Exclusion and multiplicity for stable communities in Lotka-Volterra systems
aa r X i v : . [ q - b i o . P E ] F e b EXCLUSION AND MULTIPLICITY FOR STABLECOMMUNITIES IN LOTKA-VOLTERRA SYSTEMS
WON EUI HONG AND ROBERT L PEGO
Department of Mathematical Sciences, Carnegie Mellon University,Pittsburgh, PA 15213, USA
Abstract.
For classic Lotka-Volterra systems governing many inter-acting species, we establish an exclusion principle that rules out theexistence of linearly asymptotically stable steady states in subcommu-nities of communities that admit a stable state which is internally D -stable. This type of stability is known to be ensured, e.g., by diagonaldominance or Volterra-Lyapunov stability conditions. By consequence,the number of stable steady states of this type is bounded by Sperner’slemma on anti-chains in a poset. The number of stable steady statescan nevertheless be very large if there are many groups of species thatstrongly inhibit outsiders but have weak interactions among themselves.By examples we also show that in general it is possible for a sta-ble community to contain a stable subcommunity consisting of a singlespecies. Thus a recent empirical finding to the contrary, in a study ofrandom competitive systems by Lischke and L¨offler (Theo. Pop. Biol. 115(2017) 24–34), does not hold without qualification. Contents
1. Introduction 22. Lotka-Volterra systems and notions of stability 43. Exclusion principles for stable communities 84. Bounds from Sperner’s lemma 155. Multiplicity of stable steady states 166. Linear exponential stability and instability 217. Relation to evolutionary game theory 22Acknowledgments 27References 28
E-mail address : [email protected], [email protected] . Date : February 2, 2021.2020
Mathematics Subject Classification.
Key words and phrases.
Gause’s law, multiple stable equilibria, clique, evolutionarilystable states, biodiversity, P matrices. Introduction
Lotka-Volterra systems comprise a family of classic and prototypical mod-els in population ecology. They incorporate nonlinear feedback and regula-tion mechanisms of clear biological importance in a structurally simple waythat renders them fairly amenable to mathematical analysis. Partly for thisreason they retain value and interest alongside models of greater complexityand realism [5].Based on such a model, Volterra in 1928 [40] demonstrated that twospecies exploiting a common resource cannot stably coexist. Volterra’s find-ings strongly influenced the development of competitive exclusion principlesand ecological niche theory by Gause [9], Hutchinson [19] and many others.The notion of competitive exclusion in general Lotka-Volterra competitionmodels was later investigated mathematically rather thoroughly and wasfound to be subject to a number of limitations [1, 30]. Moreover it wasdiscovered that, in principle, dynamics in such models can be arbitrarilycomplicated, admitting time-periodic and even chaotic behavior in systemswith only a few species [29, 10, 34]. Nevertheless, the concept of competitiveexclusion remains valuable and influential in ecology, as recently noted byPocheville [33].The present work is motivated by investigations regarding the numberof alternative stable steady states that a given (or typical) Lotka-Volterracompetition model may admit. Such investigations relate to a variety ofsignificant issues in ecology, such as whether a given local community ofspecies might be susceptible to invasion by a species that is not yet present,how a particular assembly of species may have come to co-exist, or whetherdifferent outcomes may have been possible based on different histories ofinvasion. See [11, 28, 4, 20, 36, 23, 21] for a small selection of papers thataddress such issues.Recently, Lischke and L¨offler [24] developed numerical methods for effi-ciently finding all the possible stable steady states in a given Lotka-Volterramodel. They carried out extensive numerical experiments to analyze a classof random competitive systems for up to 60 species, examining the effect ofrelative sizes of competition coefficients on the number and type of stableequilibria. In a small percentage of cases they find more than 30 alternativestable steady states. In addition they mention an empirical finding relatedto an exclusion principle. Loosely paraphrasing, they found that no specieswhich forms by itself a single-species stable community was ever observed tobe a member of any alternative stable community. If this were always true,then one could often greatly simplify the search for stable communities bystudying the stability of the simple single-species steady states.
Exclusion.
Below, we establish several community exclusion principlesrelated to these findings. We prove that a generalization of the empiricalLischke-L¨offler exclusion principle is valid in certain circumstances. In the
XCLUSION AND MULTIPLICITY IN LOTKA-VOLTERRA SYSTEMS 3 special case of symmetric (or diagonally symmetrizable) interspecific inter-action coefficients, it is universally the case for all Lotka-Volterra systems,whether competitive or not, that any stable community can neither containnor be contained in any other such. (See Corollary 3.2.)In the general case without symmetry, we show (Theorem 3.3) that notwo stable communities can differ by exactly one species. Furthermore, anycommunity that is “internally D -stable” does not admit any stable subcom-munity (Theorem 3.4). The general concept of D -stability has been muchstudied for constant-coefficient linear systems of differential equations ingeneral and linearized Lotka-Volterra systems in particular; see [7, 26, 22]and [18, Sec. 15.6]. Practical criteria that precisely characterize D -stabilityare not known in general, but sufficient criteria include stability due to diag-onal dominance, and Volterra-Lyapunov stability, meaning sign-definitenessof an associated quadratic form after diagonal scaling [7, Prop. 1].A still-open conjecture of Hofbauer and Sigmund states that an equilib-rium state involving all species of a Lotka-Volterra system is globally attract-ing if the interspecific interaction matrix is D -stable. Theorem 3.4 supportsthis conjecture insofar as it implies that no other equilibrium involving fewerspecies can be locally attracting.The empirical Lischke-L¨offler exclusion principle for Lotka-Volterra com-petitive systems turns out not to be valid without some qualification, how-ever. By example, we show in Section 3.5 that a single species forming astable equilibrium by itself can be contained in a larger stable community.It is plausible that such systems may be rare in typical random ensembles.If that is the case, an exclusion property for stable subcommunities may beexpected, though not guaranteed. Multiplicity.
The maximum number of stable steady states that can co-exist in Lotka-Volterra systems is an interesting quantity to consider, andcan be limited by community exclusion principles such as we study here. Ifall interspecific interactions are symmetric, or all stable states are internally D -stable, then the maximum number of stable equilibria is bounded viaSperner’s lemma for anti-chains in posets [27]; see Section 4 below. For N species with N large, this bound is approximately 2 N p /πN , which isa number somewhat smaller than 2 N , the number of all subsets of the N species, but one that still grows exponentially fast in N . We do not knowwhether the bound from Sperner’s lemma is sharp.It is true that exponentially many alternative stable subcommunities arepossible in principle, however. Particular highly symmetric examples can beconstructed similar to how cliques in graphs have been used to form stablestates in game theory [39] and continuous-time models of allele selection inpopulation genetics (replicator equations with symmetric payoff matrix) [18,p. 255].In Section 5 we describe and generalize this construction and establishquantitative criteria capable of ensuring that large numbers of alternativestable subcommunities are possible in certain Lotka-Volterra systems for N WON EUI HONG AND ROBERT L PEGO species. This can happen when many different communities can be formedconsisting of species that compete weakly with each other while stronglyinhibiting outsiders. Our criteria may have relevance for some biologicalsystems. E.g., certain recent works [6, 14] suggest that there may be com-mon patterns of interaction among the many alternative species in naturallyoccurring microbiomes. In particular, weak interactions may be predomi-nant in the microbiome of the human gut—a community comprising hun-dreds of species of bacteria—but the presence of some strongly competitiveinteractions can have a stabilizing effect [6].
Relation to evolutionary game theory.
It is well known that there is anequivalence between the dynamics of a given Lotka-Volterra system andthose of a corresponding family of replicator equations in evolutionary gametheory. A rather extensive body of work exists concerning exclusion princi-ples and multiplicity for stable states in replicator equations. Some of thefindings in this opus carry back readily to Lotka-Volterra systems. For oth-ers, their game-theoretic meaning has no evident significance in the Lotka-Volterra context. The degenerate nature of the correspondence can also getin the way.We will make a detailed comparison of our findings with correspondingresults on replicator equations in Section 7. Of special significance is thegame-theoretic notion of an evolutionarily stable state (ESS) , which has beenextensively explored following its introduction by Maynard Smith and Price[35] in an analysis of animal conflict. Each ESS is a locally attracting steadystate for replicator dynamics, but the reverse is not generally true for non-symmetric payoff matrices. The supports of ESSs are known to satisfy thesame type of exclusion principle (a non-containment property known as theBishop-Cannings theorem [2]) as we establish here for internally D -stableequilibria in Lotka-Volterra systems.We show that the ESS notion does not correspond to internal D -stabilityunder the replicator–Lotka-Volterra equivalence, however. Nor are ESSsinvariant under diagonal scalings natural to Lotka-Volterra systems. Aninteresting and extensive understanding of the multiplicity and patterns ofpossible ESSs for large numbers of strategies has been achieved; the recentpaper [3] has pointers to much relevant literature. Yet it remains unclearwhether corresponding results hold which are meaningful for Lotka-Volterrasystems.2. Lotka-Volterra systems and notions of stability
Governing equations.
Lotka-Volterra systems model the time evo-lution of the populations p i of a finite set of N species indexed by i ∈ N := { , , . . . , N } . With ′ denoting the time derivative, the governing differentialequations take the form p ′ i = p i ( a i − X j ∈ N B ij p j ) , i ∈ N . (1)
XCLUSION AND MULTIPLICITY IN LOTKA-VOLTERRA SYSTEMS 5
Here a i represents an intrinsic growth rate for species i in the limit when allpopulations are small, and B ij is a coefficient which, if positive, induces acompetitive or inhibiting effect of the presence of species j on the growth ofspecies i . Throughout this paper we take the coefficients a i and B ij to beconstant in time.Almost exclusively, our interest is in solutions of (1) belonging to thestate space R N + = { p ∈ R N : p i ≥ ∀ i } , since negative species populationsare normally not meaningful. It is convenient that this space is invariant forsolutions of system (1).Given a state p ∈ R N + , the community supporting p will refer to the setof species i for which p i >
0. Mathematically this is the support, denoted spt p = { i ∈ N : p i > } . The community supporting a solution t p ( t ) istime-invariant, since p i ( t ) is either always positive or always zero.In order to write this system in a convenient matrix-vector form, we define \ p \ = diag( p , . . . , p N )to denote the diagonal matrix with successive diagonal entries p , . . . , p N .With this notation, equation (1) takes the form p ′ = \ p \ ( a − Bp ) . (2)2.2. Equilibria, linearization, scaling.
A vector ˜ p ∈ R N + is a steady state(or equilibrium) for the system (1) if and only if a i − ( B ˜ p ) i = 0 for each i ∈ spt ˜ p . (3)We will analyze the system in block form with respect to the support com-munity I = spt ˜ p and its complement J = N \ I , via the notation p = (cid:18) p I p J (cid:19) , a = (cid:18) a I a J (cid:19) , B = (cid:18) B II B IJ B J I B J J (cid:19) . Then ˜ p J = 0, and (3) means that a I = B II ˜ p I . Thus for any community I ⊆ N , if B II is invertible then the community I supports at most onesteady state in R N + .The linearized equation of evolution for small perturbations q around thesteady state ˜ p takes the form q ′ = Aq, (4)where the constant matrix A is explicitly given by A ij = − ˜ p i B ij for i ∈ I and any j ∈ N ,a i − ( B ˜ p ) i for i / ∈ I and j = i, i / ∈ I and j = i .In block form using the diagonal-matrix notation \ p \ above, we can write A = \ a − B ˜ p \ − \ ˜ p \ B = (cid:18) −\ ˜ p I \ B II −\ ˜ p I \ B IJ \ a J − B J I ˜ p I \ (cid:19) . (5) WON EUI HONG AND ROBERT L PEGO
Diagonal scaling will sometimes be used for our analysis. If D = ( d ij )is a diagonal matrix with positive diagonal entries d ii >
0, the change ofvariables p = D ˆ p maps (2) to the systemˆ p ′ = \ ˆ p \ ( a − ˆ B ˆ p ) , ˆ B = BD, (6)having scaled columns, with ˆ B ij = B ij d jj . If d ii = ˜ p i for i ∈ I = spt ˜ p , thescaled equilibrium is uniform over I , with ˆ p i = 1 if and only if i ∈ I , whichwe write as ˆ p = I .2.3. Notions of stability.
Matrix conditions.
We recall a few standard definitions for matricesthat relate to the stability properties of the linear system (4) [7, 26, 18].
Definition 2.1.
Let A be a real N × N matrix.(1) A is stable if every eigenvalue of A has negative real part.(2) A is D -stable if DA is stable for all diagonal D > A is Volterra-Lyapunov stable (VL-stable) if there exists some diag-onal
D > DA + A T D <
0, or equivalently h x, DAx i < x ∈ R N \ { } .Here the notation S > S ≥ S <
0) for a real symmetricmatrix S means S is positive definite (resp. positive semidefinite or negativedefinite), and h· , ·i denotes the standard inner product in R N .It is known that Volterra-Lyapunov stability implies D -stability; see [7,Prop. 1]. Of course, D -stability implies stability. In case A is symmetric,the three notions are equivalent, since stability is equivalent to negativedefiniteness.The three notions are equivalent also in case A is D -symmetrizable , mean-ing D AD is symmetric for some positive diagonal D , D . For if A is stableand D = D − / D / , then the symmetric matrix S = DAD − <
0, hence2
DSD = D A + A T D <
0, thus A is VL-stable.2.3.2. Linear stability.
Our main results concern equilibria ˜ p which are sta-ble in the nondegenerate sense of being, in mathematical terms, linearlyasymptotically stable in R N . This means that q ( t ) → t → ∞ for everysolution of (4) in R N .It may seem natural instead to consider linear asymptotic stability in aweaker sense that requires perturbed populations to remain nonnegative.One could require q ( t ) → t → ∞ only for those solutions of (4) forwhich q i ≥ p i = 0, meaning ˜ p + ǫq ∈ R N + for sufficiently small ǫ > p is linearly asymptotically stable in R N + .Actually, this second notion turns out to be equivalent to the first. Proposition 2.2.
Let ˜ p be an equilibrium state for the Lotka-Volterra sys-tem (1) . Then the following are equivalent: (i) ˜ p is linearly asymptotically stable in R N . XCLUSION AND MULTIPLICITY IN LOTKA-VOLTERRA SYSTEMS 7 (ii) ˜ p is linearly asymptotically stable in R N + . (iii) A is stable. For the proof that (ii) is equivalent to (iii) see Section 6 below. Theequivalence of (i) and (iii) is well known. In Section 6 we also show thatlinear exponential instability of ˜ p in R N and in R N + are equivalent.For brevity, we say ˜ p is strictly stable if ˜ p is linearly asymptotically stable.We call I a stable community if it supports a strictly stable equilibrium ˜ p .2.4. Nonlinear stability.
There is a substantial body of literature regard-ing the nonlinear stability of Lotka-Volterra equilibria, especially with re-spect to solutions with positive population p i for every species considered,so that p ( t ) ∈ R N + = { p ∈ R N : p i > ∀ i } for all t . For example, the booksof Goh [13], Takeuchi [37] and Hofbauer and Sigmund [18] contain muchinformation. We will mainly leave aside issues concerning degenerate casesthat involve eigenvalues with zero real part.As is well known, condition (i) above ensures that the equilibrium ˜ p islocally asymptotically stable, i.e., it attracts all solutions of (1) in a smallenough neighborhood in R N . Also well known is the fact that ˜ p globallyattracts all solutions in R N + if − B is Volterra-Lyapunov stable ; see [12] and[18, p. 191].Hofbauer and Sigmund have conjectured in [18, p. 200] that ˜ p globally at-tracts all solutions in R N + if A is D -stable. To our knowledge, this conjectureremains open.2.5. Internal stability.
Given an equilibrium ˜ p with support community I , it is often natural to consider its stability with respect to solutions sup-ported by the same community. Definition 2.3.
Let ˜ p be an equilibrium state for the Lotka-Volterra system(1), and let I = spt ˜ p be its support community. We say: • ˜ p is internally stable if −\ ˜ p I \ B II is stable. • ˜ p is internally D -stable if − B II is D -stable. • ˜ p is internally VL-stable if − B II is Volterra-Lyapunov stable.We will also call the community I internally stable (resp. D - or VL-stable) if it supports some equilibrium ˜ p which is internally stable (resp. D -or VL-stable). Note that if ˜ p is internally stable, then B II is nonsingularand thus ˜ p is the unique equilibrium state supported by I , determined by˜ p I = B − II a I .If ˜ p is internally VL-stable, then it attracts all solutions of (1) havingthe same support community I . If ˜ p is internally ( D -)stable, it attracts allnearby solutions of (1) having the same support community.These notions of internal stability say nothing about the behavior of solu-tions under perturbations which introduce species external to the community I supporting ˜ p . Due to the block structure of the linearized system in (4),this behavior is evidently determined by the sign of ( a − B ˜ p ) i for i / ∈ I . WON EUI HONG AND ROBERT L PEGO
It will be convenient to consider this concept for species belonging to somegiven community
Q ⊆ N . Definition 2.4.
Let
Q ⊆ N , and let ˜ p be an equilibrium for (1) with supportcommunity I contained in Q . We say ˜ p is Q -stable if ˜ p is internally stableand a i − ( B ˜ p ) i < i ∈ Q \ I . (7)Informally, this notion ensures that the (internally stable) community I that supports ˜ p is stable against (infinitesimal) invasions by other species in Q . In particular, if we take Q = N , it is straightforward to see that we havethe following. Lemma 2.5.
Let ˜ p be an equilibrium state for system (1) . Then ˜ p is strictlystable (i.e., linearly asymptotically stable) if and only if it is Q -stable with Q = N . Exclusion principles for stable communities
Statements of main results.
Recall that a fundamental result fromthe book of Hofbauer and Sigmund [18, Sec. 15.3] states that if the fullmatrix − B is Volterra-Lyapunov stable, then the Lotka-Volterra system (1)admits a unique globally stable equilibrium state in R N + . (Also see [25]in case B is positive definite.) With weaker conditions on B , it becomespossible that the system admits many more stable equilibria, and this canhave interesting consequences for explaining the diversity and historical de-velopment of ecological communities [11, 4, 21, 14]. Thus it is interestingto identify any restriction on the composition of stable communities, suchas a competitive exclusion principle, which may follow from the nature ofinterspecific interactions.For example, one result that follows directly from the global stabilty the-orem for Volterra-Lyapunov stable matrices − B in [18, Sec. 15.3] is thefollowing: Theorem 3.1.
For any community
Q ⊆ N , if the principal submatrix − B QQ is Volterra-Lyapunov stable, then there is a unique equilibrium ˜ q ∈ R N + with support contained in Q that is Q -stable. This equilibrium ˜ q attractsall solutions of (1) with support community Q . This follows by simply restricting the equations in (1) to species i ∈ Q and setting p j = 0 for j / ∈ Q . In case the equilibrium ˜ q is given and ˜ q i > i ∈ Q , the global stability follows from an argument going back toVolterra [40, § F ( p ( t )) for the relativeentropy functional given by F ( p ) = X i ∈Q d i (˜ q i log ˜ q i p i + p i − ˜ q i ) , (8)with coefficients d i > XCLUSION AND MULTIPLICITY IN LOTKA-VOLTERRA SYSTEMS 9
The empirical finding of Lischke and L¨offler [24], if valid, would provideanother powerful example of an exclusion principle. In their extensive com-putational experiments, they found (in the present terminology) that nostable single-species community was ever a subcommunity of any other sta-ble community. As it is easy to check the stability of equilibria supportedby a single species, Lischke and L¨offler could use this principle to greatlysimplify the search for all stable communities in large systems.A quite general exclusion principle for stable communities of the Lischke-L¨offler type is in fact valid, under the condition that the interaction matrix B is D -symmetrizable. Corollary 3.2.
Suppose B is D -symmetrizable, and I is a community sup-porting a strictly stable equilibrium ˜ p for (1) . Then no other communitycontained in or containing I can support a strictly stable equilibrium ˜ q .Proof. Suppose ˜ p and ˜ q are both strictly stable and I ⊆ Q = spt ˜ q . Theneach is internally stable, and since B is D -symmetrizable, each is internallyVL-stable. In particular, − B QQ is VL-stable, so by the Theorem, ˜ q attractsall solutions with support community Q . But if ˜ q = ˜ p , this contradicts thestrict stability of ˜ p , which makes ˜ p locally asymptotically stable in R N . (cid:3) In the terminology introduced at the end of the last section, Corollary 3.2states that if B is D -symmetrizable, different stable communities cannotcompletely overlap. This strong subcommunity exclusion principle does nothold in general in the absence of symmetrizability or any special stabilityproperties. However, we find that it does always hold for communities thatdiffer by only one species.
Theorem 3.3.
No two stable communities can differ by exactly one species.I.e., if
I ⊂ N and x ∈ N \ I , then two equilibrium states with supportingcommunities I and I ∪ { x } cannot both be strictly stable. Finally, we are able to exclude complete overlap for stable communitiesunder a weaker assumption than in Theorem 3.1. In particular, the assump-tion that the larger community is internally D -stable suffices. Theorem 3.4.
Suppose Q is a community supporting an internally D -stableequilibrium ˜ q . Then no subcommunity of Q can support any different equi-librium state which is Q -stable. The proofs of Theorems 3.3 and 3.4 will be provided in subsections 3.3and 3.4 below. The notion of internal D -stabilty seems to arise naturallyfrom (4)–(5), since the stability of the block − B II is unaffected by anypositive diagonal scaling. Despite a long history of investigation, however,computationally effective criteria that completely characterize D -stabilityare presently known only for N ≤ − B II is D -stable, though, which follows from Gershgorin’s circle theorem, is the diagonal dominance condition b ii > X j ∈I\{ i } | b ij | for all i ∈ I . (9)This condition ensures − B II is VL-stable also — see Remark 5.7 below,and [26, p. 87] for a more general result.Theorem 3.4 excludes the complete overlap of a stable community by anylarger internally D -stable community, stable or not. This would appear tosupport the conjecture of Hofbauer and Sigmund [18, p. 200] regarding theglobal stability of an equilibrium with full support I = N when − B is D -stable. For if such an equilibrium is not a global attractor in R N + , then therecannot be any other strictly stable equilibrium in the system. Our presentresults leave open the possibility, however, that there could be some otherequilibrium that is degenerately (semi-)stable, or there could be an open setin R N + with non-convergent dynamics.In the most general case without symmetry, we find that an exclusionprinciple for stable sub- or super-communities does not always hold. Hereis a basic counterexample. Example 3.5 (Failure of subcommunity exclusion) . One can check that if B = , a = , (10)then the two different equilibrium states of (1) given by˜ p = , ˜ q = , with completely overlapping supports, are both strictly stable.A key property of the matrix B in this example is that −\ ˜ q \ B is stablebut not D -stable. (In particular it is not a P matrix, see subsection 3.4below.) Here the single-species equilibrium ˜ p is linearly stable in a strongsense: the matrix A in (4)–(5) is upper triangular with negative diagonal.The existence of a stable supercommunity is only possible because B is not D -stable. In subsection 3.5 below we will examine this more carefully andshow that such examples can be produced for any number of species N ≥ Exclusion for internally VL-stable states.
For the convenience ofthe reader, we prove Theorem 3.4 first in the special case when the equilib-rium ˜ q is internally VL-stable, i.e., when the principal submatrix − B QQ isVL-stable. Of course, in this case the more general result of Theorem 3.1holds, but the following proof, related to the dissipation rate of the Lyapunovfunction F ( p ) in (8), is simple and self-contained. XCLUSION AND MULTIPLICITY IN LOTKA-VOLTERRA SYSTEMS 11
Proof of Theorem 3.4 for internally VL-stable communities.
Let
Q ⊆ N bea community supporting an internally VL-stable equilibrium ˜ q , and suppose˜ p is a Q -stable equilibrium with supporting community P = spt ˜ p ⊆ Q .Note that a i − ( B ˜ p ) i ( = 0 for all i ∈ P , < i ∈ Q \ P ,while a i − ( B ˜ q ) i = 0 for all i ∈ Q . Let D be a positive diagonal matrixmaking the quadratic form of DB QQ positive definite, and let d i = D ii .Then K := X i ∈Q ˜ q i d i ( a i − ( B ˜ p ) i ) + X i ∈Q ˜ p i d i ( a i − ( B ˜ q ) i ) ≤ , while on the other hand, since 0 = ˜ q i ( a − B ˜ q ) i = ˜ p i ( a − B ˜ p ) i for all i , K = X i ∈Q ˜ q i d i (( B ˜ q ) i − ( B ˜ p ) i ) + X i ∈Q ˜ p i d i (( B ˜ p ) i − ( B ˜ q ) i )= (˜ p − ˜ q ) Q · DB QQ (˜ p − ˜ q ) Q ≥ . Thus ˜ p = ˜ q . (cid:3) Remark 3.6.
The same proof also proves that if − B QQ is any VL-stableprincipal submatrix of B , then there is at most one equilibrium with sup-porting community contained in Q that satisfies the (degenerate) condition a i − ( B ˜ q ) i ≤ i ∈ Q . This statement follows from stronger resultsproved in [18, Sec. 15.4].3.3. Exclusion for strictly stable states.
The proofs of Theorems 3.3and 3.4 make use of Schur complements. If B is a square matrix with blockrepresentation B = (cid:18) B II B IJ B J I B J J (cid:19) , and B II is invertible, the Schur complement of B II in B is defined by B/B II := B J J − B J I B − II B IJ . Since block row operations yield (cid:18) I − B J I B − II I (cid:19) (cid:18) B II B IJ B J I B J J (cid:19) = (cid:18) B II B IJ B/B II (cid:19) , evidently the Schur determinant formula holds:det B = det B II det( B/B II ) . Proof of Theorem 3.3.
Let ˜ p and ˜ q be strictly stable equilibria for the Lotka-Volterra system (1) with respective support communities I and Q = I ∪{ x } ,where x / ∈ I . Note that a I = B II ˜ p I , and due to the external stabilitycondition (7), 0 > a x − ( B ˜ p ) x = a x − B xI B − II a I . Since a Q = B QQ ˜ q Q , this is equal to( B x I ˜ q I + B xx ˜ q x ) − B x I B − II ( B II ˜ q I + B I x ˜ q x ) = ( B QQ /B II )˜ q x . Thus 0 > ( B QQ /B II ) = (det B QQ ) / (det B II ), but this contradicts the in-ternal stability conditions, which ensure both det B QQ and det B II are pos-itive. (cid:3) Proof for internally D -stable states. A key ingredient in our proofof Theorem 3.4 is that D -stable matrices enjoy a property which behavesnicely under restriction to principal submatrices and their Schur comple-ments. Firstly, it is known [7, p. 256] that for any D -stable matrix A , − A has the following P property. Definition 3.7. A ∈ R N × N is a P matrix if every principal minor of A isnonnegative. Lemma 3.8.
Schur complements in any nonsingular P matrix are also P .Proof. Let B be an N × N nonsingular P matrix with block representa-tion as above. In order to consider principal submatrices of B/B II , choose K ⊆ J and let I ′ = I ∪ K . Then (
B/B II ) KK = ( B J J − B J I B − II B IJ ) KK = B KK − B KI B − II B IK = B I ′ I ′ /B II . Thus any principal matrix of a Schurcomplement can be represented as a Schur complement. Then, by the de-terminant formula,det( B/B II ) KK = det( B I ′ I ′ /B II ) = det B I ′ I ′ / det B II ≥ . (cid:3) We will also make use of the following characterization of P matrices,observed by Fiedler and Pt´ak [8]. Theorem 3.9. (Fiedler & Pt´ak, 1966) A ∈ R N × N is a P matrix if andonly if for any nonzero x ∈ R N , there exists an index i such that x i = 0 and x i ( Ax ) i ≥ .Proof of Theorem 3.4. Without loss of generality, we assume that ˜ q is aninternally D -stable equilibrium with full support Q = N , meaning − B is D -stable. Suppose also that ˜ p is a strictly stable equilibrium with smallersupport I . Now we can analyze the external stability condition (7) for ˜ p using B and ˜ q as follows. For notational simplicity we let J = I c = N \ I .We have a − B ˜ p = (cid:18) a I a J (cid:19) − (cid:18) B II B IJ B J I B J J (cid:19) (cid:18) ˜ p I (cid:19) = (cid:18) a J − B J I ˜ p I (cid:19) , (11)hence a J − B J I ˜ p I = a J − B J I B − II a I . Since a = B ˜ q , however, we can write a J = B J I ˜ q I + B J J ˜ q J , a I = B II ˜ q I + B IJ ˜ q J , and deduce from the external stability condition (7) that, componentwise,0 > a J − B J I ˜ p I = ( B J J − B J I B − II B IJ )˜ q J = ( B/B II )˜ q J , (12) XCLUSION AND MULTIPLICITY IN LOTKA-VOLTERRA SYSTEMS 13 where
B/B II is the Schur complement of B II in B . But since B/B II inherits the P property from B , this contradicts Theorem 3.9. (cid:3) This argument yields a result that differs in a rather subtle way fromconclusions implied by Theorem 15.4.5 in the book of Hofbauer and Sigmund[18]. This theorem states that B is a P matrix (meaning all its principalminors are positive) if and only if for every a ∈ R N , the system (1) has aunique equilibrium ˜ p ∈ R N + which is “saturated,” meaning a i − ( B ˜ p ) i ≤ i . Any strictly stable equilibrium is strictly saturated, so it followsfrom [18, Thm. 15.4.5] that if Q ⊆ N and B QQ is a P -matrix, then at mostone subcommunity of Q can be Q -stable.The same proof as that of Theorem 3.4 above establishes the followingrelated exclusion principle, which relaxes the assumption on strict positivityof minors while strengthening the saturation (exterior stability) condition. If I ⊆ Q ⊆ N , let us call an equilibrium ˜ p with support I strictly Q -saturated if B II is nonsingular and a i − ( B ˜ p ) i < i ∈ Q \ I . Corollary 3.10.
If a community
Q ⊆ N supports an equilibrium ˜ q ∈ R N + and B QQ is a P matrix, then no different equilibrium ˜ p supported inside Q can be strictly Q -saturated. In particular, no ˜ p = ˜ q can be Q -stable. The internal D -stability condition in Theorem 3.4 is in principle weakerthan the VL-stability condition in Theorem 3.1. As we have indicated, it isnot known how to verify D -stability computationally in every case where itis true, when N >
4. In contrast, the assumptions in both Corollary 3.10and [18, Thm. 15.4.5] can in principle be checked by computing sufficientlymany principal minors. In practice, though, the number of minors involvedmay become prohibitively large if many species are considered.3.5.
Counterexamples to exclusion in competitive systems.
Equa-tions (1) model purely competitive interactions if all entries of the matrix B are positive. Lischke and L¨offler [24] state that in their extensive simula-tions of random competitive Lotka-Volterra systems, they never encountereda case where a single species formed a stable subcommunity of a larger one.Example 3.5 shows that this is not a universal property that holds for allcompetitive systems, but the results of [24] suggest that encountering coun-terexamples may be a rare event. In this subsection we show that one caninvent such counterexamples in systems of any size N ≥ . For a single-species community I to be stable and contained in a largerone Q , necessarily − B II <
0, and − B QQ must be stable but not D -stable.For definiteness we set I = { } , Q = N and J = Q \ I .A matrix B = B QQ , with Schur complement C = B/B II , might havethese properties if B > C has some negative diagonal element (im-plying C is not a P matrix). We can seek B in the block form B = (cid:18) b r T c C + cr T /b (cid:19) , (13) where b > c, r ∈ R N − have positive entries. (Note C = B/B II here.)In order for a state ˜ q = (˜ q , ˜ q J ) T > a = B ˜ q and all eigenvalues of \ ˜ q \ B to have positive real parts.Then in order for ˜ p = (˜ p , T to be strictly stable, it suffices by (12) that b ˜ p = a = b ˜ q + r T ˜ q J and a J − B J I ˜ p I = C ˜ q J < . (14)In Example 3.5 these conditions all hold — e.g., C ˜ q J = (cid:18) − − (cid:19) (cid:18) (cid:19) < . To construct examples for any N ≥
3, it is convenient to choose C to make B a rank-2 perturbation of ǫI for small ǫ >
0. That is, we seek to make B = ǫI + vw T + ˆ v ˆ w T , (15)where the vectors v, w, ˆ v, ˆ w ∈ R N have the block form v = (cid:18) c (cid:19) , w = (cid:18) r (cid:19) , ˆ v = (cid:18) − ˆ c (cid:19) , ˆ w = (cid:18) r (cid:19) . In this case b = 1 + ǫ and the Schur complement C = − ˆ c ˆ r T + ǫ ( I + cr T /b ) . The matrix B has the eigenvalue ǫ > N −
2, since anyvector orthogonal to both w and ˆ w is an eigenvector. It is straightforwardto show that the two remaining eigenvalues must take the form ǫ + λ where λ is an eigenvalue of the 2 × M = (cid:18) w T v w T ˆ v ˆ w T v ˆ w T ˆ v (cid:19) . (16)With the specific choices r = (1 , , . . . , T , c = 3 r, ˆ r = (1 , , . . . , T , ˆ c = 2 r − ˆ r, we find M = (cid:18) m − m − (cid:19) , m = N − . The eigenvalues of M have positive real part for all N ≥
3, since then M has positive trace 3 m and determinant 3 m − − B is stable. With the choices ˜ q = (1 , , . . . , T , a = B ˜ q , the state ˜ q becomes a strictly stable equilibrium. With ˜ p = a / (1 + ǫ ),the state ˜ p = (˜ p , T then satisfies (14) for sufficiently small ǫ >
0, since C ˜ q J = − ˆ c + O ( ǫ ) = ( − , − , . . . , − T + O ( ǫ ) < . Thus the single-species equilibrium ˜ p is also strictly stable for small ǫ > XCLUSION AND MULTIPLICITY IN LOTKA-VOLTERRA SYSTEMS 15 Bounds from Sperner’s lemma
The exclusion principles of the previous section imply bounds on thenumber of stable communities of certain types, by a well-known result fromthe combinatorial theory of posets (partially ordered sets). A poset is a set P with a binary relation ≤ satisfying reflexivity ( a ≤ a ), antisymmetry (if a ≤ b and b ≤ a , then a = b ) and transitivity (if a ≤ b and b ≤ c , then a ≤ c ). Two elements a and b in P are comparable if a ≤ b or b ≤ a . A chain in P is a subset C ⊆ P such that any two elements in C are comparable. Definition 4.1. An anti-chain in a poset P is a subset A ⊆ P such thatno two elements in A are comparable.For any set S , the collection of all subsets of S ordered by inclusionis a poset, denoted by ( P ( S ) , ⊆ ). For S = { , , } , e.g., the collection (cid:8) ∅ , { } , { , } , { , , } (cid:9) is a chain and (cid:8) { , } , { , } , { , } (cid:9) is an anti-chain. The maximal size of any anti-chain in a finite poset is boundedby the following well-known result of Sperner. See [27] for a short proof. Lemma 4.2 (Sperner’s lemma) . Let A be an anti-chain in a poset P havingN elements. Then the number of elements of A is at most (cid:0) N ⌊ N/ ⌋ (cid:1) . From Theorem 3.4 we directly infer the following.
Corollary 4.3.
For any Lotka-Volterra system (1) , no two stable subcom-munities of N = { , , . . . , N } that are internally D -stable are comparablewith respect to inclusion. The number of strictly stable equilibria that areinternally D -stable is therefore at most (cid:0) N ⌊ N/ ⌋ (cid:1) . Remark 4.4.
Note that if B is D -symmetrizable, any strictly stable stateis internally D -stable. In this case the number of strictly stable equilibria isbounded above by (cid:0) N ⌊ N/ ⌋ (cid:1) .We remark that when N is large, this bound is exponentially large in N and not so very much smaller than 2 N , the number of all subsets of N . ForStirling’s approximation says n ! ∼ √ πn ( ne ) n , thus (cid:18) N ⌊ N/ ⌋ (cid:19) ∼ N r πN . (17) Remark 4.5.
The same type of anti-chain property as described for Lotka-Volterra systems in Corollary 4.3 is well-known to hold for the supportsof evolutionarily stable states (ESSs) in evolutionary game theory. TheBishop-Cannings theorem [2, Thm. 2] implies that the support of any ESScan neither contain nor be contained in the support of any other. Thistheorem about ESSs actually provides a different collection of Lotka-Volterracommunities that enjoy the anti-chain property. We discuss this in detailbelow in Section 7. Multiplicity of stable steady states
We do not know whether the bound in Corollary 4.3 that comes fromSperner’s lemma is sharp. For certain systems whose interactions have abimodal competition structure, though, the number of stable communitiescan be exponentially large in N , and greater than 2 N/ in particular. Thisnumber is a bit larger than the square root of the bound in (17).Systems with such great numbers of stable communities may be quiterare. In the course of extensive numerical explorations of a random class ofLotka-Volterra systems, Lischke and L¨offler [24, Table 2] found that multiplestable equilibria occur in about half of their simulations involving between 2and 60 species, with about 2 percent having more than 25 stable equilibria.It appears that no more than about 40 stable equilibria in one system wereever encountered in [24]. With N = 60, though, more than 2 N/ > stable equilibria are possible in theory. Thus we are interested to investigatewhether robust conditions can be described which ensure that large numbersof stable equilibria exist.5.1. Indistinguishable species.
One property that can allow many stablecommunities to exist is that stability persists if some species in a communityis exchanged for a different species. If such a stability-preserving exchangeis possible for m different pairs of species independently, then the numberof stable communities is at least as large as 2 m .The simplest type of exchange of this kind occurs for two species x and y with identical growth rates and interaction coefficients, satisfying a x = a y , B ix = B iy , B xj = B yj , (18)for all i, j ∈ N . We call x and y indistinguishable in this case.For two such species, permuting the index labels in the Lotka-Volterrasystem (1) by swapping x and y leaves the system invariant. Thus if I is astable community that contains x but not y , then the community ˆ I obtainedby replacing x by y is also stable.We shall describe two examples which involve groups of indistinguishablespecies, permitting large numbers of stable communities. Example 5.1 (Complete indistinguishability and competitive exclusion) . In the simplest case, all species are pairwise indistinguishable, with inter-specific competition coefficients all the same, and intraspecies competitioncoefficients also all the same: B ij = ( α if j = i,β if j = i, a i = β. (19)When α > β > N strictly stable steady states˜ p with ˜ p = I for any singleton set I = { i } , i = 1 , . . . , N . Because B issymmetric, Corollary 3.2 applies. Thus, when the interspecific competitionsare stronger than the intraspecific competition, the competitive exclusion XCLUSION AND MULTIPLICITY IN LOTKA-VOLTERRA SYSTEMS 17 principle is valid. (When 0 < α < β on the other hand, B is positivedefinite and the system has a unique strictly stable equilibrium having equalpopulation densities for all species.)A much larger number of stable communities can be obtained. Supposethe set of N species can be partitioned into m disjoint subsets, each ofwhich consists of pairwise indistinguishable species, and suppose furtherthat a stable community I exists that contains exactly one member fromeach subset . Then each member of I can be exchanged with any memberindistinguishable from itself. If k , k , . . . , k m denote the number of speciesin the m different subsets, then the number of stable communities in thiscase is at least as large as the number m Y j =1 k j = k k · · · k m . (20)We will show that this is indeed possible for any partition of N , as a specialcase of the main result in the next subsection. See Example 5.5 below.5.2. Weak vs strong competition.
As mentioned in the Introduction,some recent biological studies suggest that weak interactions may predom-inate in certain naturally occurring microbiomes, but stability is enhancedby the presence of some strongly competitive interactions. In this sectionwe describe examples with this nature, having many stable communities.Our construction is motivated by a known result in evolutionary gametheory for symmetric payoff matrices related to the incidence matrix of ageneral graph. For such matrices, Cannings and Vickers [39, II] state thatthe ESSs are characterized in terms of the cliques of the graph (maximalcomplete subgraphs). In the context of continuous-time models of alleleselection in population genetics, with a symmetric fitness matrix of thistype, Hofbauer and Sigmund [18, Sec. 19.3] state that all stable rest pointsare characterized in terms of the cliques. Below we prove that a result ofthis type holds for Lotka-Volterra systems.
Example 5.2 (Friends vs rivals) . We suppose that any two different species i and j are either relatively friendly or are strong rivals . The interspecificinteraction coefficients B ij will take only three values: α (modeling friendlycompetition), β (self-inhibition), and γ (strong rivalry), and we assume α < β < γ. (21)We set a = ( a i = 1 for all i ) and B ij = α if i and j are friendly ,β if i = j,γ if i and j are rivals. (22)Evidently B is symmetric. If α = − β = γ = 0, the matrix − B is theincidence matrix for the graph whose edges connect friendly species. In this context, a clique is a maximal set of mutually friendly species.That is, a set
I ⊂ N is a clique if every two different species i, j ∈ I arefriendly, and no species k / ∈ I is friendly with all the species in I (so every k / ∈ I has some rival in I ).Under the assumptions above, in this example we have the following. Proposition 5.3.
Let
I ⊂ N be a set with m members. Then I is a stablecommunity if and only if I is a clique and c m := β + ( m − α > . (23)As a corollary, the stable communities in this example coincide exactlywith the cliques, provided we know c m > α ≥
0, meaning all interactions in the system are competitive.If α <
0, it holds if β > ( M − | α | , where M is the size of the largest clique. Proof.
First, suppose I is a clique and (23) holds. Then the state p = I β + ( m − α , with p i = ( /c m if i ∈ I ,0 otherwise, (24)is an equilibrium, and the matrix A in the linearized equation (4) has thefollowing structure: Whenever i / ∈ I we have A ij = 0 for all j = i , andmoreover, because i has at least one rival j ∈ I and A ii = a i − ( Bp ) i , c m A ii = c m − X j ∈I B ij ≤ c m − γ − ( m − α = β − γ < . (25)This is the external stability condition. On the other hand, because theblock A II = −\ p I \ B II and c m \ p I \ = I , we find that − c m A II = B II = ( β − α ) I + α T . (26)The eigenvalues of this symmetric matrix are c m (with eigenvector ) and β − α (with eigenspace orthogonal to ). Since both are positive, A II isnegative definite. Hence p is strictly stable, so I is a stable community.Conversely, suppose I is a stable community, supporting a strictly stableequilibrium p . Necessarily A II = −\ p I \ B II is stable, and so also is thesimilar (and symmetric) matrix −\ q \ B II \ q \ where q i = √ p i for all i . BySylvester’s law of inertia, B II is necessarily positive definite. Then it followsthat I contains no pair of rivals, for otherwise the indefinite matrix (cid:18) β γγ β (cid:19) would be a principal submatrix of B II .Thus I is a set of mutually friendly species, and necessarily B II has theform in (26). It follows that the eigenvalue c m > p takes theform in (24). If I is not itself a clique, then some i / ∈ I is friendly with all j ∈ I , and as in (25) we calculate that c m A ii = β − α >
0. This contradictsthe strict stability of p . Hence I is a clique. (cid:3) XCLUSION AND MULTIPLICITY IN LOTKA-VOLTERRA SYSTEMS 19
Remark 5.4.
In the example above, the stability of a given community I ofmutually friendly species persists under a slight loosening of the constraintson the interspecific interaction coefficients. Namely, one need not assumethe symmetry B ij = B ji for species i / ∈ I . It is only necessary that eachsuch species i be strongly inhibited by some member j ∈ I , having B ij > β (for this ensures A ii < B ji of species j by i is needed.In general it does not seem quite easy to count all the cliques in a graph,so we describe a class of special cases which shows that the number of stablecommunities in (20) can be achieved (cf. [18, Exercise 19.3.3]). Example 5.5 (Partitioning by rivals) . Suppose that in the preceding ex-ample, the N species can be partitioned into m disjoint and nonempty setsof mutual rivals , respectively having k , k , . . . , k m members, and any twospecies from different sets are friendly. Then clearly each clique (maximalset of mutually friendly species) has m members and is composed of onemember from each set of rivals. Moreover, the number of cliques is given by(20). Provided (23) holds for this value of m , these cliques comprise all thepossible stable communities.The maximum number of cliques in a graph of N nodes is [31] n · m − , if N = 3( m −
1) + n with n = 2 , . (27)This is therefore the maximum number of ESSs occurring in the main ex-ample considered in [39, II]. In Example 5.5 we achieve this number with m − n . For N = 60 we have m = 20 andfind 3 ≈ . × strictly stable equilibria can occur in such a system.5.3. Robust criteria for stability of cliques.
The property of beinga stable community naturally persists under sufficiently small changes inthe growth rates a i and interaction coefficients B ij . But the mathematicalnotion of “sufficiently small” leaves it unclear just how small a change isallowed. Here we aim to describe a simple and explicit set of quantitativebounds which ensure that a community I is stable, focusing on cases quali-tatively similar to Example 5.2, in which I essentially consists of a maximalset of mutually friendly species.Recall that, for given a and B , a community I is stable if it supports astrictly stable equilibrium p . This means exactly that, in the notation ofsection 2.2, the following conditions hold:(i) For all i ∈ I , a i = P j ∈I B ij p j and p i > i / ∈ I , a i < P j ∈I B ij p j and p i = 0.(iii) B II is nonsingular and A II = −\ p I \ B II is stable.For any specific case, general perturbation results for linear systems [15,Sec. 2.7] and matrix stability [17, Thm. 2.4] can be invoked to provide quan-titative bounds for changes in a and B which ensure that these propertiespersist for a perturbed equilibrium with the same support. We do not develop such results here, but instead pursue the limited aimof describing a set of systems in which interspecific competition is bimodal—either weak or strong—and that are qualitatively similar to Example 5.2,having multiple stable communities formed by cliques.For simplicity, we will consider only competitive systems for which a i > B ij ≥ i, j ∈ N . (28)For notational convenience we also suppose that a diagonal scaling as in (6)has been performed with d jj = a j /B jj , corresponding toˆ B ij = B ij a j B jj , ˆ p i = B ii p i a i for all i, j ∈ N , (29)whence a i = ˆ B ii for all i . Proposition 5.6.
Assume (28) and let α ∈ (0 , ) . Suppose C is somecollection of communities I ⊂ N for which the following hold: X j ∈I ,j = i ˆ B ij ≤ α a i for each i ∈ I , (30) X j ∈I ˆ B ij > a i − α for each i / ∈ I . (31) Then each
I ∈ C is a community that supports a strictly stable equilibriumwhich globally attracts all solutions having the same support.
Evidently, condition (30) requires that for species within I , the (total)interspecific competition is weak compared to self-inhibition, and (31) re-quires that each species not in I is strongly competed against (in total) bythe species inside I . Proof.
Let
I ∈ C have m members. A state ˆ p ∈ R N + supported by I is anequilibrium for (6) if and only ifˆ p i = F (ˆ p ) i := 1 − a i X j ∈I ,j = i ˆ B ij ˆ p j for all i ∈ I . (32)Under the given hypotheses, the function F is a strict contraction in themax norm on R m given by k v k ∞ = max i ∈I | v i | , since k F ( v ) − F ( w ) k ∞ ≤ α k v − w k ∞ . The set S = [1 − α, m ⊂ R m is mapped into itself by F , hence F hasa unique fixed point in S given by ˆ p I , where ˆ p is an equilibrium of (6)supported by I satisfying 1 − α ≤ ˆ p i ≤ i ∈ I . Condition (31)ensures that for each i / ∈ I , a i − X j ∈I ˆ B ij ˆ p j ≤ a i − (1 − α ) X j ∈I ˆ B ij < . XCLUSION AND MULTIPLICITY IN LOTKA-VOLTERRA SYSTEMS 21
Hence conditions (i) and (ii) above for a strictly stable equilibrium hold.Condition (iii) holds also because the matrix C = \ ˆ p I \ ˆ B II (similar to − A II )is diagonally dominant: Indeed, for all i ∈ I we have C ii = ˆ p i ˆ B ii = a i − X j ∈I ,j = i ˆ B ij ˆ p j ≥ a i (1 − α )since (29) and (32) hold and ˆ p j ≤
1, while X j ∈I ,j = i | C ij | = ˆ p i X j ∈I ,j = i ˆ B ij ≤ a i α < a i (1 − α ) . By Gershgorin’s theorem, every eigenvalue of C has positive real part. More-over, the diagonal dominance of C also implies A II is VL-stable (by [32,Thm. 3], or see the remark below). Hence the community I is internallyVL-stable, and the equilibrium p that it supports globally attracts all solu-tions with the same support. (cid:3) Remark 5.7.
We sketch a proof that − C is VL-stable (cf. [38]) for thereader’s convenience. Let G ij = | C ij | /C ii for i = j , and G ii = 0. Then I − G is a diagonally dominant M -matrix, with inverse P k ≥ G k whoseentries are all nonnegative. Hence q = ( I − G ) − T ∈ R N + , and it follows that C T \ q \ is diagonally dominant, for C ii q i − P j = i | C ji q j | = C ii > . Because \ q \ C is diagonally dominant too, DC + C T D > D = \ q \ .6. Linear exponential stability and instability
In this section we provide the proof of Proposition 2.2, showing thatlinear asymptotic stability in R N + is equivalent to linear asymptotic stabilityin R N . We also prove a complementary result concerning linear exponentialinstability. Proof of Proposition 2.2.
The equivalence of conditions (i) and (iii) is wellknown, and (i) implies (ii). So we only need to prove that (ii) implies (iii).Assume (ii) holds, meaning in particular that q ( t ) → t → ∞ for everysolution of (4) whose components satisfy the admissibility condition q i > i / ∈ spt ˜ p . (33)We claim that for any eigenvalue λ ∈ C of A , Re λ <
0. Let v ∈ C N be aneigenvector for λ with Re v = 0. Note we can find a vector w ∈ R N such that w > w + Re v >
0. Then the hypothesis (ii) implies that as t → ∞ , e At w → e At ( w + Re v ) → . Therefore e At Re v = Re( e λt v ) →
0, and this implies Re λ < (cid:3) Finally we note that it may seem possible a priori that an equilibriumstate on some edge or corner of R N + may be linearly exponentially unstablein R N but have an unstable manifold that does not intersect R N + . This can never happen for Lotka-Volterra systems, however, due to the structure ofthe linearized system (4). Proposition 6.1.
Let ˜ p ∈ R N + be an equilibrium state for the Lotka-Volterrasystem (1) , with supporting community I . Then the following are equivalent: (i) The system (4) has some exponentially growing solution q that sat-isfies the admissibility condition (33) . (ii) The matrix A in (4) has some eigenvalue with positive real part. (iii) With respect to the block matrix structure of A in (5) , either (a) \ ˜ p I \ B II has an eigenvalue with negative real part, or (b) a j − ( B ˜ p ) j > for some j / ∈ I .Proof. It is clear that (i) implies (ii) and that (ii) and (iii) are equivalent,so it remains to show (iii) implies (i). In the case that (b) holds, then A jj = a j − ( B ˜ p ) j > A . For any solution of (4)having q (0) = w > q i ( t ) = e A ii t w i > i / ∈ I ,and q j ( t ) grows exponentially. Thus (i) holds.Suppose now that (a) holds. Then −\ ˜ p I \ B II has some eigenvalue λ ∈ C with positive real part which is also an eigenvalue of A , and we can finda corresponding eigenvector of A having the block form v = ( v I , T , withRe v = 0. The function ˆ q ( t ) := e At Re v = Re( e λt v ) is an exponentiallygrowing solution of (4), but it is not admissible since ˆ q i ( t ) = 0 for all i / ∈ I .Thus, let w ∈ R N + be arbitrary. Then e At w is a solution of (4) that isadmissible. If this solution grows exponentially, we are done. Otherwise, q ( t ) = ˆ q ( t ) + e At w is an admissible solution that grows exponentially. (cid:3) Remark 6.2. If B II is D -symmetrizable with B II = D SD where S is symmetric, then (i) is equivalent to S having a negative eigenvalue, bySylvester’s law of inertia.7. Relation to evolutionary game theory
In evolutionary game theory, there is a substantial body of research onmultiplicity and patterns of evolutionarily stable states (ESSs) and the dy-namics of replicator equations , which bears a close comparison with theresults we have developed in this paper for Lotka-Volterra systems. For var-ious known facts about these things that we mention below, we refer to thebooks of Hofbauer and Sigmund [18] and Hadeler [16, Sec. 3.4].
Correspondence.
The dynamics of the Lotka-Volterra system (1) in R N + is well-known to correspond to those of replicator equations of the form x ′ i = x i (( Ax ) i − x T Ax ) , i = 0 , , . . . , N, (34)with x in the N -simplex ∆ N consisting of all x = ( x , x , . . . , x N ) such that x i ≥ i and P Ni =0 x = 1, via the mapping p x given by x = 1 / (1 + N X j =1 p j ) , x i = p i / (1 + N X j =1 p j ) , i = 1 , · · · , N. (35) XCLUSION AND MULTIPLICITY IN LOTKA-VOLTERRA SYSTEMS 23
This works for the payoff matrix A = (cid:18) a − B (cid:19) , (36)and after a solution-dependent nonlinear change of time variable. Notion of ESS.
An important notion in evolutionary game theory is thefollowing:
Definition 7.1.
A state y ∈ ∆ N is an evolutionarily stable state (ESS) when the following conditions are satisfied:(a) y T Ay ≥ x T Ay , for all x ∈ ∆ N (b) if x = y and y T Ay = x T Ay then y T Ax > x T Ax , for all x ∈ ∆ N An equivalent characterization is that y is an ESS if and only if y T Ax > x T Ax for all x = y near enough to y in ∆ N . (37)The condition (a) alone makes y a Nash equilibrium . It is known that anyESS is a steady state that is locally attracting (nonlinearly asymptoticallystable) for replicator dynamics. If A is symmetric, any steady state is locallyattracting if and only if it is an ESS. If A is not symmetric, however, a locallyattracting steady state of (34) need not be an ESS.In what follows, we will describe conditions that characterize Lotka-Volterra equilibria that correspond to ESSs in the way above. Our goalis to describe what stability properties such ESS-derived equilibria must ormay not have, and compare known exclusion principles for ESSs to those wehave developed in this paper. Symmetries.
A few relevant facts are the following: The correspondenceholds and the mapping p x can be reversed under the proviso that x = 0.Replicator dynamics are known to be invariant under two kinds of trans-formations, one that modifies all entries in any column of A by adding aconstant b i , and one that scales by a positive diagonal matrix D :(i) A A + b T and x x with the same time scale,(ii) A AD and x D − x/ ( T D − x ) with a nonlinear time change.Using a transformation of type (i), one can map any replicator equationin ∆ N with x = 0 to an N -component Lotka-Volterra system. We note,however, that these correspondences do not generally allow symmetric A tocorrespond with symmetric B in (1) or vice versa.Meanwhile, recall from (6) that Lotka-Volterra systems are invariant un-der a positive diagonal scaling on B : B BD and p D − p with the same time scale. (38)One can expect that internal stability (see Def. 2.3) of equilibria of Lotka-Volterra systems will be conserved through the transformation, and it istrue indeed. In replicator equations, however, a transformation of type(ii) can disrupt an ESS. In other words, when y is an ESS, an image ˆ y = D − y/ ( T D − y ), under a transformation of type (ii), is a Nash equilibriumbut might not be an ESS. We provide an example regarding this issue below. Relation to Lotka-Volterra. If y = ( y , y , . . . , y N ) is an ESS with y > q = y − ( y , . . . , y N ) (39)for the Lotka-Volterra system (1) obtained by reducing A to the form (36)by a transformation of type (i) above. One can readily check that theNash equilibrium condition (a) corresponds to the condition that, for all i = 1 , . . . , N ,( a − Bq ) i = 0 if q i > , ( a − Bq ) i ≤ q i = 0 . (40)A state q satisfying these conditions is called a saturated fixed point in [18].The ESS condition (37) translates to mean that (cid:18) T p T q q − p (cid:19) T ( a − Bp ) > p = q near q in R N + .Substituting p = q + r , this is equivalent to saying that for all small enough r with q + r ∈ R N + ,0 < (cid:18)(cid:18) I − q T T q (cid:19) r (cid:19) T ( Br + Bq − a ) . (41)Substituting r = ( I + q T ) v , one then finds the following characterization. Lemma 7.2.
A state y ∈ ∆ N with y > is an ESS for A in the form (36) if and only if for all nonzero v ∈ R N small enough we have < v T B ( I + q T ) v + v T ( Bq − a ) if v i ≥ whenever q i = 0 . (42)From this characterization we can infer the following. If q i > i , then Bq = a and it is necessary and sufficient for y to be an ESS thatthe symmetric part of B + a T is positive definite. (Or equivalently, thesymmetric part of B + a T is positive definite.)In general, if q i = 0 for some i , let I = spt q , then since ( Bq − a ) I = 0,necessarily 0 < v T I ( B II + a I T I ) v I for all nonzero v ∈ R N , (43)meaning the symmetric part of B II + a I T I is positive definite. This impliesthe symmetric part of B II is positive definite on the block subspaces ofdimension |I| orthogonal to both I and a I .It is natural to ask how (42) is related to internal stability in Lotka-Volterra equation. Considering that the ESS property brings nonlinear as-ymptotic stability, we cannot expect that −\ q I \ B II to be exponentiallyunstable; in particular (iii)(a) of proposition 6.1 should not hold. Combinedwith the external stability that the Nash condition provides, we have thefollowing implication for the image of an ESS in the Lotka-Volterra system. XCLUSION AND MULTIPLICITY IN LOTKA-VOLTERRA SYSTEMS 25
Theorem 7.3.
Let y and q be equilibria of the replicator and Lotka-Volterraequations, respectively, that are equivalent in the sense of (39) . If y is anESS, then q is an internally stable, saturated fixed point.Proof. We already checked the saturated condition in (40). Note that in(41), spt r ⊆ spt q if and only if spt v ⊆ spt q . So without loss of generality,we can assume that q is of full support. Let’s define C = ( I + q T ) − B andrewrite (41) as follows:0 < r T Cr = r T (cid:18) I − q T T q (cid:19) Br, for all nonzero r ∈ R N . In order to check internal stability of q , let’s examine \ q \ B = \ q \ ( I + q T ) C = ( \ q \ + qq T ) C. Since \ q \ + qq T is (symmetric) positive definite, we can find a positive definitematrix R such that R = \ q \ + qq T . Then, \ q \ B ∼ R − \ q \ BR = RCR.
Note − RCR is Volterra-Lyapunov stable since r T RCRr = ( Rr ) T C ( Rr ) > − RCR is stable. Thus by similarity of \ q \ B and RCR ,we can conclude that −\ q \ B is stable, i.e., q is internally stable. (cid:3) Relation to strict stability.
We would like to point out that Theorem 7.3is sharp in the sense that the ESS property neither implies nor is impliedby strict stability of the corresponding equilibrium in the Lotka-Volterrasystem. The following examples not only support this but also bring outthe problematic lack of an intrinsic dynamical nature for the ESS property.
Example 7.4.
For N = 2, let 0 < α < β > α and B = (cid:18) − αβ (cid:19) , q = , a = Bq.
Then − B is D -stable, and strictly stable (but not VL-stable). The definite-ness condition in Lemma 7.2 fails to hold, however, since for v T = (1 , − v T B ( I + q T ) v = 1 + α − β < . Therefore the corresponding state y = (1 , , ∈ ∆ for the replicatorsystem is not an ESS for the corresponding matrix A in (36).However, if we consider˜ B = BD, ˜ a = a, where D = (cid:18) − α +2 βα (cid:19) , we can check that (42) holds with ˜ q = D − q , i.e., for all nonzero v ∈ R N ,0 < v T ( ˜ B + a T ) v = v T (cid:18) − α − β β β (cid:19) v. This implies ˜ y = α α +2 β (1 , , − α +2 βα ) ∈ ∆ , which corresponds to ˜ q , is anESS. Note that ˜ y is dynamically equivalent to y .Moreover, since an internal ESS is a global attractor, we can see that theLotka-Volterra steady state state q = is a global attractor in R .We can summarize the implications of this example as follows: • the converse of Theorem 7.3 is false, • the image of an ESS under a transformation of type (ii) might notbe an ESS, • Lemma 7.2 can be used to prove global stability of an internal equi-librium in a Lotka-Volterra system which is not VL-stable.The next example shows that the conclusions that Theorem 7.3 ensuresfor the Lotka-Volterra image of an ESS are sharp in the sense that we cannotexpect strict stability of the corresponding equilibrium in general.
Example 7.5.
We describe an example with N = 3 of a non-strictly stablesteady state q that corresponds to an ESS. Take B = , a = , q = . (44)Then a = Bq and the condition in Lemma 7.2 reduces to saying that for allnonzero v = ( v , v , v ) with v , v ≥ < v T ( B + T ) v = 2( v + v + v ) + 2 v v . (45)This is indeed true, so q does correspond to an ESS y ∈ ∆ for the payoffmatrix in (36). The matrix A coming from (4), the linearized Lotka-Volterrasystem about ˜ p = q , takes the form A = − − −
10 0 00 0 0 . This linearization is degenerate and q is not strictly stable (linearly asymp-totically stable). We can note also that the matrix B ( I + q T ) is symmetricbut not positive definite, despite the validity of (45) when v , v ≥ Relation to stability of cliques.
One last comparison we will make isbetween our result on the stability of cliques in our graph-based Example 5.2and the characterization of ESSs in terms of cliques by Cannings and Vickers[39] for payoff matrices with the same graph-based structure.When the Lotka-Volterra growth rates a i are all the same, there is a differ-ent map between Lotka-Volterra solutions and the replicator equations [18,Exercise 7.5.2]. Namely, this is the projection map p x ∈ ∆ N given by x i = p i / N X j =1 p j , i = 1 , · · · , N, (46) XCLUSION AND MULTIPLICITY IN LOTKA-VOLTERRA SYSTEMS 27 together with a nonlinear time change, taking the payoff matrix A simplyas − B .Theorem 1 of Cannings and Vickers states, in our present terminology,that if B = − A is as in Example 5.2 above, so (21)–(22) hold, then thereis an ESS with support T ⊂ N if and only if T is a clique. Moreover, suchan ESS must take the form y = T / | T | . These ESSs comprise all the stableequilibria in the replicator equation in this case.But as Proposition 5.3 shows, a clique T with m = | T | members supportsa strictly stable state p under Lotka-Volterra dynamics if and only if theadditional condition β + ( m − α > α = − β = 0, in which case (23) neverholds and no strictly stable states exist. (Any Lotka-Volterra solution withsupport inside a clique will be unbounded in time, in fact.) Replicatordynamics remain invariant under adding the same constant to all entries of A , though. So after a suitable change of α , β , γ the ESSs and strictly stableLotka-Volterra states can all correspond. Remark 7.6 (The Cannings-Vickers characterization of ESSs) . Here weaddress an issue in the proof of Theorem 1 in [39] and indicate a clarification.In the proof that the support T of an ESS must be a clique, Cannings andVickers state that “if T is not a clique then there is a clique T ∗ containing T ,or contained in it.” As a general statement about graphs, this is not true—E.g., the set T = { , , } in the following graph has no super- or sub-graphthat is a clique: 1 234 5One can conclude T is a clique by arguing as follows instead. Suppose T supports an ESS but not a clique. If T is complete, we can find a clique T ∗ that strictly contains T , which yields a contradiction with the exclusionprinciple. If T is not complete, there exists a complete T ∗ that is maximalas a subgraph of T . Let y be an ESS supported by T and let x = | T ∗ | T ∗ .Since T ∗ ⊂ T , y T Ay = x T Ay so from Definition 7.1(b), y T Ax > x T Ax musthold. On the other hand, because T ∗ is maximal in T ,( Ax ) i ( = | T ∗ | ( | T ∗ | − i ∈ T ∗ , ≤ | T ∗ | ( | T ∗ | − i ∈ T \ T ∗ . This implies x T Ax > y T Ax , contradicting the observation we just made. Acknowledgments
This material is based upon work supported by the National Science Foun-dation under grant DMS 1812609.
References [1]
R. Armstrong and R. McGehee , Competitive exclusion , Amer. Natur., 115 (1980),pp. 151–170.[2]
D. T. Bishop and C. Cannings , Models of animal conflict , Advances in AppliedProbability, 8 (1976), pp. 616–621.[3]
I. M. Bomze and W. Schachinger , Constructing patterns of (many) ESSs undersupport size control , Dyn. Games Appl., 10 (2020), pp. 618–640.[4]
T. J. Case and R. G. Casten , Global stability and multiple domains of attractionin ecological systems , Amer. Natur., 113 (1979), pp. 705–714.[5]
P. Chesson , Mechanisms of maintenance of species diversity , Annual review of Ecol-ogy and Systematics, 31 (2000), pp. 343–366.[6]
K. Z. Coyte, J. Schluter, and K. R. Foster , The ecology of the microbiome:networks, competition, and stability , Science, 350 (2015), pp. 663–666.[7]
G. Cross , Three types of matrix stability , Linear Algebra and its Applications, 20(1978), pp. 253 – 263.[8]
M. Fiedler and V. Pt´ak , Some generalizations of positive definiteness and mono-tonicity. , Numerische Mathematik, 9 (1966/67), pp. 163–172.[9]
G. F. Gause et al. , Experimental analysis of vito volterra’s mathematical theory ofthe struggle for existence , Science, 79 (1934), pp. 16–17.[10]
M. E. Gilpin , Limit cycles in competition communities , The American Naturalist,109 (1975), pp. 51–60.[11]
M. E. Gilpin and T. J. Case , Multiple domains of attraction in competition com-munities , Nature, 261 (1976), pp. 40–42.[12]
B. S. Goh , Global stability in many-species systems , The American Naturalist, 111(1977), pp. 135–143.[13]
B.-S. Goh , Management and Analysis of Biological Populations , Elsevier, 1980.[14]
J. E. Goldford, N. Lu, D. Baji´c, S. Estrela, M. Tikhonov, A. Sanchez-Gorostiaga, D. Segr`e, P. Mehta, and A. Sanchez , Emergent simplicity in mi-crobial community assembly , Science, 361 (2018), pp. 469–474.[15]
G. H. Golub and C. F. Van Loan , Matrix Computations , Johns Hopkins Stud-ies in the Mathematical Sciences, Johns Hopkins University Press, Baltimore, MD,third ed., 1996.[16]
K. P. Hadeler , Topics in Mathematical Biology , Lecture Notes on MathematicalModelling in the Life Sciences, Springer, Cham, 2017.[17]
G. Hewer and C. Kenney , The sensitivity of the stable Lyapunov equation , SIAMJ. Control Optim., 26 (1988), pp. 321–344.[18]
J. Hofbauer and K. Sigmund , Evolutionary Games and Population Dynamics ,Cambridge University Press, Cambridge, 1998.[19]
G. E. Hutchinson , Population studies: Animal ecology and demography—concludingremarks , Cold Spring Harbor Symposia on Quantitative Biology, 22 (1957), pp. 415–427. Reprinted in: Bull. Math. Biol. 53 (1991) 193-213.[20]
T. C. Ings, J. M. Montoya, J. Bascompte, N. Bl¨uthgen, L. Brown, C. F.Dormann, F. Edwards, D. Figueroa, U. Jacob, J. I. Jones, et al. , Ecologicalnetworks–beyond food webs , Journal of Animal Ecology, 78 (2009), pp. 253–269.[21]
G. Kokkoris, A. Troumbis, and J. Lawton , Patterns of species interactionstrength in assembled theoretical competition communities , Ecology Letters, 2 (1999),pp. 70–74.[22]
O. Y. Kushel , Unifying matrix stability concepts with a view to applications , SIAMRev., 61 (2019), pp. 643–729.[23]
R. Law and R. D. Morton , Alternative permanent states of ecological communities ,Ecology, 74 (1993), pp. 1347–1361.
XCLUSION AND MULTIPLICITY IN LOTKA-VOLTERRA SYSTEMS 29 [24]
H. Lischke and T. J. L¨offler , Finding all multiple stable fixpoints of n-speciesLotka–Volterra competition models , Theoretical Population Biology, 115 (2017),pp. 24–34.[25]
H. Liu, W. Cai, and N. Su , Entropy satisfying schemes for computing selectiondynamics in competitive interactions , SIAM J. Numer. Anal., 53 (2015), pp. 1393–1417.[26]
D. O. Logofet , Stronger-than-Lyapunov notions of matrix stability, or how “flow-ers” help solve problems in mathematical ecology , Linear Algebra Appl., 398 (2005),pp. 75–100.[27]
D. Lubell , A short proof of Sperner’s lemma , J. Combinatorial Theory, 1 (1966),p. 299.[28]
R. M. May , Thresholds and breakpoints in ecosystems with a multiplicity of stablestates , Nature, 269 (1977), pp. 471–477.[29]
R. M. May and W. J. Leonard , Nonlinear aspects of competition between threespecies , SIAM J. Appl. Math., 29 (1975), pp. 243–253.[30]
R. McGehee and R. A. Armstrong , Some mathematical problems concerningthe ecological principle of competitive exclusion , J. Differential Equations, 23 (1977),pp. 30–52.[31]
J. W. Moon and L. Moser , On cliques in graphs , Israel J. Math., 3 (1965), pp. 23–28.[32]
P. J. Moylan , Matrices with positive principal minors , Linear Algebra Appl., 17(1977), pp. 53–58.[33]
A. Pocheville , The ecological niche: History and recent controversies , in Handbookof Evolutionary Thinking in the Sciences, T. Heams, P. Huneman, G. Lecointre, andM. Silberstein, eds., Springer Netherlands, Dordrecht, 2015, pp. 547–586.[34]
S. Smale , On the differential equations of species in competition , J. Math. Biol., 3(1976), pp. 5–7.[35]
J. M. Smith and G. R. Price , The logic of animal conflict , Nature, 246 (1973),pp. 15–18.[36]
Y. M. Svirezhev , Nonlinearities in mathematical ecology: Phenomena and models:Would we live in Volterra’s world? , Ecological Modelling, 216 (2008), pp. 89–101.[37]
Y. Takeuchi , Global dynamical properties of Lotka-Volterra systems , World ScientificPublishing Co., Inc., River Edge, NJ, 1996.[38]
L. Tartar , Une nouvelle caract´erisation des M matrices , Rev. Fran¸caise Informat.Recherche Op´erationelle, 5 (1971), pp. 127–128.[39] G. T. Vickers and C. Cannings , Patterns of ESSs. I, II , J. Theoret. Biol., 132(1988), pp. 387–408, 409–420.[40]