The fitness-density covariance (also known as *growth-density covariance*) is a coexistence mechanism that can allow similar species to coexist because they are in different locations.[1] This effect will be the strongest if species are completely segregated, but can also work if their populations overlap somewhat. If a fitness-density covariance is operating, then when a species becomes very rare, its population will shift to predominantly locations with favorable conditions (e.g., less competition or good habitat). Similarly, when a species becomes very common, then conditions will worsen where they are most common, and they will spread into areas where conditions are less favorable. This negative feedback can help species avoid being driven extinct by competition, and it can prevent stronger species from becoming too common and crowding out other species.
Along with storage effects and relative nonlinearities, fitness-density covariances make up the three variation-dependent mechanisms of modern coexistence theory.[2]
Here, we will consider competition between n species. We will define Nxj(t) as the number of individuals of species j at patch x and time t, and λxj(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. λxj(t) will be determined by many things, including habitat, intraspecific competition, and interspecific competition at x. Thus, if there are currently Nxj(t) individuals at x, then they will contribute Nxj(t)λxj(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),
\tilde{λ}(t)
\tilde{λj}(t)=\overline{Nj(t+1)}/\overline{Nj(t)}
\tilde{λ}(t)=
1 | |
X |
\sumX
Nxj(t)λxj(t) | |
\overline{Nj(t) |
\nuxj=Nxj(t)/\overline{Nj(t)}
\tilde{λj}(t)=
1 | |
X |
\sumX\nuxjλxj(t)=\overline{\nuxjλxj(t)}.
\overline{XY}=\overline{X}~\overline{Y}+cov(X,Y)
\tilde{λj}(t)=\overline{\nuxj
\nuxj=Nxj(t)/\overline{Nxj(t)}
\tilde{λj}(t)=\overline{λxj(t)}+cov(\nuxj,λxj(t)).
\tilde{λj}(t)
\overline{λxj(t)}
\tilde{λj}(t)=\overline{λxj(t)}
To analyze how species coexist, we perform an invasion analysis. 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
\tilde{λr}(t)=1
Because
\tilde{λr}(t)=1
\tilde{λi}(t)
\tilde{λi}(t)=\tilde{λi}(t)-
1 | |
n-1 |
\sum(\tilde{λr}(t)-1),
\tilde{λj}(t)
\tilde{λi}(t)=[\overline{λxi(t)}+cov(\nuxi,λxi(t))]-
1 | |
n-1 |
\sum[\overline{λxr(t)}+cov(\nuxr,λxr(t))-1].
\tilde{λi}(t)=1+\left[\overline{λxi(t)}-
1 | |
n-1 |
\sum\overline{λxr(t)}\right]+\Delta\kappa.
\Delta\kappa=cov(\nuxi,λxi(t))-
1 | |
n-1 |
\sumcov(\nuxr,λxr(t))
\left[\overline{λxi(t)}-
1 | |
n-1 |
\sum\overline{λxr(t)}\right]
Thus, if Δκ is positive, then the invader's population is more able to build up its population in good areas (i.e., νxi is higher where λxi(t) is large), compared to the residents. This can occur if the invader builds up in good areas (i.e., cov(νxi, λxi(t)) is very positive) or if the residents are forced into poor areas (i.e., cov(νxr, λxr(t)) is less positive, or negative). In either case, species gain an advantage when they are invaders, a key point of any stabilizing mechanism.