Per Unit Mass Growth Dynamics

Suppose that each unit of herbivore mass (H) consumes fh units of resource mass (P), and that rh (respiration rate) units are required to sustain life, then fh - rh units will be available for growth and reproduction. The effects of temperature and levels of food on fh are illustrated in Figure 16.5a. Consumption (f) depends on temperature but also decreases with level of resource (the thinner lines). Also shown is the respiration rate rh (i.e. the Q10 rule), which increases with temperature but is largely independent of resource level. The extra costs of searching at low levels of resource could affect r, but the effect is ignored here.

Hence, if 8h units of offspring can be produced per unit of food in excess of respiration, then the instantaneous per unit mass rate of change of H is:

in which dH/Hdt is a hump-shaped function on temperature that equals zero at the lower thermal threshold for feeding and when fh equals rh at a high temperature (Fig. 16.5b, Gutierrez, 1992). If both sides are divided by the

maximum growth rate under non-limiting conditions, a humped growth index with respect to temperature, level of food and intraspecific competition arises. As the levels of resource decrease or competition increases, the maximum growth rate (i.e. the optimum) falls and the function shifts to the left (the thinner lines in Fig. 16.5b). Growth rate indices on temperature and high resource abundance for three hypothetical species in a food chain are illustrated in Fig. 16.5c.

Among the essential resources required by plants are light, minerals (e.g. N, K, P) and water, while animals may require, among other things, vitamins, micronutrients and water. There is a plethora of abiotic factors that may affect the growth rates of organisms. For example, soil pH, wind, saturation deficits or lack of mates, nesting sites or territories may affect growth rates of different species across trophic levels. A degree of imagination is needed to make some of these important analogies. Shortfalls of essential resources and levels of abiotic factor may affect / and their acquisition is modelled using variants of equation 16.3 (Gutierrez et al, 1988, 1993). The parameter a7 must reflect the apparency of the jth resource to the species. Rearranging the terms of equation 16.3 gives the supply/demand ratio //dj for the jth factor. The right-hand side is the probability of acquiring dj units of resource j. The effect of shortfalls of all j = 1,...,J factors on a species may be viewed as compounded resource acquisition probabilities (f*):

The effects of all resource shortfalls (equation 16.5) may be included in equation 16.4 as follows:

dH Hdt

Substituting the full model in equation 16.2 for fh in equation 16.5 yields:

This model may be normalizing by dividing both sides by the maximum growth rate, yielding a hump-shaped growth index (0 £ GI < 1) on temperature, ranging from zero to unity. This model is akin to the Fitzpatrick—Nix model, but its origins are quite different. The shortfalls of all other factors are included in f h (Gutierrez et al., 1994).

In a metapopulation context, the favourability of conditions for growth at different locations may be quite variable, but the same model (equation 16.7) may be used to estimate the growth index value at each site and the isolines of favourability over a large geographical area may be displayed using GIS.

To examine top-down effects, equation 16.6 may be cast as a tritrophic population model (equation 16.7) (Gutierrez and Baumgartner, 1984; Gutierrez et al., 1994) by including predator mass (C) in equation 16.7 and rearranging terms:

dH ~dt

' a h p N

' acH ^

q h

f h5 h

1 - e 5 H

- rh

H -f * 5 c

1 - e

V 0

This model could be evaluated at equilibrium with temperature related parameters substituted to evaluate the effects of climate change on regulation (Gutierrez et al., 1994). Gutierrez and colleagues have called equation 16.8 the 'metabolic pool' approach to modelling trophic interrelationships because the approach seeks to model behavioural (a) and weather-driven physiological processes (5 and r) that link and drive the trophic dynamics of species. These results could be plotted for a region using GIS. This model has been used extensively to model diverse tritrophic systems with and without age structure (Gutierrez and Wang, 1977, Gutierrez et al., 1988, 1994; Gutierrez, 1992, 1996).

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