break the CEP

In Equation 4, the explicit form of function is not specified. We assume that the dynamics of the resources follow the same construction principle as that in the classical MacArthur's consumer-resource model (MacArthur, 1970, Chesson, 1990). Then,

Biotic resources take logistic population growth in the absence of consumers, hence represents the intrinsic growth rate (with dimension 1/time) and represents the carrying capacity. Abiotic resources are supplied externally, then stands for the external resource supply rate (with dimension mass/time) and is the steady state resource abundance in the absence of consumers.

Using dimensional analysis, we make all parameters dimensionless (see Supplemental Information Sec.VII for details). For convenience, below we still use the same parameter notations, yet they are all dimensionless. Actually, and are two reducible parameters (see Supplemental Information Sec.VII for details); for convenience, we set and for biotic resource cases, whereas and for abiotic resource cases. In the numerical simulations of the P-T Model (Figures 2A and 2B), we find that two consumer species can achieve steady coexistence when there is only one type of resource (Figures 2D, 2F, and S5D), which naturally breaks the CEP.

When the abundance of resources are much larger than that of consumers (i.e., ), which generally hold in most natural ecosystems, the steady-state population of the consumer species and resource species can be analytically calculated (see Supplemental Information Sec.V.B for details). In Figures 2F and S6, we show that the steady-state analytical results of both biotic and abiotic resource cases agree well with numerical results.

Interestingly, there are several types of coexistence trajectories in phase space within the scenario of P-T Model, which involves chasing pair and triplet formed between consumers of the same species. When the resource is abiotic, there is only one type of behavior: the coexistence state is globally attracting as long as the initial abundances of these species are non-zero, as shown in Figure 3A. However, in the case that the resource is biotic, the coexistence state can be either globally attracting (Figures 3B and 3C) or unstable, leading to a limit cycle (Figure 3D) (see Figure S5B for the oscillating coexistence in time series). In some cases, the oscillations damps, and ends in the globally attracting fixed point, as shown in Figure 3B.

Figure 3

Coexistence of Two Consumer Species () and One Resource Species () within the P-T Model

(A–D) Different types of coexistence trajectories in the state space. (A) Abiotic resource case, the coexistence state (green dot) is globally attracting. (B–D) Biotic resource cases, green dot marks the fixed point. (B), (C) The coexistence state is globally attracting. (D) The coexistence state is unstable; all trajectories attract to a stable limit cycle.

(E–H) Stable coexistence region in the P-T Model. (i = 1, 2) is the only different parameter between consumer species and , and , the relative difference in mortality rate, measures the competitive differences between the two consumer species. in (E) and (F) is a dimensionless multiplier that to tune the capture rate and escape rate parameters for the two-consumer species in each scenario. in (G) and (H) is a dimensionless multiplier that to tune (i = 1,2), the capture rate in the chasing triplet for the two consumer species. The regions below the blue surface and above the red surface are stable coexistence region, whereas the regions below the red surface and above are the regions for unstable fixed point, where trajectories typically end in a limit cycle (see also Figure S8 and Supplemental Information Sec.V.D). (E) and (G) Biotic resource cases. (F) and (H) Abiotic resource cases. (A)–(D) were simulated from Equations 4 and 5. (E)–(H) were calculated from Equations 4 and 5.

See Table S1 for simulation and calculation details. See also Figure S8.

We further considered scenarios involving interspecific group (Bshary et al., 2006, Vail et al., 2013), specifically, two variants of the P-T Model, where the chasing triplet is formed between different species of consumers (Figure S4A) or either between the same or different species (Figure S4B). In both Variants, two consumer species can coexist either steadily (Figures S7A and S7C) or with sustained oscillations (Figures S7B and S7D) when there is only one type of resource species (see Supplemental Information Sec.V.C for details).

To verify that our findings are not due to accidental model parameters, we systematically studied the parameter space for stably steady coexistence. We found that, for both the P-T Model and its two variants, regardless of biotic or abiotic resources, there exists a non-zero parameter space where the two consumer species can stably steadily coexist with one type of resource species (see Figures 3E–3H and S8 and Supplemental Information Sec.V.D for details), demonstrating that the violation of CEP is not due to a special set of model parameters. Note that the violation of CEP in the case of actually implies that it will be violated for more general cases with (see Supplemental Information Sec.VI for details).

Intuitive Explanation of Why Forming Chasing Pairs and Chasing Triplets Can Break the CEP

A simple explanation is that a resource within a chasing pair can be effectively regarded as another species when forming a chasing triplet (e.g., within in Figure 2B), although in essence, they still remain the same identity. In the P-T Model, consumer species can get resource in two ways, potentially with different effective capture rates in a chasing pair or a chasing triplet. As a rough estimate, the resource abundance in a chasing pair () is proportional to , whereas in a chasing triplet () it is proportional to . Here and are the total populations of resource and consumer , respectively. Thus, when becomes higher, it obtains a higher fraction of resource from a chasing triplet. If the effective capture rate in a chasing triplet is lower than that in a chasing pair, implying a wasteful or redundant foraging, then this gives rise to an auto-suppression on the intraspecific growth rate, which can facilitate species coexistence (Chesson, 2000). Without loss of generality, let's assume that is more competitive than , in obtaining the resource from either a chasing pair or triplet; however, can be more effective in obtaining the resource from chasing pair than from chasing triplet owing to redundant foraging. When the population of is larger than that of , higher fraction of is involved in the less effective chasing triplet foraging, which may lead to an overall balanced competitiveness between the two consumer species at such population densities (with ) and thus facilitates species coexistence. If this redundant foraging hypothesis is correct, then the less effective the chasing triplet foraging is, the easier it is for the two consumer species to coexist. In other words, the two consumer species can coexist with a larger competitiveness difference (see Supplemental Information section V.D and Figure S8C for details). To test this redundant foraging hypothesis, we set to be the only different parameter between consumer species and , and specifically tune , the capture rate in a chasing triplet with a multiplier : , where and is the capture rate in a chasing pair. In Figures 3G and 3H, our systematical numerical results show that decreasing the capture rate in a chasing triplet indeed promotes species coexistence, where measures the competitiveness difference between the two consumer species. The supremum of Δ peaks at and it decreases with increasing for both biotic and abiotic resource cases (Figures 3G and 3H). These results fully support the redundant foraging hypothesis.

To offer a more quantitative explanation, we consider the functional forms of population dynamics at steady state. In the classical proof of CEP, in the case of and (Figures 1A–1C), if both consumers species can coexist at steady state, the abundance of the resource species needs to satisfy two equations ( (i = 1, 2)) simultaneously. This is equivalent to requiring that two parallel planes share a common point, which is typically impossible (Figure 4A). In the presence of chasing pairs, as shown in Figure 4B, the requirement for steady coexistence corresponds to parallel surfaces ( (i = 1, 2)), see Supplemental Information Sec.IV-V for details). In the presence of both chasing pairs and chasing triplets, as shown in Figure 4C, the requirement for steady coexistence corresponds to three non-parallel surfaces (i = 1, 2), (see Supplemental Information Sec.V for details) to cross at one point, which can in principle happen and hence breaks the CEP. To verify the intuitive explanation, we resort to numerical solutions. Figures 4D, 4E, and S9 show the results, where the yellow, green, and blue surfaces are the exact solutions. The parallel green and blue surfaces in the cases of only chasing pairs are verified with Figure 4D, whereas the three non-parallel surfaces in scenarios involving both chasing pairs and chasing triplets are verified with Figures 4E and S9.

Figure 4

Intuitive Explanation of why the Formation of Chasing Triplet Can Break the CEP

For simplicity, we consider the case of and .

(A) In the classical proof, the green plane and blue plane are parallel to each other and thus do not have a common point.

(B) In the model involving chasing pairs, the green surface and blue surface are still parallel to each other and thus still do not have a common point (see Figure S3C, Supplemental Information Sec.IV-V for details).

(C) In the model involving both chasing pairs and chasing triplets, the yellow, green, and blue surfaces are not parallel to each other and thus the green and the blue ones can have an intersection curve (shown in dashed purple), whereas the three surfaces can intersect at one point (shown in red) and thus facilitate coexistence.

(D and E) Demonstration of the intuitive explanation with numerical solutions. (D) In the scenario involving only chasing pair, numerical solutions confirm that the green surface and blue surface are parallel to each other. (E) In scenarios involving both chasing pair and triplet (P-T Model), numerical solutions confirm that the yellow, green, and blue surfaces are not parallel to each other and definitely can have a common point (marked with red dot, see Figure 2D for time series). (D) was calculated from Equation 3, where ; (E) was calculated from Equations 4 and 5.

See Table S1 for simulation details. See also Figures S9 and S10.

To provide deeper insights into the break of CEP, we argue that the competitive exclusion (i.e., at steady state) in the classical proof of CEP or the scenario involving only chasing pairs stems from the symmetry constraint of the equation form. In those scenarios, for and , there exists a variable satisfying the symmetry constraint that (, where is an unspecified function) for the steady-state population dynamics. In the classical proof (see Equation 1); in the scenario involving only chasing pairs, (see Equations 3 and S13). The existence of directly leads to parallel planes/surfaces (see Figures 4A, 4B, 4D, and S3C) and thus precludes consumer species coexistence. However, scenario involving both chasing pairs and chasing triplets or even higher-order terms (e.g., quadruplets, quintuplets) breaks the symmetry constraint in the equation form so that the variable does not exist; otherwise, there cannot be any intersection points in Figure 4E or Figure S9 (see Supplemental Information Sec.V.A.2 for details). This symmetry breaking enables the break of the CEP.

Discussion

Over the past several decades, various mechanisms have been proposed to overcome the constraint on biodiversity set by CEP. Mechanisms such as temporal or spatial factors, self-organized dynamics, cross-feeding, and predator interference are likely to play significant role in maintaining the biodiversity in nature. Here, within the original context of CEP, by considering the details of the consumption process, especially the formation of chasing pairs and triplets inspired by the prevalent phenomenon of pack hunting, our mechanism naturally breaks the constraint of the CEP. Furthermore, we show that there are non-special parameter sets (of non-zero measure) that break the CEP in all scenarios involving different forms of chasing triplets. Meanwhile, we notice that breaking CEP is parameter dependent (see Figure S10 and Supplemental Information Sec.V.E): for certain parameters, there is no feasible fixed point for coexistence, or the fixed point can be unstable (for biotic resource cases), which may end in a limit cycle.

The coexistence predicted in our model is testable in experiments. Both macro- and microscopic ecosystems involving pack hunting are potential candidates. For microbial ecosystems, it has been reported that some microorganisms, such as Myxococcus xanthus (Berleman and Kirby, 2009), ameba (Geisen et al., 2015), and Lysobacter (Seccareccia et al., 2015), can feed on other microorganisms through pack hunting. But the caveat is that microbial communities are typically shaped by metabolic cross-feeding (Goldford et al., 2018), which is not considered in our model. Hence special attention needs to be paid to disentangle the impacts of cross-feeding and pack hunting in breaking the CEP of microbial communities.

Limitations of the Study

This study shows that, by forming chasing pairs and chasing triplets among the consumers and resources in the consumption process, the CEP can be naturally violated. However, several limitations should be paid attention. First, there are non-special parameter settings that can break the CEP in scenarios involving chasing triplets. Yet, this is not true for certain parameter settings, especially when there is large competitiveness difference between consumer species. Second, our model framework does not consider other factors that can also promote biodiversity in nature such as temporal or spatial factors and cross-feeding. Special attention needs to be paid to disentangle these confounding factors in future experimental validations.

Methods

All methods can be found in the accompanying Transparent Methods supplemental file.