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Understanding the impact of third-party species on pairwise coexistence [1]

['Jie Deng', 'Department Of Civil', 'Environmental Engineering', 'Mit', 'Cambridge', 'Massachusetts', 'United States Of America', 'Washington Taylor', 'Center For Theoretical Physics', 'Serguei Saavedra']

Date: 2022-11 

The persistence of virtually every single species depends on both the presence of other species and the specific environmental conditions in a given location. Because in natural settings many of these conditions are unknown, research has been centered on finding the fraction of possible conditions (probability) leading to species coexistence. The focus has been on the persistence probability of an entire multispecies community (formed of either two or more species). However, the methodological and philosophical question has always been whether we can observe the entire community and, if not, what the conditions are under which an observed subset of the community can persist as part of a larger multispecies system. Here, we derive long-term (using analytical calculations) and short-term (using simulations and experimental data) system-level indicators of the effect of third-party species on the coexistence probability of a pair (or subset) of species under unknown environmental conditions. We demonstrate that the fraction of conditions incompatible with the possible coexistence of a pair of species tends to become vanishingly small within systems of increasing numbers of species. Yet, the probability of pairwise coexistence in isolation remains approximately the expected probability of pairwise coexistence in more diverse assemblages. In addition, we found that when third-party species tend to reduce (resp. increase) the coexistence probability of a pair, they tend to exhibit slower (resp. faster) rates of competitive exclusion. Long-term and short-term effects of the remaining third-party species on all possible specific pairs in a system are not equally distributed, but these differences can be mapped and anticipated under environmental uncertainty.

It is debated whether the frequency with which two species coexist in isolation or within a single environmental context is representative of their coexistence expectation within larger multispecies systems and across different environmental conditions. Here, using analytical calculations, simulations, and experimental data, we show why and how third-party species can provide the opportunity for pairwise coexistence regardless of whether a pair of species can coexist in isolation across different environmental conditions. However, we show that this opportunity is not homogeneously granted across all pairs within the same system. We provide a framework to understand and map the long-term and short-term effects that third-party species have on the coexistence of each possible subset in a multispecies system.

Here, we follow a geometric and probabilistic analysis based on nonlinear dynamics at equilibrium to estimate the effect of third-party species (the larger multispecies system) on the coexistence of a pair (community) of species under unknown environmental conditions using only information from pairwise species interactions. We estimate these system-level effects over the long term (using analytical calculations) and over the short term (using simulations and available experimental data on gut microbiota). Building on generalized Lotka-Volterra (gLV) dynamics [ 8 ], we provide a geometric understanding regarding whether the conditions limiting the coexistence of a pair of species in isolation (the species pool consists only of this pair) remain with third-party species (when the species pool increases). Next, we derive a system-level indicator of the extent to which third-party species can affect the probability of coexistence of a pair relative to its probability in isolation. Then, we compare the numerical and experimental effects to the analytical expectation in order to distinguish the role that third-party species play in shaping pairwise coexistence under short-term and long-term behavior, respectively. Finally, we show how our theory can be used to provide a cartographic representation [ 38 ] of the long-term and short-term effects acting on each pair within a multispecies system under environmental uncertainty. While our focus is on pairwise coexistence (following traditional work in ecology), our methodology and results are scalable to any subset (community) dimension.

In this line, ecological research has suggested that species embedded into larger multispecies systems (more than two species) may experience higher-order effects, i.e., the effect of species i on the per capita growth rate of species j might itself depend on the abundance of a third species k due to either compensatory effects, supra-additivity, trait-mediated effects, functional effects, meta-community effects, or indirect effects [ 8 , 28 – 31 ]. Unfortunately, additional research has shown that it is virtually impossible to derive those potential effects since many parameter combinations can equally explain the observed ecological dynamics—a problem known as structural identifiability [ 32 , 33 ]. In fact, many of the interactions measured under multispecies systems are not direct effects, as studies often believe [ 15 , 16 ], but chains of direct effects [ 34 , 35 ]. Moreover, measurements of species interactions are expected to change as a function of the type of data (i.e., snapshot or time), experiment (e.g., long term or short term), perturbations (e.g., pulse or press), and dimensionality, leading to inconsistencies and inference issues [ 32 , 36 , 37 ]. Thus, it is unclear whether the frequency with which two or more species coexist in isolation is representative of their coexistence expectation within larger multispecies systems and across different environments.

Because the environmental conditions acting on species are typically unknown and diverse in natural settings, the majority of theoretical work has been centered on deriving the fraction of conditions (set of parameter values) leading to the persistence of a community (formed by either two or more species) [ 8 , 21 – 23 ]. Yet, species are seldom in isolation and the coexistence of a specific combination of multiple species is expected to be rare in a random environment [ 5 , 24 , 25 ]. Moreover, the methodological and philosophical question has always been whether we can observe the entire community and, if not, what the conditions are under which an observed subset of the community can persist as part of a larger multispecies system. The answer to this question, however, may also be context-dependent due to the presence and absence of third-party species, which can be acting as ecosystem engineers [ 26 , 27 ]. Thus, it is unclear how far knowledge of pairwise coexistence in isolation (or full community persistence) can take us while investigating ecological systems, and how we should compare the effects of third-party species on the coexistence probability of a pair (or community) of species.

The persistence of virtually all living organisms on Earth depends to a greater or lesser extent on the presence of other living organisms and on the environmental (abiotic and biotic) conditions present in a given place and time [ 1 , 2 ]. This observation has established a rich research program in quantifying pairwise interactions and their impact on pairwise coexistence across environmental gradients [ 3 – 8 ]. In this line, observational work has been focused on finding the frequency of occurrence of multiple pairs of species across different environments either in isolation or within different systems of multiple species [ 9 – 16 ]. For example, studies have long debated whether there exists in nature pairs of species whose niches forbid their coexistence regardless of the environment—known as the checkerboard hypothesis [ 11 , 17 ]. Instead, experimental studies have shown that pairwise coexistence strongly depends on the details of both the system and the environment [ 10 , 12 , 18 , 19 ]. In fact, it has been shown that the coexistence expectations that may operate under controlled conditions (or unique environments) [ 12 , 14 ] do not necessarily operate under uncontrolled conditions (or diverse environments) [ 5 , 20 ]. These observational results have shown that in order to implement successful interventions in ecological communities, it is essential to develop a testable theory to be able to explain why and how emergent processes at the system level can affect the possibility that a given pair (or community) of species will coexist under different abiotic and biotic environments.

Methods

Traditional indicators of pairwise coexistence Our theoretical framework is based on the tractability of the gLV dynamics [33], where the per-capita growth rate of species is written as , where the vector represents the density of all species in the system . The matrix corresponds to the interaction matrix dictating the internal structure of the system, i.e., the per-capita effect of species j on an individual of species i. The effective parameters consist of the phenomenological effect of the internal (e.g., intrinsic growth rate), abiotic (e.g., temperature, pH, nutrients), and biotic factors (e.g., unknown species) acting on the per-capita growth rate of each particular species. That is, we consider that the effective growth rate θ represents the total additional effect of all the unknown potential factors as a function of the environmental conditions acting on each species independently. Note that pairwise effects a ij from a given pool of species are assumed to be known. Our essential assumptions are thus that the effective growth rates θ are environmentally determined (i.e., change with the environment), but a pairwise interaction a ij is a property only of the species pair (i.e., invariant under change of environment or addition of further species). Under steady-state dynamics (i.e., f = 0), the necessary (but not sufficient) condition for species coexistence is the existence of a feasible equilibrium (i.e., N* = A−1 θ > 0) [22]. We are interested here in globally stable systems; feasible and globally stable systems fulfill the necessary and sufficient conditions for coexistence [22]. The combinations of the effective growth rates θ compatible with the feasibility of a multispecies system (known as the feasibility region D F ) can be described as: where a j is the jth column vector of the interaction matrix A. Additionally, assuming no a priori information about how the environmental conditions affect the effective growth rate [39], the feasibility region can be normalized by the volume of the entire parameter space [25] as where is the -dimensional closed unit sphere in dimension . Geometrically, the feasibility region is the convex hull of the spanning vectors in -dimensional space, and thus always corresponds to an -dimensional solid angle that is contained within a single hemisphere (e.g., an acute angle in the case ). Thus, the feasibility that is normalized by the entire -dimensional space cannot be higher than 0.5. Ecologically, this normalized feasibility region can be interpreted as the probability of feasibility of a system characterized by known pairwise interactions A under environmental uncertainty (all possible effective growth rates are equally likely to happen). The measure can be efficiently estimated analytically [25, 40] or using Monte Carlo methods (S4 Appendix). Moreover, is robust to gLV dynamics with linear functional responses [25], a large family of nonlinear functional responses [39, 41], gLV stochastic dynamics [39, 41], and as a lower bound for complex polynomial models in species abundances [33]. represents the feasibility of all species together. That is, if a system (i.e., a given pool) is composed of 2 species or of 10 species, represents the probability of feasibility of the full set of 2 or 10 species, respectively [25]. In the case where , corresponds to the traditional indicator of pairwise coexistence in isolation, which is proportional to what is typically known as niche overlap in gLV competition systems [42, 43]. It is known that will tend to decrease as a function of the dimension of the system [23, 33]: if . Thus, focusing on a given pair of species within a system , will likely underestimate the probability of feasibility for the pair, whereas using only the feasibility of the pair in isolation can be potentially misleading. For example, considering a pair that has a rather small feasibility region in isolation ( , e.g., almost identical niches) will prompt us to conclude that the coexistence of this pair should be forbidden across all possible environmental conditions—as suggested by the checkerboard hypothesis [11, 14, 17]. But, can this pair of species coexist more easily with additional third-party species [44, 45]? And will different pairs within the same system of multiple species be equally benefited (or affected)?

Looking at traditional indicators from a systems perspective To understand the extent to which provides a reliable indicator of the possibility of coexistence of a pair (or community) within a larger multispecies system characterized by a pairwise interaction matrix A, we study geometrically how the feasibility conditions θ change when a pair of species is in isolation and with third-party species. We approach this question from several different perspectives. First, we start by asking to what extent can third-party species act as ecosystem engineers and modify the environment into a more suitable habitat [26, 27]? In particular, for a given pair i, j within a fixed community containing third-party species, we consider the range of environmental conditions associated with the two parameters θ i , θ j such that it is possible for the pair i, j to exist for some set of values for the remaining θ k ’s. This gives a sense of the potential environmental range of this pair in the context of a fixed community. In the following subsection we explore the more precise question of what the probability of coexistence is when we average over all possible values of all the θ k ’s. Mathematically, the projection region of the feasibility region D F of a multispecies system onto the 2-dimensional parameter space of a pair of species can be represented by the conical hull, i.e., the set of all conical combinations of the projection of spanning rays: where b j is the jth column vector of B, and it contains a subset of elements corresponding to the pair in the column vector a j of the interaction matrix A. In other words, B consists of the row vectors a i of A that correspond to the pair (i.e., ). For example, for the pair of species 1 and 3 (i.e., ), for each column vector , in A, we have in B. Then, we can obtain the normalized projection by the following ratio of volumes: Recall that is the feasibility of pair in isolation (using the corresponding 2-dimensional sub-matrix A[i, j; i, j]). This projection region captures the first ecological question raised above: what is the range of θ i , θ j under which it is possible for species i, j to coexist for a given matrix A and some values of the remaining parameters θ k , k ≠ i, j. If the projection region is not the entire 2-dimensional parameter space θ of the pair, the projection ranges within (Fig 1A provides an illustration). Otherwise, , indicating that the region (θ 1 , θ 2 ) incompatible with the possible coexistence of the pair disappears. In other words, in these cases pairwise coexistence can be possible under any combination of (θ i , θ j ). The realization of such coexistence, however, still depends on the other θ values associated with the third-party species in the system. In fact, per definition ), the entire projection region ) defines equal or weaker constraints on the plane (θ i , θ j ) for the coexistence of the pair. Thus, the amount of projection contribution increasing the feasibility region on the plane (θ i , θ j ) can be defined as (1) Because the combination of spanning vectors increases as the dimension of the system grows, the projection is expected to approach 1 in large multispecies systems (Fig 1D, S3, S4, and S5 Figs). PPT PowerPoint slide

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TIFF original image Download: Fig 1. Geometric perspective of the effect of third-party species on pairwise coexistence. For a fictitious system of three species, Panel A shows the geometrical representation of the feasibility region ) (i.e., pink cone spanned by a 1 , a 2 , a 3 ) of an illustrative three-species system characterized by the interaction matrix in the three-dimensional space of effective growth rates (θ 1 , θ 2 , θ 3 ). The panel also shows the projection ) of the spanning rays onto the two-dimensional plane (θ 1 , θ 2 ) of the pair (i.e., light pink and dark pink disk spanned by b 2 , b 3 ). The feasibility of pair {1, 2} in isolation (i.e., a 2-species pool) ) is given by b 1 , b 2 (i.e., dark pink disk), and becomes a fraction of the projection corresponding to the possible coexistence of species 1 and 2 only. Panel B illustrates the geometric partition of the analytical feasibility regions of all possible compositions in the previous three-species system. For the same system, the points in Panel C show the resulting species composition (species with final abundances N < 10−6 are considered statistically extinct) from 10,000 simulations over a short finite time interval (total time = 200, stepsize = 0.01) conducted by the Runge-Kutta method. Note that these results are a function of the time span of simulations (see S7 Fig), although the qualitative effects are similar for other choices of time span. The black lines correspond to the borders of the analytical feasibility regions in Panel B. Panel D shows the distribution of system-level effects calculated on a fixed hypothetical pair of symmetrically competing species when assembled with different random third species. The distributions are generated using 50 random third species (see text for details). Each point in the distributions corresponds to the same pair with a different set of third-party species. For reference, the dashed line shows the value of one: the relative feasibility in isolation. Panels E-F show the positive and negative association of the long-term effect with the short-term and buffering effects, respectively (Panel E: Pearson’s product-moment correlation = 0.995, p- value < 0.001; Panel F: Pearson’s product-moment correlation = −0.707, p- value < 0.001). These results are complemented by a perturbative analysis for weakly interacting systems in S5 Appendix. https://doi.org/10.1371/journal.pcbi.1010630.g001

Analytical system-level effects While the projection contribution, which measures the possibility of pairwise coexistence, is always beneficial to the likelihood of pairwise coexistence, it is measured in the two-dimensional (θ i , θ j ) plane of the pair {i, j} in question. The actual probability of pairwise coexistence, however, depends on the θ values of the rest of the species in the community. To find long-term analytical indicators of the effect of third-party species on pairwise coexistence, first we propose to do a geometric partition of the parameter space of θ (a closed unit sphere) into non-invadable feasibility regions of all possible species compositions within a multispecies system. These regions are disjoint when there is a global stable equilibrium. Formally, in a multispecies system with an interaction matrix A and a stable global equilibrium, the feasibility region of a composition (a subset of species found in the system, i.e., ) can be defined as: where e k is a unit vector whose kth entry is 1 and 0 elsewhere. In fact, for a composition , we only need the subset of column vectors a j of A corresponding to the composition (i.e., ) instead of the whole matrix. For example, for the species composition , only the 1st, 2nd, and 3rd column vectors of A (i.e., the sub-matrix [a 1 a 2 a 3 ]) are needed to define its feasibility region. Furthermore, we can define the normalized feasibility of a composition by is a function of the sub-matrix formed only by the interactions between the species in (i.e., ); while is a function of this sub-matrix augmented by the system-level effects of the composition on the third-party species in the system. In general, only when . Note that when the full community admits a globally stable equilibrium, non-invadable equilibria of any subsystems are also stable [46]. Following our geometric partition, we define the probability of feasibility of a pair of species in a larger multispecies system as the sum of all the feasibility regions where the pair forms part of the species composition ( ): Because the proposed probability of feasibility is related to the solution of the system at equilibrium, we estimate analytically the long-term effect of a multispecies system on a pair by (2) which represents the extent to which third-party species can modify the probability of pairwise coexistence relative to the probability of the pair in isolation allowing infinite time (Fig 1B). Note that the long-term effect given by Eq (2) describes the effect of the larger multispecies system on the probability that the composition arises as a community of the multispecies system, while the projection contribution given by Eq (1) denotes the effect of the larger multispecies system in weakening constraints on the range of parameters allowing possible coexistence of composition . The actual probability of pairwise coexistence depends on averaging over the effective growth rates θ of the rest of the species in the community. The long-term effect of Eq (2) takes that into account and therefore, measures the relative probability of a pairwise equilibrium being feasible within a specific multispecies community. Note also that while the projection contribution always expands the possible range of coexistence for a given pair, the long-term effect that measures the average coexistence range is often (roughly half of all cases) decreased; these statements are completely consistent since the probability of the additional θ k ’s taking values that significantly expand the pairwise coexistence range can be nonzero but quite small compared to the probability of contracting the coexistence range, and the projection contribution only incorporates the former.

Numerical system-level effects While the long-term behavior of a multispecies system provides an expectation of the effect of third-party species on a pair of species under an ideal setting; in practice, the study of pairwise coexistence in natural or experimental settings depends on the time scale of investigation, extinction thresholds, and the finite number of replicates. This implies that transient dynamics may play an additional role and observed coexistence rates may be different from the long-term expectations [12, 47]. Thus, to provide a practical short-term perspective regarding the effect of third-party species on the probability of pairwise coexistence, we complement the analytical calculations of feasibility with numerical calculations. We simulate gLV dynamics over a given time interval across an arbitrary number of repetitions with arbitrary initial conditions (picking from a uniform distribution between 0 and 1 for each species, but starting with the same abundance for all species yields similar results) and classify species with a final abundance N < 10−6 (results are robust to different thresholds) as statistically extinct (Fig 1C). The time interval 200 is chosen for consistency with the experimental data in S1 Appendix. That is, while correlations between short-term and experimental effects were in general good, correlations using a total time of 200 were the strongest (S10 Fig and S11 Fig), given the specific time parameters of the experiments. Hence, for consistency, we simply set all time intervals of simulations to 200. For multispecies systems, the vector of effective growth rates θ is sampled randomly from the closed unit sphere. Then, to control for confounding factors, we take the two values in the sampled vector θ that correspond to the pair as the effective growth rates for the pair in isolation. All simulations are conducted by the Runge-Kutta method. We quantify the short-term effect of a multispecies system on a pair of species by the ratio of the simulated frequencies of pairwise coexistence with third-party species and the simulated frequencies in isolation: (3) The simulated coexistence frequencies are always greater than or equal to the analytical, i.e., there is a short-term buffering effect and at , because we always start with all species present in the system and the global equilibrium is assumed to be stable. Only after a sufficiently long time do the simulated and analytical frequencies become the same (S7 Fig). To study whether the equilibration between short-term and long-term effects happens more quickly with third-party species or in isolation, we quantify the buffering effect of a simulated system by the ratio between the short-term and long-term effects (4) which represents the transient benefit (if >1) or disadvantage (if <1) to a pair of species after a finite time simulation of the system dynamics relative to the expected long-term behavior in the analytical calculations. For a given pair, the value of the buffering effect can change according to the chosen criteria for the simulations (S10 Fig), particularly the time span of the simulation. Yet, if the two simulated frequencies (within systems and in isolation )) are greater than their analytical counterparts (within systems and in isolation )) in equal proportion. On the other hand, if is greater (resp. less) than 1, the buffering effect is stronger with third-party species (resp. in isolation).

Comparing analytical and numerical effects for a given pair To illustrate how the long-term (analytical) and short-term (numerical) effects act on a given pair with different third-party species, Fig 1D shows the distribution of analytical and numerical system-level effects calculated on a hypothetical pair of symmetrically competing species (a 12 = a 21 = 0.22) when assembled together with different third species. Here, the interactions associated with the third species are chosen randomly from a normal distribution with μ = 0 and σ = 1 (all diagonal values are set to a ii = 1). Note that for such multispecies systems there is generally a stable global equilibrium, and . First, in this example, the projection contribution (first pink distribution) displays a large range of values including its maximum 1, confirming that the necessary conditions in (θ 1 , θ 2 ) for pairwise coexistence in isolation are substantially expanded with the third-party species. Second, the long-term effects (second blue distribution) as well as the short-term effects (third orange distribution) show that the probability of pairwise coexistence with third-party species can either decrease or increase compared to the probability in isolation. Yet, both distributions are approximately centered at 1, especially for the analytical case (95% confidence intervals for the mean of analytical effects [0.956, 1.027]). In fact, Fig 1E shows that both effects are positively associated, but not perfectly correlated, as expected due to transient behavior. We have considered a wide variety of such systems and find that this property is similar for any pair of species under various numbers of randomly-assembled third-party species (S3 Fig, S4 Fig, and S5 Fig). This reveals that the probability of pairwise coexistence in isolation is very close to the expected probability across all possible (abiotic and biotic) conditions under both short-term and long-term behaviors. This result is confirmed and further illustrated by a perturbative analysis for weakly interacting systems in S5 Appendix, where the center of the probability distribution of the analytical long-term effect varies from 1 only at fourth order in the interactions. Lastly, for this particular pair of competing species, the buffering effects (fourth green distribution in Fig 1D) are predominantly greater than 1, showing that the short-term effects tend to be stronger (relative to the long-term effects) with third-party species than in isolation. More generally, Fig 1F shows that negative (resp. positive) long-term effects of third-party species on pairwise coexistence are associated with a stronger (resp. weaker) buffering effect with third-party species than in isolation. In other words, short-term effects tend to be higher (resp. lower) than long-term effects if these long-term effects are less (resp. greater) than one. These results also hold for cases where the environmental effects are constrained to specific regions of the parameter space (S3 Appendix, S6 Fig). All these results are also confirmed by the perturbative analysis for weakly interacting systems in S5 Appendix.

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