Fitness-density covariance

Here, we will consider competition between n species.^[1] We will define N_xj(t) as the number of individuals of species j at patch x and time t, and λ_xj(t) the fitness (i.e., the per-capita contribution of individual to the next time period through survival and reproduction) of individuals of species js at patch x and time t.^[1] λ_xj(t) will be determined by many things, including habitat, intraspecific competition, and interspecific competition at x. Thus, if there are currently N_xj(t) individuals at x, then they will contribute N_xj(t)λ_xj(t) individuals to the next time period (i.e., t+1). Those individuals may stay at x, or they may move; the net contribution of x to next year's population will be the same.

With our definitions in place, we want to calculate the finite rate of increase of species j (i.e., its population-wide growth rate), ${\displaystyle {\tilde {\lambda }}(t)}$. It is defined such that ${\displaystyle {\tilde {\lambda _{j}}}(t)={\overline {N_{j}(t+1)}}/{\overline {N_{j}(t)}}}$, where each average is across all space.^[1] In essence, it is the average fitness of members of species j in year t. We can calculate N_j(t+1) by summing N_xj(t)λ_xj(t) across all patches, giving

${\displaystyle {\tilde {\lambda }}(t)={\frac {1}{X}}\sum _{X}{\frac {N_{xj}(t)\lambda _{xj}(t)}{\overline {N_{j}(t)}}},}$

where X is the number of patches. Defining ${\displaystyle \nu _{xj}=N_{xj}(t)/{\overline {N_{j}(t)}}}$ as species j's relative density at x, this equation becomes

${\displaystyle {\tilde {\lambda _{j}}}(t)={\frac {1}{X}}\sum _{X}\nu _{xj}\lambda _{xj}(t)={\overline {\nu _{xj}\lambda _{xj}(t)}}.}$

Using the theorem that ${\displaystyle {\overline {XY}}={\overline {X}}~{\overline {Y}}+{\mbox{cov}}(X,Y)}$, this simplifies to

${\displaystyle {\tilde {\lambda _{j}}}(t)={\overline {\nu _{xj}}}{\overline {\lambda _{xj}(t)}}+{\mbox{cov}}(\nu _{xj},\lambda _{xj}(t)).}$

Since ${\displaystyle \nu _{xj}=N_{xj}(t)/{\overline {N_{xj}(t)}}}$, its average will be 1. Thus,

${\displaystyle {\tilde {\lambda _{j}}}(t)={\overline {\lambda _{xj}(t)}}+{\mbox{cov}}(\nu _{xj},\lambda _{xj}(t)).}$

Thus, we have partitioned ${\displaystyle {\tilde {\lambda _{j}}}(t)}$ into two key parts: ${\displaystyle {\overline {\lambda _{xj}(t)}}}$ calculates the fitness of an individual, on average in any given site. Thus, if species are distributed uniformly across the landscape, ${\displaystyle {\tilde {\lambda _{j}}}(t)={\overline {\lambda _{xj}(t)}}}$. If, however, they are distributed non-randomly across the environment, then cov(ν_xj, λ_xj(t)) will be non-zero. If individuals are found predominantly in good sites, then cov(ν_xj, λ_xj(t)) will be positive; if they are found predominantly in poor sites, then cov(ν_xj, λ_xj(t)) will be negative.

To analyze how species coexist, we perform an invasion analysis.^[2] In short, we remove one species (called the "invader") from the environment, and allow the other species (called the "residents") to come the equilibrium (so that ${\displaystyle {\tilde {\lambda _{r}}}(t)=1}$ for each resident). We then determine if the invader has a positive growth rate. If each species has a positive growth rate as an invader, then they can coexist.

Because ${\displaystyle {\tilde {\lambda _{r}}}(t)=1}$ for each resident, we can calculate the invader's growth rate, ${\displaystyle {\tilde {\lambda _{i}}}(t)}$, as

${\displaystyle {\tilde {\lambda _{i}}}(t)={\tilde {\lambda _{i}}}(t)-{\frac {1}{n-1}}\sum ({\tilde {\lambda _{r}}}(t)-1),}$

where n-1 is the number of residents (since n is the number of species), and the sum is over all residents (and thus represents an average).^[1] Using our formula for ${\displaystyle {\tilde {\lambda _{j}}}(t)}$, we find that

${\displaystyle {\tilde {\lambda _{i}}}(t)=[{\overline {\lambda _{xi}(t)}}+{\mbox{cov}}(\nu _{xi},\lambda _{xi}(t))]-{\frac {1}{n-1}}\sum [{\overline {\lambda _{xr}(t)}}+{\mbox{cov}}(\nu _{xr},\lambda _{xr}(t))-1].}$

This rearranges to

${\displaystyle {\tilde {\lambda _{i}}}(t)=1+\left[{\overline {\lambda _{xi}(t)}}-{\frac {1}{n-1}}\sum {\overline {\lambda _{xr}(t)}}\right]+\Delta \kappa .}$

where

${\displaystyle \Delta \kappa ={\mbox{cov}}(\nu _{xi},\lambda _{xi}(t))-{\frac {1}{n-1}}\sum {\mbox{cov}}(\nu _{xr},\lambda _{xr}(t))}$

is the fitness-density covariance, and ${\displaystyle \left[{\overline {\lambda _{xi}(t)}}-{\frac {1}{n-1}}\sum {\overline {\lambda _{xr}(t)}}\right]}$ contains all other mechanisms (such as the spatial storage effect).^[1]

Thus, if Δκ is positive, then the invader's population is more able to build up its population in good areas (i.e., ν_xi is higher where λ_xi(t) is large), compared to the residents. This can occur if the invader builds up in good areas (i.e., cov(ν_xi, λ_xi(t)) is very positive) or if the residents are forced into poor areas (i.e., cov(ν_xr, λ_xr(t)) is less positive, or negative). In either case, species gain an advantage when they are invaders, a key point of any stabilizing mechanism.