Talk:Relative nonlinearity

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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 N_j 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 $a_{j}F-d$, where a_j 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

${\overline {r_{j}}}\approx \phi _{j}({\overline {F}})+{\frac {1}{2}}\sigma _{F}^{2}\phi _{j}({\overline {F}})''$,

where $\sigma _{F}^{2}$ 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

${\overline {r_{i}}}\approx \left(\phi _{i}({\overline {F}})+{\frac {1}{2}}\sigma _{F}^{2}\phi _{i}({\overline {F}})''\right)-\left(\phi _{r}({\overline {F}})+{\frac {1}{2}}\sigma _{F}^{2}\phi _{r}({\overline {F}})''\right)$.

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,

$\Delta N_{i}={\frac {1}{2}}\sigma _{F}^{2}\left(\phi _{i}({\overline {F}})''+\phi _{r}({\overline {F}})''\right)$.

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)