One way to think about trophic relationships is to consider how individual organisms acquire resources from the lower trophic level and allocate them to growth and reproduction. For example, if each consumer in a trophic chain consumes f units of resource and requires u units to sustain life (i.e., for basal metabolism) then f - u units will be available for growth and/or reproduction (biomass accretion). Consequently, if c offspring can be produced per unit of food, then the number of offspring produced per consumer is c(f - u), and the instantaneous rate of change of the population of consumers, H, will be
where the subscript h identifies parameters associated with species H. Now if this consumer is attacked by a population of carnivores in the next higher trophic level, then the biomass consumed by that carnivore population will be given by the per-capita consumption rate of carnivores, fc, times their density, C, so that we can write the following general equation for a food chain of any length
Because the rate of food consumption, f, is given by the functional response, we could insert the Michaelis-Menten-Holling or the Ivlev-Watt equations for f. For the present the Ivlev-Watt functional response is inserted for fh and fc to yield an explicit model for a food chain involving plant, P, herbivore, H, and carnivore, C,
Models of this form were first proposed by Gutierrez and Baumgärtner (1984, page 932) and have been used extensively to model diverse systems with and without age structure (Gutierrez 1992, 1996, Gutierrez et al. 1994). It can be shown that this model fulfills all the conditions for a credible food chain model (see Gutierrez et al. 1994, Berryman et al. 1995). It is particularly important to note that this model is truly general in that one can write a single equation for any population in a food chain of any length, thus fulfilling condition 5. For example, if Xi is the density of the ith population in a food chain of arbitrary length, then the model for any population in that food chain is
In this model, each trophic level acquires biomass or energy from the next lower trophic level through their functional responses and surrenders biomass to the next higher trophic level through the functional responses of its predators. In the simplest form of the model, the consumer first uses energy to sustain life (i.e., u = basal or maintenance metabolism), and then uses the surplus to grow and reproduce (i.e., f - u = biomass accretion via the constant of conversion, c). This has been called the "metabolic pool" model because it describes the physiological process of resource conversion into consumer biomass (Gutierrez et al. 1994). More complicated metabolic pool models may be necessary if the energy has to be partitioned among competing sinks, such as the root, stem, leaves and fruit of plants. It is important to remember that the metabolic pool model is really a biomass conversion or energetic model, in the sense that it describes the process of energy acquisition and allocation as well as the flow of energy through the food chain (Berryman 1999, Berryman and Gutierrez 1999). Parameters of the model are usually estimated independently by consumption and respiration experiments, and the model is readily extended to cases where multiple resources are involved (see Gutierrez et al. 1994). Finally, this is also an instantaneous equation and must be integrated before application to real data.
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©1998 Alan A. Berryman