Relative nonlinearity

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

${\displaystyle {\frac {dN_{j}}{dt}}=\phi _{j}(F)N_{j}}$,

where N_j is species j's population density, and ${\displaystyle \phi _{j}(F)}$ is some function of the density-dependent factor F. For example, under a Monod chemostat model, F would be the resource density, and ${\displaystyle \phi _{j}(F)}$ would be ${\displaystyle a_{j}F-d}$, 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 McGehee [1][cite Armstrong], ${\displaystyle \phi _{j}(F)}$ 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, ${\displaystyle r_{j}={\frac {1}{N_{j}}}{\frac {dN_{j}}{dt}}}$, using a Taylor series approximation as

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

where ${\displaystyle {\overline {F}}}$ 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

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

where ${\displaystyle \sigma _{F}^{2}}$ is the variance of F. This occurs because the average of ${\displaystyle (F-{\overline {F}})}$ is 0, and the average of ${\displaystyle (F-{\overline {F}})^{2}}$ 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.[2]

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

${\displaystyle {\overline {r_{i}}}={\overline {r_{i}}}-{\overline {r_{r}}}}$

.[3] Substituting in our above formula for average growth, we see that

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

We can rearrange this to

${\displaystyle {\overline {r_{i}}}\approx \left(\phi _{i}({\overline {F}})-\phi _{r}({\overline {F}})\right)+\Delta N_{i}}$,

where ${\displaystyle \Delta N_{i}}$ quantifies the effect of relative nonlinearity,

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

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, ${\displaystyle \Delta N_{i}}$ will be positive, and thus help species i to invade, if ${\displaystyle \phi _{i}({\overline {F}})''>\phi _{r}({\overline {F}})''}$ (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 ${\displaystyle \phi _{i}({\overline {F}})''<\phi _{r}({\overline {F}})''}$.

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., ${\displaystyle \Delta N_{j}+\Delta N_{k}>0}$ for species j and k).[4] The ${\displaystyle \phi _{j}({\overline {F}})}$ terms will generally not change much when the invader changes, but the variation in F will. For the sum of the ${\displaystyle \Delta N_{i}}$ terms to be positive, the variation in F must be larger when the species with the more positive (or less negative) ${\displaystyle \phi _{j}({\overline {F}})''}$ is the invader.