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A model is developed for the coexistence and exclusion of species over a region of similar habitable patches. Since the balance of local extinction and colonization would leave some patches unoccupied even without competitors, species may coexist even when all the patches are the same. Regional competition coefficients are found when species affect the local extinction or migration rates of each other. Rare species can regulate each other and even exclude other species completely.Many environments have a patchy, island-like pattern of occurrence. It is generally assumed that species that occur on a small fraction of the available patches will have little effect on each other because their co-occurrence would be an exceedingly rare event. However, we have recently obtained evidence of competition between rare species. A group of predaceous ants recorded by Gregg (1) in Colorado, all of which are rare, showed a lower than expected microhabitat overlap. MacArthur and Pianka (2) predicted reduced microhabitat overlap for 'searching' predators competing for the same prey; our data seemed to confirm MacArthur and Pianka's optimization model. The problem is not why the ants reduced competition by the optimization proposed by MacArthur and Pianka, but rather, how there could be any significant competition to reduce. We will use an immigrationextinction model developed by Levins (3,4) for predicting the number of islands, or island-like habitats, occupied by a species, and we will allow competition to affect either the migration or extinction rate. In contrast to traditional competition theory, the focus of our attention will be on changes in the number of populations of a species, rather than on the sizes of the local populations.We will first present the migration-extinction model for a species in the absence of competitors, then the possibility of significant competitive effects on a rare species by other species, conditions for coexistence of two species, the effect of environment on coexistence, and mechanisms for the avoidance of competition. Finally, we will discuss situations where the model might be applicable.Let N be the number of local populations, T the total number of sites, x the extinction rate per population, and m' the rate of migration from one given site to another given (2)At equilibrium:which is the proportion of sites occupied in the absence of competitors. For a rare species, a ten-fold change in p from 0.01 to 0.1 requires only about a 10% change in x/m, from 0.99 to 0.90. Finally, p is more sensitive to differences in the parameters x and m if these are separately small:Thus, small differences in m and x can make one species common and a species with a similar biology rare. A standard explanation is that species are rare because suitable habitats are rare. While this may be true for many situations, the arguments above indicate that this is not necessarily the case, and we are justified in looking for competition between initially rare species. COMPETITION AFFECTING EXTINCTION RATEThe si...

Suppose now we are at optimum geometric design. Then, if our vector could suffer arbitrary variations, we could lower all the undesirable quantities by simply decreasing the first two components, keeping the second two components fixed. Such a possibility is in contradiction to our assumption of being at a minimum. We conclude that in the vicinity of the optimum design the vector (LFe, Lcu, AFe, Ace) cannot be varied in an arbitrary manner. In mathematical language, the appropriate Jacobian must vanish: a (LFe, Lcu, AFe, Acu) = 0. (4) aO(a, b, a', b') Substitution of relations (3) into (4), and making use of the definitions (1), leads directly to our result (2). Note added in proof: The authors are indebted to the engineers of our Power Transformer Division for constructive comments. Mr. J. H. McWhirter has pointed out that the rectangularity law may be generalized to the case of fixed clearances between the copper conductors within the iron window. Dr. Stein has pointed out that Reed has obtained (eq. (2)) for that particular design which minimizes the total transformer losses [Reed, E. G., Essentials of Transformer Practice