Relative nonlinearity is a coexistence mechanism that maintains species diversity via differences in the response to and effect on variation in resource density or some other factor mediating competition. Relative nonlinearity depends on two processes: 1) species have to differ in the curvature of their responses to resource density and 2) the patterns of resource variation generated by each species must favor the relative growth of another species. In its most basic form, one species grows best under equilibrium competitive conditions and another performs better under variable competitive conditions. Like all coexistence mechanisms, relative nonlinearity maintains species diversity by concentrating intraspecific competition relative to interspecific competition. Because resource density can be variable, intraspecific competition is the reduction of per-capita growth rate under variable resources generated by conspecifics (i.e. individuals of the same species). Interspecific competition is the reduction of per-capita growth rate under variable resources generated by heterospecifics (i.e. individuals of a different species). Like some other coexistence mechanisms (see the storage effect), relative nonlinearity can allow coexistence of at least two species on a single resource.
Functional components
Differential nonlinear responses to resources Relative nonlinearity requires that species differ in the curvature of their fitness response ϕ i {\textstyle \phi _{i}} to some competitive factor, F, like resource density. The nonlinearity of a response to competition is the second derivative of the per-capita growth rate with respect to the competitive factor ϕ i ′ ′ ( F ) {\textstyle \phi _{i}^{\prime \prime }(F)} , which is zero if the growth response is linear, positive if the response is accelerating (convex), and negative if the response is decelerating (concave). For competition between two species, the greater the difference in the curvatures of their response to changes in a competitive factor, the greater the differences in their overall specialization on competitive factor variation. For example, by Jensen's inequality, compared to constant resource density, variation in a competitive factor has no effect on species with zero curvature, positive effects on species with positive curvature, and negative effects on species with negative curvature. Thus, ϕ i ′ ′ ( F ) {\textstyle \phi _{i}^{\prime \prime }(F)} indicates a species response to variation in competitive factors, a dimension of competition that can be partitioned. Competitive factors are best thought of as dimensions of the environment that are jointly used by more than one species and contribute to a reduction in performance of individuals when used. For example, space is a common competitive factor for trees because many species require space for new trees to grow and the reduction in space reduces opportunities for other species to capture that space and grow. Resources and predators have similar properties and count as competitive factors. For competition between two species for a single shared resource, it is easy enough to think of the competitive factor as the reduction in species density due to consumption. In the absence of resource consumption, resources will tend to be at some equilibrium value, K. Thus, the competitive factor for our example is F = K − R {\textstyle F=K-R} for any value of R. The original demonstration of relative nonlinearity was in a consumer-resource model with differences in functional responses of the two species. One species has a Type I functional response and has zero curvature. The second species has a Type II functional response - which occurs when individuals must spend time handling resources before moving on to the next resource - and has negative curvature. Because the second species is limited by time when capturing resources, it is unable to exploit resources at high density compared to its competitor. If the Type II functional response species does better under average conditions than the species with a Type I functional response, the species differ in their response to equilibrium and variable resource density.
Differential effect on resource variation Not only must species respond differently to variation in competition, species must also affect variation in competition differently. Given these two processes, differential effects on and response to resource variation, species may coexist via relative nonlinearity.
Mathematical derivation Here, we will show how relative nonlinearity can occur between two species. We will start by deriving the average growth rate of a single species. Let us assume that each species' growth rate depends on some density-dependent factor, F, such that
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