Effects of Structured Interaction Matrices on Spatial Multispecies Communities

Clare AbreuPhysics Department, Massachusetts Institute of Technology

(Dated: May 13, 2016)

Modeling multispecies communities is a challenge of modern theoretical ecology. Although highlyinteracting, species-rich ecosystems abound in nature, attempts to find stable theoretical solutionsof such systems has been difficult. We employ a recently suggested algorithm to create structuredinteraction matrices for such communities, and simulate the resulting populations on a spatial gridwith nearest-neighbor interactions and randomly located migration events. We find that commu-nities generated by the structured matrices are more stable over time than ones that emerge fromunstructured interactions. We also examine phylogenetic distance between individuals and concludethat these stable communities are locally more homogeneous than might have been expected.

I. INTRODUCTION

In naturally occuring ecosystems, data that describesthe structure of the community is easily collected, butthe underlying dynamics are harder to determine. Ac-cordingly, ecologists seek tools that connect structure todynamics.

Some of these tools include models such as Lotka-Volterra, and rules like Taylor’s power law [6]. In thisstudy, we use models inspired by Lotka-Volterra and vari-ations of Taylor’s power law to measure the stability ofmultispecies communities.

In the next section, we give background information onthe models. The following section describes our imple-mentation of them as a spatial simulation. Subsequently,we describe our results, before framing them within ex-isting theory in the final section.

II. LOTKA-VOLTERRA MODEL

The classical model of Lotka (1932) and Volterra(1926) assumes that species diminish each other’s growthrates by direct interference: a relative increase of onespecies (xj) causes another species (xi) to decrease:

dxidt

= rixi

(Ki − xi −n∑

j=1

αijxj

Ki

)(1)

The model has been criticized for its simplicity andemphasis on competition over all other types interactions[1]. Yet it remains a useful tool that is often used tomodel inter- and intraspecies dynamics.

The condition for coexistence of two species can bereduced to the values of the interspecific competition co-efficients. Assuming carrying capacities K and single-species growth rates r equal unity, stable coexistence re-sults if αij , αji < 1.

Higher numbers of species can stably coexist, but thenecessary conditions are more difficult to find. As shownby Robert May in 1972 [2], even small amounts of nicheoverlap between species causes the system to transition

to an unstable state. This means that stable coexistencerequires most of the competition coefficents to be nearor equal to zero (or that the interaction matrix has alow connectivity [3]). Such an assumption is unrealistic,because all members of a multispecies ecosystem mustcompete to some degree for shared resources.

A. Interaction Matrix Algorithms

In order to study the stability of large, highly inter-acting multispecies communities, we make use of twoalgorithms that address May’s conundrum. The first[4], which we will call the structured algorithm, gen-erates structured interaction matrices. It assumes theDarwinian concept of higher niche overlap between moreclosely related species. The second [3], to be referred toas the unstructured algorithm, uses unstructured inter-action matrices. It mirrors May’s instability transitionby showing that varying the connectivity of an interac-tion matrix changes the probability distribution of themultispecies population.

1. The Structured Algorithm

Shtilerman et. al. claim to solve May’s coexistenceproblem with the following algorithm:

1: A “daughter” species evolves from a randomly chosen(but weighted by fitness) “mother” species.

2: The daughter species inherits slightly modified ver-sions of the mother’s competition coefficients:

αdaughter,j = αmother,j(1 − γ) + γε (2)

where γ ≪ 1 and ε is drawn from a Gamma dis-tribution of mean 1 and variance 1.

3: The daughter and mother species are connected bysignificant niche overlap:

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αmother,daughter = h(1 − γ) + γε (3)

where again, ε is a random variable and h ≤ 1 butshould be high enough to reflect the strong overlap(≈ 0.9).

The process repeats itself until an interaction ma-trix is complete. The structure of the matrix is block-diagonal, as seen in Figure 1, with strongly interacting“sub-communities” that are weakly connected to eachother.

2. The Unstructured Algorithm

The connectivity, C, of the interaction matrix is theonly given quantity in this algorithm, from which all elsefollows. The matrix is composed of numbers randomlydistributed between 0 and 1. If αij > 0, species i isconnected to species j, and αji > 0 as well, although thetwo numbers do not have to be close. At each time step,the following occurs:

1: With probability µ, a randomly chosen site in theecosystem, occupied by a species B, is replaced bya species A randomly chosen from the species pool.

2: With probability 1−µ, two randomly chosen individ-uals, A and B, interact. If αAB > αBA, A spreads,replacing B. If αAB = αBA, nothing happens.

For small values of µ (≈ 0.001) and SN (number of

species divided by number of individuals), the probabilitydistribution transitions from Gaussian to log-normal topower law as C increases.

This transition is equivalent to May’s instability tran-sition, because in this well-mixed environment, highlyconnected interaction matrices will cause the strongestspecies to outcompete the rest of the population, givenenough time. Species that are unconnected (αij = αji =0) can coexist.

III. SIMULATIONS IN SPACE

The well-mixed nature of the unstructured algorithm iswhat allows for the probably density of the population todepend solely on the connectivity of the interaction ma-trix. Furthermore, above a certain level of connectivity,the algorithm makes no distinctions: all highly connectedmatrices yield power law distributions.

We were curious about the effects of a structured ma-trix, where the distribution might depend not only on thenumber of nonzero components, but also on their loca-tion. We needed more details about the population thanits probability distribution, and thus decided to analyzethe spatial distribution of the population.

FIG. 1. Left: interaction matrices were generated us-ing the structured algorithm. Right: by tracking phylo-genetic distance during mother-daughter speciation events,we rearranged the matrices to reveal block-diagonal “sub-communities.”

FIG. 2. A structured ecosystem after 10,000 generations.Left: ordered by species. Right: ordered by phylogeneticdistance.

To create a spatial simulation, we first built matricesusing the stuctured algorithm, and then modified the un-structured algorithm to include only nearest-neighbor in-teractions. The random migration events did not change.

Ecosystems were constructed using both structuredand unstuctured interactions. After 10, 000 generationsof simulation (N time steps per generation), we used avariation of Taylor’s power law to evaluate the spatialand temporal stability of these ecosystems, which we de-scribe in the next section.

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FIG. 3. We find a good fit with the multispecies temporalversion of Taylor’s law. The mean slope is 1.087.

FIG. 4. The unstructured ecosystems also show a good fit,with a mean slope of 1.23,

IV. RESULTS

Taylor’s power law [6] relates the mean (m) and vari-ance (V ) of a species abundance as:

V = amb (4)

If b = 1, the population distribution is Poisson-like,and is distributed randomly. Values of b greater than1 indicate aggregation, and are more commonly seen innature. Values less than 1 indicate uniform distributions,and are very rare.

The rule is well established and widely used in ecol-ogy, with some believing it should be considered morefundamental than the model of logistic growth [5].

Many extensions of the rule have been developed, andwe use one that was recently suggested for multispeciespopulation densities over time [5]:

Vs = ambs

s = 1, 2, ...S(5)

where ms is the mean population abundance of onespecies over time, and S is the total number of species.In proposing this variation, its author [5] argued that thelarger the value of b, the more temporally aggregated andless stable is the multispecies population. As b increases,varations in the aggregation increase, which could leadto extinction of some species. For b ≤ 1, the communityaggregation over time is essentially constant, or “tempo-rally uniform.”

We applied this rule to our stuctured and unstructuredecosystems, as seen in Figures 3 and 4. We achieved agood fit with the model, with values of b greater than 1in both cases.

Interestingly, the unstructured interactions yield ahigher value of b than the structured interactions (a meanof 1.23 vs. 1.09), indicating that the latter is more tem-porally stable.

Values of b near 1 may also be interpreted as reflec-tive of near-neutral interactions among species [5]. Wemight have expected this, because the block-diagonal“sub-communities” in the structured matrices, whilehighly interacting within, interact weakly among eachother. This structure can be seen in Figure 2, whichdepicts an ecosystem ordered both by species and phy-logenetic distance. The phylogentic ordering exposes thesub-community clustering: each color corresponds to asub-community, with some sub-communities inhabiting(dominating) more space than others.

V. DISCUSSION

The aforementioned niche-overlap hypothesis says thatcompetition between more closely related species shouldbe stronger. Continuing with this argument, some con-clude that local communities should exhibit phylogeneticoverdispersion, with species being less related on averagethan if drawn randomly from the global species pool.

Shtilerman et. al. conjecture that their model chal-lenges this conclusion [4]. While members of the sub-communities do compete strongly with each other, thevariance of their competition coefficients is smaller than

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that of the entire matrix. Fitness differences are de-fined as the level of variance of these coefficents. There-fore, while competition may push members of a sub-community apart in space, their smaller fitness differ-ences pull them back together.

Having access to our spatially simulated ecosystemsallowed us to directly measure local versus global phy-logenetic distance. We divided the ecosystems into 300sub-populations and measured the averaged phylogeneticdistance between many random pairs, and did the same

throughout the global population. The results across allecosystems show that globally, individuals are about 20%more distantly related than locally, confirming the pre-diction of Shtilerman et. al.

Our findings show that the structured interactions pro-posed by Shtilerman et. al. lead to ecosystems that aretemporally more stable than those generated by naive,unstructured interactions, and locally more homogeneousthan expected. The structured algorithm gives rise tomultispecies communities with a stable mechanism of in-teraction between balanced sub-communities.

[1] May, Robert & McLean, Angela. Theoretical Ecology. Ox-ford University Press, 2007.

[2] May, Robert. Will a Large Complex System be Stable? Na-ture, 1972.

[3] Sole, Ricard et. al. Scaling in a network model of a mul-tispecies ecosystem. Physica A: Statistical Mechanics andits Applications, 2000.

[4] Shtilerman, Elad et. al. Emergence of structured communi-

ties through evolutionary dynamics. Journal of TheoreticalBiology, 2015.

[5] Ma, Zhanshan. Power law analysis of the human micro-biome. Molecular Ecology, 2015.

[6] Taylor, L. R. Aggregation, Variance, and the Mean. Na-ture, 1961.