In the late 1990s, “
metabolic theory of ecology” based on allometric scaling of metabolic rates with body mass has experienced a renaissance (Brown et al.
2004; Sibly et al.
2012). Metabolism is the biochemical processes of uptake, transformation, and allocation of energy and materials within a living organism (Brown et al.
2004; Sibly et al.
2012). The metabolic rates are the fundamental biological rates and often described as power functions of body size (body mass) called allometric relations (Peters
1983). Studies of the relations have a long history (Peters
1983; Whitfield
2006) including the contribution by Huxley (
1932) and the famous 3/4 (three quarters) law by Kleiber (
1932). However, the meeting of a physicist, Geoffrey West, with two ecologists, Brian Enquist and James Brown at the Santa Fe Institute in 1995 ignited the enthusiastic search for the mechanisms and extensions to wider biological phenomena (Whitfield
2006).
Emmerson and Raffaelli (
2004) estimated per capita interaction strengths between prey and predators (coefficients of the Lotka-Volterra model) from an allometric relation with the predator–prey size ratio and showed that the maximum real part of the eigenvalues of the community matrix is always negative and that the real complex food web is stable. Yodzis and Innes (
1992) already estimated mass-specific metabolic rates through the allometric relations with the negative quarter power of body size, constructed the so-called bioenergetic model including nonlinear functional responses of predators (Brose et al.
2006), and analyzed the model. Otto et al. (
2007) parametrized a bioenergetic model of food chains with three trophic levels. They found that almost all of tri-trophic food chains across five natural food webs exhibit body mass ratios between top and intermediate species and between intermediate and basal species within the persistence domain predicted from the dynamic model. By two types of random rewiring of food webs, the first preserving the body masses of the species and the total number of links and the second preserving the body mass and number of links for each species, they concluded that body mass and allometric degree distributions in natural food webs mediate the consistency.
Williams and Martinez (
2004) and Martinez et al. (
2006) generalized the bio-energetic model to
n species and arbitrary functional responses. They described the functional response in terms of the (1 +
q)-th power of body size
B and examined stability of the model for values of a control parameter
q between 0 and 1 (
q = 0 and 1 respectively corresponds to the type II and III functional response). Williams and Martinez (
2004) examined dynamics of the three species food chain model studied by McCann and Hastings (
1997) and found that a dramatic change occurred as
q increased. When
q = 0, the system with the type II functional response exhibited chaos, but the system was stabilized through period-doubling reversals as
q increased and finally reached a stable stationary solution when
q ≈ 0.2. They named the functional response corresponding to
q = 0.2 the “type II.2” response. Martinez et al. (
2006) built model food webs of large size assuming the initial topological structure of the random, modified cascade, and niche webs (Williams and Martinez
2000). They suggested that the hierarchical ordering of the cascade model and the contiguous niches allowing cannibalism and looping in the niche model enhanced the persistence of populations and that both predator interference and type II.2 functional response showing respective decelerated and accelerated responses on rare and abundant resources had stabilizing effects even in large networks with many species. However, relative persistence, or the ratio of the final to initial number of species decreased linearly with the increasing initial network size and connectance, thus qualitatively replicating May’s results (Dunne et al.
2005). Brose et al. (
2006) analyzed a similar model with the initial structure following the modified cascade, niche or nested hierarchy model assuming that the mass-specific metabolic rates followed negative-quarter power law relationships with the predator–prey body mass ratios. In their simulations, most populations exhibited chaotic or limit cycle dynamics, and few populations reached equilibrium. They found that body mass ratios are the most important determinant of population and community stability and that food web stability (equivalent to the relative persistence in Martinez et al.
2006) consistently increases with the predator–prey body mass ratios. In their case, the relation between species richness and species’ probability of persistence is negative when body size ratios are smaller than 10
1 but positive at ratios of 10
2 and above. Thus, their results partially reproduce and partially refute the classic result (May
1972). They suggested that average body mass ratios well above unity, usual in natural systems, yield neutral or positive diversity–stability relationships.