Relative nonlinearity of competition

ecology tutorial

community ecology

coexistence

theoretical ecology

Relative nonlinearity in R: how curvature in a species’ response to a fluctuating limiting factor is a second fluctuation-dependent way to coexist.

Author

Tidy Ecology

Published

2026-08-02

The storage effect is one way a fluctuating environment can hold competitors together. There is a second, and it is easy to miss because it hides inside a piece of school algebra: the average of a curved function is not the function of the average. When two species differ in how curved their response to a fluctuating limiting factor is, that difference alone can let them coexist. This is relative nonlinearity, the subtler of the two fluctuation-dependent mechanisms.

Jensen’s inequality, made ecological

Suppose a species’ per-capita growth depends on some fluctuating quantity: a shared resource, the intensity of competition, a limiting factor that rises and falls from year to year. If that dependence is a straight line, fluctuations wash out: the good years and the bad years cancel, and the average growth is exactly the growth at the average condition. If the dependence curves upward, they do not cancel. The species gains more in the good years than it loses in the bad, so fluctuations lift its average growth above the constant-environment value. That is Jensen’s inequality, and the gap it opens is the engine here.

Rstates <- c(0.5, 1.5)                             # limiting factor swings around a mean of 1
gL <- function(R) 0.8*R - 0.8                      # linear response: no curvature
gV <- function(R) 0.8*R + 0.6*(R - 1)^2 - 0.8      # convex response: curves upward
# at the mean condition the two species are identical; compare their averages under fluctuation
c(linear_mean   = mean(sapply(Rstates, gL)),
  convex_mean   = mean(sapply(Rstates, gV)),
  linear_atmean = gL(1), convex_atmean = gV(1))

  linear_mean   convex_mean linear_atmean convex_atmean 
 5.551115e-17  1.500000e-01  0.000000e+00  0.000000e+00

The two species are built to be identical at the average condition: both have growth 0.00 when the limiting factor sits at its mean. Under fluctuation they part company. The linear species averages 0.000, unchanged, because a straight line has nothing to exploit. The convex species averages +0.150, lifted above zero purely by the variance. The size of that lift is set by the curvature and the variance together, and Jensen’s inequality gives it in closed form as half the second derivative times the variance.

gpp <- 1.2                                         # second derivative of the convex responder
varR <- mean((Rstates - 1)^2)                      # variance of the limiting factor
c(realised_gain = mean(sapply(Rstates, gV)) - gV(1),
  jensen_prediction = 0.5 * gpp * varR)

    realised_gain jensen_prediction 
             0.15              0.15

The realised gain and the prediction agree at +0.150: the convex species’ bonus is exactly half its curvature times the variance of the limiting factor. Nothing about this depended on the mean; two species with identical average responses can have completely different average growth once the environment varies, as long as their curvatures differ.

rg <- seq(0.3, 1.7, 0.01)
dl <- rbind(data.frame(R = rg, g = sapply(rg, gL), sp = "linear"),
            data.frame(R = rg, g = sapply(rg, gV), sp = "convex"))
pts <- data.frame(R = 1, g = c(EL, EV), sp = c("linear", "convex"))
ggplot(dl, aes(R, g, colour = sp)) +
  geom_hline(yintercept = 0, colour = col_ref, linewidth = 0.4) +
  geom_vline(xintercept = 1, colour = col_ref, linetype = "dotted") +
  geom_line(linewidth = 1) +
  geom_point(data = pts, aes(R, g, colour = sp), size = 2.6, shape = 18) +
  scale_colour_manual(values = c("linear" = col_lin, "convex" = col_con)) +
  labs(x = "Limiting factor", y = "Per-capita growth", colour = NULL,
       title = "Curvature turns variance into a growth bonus")

Two growth curves against a limiting factor, one straight and one curving upward, both through zero at the mean; markers show the averaged convex value sits above zero while the linear average stays at zero. — Figure 1: Growth against a fluctuating limiting factor. Both responses pass through zero at the mean (dotted). Averaging over the two environmental states leaves the linear species at zero but lifts the convex species above it, the Jensen gap.

From a bonus to coexistence

A growth bonus is not yet coexistence; coexistence needs each species to gain when it is rare. Relative nonlinearity delivers that through a two-step link that is worth stating plainly. A species tends to generate variance in the very factor it responds to nonlinearly, and it experiences the most variance when it is common. So the convex, variance-loving species, when rare, sits in an environment made variable by the other species and reaps the Jensen bonus; when common, it damps that variance itself and loses the bonus. Run the invasion test at the mean condition and under fluctuation.

# invader growth when the limiting factor is held constant vs when it fluctuates
c(constant_linear = gL(1), constant_convex = gV(1),
  fluct_linear = mean(sapply(Rstates, gL)), fluct_convex = mean(sapply(Rstates, gV)))

constant_linear constant_convex    fluct_linear    fluct_convex 
   0.000000e+00    0.000000e+00    5.551115e-17    1.500000e-01

Held constant, both invaders sit at 0.00: a dead heat, no mechanism, and the slightest average difference would decide the whole community. Let the factor fluctuate and the convex species invades at +0.150 while the linear species stays at 0.000. The curvature difference, and nothing else, has opened a route in for the species that would otherwise be shut out.

di <- rbind(data.frame(sp = "linear", env = "constant",    r = gLbar),
            data.frame(sp = "convex", env = "constant",    r = gVbar),
            data.frame(sp = "linear", env = "fluctuating", r = EL),
            data.frame(sp = "convex", env = "fluctuating", r = EV))
di$env <- factor(di$env, levels = c("constant", "fluctuating"))
ggplot(di, aes(env, r, fill = sp)) +
  geom_col(position = position_dodge(width = 0.65), width = 0.6) +
  geom_hline(yintercept = 0, colour = col_ref, linewidth = 0.4) +
  scale_fill_manual(values = c("linear" = col_lin, "convex" = col_con)) +
  labs(x = NULL, y = "Invader growth rate", fill = NULL,
       title = "Fluctuations let the more nonlinear species in")

A grouped bar chart of invader growth for a linear and a convex species under a constant and a fluctuating environment; only the convex species under fluctuation rises above zero. — Figure 2: Invader growth for the two species. In a constant environment both are tied at zero, no coexistence force; once the limiting factor fluctuates the convex, variance-loving species gains a positive growth rate while the linear one does not.

The honest boundary

Relative nonlinearity is real but it is delicate, and it deserves more caveats than the storage effect. The sign of the effect depends on which species has the more convex response and on which species generates more variance in the limiting factor, and these can point the same way or opposite ways; relative nonlinearity can just as easily destabilise a pair as stabilise it. It usually produces a weaker coexistence than the storage effect, and in models with a single fluctuating resource it often needs the fluctuations to be self-generated, through consumer-resource cycles, rather than imposed from outside. The lesson to carry is narrow and solid: curvature plus variance changes average growth, two species that differ in curvature therefore differ in how fluctuations treat them, and that difference is a coexistence mechanism in its own right. Whether it helps or hurts a particular pair is a question the invasion growth rate has to answer case by case.

References

Chesson P 2000. Annual Review of Ecology and Systematics 31:343-366 (10.1146/annurev.ecolsys.31.1.343).
Armstrong RA, McGehee R 1980. The American Naturalist 115(2):151-170 (10.1086/283553).

--- title: "Relative nonlinearity of competition" description: "Relative nonlinearity in R: how curvature in a species' response to a fluctuating limiting factor is a second fluctuation-dependent way to coexist." date: "2026-08-02 11:00" categories: [ecology tutorial, R, community ecology, coexistence, theoretical ecology] image: thumbnail.png image-alt: "A convex growth response to a fluctuating resource sitting above its constant-environment value, the Jensen gap that relative nonlinearity exploits." --- ```{r setup} #| include: false knitr::opts_chunk$set(echo = TRUE, message = FALSE, warning = FALSE, dev = "png", dpi = 120, fig.path = "figures/", fig.width = 7, fig.height = 4.4) library(ggplot2) theme_te <- theme_minimal(base_size = 12) + theme(panel.background = element_rect(fill = "white", colour = NA), plot.background = element_rect(fill = "white", colour = NA), panel.grid.minor = element_blank(), plot.title = element_text(face = "bold", size = 12)) theme_set(theme_te) col_lin <- "#b5651d"; col_con <- "#3a6b52"; col_ref <- "#555555" Rstates <- c(0.5, 1.5) # fluctuating limiting factor, mean 1 Rbar <- mean(Rstates) gL <- function(R) 0.8*R - 0.8 # linear responder gV <- function(R) 0.8*R + 0.6*(R - 1)^2 - 0.8 # convex responder, same value and slope at the mean EL <- mean(sapply(Rstates, gL)); EV <- mean(sapply(Rstates, gV)) gLbar <- gL(Rbar); gVbar <- gV(Rbar) gpp <- 1.2 # second derivative of the convex responder varR <- mean((Rstates - Rbar)^2) jensen <- 0.5 * gpp * varR ``` The [storage effect](../the-storage-effect/) is one way a fluctuating environment can hold competitors together. There is a second, and it is easy to miss because it hides inside a piece of school algebra: the average of a curved function is not the function of the average. When two species differ in how curved their response to a fluctuating limiting factor is, that difference alone can let them coexist. This is relative nonlinearity, the subtler of the two fluctuation-dependent mechanisms. ## Jensen's inequality, made ecological Suppose a species' per-capita growth depends on some fluctuating quantity: a shared resource, the intensity of competition, a limiting factor that rises and falls from year to year. If that dependence is a straight line, fluctuations wash out: the good years and the bad years cancel, and the average growth is exactly the growth at the average condition. If the dependence *curves upward*, they do not cancel. The species gains more in the good years than it loses in the bad, so fluctuations lift its average growth above the constant-environment value. That is Jensen's inequality, and the gap it opens is the engine here. ```{r jensen} Rstates <- c(0.5, 1.5) # limiting factor swings around a mean of 1 gL <- function(R) 0.8*R - 0.8 # linear response: no curvature gV <- function(R) 0.8*R + 0.6*(R - 1)^2 - 0.8 # convex response: curves upward # at the mean condition the two species are identical; compare their averages under fluctuation c(linear_mean = mean(sapply(Rstates, gL)), convex_mean = mean(sapply(Rstates, gV)), linear_atmean = gL(1), convex_atmean = gV(1)) ``` The two species are built to be identical at the average condition: both have growth `r sprintf("%.2f", gLbar)` when the limiting factor sits at its mean. Under fluctuation they part company. The linear species averages `r sprintf("%.3f", EL)`, unchanged, because a straight line has nothing to exploit. The convex species averages `r sprintf("%+.3f", EV)`, lifted above zero purely by the variance. The size of that lift is set by the curvature and the variance together, and Jensen's inequality gives it in closed form as half the second derivative times the variance. ```{r jensen-check} gpp <- 1.2 # second derivative of the convex responder varR <- mean((Rstates - 1)^2) # variance of the limiting factor c(realised_gain = mean(sapply(Rstates, gV)) - gV(1), jensen_prediction = 0.5 * gpp * varR) ``` The realised gain and the prediction agree at `r sprintf("%+.3f", jensen)`: the convex species' bonus is exactly half its curvature times the variance of the limiting factor. Nothing about this depended on the mean; two species with identical average responses can have completely different average *growth* once the environment varies, as long as their curvatures differ. ```{r fig-jensen} #| fig-cap: "Growth against a fluctuating limiting factor. Both responses pass through zero at the mean (dotted). Averaging over the two environmental states leaves the linear species at zero but lifts the convex species above it, the Jensen gap." #| fig-alt: "Two growth curves against a limiting factor, one straight and one curving upward, both through zero at the mean; markers show the averaged convex value sits above zero while the linear average stays at zero." rg <- seq(0.3, 1.7, 0.01) dl <- rbind(data.frame(R = rg, g = sapply(rg, gL), sp = "linear"), data.frame(R = rg, g = sapply(rg, gV), sp = "convex")) pts <- data.frame(R = 1, g = c(EL, EV), sp = c("linear", "convex")) ggplot(dl, aes(R, g, colour = sp)) + geom_hline(yintercept = 0, colour = col_ref, linewidth = 0.4) + geom_vline(xintercept = 1, colour = col_ref, linetype = "dotted") + geom_line(linewidth = 1) + geom_point(data = pts, aes(R, g, colour = sp), size = 2.6, shape = 18) + scale_colour_manual(values = c("linear" = col_lin, "convex" = col_con)) + labs(x = "Limiting factor", y = "Per-capita growth", colour = NULL, title = "Curvature turns variance into a growth bonus") ``` ## From a bonus to coexistence A growth bonus is not yet coexistence; coexistence needs each species to gain when it is rare. Relative nonlinearity delivers that through a two-step link that is worth stating plainly. A species tends to generate variance in the very factor it responds to nonlinearly, and it experiences the most variance when it is common. So the convex, variance-loving species, when rare, sits in an environment made variable by the other species and reaps the Jensen bonus; when common, it damps that variance itself and loses the bonus. Run the invasion test at the mean condition and under fluctuation. ```{r invasion} # invader growth when the limiting factor is held constant vs when it fluctuates c(constant_linear = gL(1), constant_convex = gV(1), fluct_linear = mean(sapply(Rstates, gL)), fluct_convex = mean(sapply(Rstates, gV))) ``` Held constant, both invaders sit at `r sprintf("%.2f", gVbar)`: a dead heat, no mechanism, and the slightest average difference would decide the whole community. Let the factor fluctuate and the convex species invades at `r sprintf("%+.3f", EV)` while the linear species stays at `r sprintf("%.3f", EL)`. The curvature difference, and nothing else, has opened a route in for the species that would otherwise be shut out. ```{r fig-invasion} #| fig-cap: "Invader growth for the two species. In a constant environment both are tied at zero, no coexistence force; once the limiting factor fluctuates the convex, variance-loving species gains a positive growth rate while the linear one does not." #| fig-alt: "A grouped bar chart of invader growth for a linear and a convex species under a constant and a fluctuating environment; only the convex species under fluctuation rises above zero." di <- rbind(data.frame(sp = "linear", env = "constant", r = gLbar), data.frame(sp = "convex", env = "constant", r = gVbar), data.frame(sp = "linear", env = "fluctuating", r = EL), data.frame(sp = "convex", env = "fluctuating", r = EV)) di$env <- factor(di$env, levels = c("constant", "fluctuating")) ggplot(di, aes(env, r, fill = sp)) + geom_col(position = position_dodge(width = 0.65), width = 0.6) + geom_hline(yintercept = 0, colour = col_ref, linewidth = 0.4) + scale_fill_manual(values = c("linear" = col_lin, "convex" = col_con)) + labs(x = NULL, y = "Invader growth rate", fill = NULL, title = "Fluctuations let the more nonlinear species in") ``` ## The honest boundary Relative nonlinearity is real but it is delicate, and it deserves more caveats than the storage effect. The sign of the effect depends on which species has the more convex response *and* on which species generates more variance in the limiting factor, and these can point the same way or opposite ways; relative nonlinearity can just as easily *destabilise* a pair as stabilise it. It usually produces a weaker coexistence than the storage effect, and in models with a single fluctuating resource it often needs the fluctuations to be self-generated, through consumer-resource cycles, rather than imposed from outside. The lesson to carry is narrow and solid: curvature plus variance changes average growth, two species that differ in curvature therefore differ in how fluctuations treat them, and that difference is a coexistence mechanism in its own right. Whether it helps or hurts a particular pair is a question the [invasion growth rate](../partitioning-invasion-growth/) has to answer case by case. ## Related tutorials - [The storage effect and coexistence](../the-storage-effect/) - [Partitioning the invasion growth rate](../partitioning-invasion-growth/) - [Checking a storage-effect analysis](../checking-a-storage-effect-analysis/) - [Stochastic population growth](../stochastic-population-growth/) ## References - Chesson P 2000. Annual Review of Ecology and Systematics 31:343-366 (10.1146/annurev.ecolsys.31.1.343). - Armstrong RA, McGehee R 1980. The American Naturalist 115(2):151-170 (10.1086/283553).