Qualitative Stability in Model Ecosystems

We discuss deductions that can be made as to the stability of multispecies communities, knowing only the structure of the food web, i.e., knowing only the signs of the elements in the interaction matrix. Systems which are stable in these circumstances are called qualitatively stable and have been treated in economic and other contexts. The discussion touches upon general aspects of the relation between complexity and stability in multispecies systems, and in particular suggests that on stability grounds predator-prey bonds should be more common than mutualistic ones. This result is not intuitively obvious, but is a feature of many real-world ecosystems. A wide variety of mathematical models have recently been studied, with a view to elucidating general features of the relation between complexity and stability in multispecies communities. Roughly speaking, complexity may be measured by the number and nature of the individual links in the trophic web, and stability by the tendency for relatively small population perturbations to damp out, returning the system to its equilibrium configuration (other usages of the terms are, of course, possible). Thus Gardner and Ashby (1970) and May (1972) have studied the stability character of large, complex ecosystem models in which the trophic web links are assembled (connected) at random; the stability is an interesting function of the number of species and the level of connectance. Drawing upon Kauffman's ( 1970a, b ) and others' analytic and computer work, Levins (1970') argues that "the dynamics of a broad class of complex systems will result in simplification through instability." May ( 197l a ) has reviewed work on the dynamical stability of multispecies generalizations of the familiar Lotka-Volterra predator-prey models, and many people have made numerical systems-analysis studies of the stability properties of specific multispecies systems (e.g.. reviews by Watt 1968, May 1971b). In one form or another, all this work makes assumptions about the magnitudes of the interactions between species in the community. The present note sets out some things that can be said, knowing only the topological structure of the trophic web, i.e., knowing only the signs (+, -, or 0) of the interactions between the various species. More specifically, consider a community with n populations, N,( t ) , labelled by the index i = 1, 2 , . . . , n. The dynamics of the community may in general be described by some nonlinear set of firstorder differential equations. The possible equilibrium or time-independent populations, N,*, in such a system are found by setting all the growth rates zero, Received July 24, 1972; accepted September 5, 1972. Present address: Department of Biology, Princeton University, Princeton, NJ 08540. and solving the consequent algebraic equations. To study the stability of the equilibrium community, one writes where the quantities x,(t) measure the initially relatively small perturbations to the equilibrium configuration. Then, expanding the n nonlinear population equations about the equilibrium point, the dynamics of small disturbances are described by a set of n linear first-order differential equations, whose structure is summarized by an n X n matrix of interaction coefficients: Alternatively, in matrix notation, Here x is the n x 1 column matrix of the x,, and A is the n x n "interaction matrix" or "community matrix" (Levins 1968), whose elements a,, describe the effect of species j upon species i near equilibrium. The necessary and sufficient condition for the equilibrium point to be stable, in the above sense, is that all the eigenvalues of the interaction matrix (which can be found by turning the handle of some well-defined mathematical machinery) have negative real parts. For a fuller account, see Maynard Smith (1968) or Rosen (1 970). To follow the theme of the present note, such detail is not necessary. This matrix A is clearly a quantity of direct biological significance. A diagram of the trophic web immediately shows which elements a,, are zero (no web link); the type of interaction sets the sign of the non-zero elements; and the details of the interactions determine the magnitude of these elements. The sign structure of this n X n matrix is directly tied to Odum's (1953) scheme which classifies interactions between species in terms of the signs of the effects produced. He characterizes the effect of species j upon species i as positive, neutral, or negative (that is, a, , +, 0, or ) depending on whether the 639 Late Spring 1973 ECOSYSTEM STABILITY population of species i is increased, is unaffected, or is decreased by the presence of species j . Thus for the pair of matrix elements abj and a,i we can construct a table of all possible interaction types: Effect of species j on i (i.e., sign of ao) I + 0 Effect of species + ++ +O +i on j 0 o+ 00 0(i.e., sign of aj,) -+ -0 -Apart from complete independence, there are five distinguishably different categories of interaction between any given pair of species, namely commensalism (+0), amensalism (-0) mutualism or symbiosis (++), competition (--), and general predatorprey (+-) including plant-herbivore, parasite-host, and so on. For a more thorough exposition, see Williamson (1972, Ch. 9). In short, on the one hand, this matrix summarizes the biology of the situation and encapsulates the character of the species' interactions, and, on the other hand, it gives the system's stability properties. What can be said, knowing only the sign of the individual matrix elements (+, -, or 0) and nothing else? In general. if a matrix can be shown to be necessarily stable (i.e., all eigenvalues having negative real parts), altogether independent of the actual magnitude of the non-zero elements, the matrix is called "qualitatively stable." This in an important subject in mathematical economics, where often no quantitative information is available (Quirk and Ruppert 1965, Maybee and Quirk 1969). The situation in ecology is similar. The sign of the interaction matrix elements can often be found by inspecting the food web diagram, even in the total absence of any quantitative data. The necessary and sufficient conditions for a matrix to be qualitatively stable are set out in the next section. If the signs (+, -, or 0 ) of the various matrix elements satisfy these detailed criteria, then the system is stable. If the criteria are not obeyed, nothing can be said: the matrix may be stable or unstable, depending on the actual magnitudes of the matrix elements. Usually this set of mathematically rigorous qualitative stability criteria will not apply exactly to complicated real-world situations, but even so they are useful in suggesting general tendencies. Some particular examples and consequences are discussed below. Of the conditions the web (or its matrix) must satisfy to be qualitatively stable, one worth remarking is that reciprocal pairs of elements ai, and a,i must either be of opposite sign, or at least one be zero. That is to say, the interspecific relationships prey-predator, commensalism, amensalism are all compatible with qualitative stability, but competition and mutualism or symbiosis are not. This is a mathematically rigorous statement, and it may be plausibly extended into the broader, if rougher, statement that competition or mutualism between two species is less conducive to overall web stability than is a predator-prey relationship. This is an interesting result, for it suggests that on stability grounds we would expect strong predatorprey bonds to be more common than mutualistic ones. It is a result which is not intuitively obvious, yet is a feature of many real-world ecosystems. QUALITATNE CONDITIONS STABILITY In mathematical terms, the necessary and sufficient conditions for an n X n matrix A, with elements q i , to be qualitatively stable are (Quirk and Ruppert 1965) :