Soil food webs in agricultural ecosystems - Microbial Ecology in Sustainable Agroecosystems - page 70

n

dX
dt

∑

i

= b X − f X X ( )

(4.2)

i i i j

i = 1

where X i and X j represent the population densities of the primary producer and consum-
ers, respectively; b i is the specific growth rate of the primary producer; and f ( X i ) represents
the functional response of the interaction between (species i ≠ j ) or within the populations
(species i = j ). As such, the Jacobian matrix is an n × n matrix whose elements (α ij , with
units of “per time”) are the partial derivatives of the equations describing the growth and
dynamics of the functional groups at or near equilibrium ( X i * ) (May, 1973):

α ij = [δ( dX * i / dt )/δΧ j ] (4.3)

The diagonal elements (α ii ) of the n × n Jacobian matrix depict the degree of self-regula-
tion or intraspecific competition, while the paired off-diagonal elements represent the per-
capita effects of predators on prey (α ij ) and prey on predators (α ji ). For our purposes, it will
be important to estimate the magnitudes of the interaction strengths to understand how
agricultural management practices affect structural and functional aspects of food webs
and in turn their stability. The off-diagonal interaction strengths can be estimated by link-
ing the rate equations used to model the population dynamics of the functional groups
with estimates of the feeding rates among the functional groups that make up the energy
flux description. To do this requires that we make two assumptions. First, we assume that

Phytophagous
Nematodes

Collembolans

Predaceous
Mites

Cryptostigmati
Mites

Roots

Noncrypto
stigmatic Mites

Saprophytic
Fungi

Nematode
Feeding Mites

Fungivorous
Nematodes

Enchytraeids

Bacteriophagous

Nematodes

Predaceous
Nematodes

Detritus Bacteria Flagellates

Amoebae

Bacteriophagous

Mites

Figure 4.4 A partial functional food web description for the shortgrass steppe soil food web from
the Central Plains Experimental Range (CPER), Nunn, Colorado. The description is based on the
connectedness and energy flux descriptions. The arrows that depicted carbon fluxes in the energy
flux description are split to reflect the negative (predator-to-prey) and positive (prey-to-predator)
interaction strengths (y −1 ) associated with each trophic interaction. A subset of the interactions is
quantified here. Estimates of all interaction strengths are presented in Figure 4.5.