Talk:Relative nonlinearity

[Untitled]
I'm working on Mathematical derivation section, similar to the one for the storage effect and fitness-density covariance. This is still a work in progress, and needs things like citations (such as Chesson 1994 and Armstrong & McGhee 1980).Simonmstump (talk) 17:18, 28 September 2016 (UTC)

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
 * $$ \frac{dN_j}{dt} = \phi_j(F)N_j $$,

where Nj is species j's population density, and $$\phi_j(F)$$ is some function of the density-dependent factor F. For example, under a Monod chemostat model, F would be the resource density, and $$\phi_j(F)$$ would be $$a_jF - d$$, where aj is the rate that species j can uptake the resource, and d is its death rate. In a classic paper by Armstrong and McGhee [cite Armstrong], $$\phi_j(F)$$ was the a Type I functional response for one species and a Type II functional response for the other. We can approximate the per-capita growth rate, $$r_j = \frac{1}{N_j} \frac{dN_j}{dt}$$, using a Taylor series approximation as
 * $$ r_j \approx \phi_j(\overline{F}) + (F - \overline{F}) \phi_j(\overline{F})' + \frac{1}{2} (F - \overline{F})^2 \phi_j(\overline{F})'' $$,

where $$\overline{F}$$ is the average value of F. If we take the average growth rate over time (either over a limit cycle, or over an infinite amount of time), then it becomes
 * $$\overline{r_j} \approx \phi_j(\overline{F}) + \frac{1}{2} \sigma^2_F \phi_j(\overline{F})'' $$,

where $$\sigma^2_F$$ is the variance of F. This occurs because the average of $$(F - \overline{F})$$ is 0, and the average of $$(F - \overline{F})^2$$ is the variance of F. Thus, we see that a species' average growth rate is helped by variation if Φ is convex, and it is hurt by variation if Φ is concave.

We can measure the effect that relative nonlinearity has on coexistence using an invasion analysis. To do this, we set one species' density to 0 (we call this the invader, with subscript i), and allow the other species (the resident, with subscript r) is at a long-term steady state (e.g., a limit cycle). If the invader has a positive growth rate, then it cannot be excluded from the system. If both species have a positive growth rate as the invader, then they can coexist [cite].

Though the resident's density may fluctuate, its average density over the long-term will not change (by assumption). Therefore, $$\overline{r_r} = 0$$. Because of this, we can write the invader's density as
 * $$ \overline{r_i} = \overline{r_i} - \overline{r_r}$$.

Substituting in our above formula for average growth, we see that
 * $$\overline{r_i} \approx \left(\phi_i(\overline{F}) + \frac{1}{2} \sigma^2_F \phi_i(\overline{F}) \right) - \left(\phi_r(\overline{F}) + \frac{1}{2} \sigma^2_F \phi_r(\overline{F}) \right) $$.

We can rearrange this to
 * $$\overline{r_i} \approx \left(\phi_i(\overline{F}) + \phi_r(\overline{F}) \right) + \Delta N_i$$,

where $$\Delta N_i$$ quantifies the effect of relative nonlinearity,
 * $$\Delta N_i = \frac{1}{2} \sigma^2_F \left( \phi_i(\overline{F})+ \phi_r(\overline{F}) \right) $$.

Thus, we have partition the invader's growth rate into two components. The left term represents the variation-independent mechanisms, and will be positive if the invader is less hindered by a shortage of resources. Relative nonlinearity, $$\Delta N_i$$ will be positive, and thus help species i to invade, if $$\phi_i(\overline{F})> \phi_r(\overline{F})$$ (i.e., if the invader is less harmed by variation than the resident). However, relative nonlinearity will hinder species i's ability to invade if $$\phi_i(\overline{F}) < \phi_r(\overline{F})$$.

Under most circumstances, relative nonlinearity will help one species to invade, and hurt the other. It will have a net positive impact on coexistence if its sum across all species is positive (i.e., $$ \Delta N_j + \Delta N_k > 0$$ for species j and k). The $$\phi_j(\overline{F})$$ terms will generally not change much when the invader changes, but the variation in F will. For the sum of the $$\Delta N_i$$ terms to be positive, the variation in F must be larger when the species with the more positive (or less negative) $$\phi_j(\overline{F})''$$ is the invader.

Simonmstump (talk) 19:26, 23 November 2016 (UTC)