Limits to Biodiversity (Species Packing) curved lines, which may be interpreted as social interactions of various sorts Also, changing the sign of the interactions could crudely mimic predation or symbiosis Loss of organisms across the ‘‘container’’ boundary could be thought of as mimicking predation Expansion of the Theory to Multiple Species The more general question is: How many species can be introduced into a space? Multiple species packing theory is mathematically more difficult because a multispecies theory cannot be conveniently represented as isoclines on a two-dimensional phase space If a third species were to be added, the phase space would have to be three-dimensional and isoplanes would be needed instead of isoclines For coexistence of three competing species, the three isoplanes would have to intersect somewhere in the region below the intersection of any two of the species isoclines In the case of multiple species packing, there can be a region in the multidimensional phase space in which resources are divided in such a way that all species can share, but only if each species is relatively strongly selflimiting and relatively mild in its effects on the other species present The multiple species theory requires clarification of the notion of ecological niche The ecological niche of a species is often considered as the full set of measurements of environmental properties that are relevant to the survival and persistence of that population Hutchinson and his students developed this into an image of a multidimensional hyperspace If two hyperspaces not contain common dimensions, then the occurrences of the species concerned are independent of each other As more and more dimensions are identical, or very similar, the stronger are the competitive interactions between the two species Competition is one partial explanation for replacement of similar species along gradients in such physical features as temperature and salinity How Similar can Coexisting Species be? MacArthur and Levins asked: What might be expected if a third species invades a space in which two species are already resident? They simplified the situation by considering that all species concerned interact through competition on a single niche dimension, or at least a linear representation of a multidimensional niche They also assumed simple distribution of resources along that dimension If, for example, the niche dimension is some particular prey, one of the species might eat large specimens while the other eats only small ones They considered that the niche of any species would have a shape on any single niche dimension If the niche can be represented by a straight line, the probability of an individual finding a bit of resource at some point on the line can be represented as an ordinate, generating a two-dimensional shape If the shapes for several species overlap, then the intensity of competition is measurable in terms of the probability of the several species striving for the same resourceFthe product of the individual species-specific probabilities 645 They could then consider species packing along a niche dimension Given their assumptions, packing can be closer on any one dimension as the number of niche dimensions increases If species are arranged in a one-dimensional array along a line, the closest stable packing is admissible when the niche spaces are rectangular and all K values are equal These conclusions were strongly dependent on their assumptions More general conclusions were that closer packing is possible if niche dimensionality is high, niche breadth is small, and the environment is predictable and has high productivity Centers of adjacent niches are in a multiplicative series of numerical value approximately 1.1 If the single linear dimension of competition is considered more realistically, so that there is a multiplicity of niche dimensions, MacArthur and Levins concluded that interspecific competition could be less important In fact, it is sometimes considered that multispecies packing involves as many niche dimensions as there are species Tilman considers that this is only valid if the organisms are not fixed spatially (i.e., they are free to move about in the container) Various authors, by slightly modifying the mathematical assumptions, could reach a variety of conclusions about the packing of niches For example, it was possible to plausibly argue that niches could sometimes be packed infinitely closely MacArthur and Levins suggested, as a matter of convenience, that the shapes of niches projected onto a single niche dimension might be thought of as approximating normal distributions Other shapes are, of course, imaginable Species packing became a rich field for construction of theories, which were not necessarily connected to specific biological observations and more These theories each make slightly modified assumptions or focus on special cases of previous theories For example, Roughgarden and Feldman considered the significance of shape differences as if the shapes could be empirically demonstrated Distributions with ‘‘thick’’ tails permitted more species to coexist than those with thinner tails There were also studies of multispecies biological situations under natural or artificial circumstances, which used the term ‘‘species packing.’’ In these the connection with species packing theory was sometimes explicit but very often the general conclusions of theory were presented in the context of field data without explicit theoretical derivation This appears to be due to the fact that explicit formal theory production becomes very difficult once more realistic elements of natural complexity are involved This is typical of mathematical models, at least prior to the recent development of powerful computing machinery Sometimes the models were sufficiently complex, combining ecological and evolutionary predictions, that selfcontradictory assertions were made within the same theoretical argument and were not noticed for years For example, one study published in 1983 assumed absence of any genetic heterogeneity and then proceeded to discuss the genetic effects of selection, as noted by Taper and Case Niche Separation and Anatomical Difference The exact ratio of minimum possible distinction between adjacent species can be derived from special, simplified, and