Fitness-density covariance

The fitness-density covariance is a coexistence mechanism that can allow similar species to coexist because they are in different locations. 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. 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.

Mathematical derivation

Here, we will consider competition between n species. We will define N_xj as the number of individuals of species j at patch x and time t, and λ_xj the fitness of individuals of species js at patch x and time t. λ_xj will be determined by many things, including habitat, intraspecific competition, and interspecific competition at x. Thus, if there are currently N_xj individuals at x, then they will contribute N_xjλ_xj individuals to the next time period. 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,. It is defined such that, where each average is across all space. In essence, it is the average fitness of members of species j in year t. We can calculate N_j by summing N_xjλ_xj across all patches, giving
where X is the number of patches. Defining as species j's relative density at x, this equation becomes
Using the theorem that, this simplifies to
Since, its average will be 1. Thus,
Thus, we have partitioned into two key parts: calculates the fitness of an individual, on average in any given site. Thus, if species are distributed uniformly across the landscape,. If, however, they are distributed non-randomly across the environment, then cov will be non-zero. If individuals are found predominantly in good sites, then cov will be positive; if they are found predominantly in poor sites, then cov will be negative.
To analyze how species coexist, we perform an invasion analysis. In short, we remove one species from the environment, and allow the other species to come the equilibrium. 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 for each resident, we can calculate the invader's growth rate,, as
where n-1 is the number of residents, and the sum is over all residents. Using our formula for, we find that
This rearranges to
where
is the fitness-density covariance, and contains all other mechanisms.
Thus, if Δκ is positive, then the invader's population is more able to build up its population in good areas, compared to the residents. This can occur if the invader builds up in good areas or if the residents are forced into poor areas. In either case, species gain an advantage when they are invaders, a key point of any stabilizing mechanism.