Fitness-density covariance: modifying population dynamics with spatial scale; a mechanism of community stability; a mechanism of diversity maintenance. There is another name for exactly the same thing: “growth-density covariance.”

Fitness-density covariance captures important aspects of variation of population density in space.  If population density is higher in some places than others, what does that mean for the growth of the population summed over the entire area? The theory of fitness-density covariance shows that this variation matters if fitness of the population varies in a space in a manner that is correlated with variation in population density.  Fitness-density covariance measures exactly how much it matters. The concept can be used with data from nature, and with models. The concept captures the importance of spatial density variation in situations with any complexity.  A spatial pattern may be extremely complicated, but the bottom line is simply how much this complicated pattern affects fitness-density covariance.

Fitness-density covariance is, in simplest terms, the spatial covariance between the fitness of an individual and the local population density that it experiences.  So what is fitness?  Fitness can be defined in discrete-time or continuous time.  Let us first of all examine it in discrete time. An individual at a given location x in space, at time t, will over one unit of time change its state.  For example, it may die, or it may give birth to a number or other offspring, or it may do all of these.  Fitness as a function of x and t is equal to probability that the individual survives to time t + 1, plus the expected number of offspring attributed to it. Offspring are attributed to an individual according to genomic contributions.  For asexual reproduction, genomic contributions are unity. For sexual reproduction, selfed offspring also have unit genomic contributions, but outcrossed offspring are counted as half.  The expectation is the statistically predicted number based on the conditions at location x and time t.

Ifis the number of individuals at location x at time t, then the expected number of survivors and offspring arising from them at time t + 1 is lambda*N.  These individuals need not still be at location x. In general, some of them will move. Thus, N(x, t+1) is not lambda N(x,t) . However, if we assume for the moment that expected and actual numbers of survivors and offspring are the same, then
sum N equals sum lambdaN    
where the summation is over all spatial locations. The assumption here is that individuals are not lost from the system.  With a few definitions, we can now simply derive our fundamental equation.  First, we define the total population as n equals sum nxt, then population density as the average number of individuals per location: nbar equals sumN over k, where k is the number of spatial locations, similarly, lambda bar, the spatial average fitness, and finally, the relative density as nu. We can now state our fundamental equations:
 fundamental equations   .

The quantity fitness-density covarianceis fitness-density covariance.  We can introduce another concept here to help understand this equation. This is the concept of individual average fitness,lambda tilde.  Individual average fitness is defined as N_t+1 over N_t, i.e. simply the number of individuals at time t + 1, divided by the number at the previous time.  Average individual fitness by definition relates the numbers of individuals at different times, i.e.
  
These features mean that
   lambda tilde = lambda bar + cov,
i.e. that individual average fitness is equal to the spatial average of fitness plus fitness-density covariance.  

These equations show that averaging fitness in space is not going to give the right answer for the rate for the growth of the population.  Growth of the population is instead given by average individual fitness, lambda tilde. This differs from the average by an amount equal to fitness-density covariance, i.e. by an amount equal to fitness-density covariance.  We can also think of this in a little different way.  We can write
    lambda tilde as a weighted average,
where both expressions can be seen to be a weighted average of the individual fitness,, over space, with the weights being the local population density.  This is the way this concept was first introduced by Chesson and Murdoch (1986). 
  
Continuous-time development of fitness-density covariance can be found in Chesson et al (2005).  Chesson and Murdoch (1986), Hassell et al (1991) and Chesson et al (2005)show how fitness-density covariance affects stability of host-parasitoid relationships.  The effect of fitness-density covariance on host-parasitoid dynamics is explained in most detail in Chesson et al (2005).  The effects of fitness-density covariance on species coexistence are explored in  Chesson (2000), where the concept is first formally introduced, and in Chesson et al (2005) and Snyder and Chesson (2003). 

One point in the development above equated expected and actual numbers of survivors and offspring.  When is this justified?  As discussed in Chesson (1981) and Chesson (2000), this is justified in the development above when the total population size is large.  The difference between actual and expected numbers is due to demographic stochasticity. It can be neglected  in the description of the total population size if the total population is large.  For the calculation of fitness density covariance, the value of can always be the expected numbers not the actual numbers of survivors and offspring.  However, demographic stochasicity cannot be neglected in the calculation of the local population sizes,. Demographic stochasiticity will affect this value, and normallywill be a function of it. In this way, demographic stochasiticity will affect fitness-density covariance through its effects on local population densities. Ignoring demographic stochasicity in the calculation of local population densities is thus only justified if local population densities are large, or does not depend on local population density.

References

Chesson, P.L.1981. Models for spatially distributed populations: the effect of within-patch variability.  Theor. Pop. Biol. 19, 288-325.

Chesson, P.L., Murdoch, W.W. 1986.  Aggregation of risk: relationships among host-parasitoid models. American Naturalist 127, 696 - 715.

Chesson, P. 2000. General theory of competitive coexistence in spatially varying environments. Theoretical Population Biology 58, 211-237

Chesson, P., Donahue, M., Melbourne, B., Sears, A. 2005. Scale transition theory for understanding mechanisms in metacommunities.  In Holyoak, M, Leibold, M.A., Holt, R.D., eds, Metacommunities: spatial dynamics and ecological communities, pp 279-306.    

Hassell, M.P., May, R.M., Pacala, S.W., Chesson, P.L. 1991. The persistence of host-parasitoid associations in patchy environments. I. A general criterion.  American Naturalist 138, 568-583.

Snyder, R.E. and Chesson, P. 2003. Local dispersal can facilitate coexistence in the presence of permanent spatial heterogeneity. Ecology Letters 6,301–309.