Exclusion and multiplicity for stable communities in Lotka-Volterra systems

Lotka-Volterra systems model the time evolution of the populations $p_{i}$ of a finite set of $N$ species indexed by $i\in\underline{N}:=\{1,2,\ldots,N\}$. With ^′ denoting the time derivative, the governing differential equations take the form

Here $a_{i}$ represents an intrinsic growth rate for species $i$ in the limit when all populations are small, and $B_{ij}$ is a coefficient which, if positive, induces a competitive or inhibiting effect of the presence of species $j$ on the growth of species $i$. Throughout this paper we take the coefficients $a_{i}$ and $B_{ij}$ to be constant in time.

Almost exclusively, our interest is in solutions of (1) belonging to the state space ${\overline{{\mathbb{R}^{N}_{+}}}}=\{p\in\mathbb{R}^{N}:p_{i}\geq 0\;\forall i\}$, since negative species populations are normally not meaningful. It is convenient that this space is invariant for solutions of system (1).

Given a state $p\in{\overline{{\mathbb{R}^{N}_{+}}}}$, the community supporting $p$ will refer to the set of species $i$ for which $p_{i}>0$. Mathematically this is the support, denoted $\operatorname{\mathbf{spt}}p=\{i\in\underline{N}:p_{i}>0\}$. The community supporting a solution $t\mapsto p(t)$ is time-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

to denote the diagonal matrix with successive diagonal entries $p_{1},\ldots,p_{N}$. With this notation, equation (1) takes the form

A vector $\tilde{p}\in{\overline{{\mathbb{R}^{N}_{+}}}}$ is a steady state (or equilibrium) for the system (1) if and only if

We will analyze the system in block form with respect to the support community ${\mathcal{I}}=\operatorname{\mathbf{spt}}\tilde{p}$ and its complement ${\mathcal{J}}=\underline{N}\setminus{\mathcal{I}}$, via the notation

Then $\tilde{p}_{\mathcal{J}}=0$, and (3) means that $a_{\mathcal{I}}=B_{{\mathcal{I}}{\mathcal{I}}}\tilde{p}_{\mathcal{I}}$. Thus for any community ${\mathcal{I}}\subseteq\underline{N}$, if $B_{{\mathcal{I}}{\mathcal{I}}}$ is invertible then the community ${\mathcal{I}}$ supports at most one steady state in ${\overline{{\mathbb{R}^{N}_{+}}}}$.

The linearized equation of evolution for small perturbations $q$ around the steady state $\tilde{p}$ takes the form

where the constant matrix $A$ is explicitly given by

In block form using the diagonal-matrix notation $\backslash p\backslash$ above, we can write

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 of variables $p=D\hat{p}$ maps (2) to the system

having scaled columns, with $\hat{B}_{ij}=B_{ij}d_{jj}$. If $d_{ii}=\tilde{p}_{i}$ for $i\in{\mathcal{I}}=\operatorname{\mathbf{spt}}\tilde{p}$, the scaled equilibrium is uniform over ${\mathcal{I}}$, with $\hat{p}_{i}=1$ if and only if $i\in{\mathcal{I}}$, which we write as $\hat{p}=\mathds{1}_{\mathcal{I}}$.

There is a substantial body of literature regarding the nonlinear stability of Lotka-Volterra equilibria, especially with respect to solutions with positive population $p_{i}$ for every species considered, so that $p(t)\in{\mathbb{R}^{N}_{+}}=\{p\in\mathbb{R}^{N}:p_{i}>0\;\forall i\}$ for all $t$. For example, the books of Goh [13], Takeuchi [37] and Hofbauer and Sigmund [18] contain much information. We will mainly leave aside issues concerning degenerate cases that involve eigenvalues with zero real part.

As is well known, condition (i) above ensures that the equilibrium $\tilde{p}$ is locally asymptotically stable, i.e., it attracts all solutions of (1) in a small enough neighborhood in $\mathbb{R}^{N}$. Also well known is the fact that $\tilde{p}$ globally attracts all solutions in ${\mathbb{R}^{N}_{+}}$ if $-B$ is Volterra-Lyapunov stable; see [12] and [18, p. 191].

Hofbauer and Sigmund have conjectured in [18, p. 200] that $\tilde{p}$ globally attracts all solutions in ${\mathbb{R}^{N}_{+}}$ if $A$ is $D$-stable. To our knowledge, this conjecture remains open.

Given an equilibrium $\tilde{p}$ with support community ${\mathcal{I}}$, it is often natural to consider its stability with respect to solutions supported by the same community.

We will also call the community ${\mathcal{I}}$ internally stable (resp. $D$- or VL-stable) if it supports some equilibrium $\tilde{p}$ which is internally stable (resp. $D$- or VL-stable). Note that if $\tilde{p}$ is internally stable, then $B_{{\mathcal{I}}{\mathcal{I}}}$ is nonsingular and thus $\tilde{p}$ is the unique equilibrium state supported by ${\mathcal{I}}$, determined by $\tilde{p}_{\mathcal{I}}=B_{{\mathcal{I}}{\mathcal{I}}}^{-1}a_{\mathcal{I}}$.

If $\tilde{p}$ is internally VL-stable, then it attracts all solutions of (1) having the same support community ${\mathcal{I}}$. If $\tilde{p}$ is internally ($D$-)stable, it attracts all nearby solutions of (1) having the same support community.

These notions of internal stability say nothing about the behavior of solutions under perturbations which introduce species external to the community ${\mathcal{I}}$ supporting $\tilde{p}$. Due to the block structure of the linearized system in (4), this behavior is evidently determined by the sign of $(a-B\tilde{p})_{i}$ for $i\notin{\mathcal{I}}$. It will be convenient to consider this concept for species belonging to some given community ${\mathcal{Q}}\subseteq\underline{N}$.

Informally, this notion ensures that the (internally stable) community ${\mathcal{I}}$ that supports $\tilde{p}$ is stable against (infinitesimal) invasions by other species in ${\mathcal{Q}}$. In particular, if we take ${\mathcal{Q}}=\underline{N}$, it is straightforward to see that we have the following.

Recall that a fundamental result from the book of Hofbauer and Sigmund [18, Sec. 15.3] states that if the full matrix $-B$ is Volterra-Lyapunov stable, then the Lotka-Volterra system (1) admits a unique globally stable equilibrium state in ${\overline{{\mathbb{R}^{N}_{+}}}}$. (Also see [25] in case $B$ is positive definite.) With weaker conditions on $B$, it becomes possible that the system admits many more stable equilibria, and this can have interesting consequences for explaining the diversity and historical development of ecological communities [11, 4, 21, 14]. Thus it is interesting to identify any restriction on the composition of stable communities, such as a competitive exclusion principle, which may follow from the nature of interspecific interactions.

For example, one result that follows directly from the global stabilty theorem for Volterra-Lyapunov stable matrices $-B$ in [18, Sec. 15.3] is the following:

This follows by simply restricting the equations in (1) to species $i\in{\mathcal{Q}}$ and setting $p_{j}=0$ for $j\notin{\mathcal{Q}}$. In case the equilibrium $\tilde{q}$ is given and $\tilde{q}_{i}>0$ for all $i\in{\mathcal{Q}}$, the global stability follows from an argument going back to Volterra [40, §10.2] using the strict monotonicity of $F(p(t))$ for the relative entropy functional given by

with coefficients $d_{i}>0$ determined by VL-stability. See also [12, 25].

The empirical finding of Lischke and Löffler [24], if valid, would provide another powerful example of an exclusion principle. In their extensive computational experiments, they found (in the present terminology) that no stable single-species community was ever a subcommunity of any other stable community. As it is easy to check the stability of equilibria supported by a single species, Lischke and Löffler could use this principle to greatly simplify the search for all stable communities in large systems.

A quite general exclusion principle for stable communities of the Lischke-Löffler type is in fact valid, under the condition that the interaction matrix $B$ is $D$-symmetrizable.

In the terminology introduced at the end of the last section, Corollary 3.2 states that if $B$ is $D$-symmetrizable, different stable communities cannot completely overlap. This strong subcommunity exclusion principle does not hold in general in the absence of symmetrizability or any special stability properties. However, we find that it does always hold for communities that differ by only one species.

Finally, we are able to exclude complete overlap for stable communities under a weaker assumption than in Theorem 3.1. In particular, the assumption that the larger community is internally $D$-stable suffices.

The proofs of Theorems 3.3 and 3.4 will be provided in subsections 3.3 and 3.4 below. The notion of internal $D$-stabilty seems to arise naturally from (4)–(5), since the stability of the block $-B_{{\mathcal{I}}{\mathcal{I}}}$ is unaffected by any positive diagonal scaling. Despite a long history of investigation, however, computationally effective criteria that completely characterize $D$-stability are presently known only for $N\leq 4$ [7, 26]. One simple criterion that is sufficient to ensure the matrix $-B_{{\mathcal{I}}{\mathcal{I}}}$ is $D$-stable, though, which follows from Gershgorin’s circle theorem, is the diagonal dominance condition

This condition ensures $-B_{{\mathcal{I}}{\mathcal{I}}}$ 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 any larger internally $D$-stable community, stable or not. This would appear to support the conjecture of Hofbauer and Sigmund [18, p. 200] regarding the global stability of an equilibrium with full support ${\mathcal{I}}=\underline{N}$ when $-B$ is $D$-stable. For if such an equilibrium is not a global attractor in ${\mathbb{R}^{N}_{+}}$, then there cannot be any other strictly stable equilibrium in the system. Our present results leave open the possibility, however, that there could be some other equilibrium that is degenerately (semi-)stable, or there could be an open set in ${\mathbb{R}^{N}_{+}}$ with non-convergent dynamics.

In the most general case without symmetry, we find that an exclusion principle for stable sub- or super-communities does not always hold. Here is a basic counterexample.

A key property of the matrix $B$ in this example is that $-\backslash\tilde{q}\backslash B$ is stable but not $D$-stable. (In particular it is not a $P_{0}$ matrix, see subsection 3.4 below.) Here the single-species equilibrium $\tilde{p}$ is linearly stable in a strong sense: 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 and show that such examples can be produced for any number of species $N\geq 3$.

For the convenience of the reader, we prove Theorem 3.4 first in the special case when the equilibrium $\tilde{q}$ is internally VL-stable, i.e., when the principal submatrix $-B_{{\mathcal{Q}}{\mathcal{Q}}}$ is VL-stable. Of course, in this case the more general result of Theorem 3.1 holds, but the following proof, related to the dissipation rate of the Lyapunov function $F(p)$ in (9), is simple and self-contained.

The proofs of Theorems 3.3 and 3.4 make use of Schur complements. If $B$ is a square matrix with block representation

and $B_{\mathcal{II}}$ is invertible, the Schur complement of $B_{\mathcal{II}}$ in $B$ is defined by

Denoting the identity matrix by $I$, block row operations yield

evidently the Schur determinant formula holds:

A key ingredient in our proof of Theorem 3.4 is that $D$-stable matrices enjoy a property which behaves nicely under restriction to principal submatrices and their Schur complements. Firstly, it is known [7, p. 256] that for any $D$-stable matrix $A$, $-A$ has the following $P_{0}$ property.

We will also make use of the following characterization of $P_{0}$ matrices, observed by Fiedler and Pták [8].

This argument yields a result that differs in a rather subtle way from conclusions 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 principal minors are positive) if and only if for every $a\in\mathbb{R}^{N}$, the system (1) has a unique equilibrium $\tilde{p}\in{\overline{{\mathbb{R}^{N}_{+}}}}$ which is “saturated,” meaning $a_{i}-(B\tilde{p})_{i}\leq 0$ for all $i$. Any strictly stable equilibrium is strictly saturated, so it follows from [18, Thm. 15.4.5] that if ${\mathcal{Q}}\subseteq\underline{N}$ and $B_{{\mathcal{Q}}{\mathcal{Q}}}$ is a $P$-matrix, then at most one subcommunity of ${\mathcal{Q}}$ can be ${\mathcal{Q}}$-stable.

The same proof as that of Theorem 3.4 above establishes the following related exclusion principle, which relaxes the assumption on strict positivity of minors while strengthening the saturation (exterior stability) condition. If ${\mathcal{I}}\subseteq{\mathcal{Q}}\subseteq\underline{N}$, let us call an equilibrium $\tilde{p}$ with support ${\mathcal{I}}$ strictly ${\mathcal{Q}}$-saturated if $B_{{\mathcal{I}}{\mathcal{I}}}$ is nonsingular and $a_{i}-(B\tilde{p})_{i}<0$ for all $i\in{\mathcal{Q}}\setminus{\mathcal{I}}$.

The internal $D$-stability condition in Theorem 3.4 is in principle weaker than the VL-stability condition in Theorem 3.1. As we have indicated, it is not known how to verify $D$-stability computationally in every case where it is true, when $N>4$. In contrast, the assumptions in both Corollary 3.10 and [18, Thm. 15.4.5] can in principle be checked by computing sufficiently many principal minors. In practice, though, the number of minors involved may become prohibitively large if many species are considered.

Equations (1) model purely competitive interactions if all entries of the matrix $B$ are positive. Lischke and Löffler [24] state that in their extensive simulations of random competitive Lotka-Volterra systems, they never encountered a 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 all competitive systems, but the results of [24] suggest that encountering counterexamples may be a rare event. In this subsection we show that one can invent such counterexamples in systems of any size $N\geq 3.$

For a single-species community ${\mathcal{I}}$ to be stable and contained in a larger one ${\mathcal{Q}}$, necessarily $-B_{{\mathcal{I}}{\mathcal{I}}}<0$, and $-B_{{\mathcal{Q}}{\mathcal{Q}}}$ must be stable but not $D$-stable. For definiteness we set ${\mathcal{I}}=\{1\}$, ${\mathcal{Q}}=\underline{N}$ and ${\mathcal{J}}={\mathcal{Q}}\setminus{\mathcal{I}}$.

A matrix $B=B_{{\mathcal{Q}}{\mathcal{Q}}}$, with Schur complement $C=B/B_{{\mathcal{I}}{\mathcal{I}}}$, might have these properties if $B_{11}>0$ and $C$ has some negative diagonal element (implying $C$ is not a $P_{0}$ matrix). We can seek $B$ in the block form

where $b>0$ and $c,r\in\mathbb{R}^{N-1}$ have positive entries. (Note $C=B/B_{{\mathcal{I}}{\mathcal{I}}}$ here.)

In order for a state $\tilde{q}=(\tilde{q}_{1},\tilde{q}_{\mathcal{J}})^{T}>0$ to be a strictly stable equilibrium, we require $a=B\tilde{q}$ and all eigenvalues of $\backslash\tilde{q}\backslash B$ to have positive real parts. Then in order for $\tilde{p}=(\tilde{p}_{1},0)^{T}$ to be strictly stable, it suffices by (21) that

In Example 3.5 these conditions all hold — e.g.,

To construct examples for any $N\geq 3$, it is convenient to choose $C$ to make $B$ a rank-2 perturbation of $\epsilon I$ for small $\epsilon>0$. That is, we seek to make

where the vectors $v,w,\hat{v},\hat{w}\in\mathbb{R}^{N}$ have the block form

In this case $b=1+\epsilon$ and the Schur complement

The matrix $B$ has the eigenvalue $\epsilon>0$ with multiplicity $N-2$, since any vector orthogonal to both $w$ and $\hat{w}$ is an eigenvector. It is straightforward to show that the two remaining eigenvalues must take the form $\epsilon+\lambda$ where $\lambda$ is an eigenvalue of the $2\times 2$ matrix

With the specific choices

we find

The eigenvalues of $M$ have positive real part for all $N\geq 3$, since then $M$ has positive trace $3m$ and determinant $3m-4$.

Thus the matrix $-B$ is stable. With the choices $\tilde{q}=(1,1,\ldots,1)^{T}$, $a=B\tilde{q}$, the state $\tilde{q}$ becomes a strictly stable equilibrium. With $\tilde{p}_{1}=a_{1}/(1+\epsilon)$, the state $\tilde{p}=(\tilde{p}_{1},0)^{T}$ then satisfies (23) for sufficiently small $\epsilon>0$, since

Thus the single-species equilibrium $\tilde{p}$ is also strictly stable for small $\epsilon>0$.

One property that can allow many stable communities to exist is that stability persists if some species in a community is exchanged for a different species. If such a stability-preserving exchange is possible for $m$ different pairs of species independently, then the number of 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

for all $i,j\in\underline{N}$. We call $x$ and $y$ indistinguishable in this case.

For two such species, permuting the index labels in the Lotka-Volterra system (1) by swapping $x$ and $y$ leaves the system invariant. Thus if ${\mathcal{I}}$ is a stable community that contains $x$ but not $y$, then the community $\hat{\mathcal{I}}$ obtained by replacing $x$ by $y$ is also stable.

We shall describe two examples which involve groups of indistinguishable species, permitting large numbers of stable communities.

A much larger number of stable communities can be obtained. Suppose the set of $N$ species can be partitioned into $m$ disjoint subsets, each of which consists of pairwise indistinguishable species, and suppose further that a stable community ${\mathcal{I}}$ exists that contains exactly one member from each subset. Then each member of ${\mathcal{I}}$ can be exchanged with any member indistinguishable from itself. If $k_{1},k_{2},\ldots,k_{m}$ denote the number of species in the $m$ different subsets, then the number of stable communities in this case is at least as large as the number

We will show that this is indeed possible for any partition of $N$, as a special case of the main result in the next subsection. See Example 5.5 below.

As mentioned in the Introduction, some recent biological studies suggest that weak interactions may predominate in certain naturally occurring microbiomes, but stability is enhanced by the presence of some strongly competitive interactions. In this section we describe examples with this nature, having many stable communities.

Our construction is motivated by a known result in evolutionary game theory for symmetric payoff matrices related to the incidence matrix of a general graph. For such matrices, Cannings and Vickers [39, II] state that the ESSs are characterized in terms of the cliques of the graph (maximal complete subgraphs). In the context of continuous-time models of allele selection in population genetics, with a symmetric fitness matrix of this type, Hofbauer and Sigmund [18, Sec. 19.3] state that all stable rest points are characterized in terms of the cliques. Below we prove that a result of this type holds for Lotka-Volterra systems.