Some Methods for Calculating Competition Coefficients from Resource-Utilization Spectra

When relative frequencies of resource kinds in the diet are known, the competition coefficient giving the effect of competitor j on i may be computed as \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{wasysym} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document}$$\alpha_{ij}=\left(\frac{T_{j}}{T_{i}}\right)\left[\frac{{\sum\limits_{k=1}^{m}}(d_{ik}/f_{k})\:(d_{jk}/f_{k})\:b_{ik}}{\sum\limits_{k=1}^{m}(d_{ik}/f_{k})^{2}\:b_{ik}}\right],$$\end{document} where T_j/T_i= the ratio of the number of items consumed by an individual of competitor j to that consumed by an individual of competitor i, measured over an interval of time that includes all regular fluctuations in consumption for both species; d_ik = the frequency of resource k in the diet of competitor i (and similarly for d_jk); f_k = the standing frequency of resource k in the environment; b_ik = the net calories gained by an individual of competitor i from an item of resource k, or more approximately the calories contained in an item of resource k, or still more approximately the weight or volume of an item of resource k; and the summations are taken over all resources eaten by at least one of the competing species. The coefficient follows from MacArthur's (1968) consumer-resource system when the ratio of the carrying capacity to intrinsic rate of increase is constant for all resources. When relative frequencies of time spent foraging in habitat kinds are known, the competition coefficient may be computed as \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{wasysym} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document}$$\alpha_{ij}=\left(\frac{T_{j}}{T_{i}}\right)^{\prime} \frac{\sum\limits^{m}_{k=1}p_{ik}p_{jk}b_{ik}}{\sum\limits^{m}_{k=1}p_{ik}{}^2b_{ik}}$$\end{document} where (T_j/T_i)' = the ratio of the total time spent searching for food by an individual of competitor j in all habitats to that spent by an individual of competitor i; b_ik = as above, except resource k is the average food item in habitat k; and summations are taken as before. This coefficient, with the same resource restrictions and assuming equal consumption rates per unit search time for the competitor species, follows also from MacArthur's system. It equals the Levins-MacArthur α (eq. [3]) when it is assumed or known that (T_j/T_i)' = 1 and the b 's are equal.