Robert M. May

On the theory of niche overlap

Abstract The factors which are likely to limit niche overlap are studied for a class of idealized biological communities in which several species compete on a onedimensional continuum of resources, e.g., food size. Although the shape of the resource spectrum restricts the composition of possible equilibrium communities, and although this restriction is increasingly stringent as niche overlap increases, such considerations do not in principle put a limit to niche overlap. In analysing the stability of these equilibrium communities, two qualitatively different circumstances need be distinguished. In a strictly unvarying (deterministic) environment, stability sets no limit to the degree of overlap, short of complete congruence. However, in a randomly fluctuating (stochastic) environment, community stability requires that the average food sizes for species adjacent on the resource spectrum must differ by an amount roughly equal to the standard deviation in the food size taken by either individual species. This limit to species packing has a weak (logarithmic) dependence on the degree of environmental variance. This mathematical result is very robust, as is shown by considering, inter alia, a wide range of resource spectrum shapes, and a variety of shapes for the functions describing how the species utilize the resource. The effect of including additional resource dimensions is estimated. The general conclusion from the model is that there is an effective limit to niche overlap in the real world, and that this limit is insensitive to the degree of environmental fluctuation, unless it be very severe. This conclusion seems in accord with an increasing body of field data.

Stability in multispecies community models

Abstract First, we consider a simple mathematical model for a many-predator-many-prey system, and show it to be in general less stable, and never more stable, than the analogous one-predator-one-prey community. This result would seem to caution against any simple belief that increasing population stability is a mathematical consequence of increasing multispecies complexity. Second, we take up the question of the relation between stability in any one trophic level by itself and stability of the total trophic web. We find that in a simple mathematical model the criteria for isolated single-level stability and for total web stability are not identical, but they tend to be similar, so that usually stability (or instability) at one level goes with stability (or instability) of the whole. It is possible, however, to construct examples where either stability at one level occurs in an overall unstable system, or alternatively instability at one level goes with total web stability. This model points the way to a synthesis of the diverse views that have been expressed on this subject.

Interspecific competition, predation and species diversity: A comment

Abstract Motivated by a conjecture due to Paine (1966) , Parrish & Saila (1970) recently investigated a simple mathematical model for a two prey-one predator system, seeking to show that the three-species system can be stable in circumstances where the two-prey system would in the absence of predation be unstable with respect to competition. Parrish & Saila in fact did not find such a three-species system, although they did show that the doomed prey species may persist longer with predation. We show that such three-species stable systems can be constructed under the model, and give examples illustrating the working of Paines conjecture.

Mass and energy flow in closed ecosystems: A comment

Recently Ulanowicz (1972) has proposed a formal framework in which, for a closed ecosystem, the parameters in the equations of population dynamics are related to the parameters of energy flow. In general these relations are not determinate, but Ulanowicz illustrates the method by deriving a sufficient stability condition for the simple three-species straight-chain closed system. We consider a much wider class of special webs for which Ulanowiczs relations are effectively determinate, namely webs which may be articulated into N trophic levels, with the species in any one level being linked in an arbitrary way to those in the levels immediately above and below, and with all species within any one level having a common respiration rate and energy content per unit biomass. The necessary and sufficient condition for ecosystem stability is shown to be that energy content per unit biomass increase in a hierarchical fashion as one climbs the trophic ladder. Some more fragile results are given, and discussed, for the equilibrium biomass ratios between various trophic levels.

Test Ion Energy Loss in a Plasma with a Magnetic Field

The energy loss by a test ion in a plasma in the presence of a homogeneous magnetic field is considered, and numerical results for the magnetic field corrections to the rate of energy loss are presented. This is done for strong fields and for weak fields (i.e., for the electron cyclotron frequency greater than, and less than, the plasma frequency), and for all values of x, the ratio between test ion speed and plasma electron thermal speed; previous work is confined to x ≫ 1 .

Magnetic Properties of Charged Ideal Quantum Gases in n Dimensions

We consider the mathematical model of an ideal gas of charged bosons or fermions in an n‐dimensional space, treating n as a continuous variable. The investigation shows the extent to which the magnetic behavior depends on the dimensionality of the system. In particular, the charged Bose gas in a homogeneous magnetic field does not condense unless n > 4, in contrast to the field‐free gas which condenses for n > 2: however so long as n > 2 and T 4; but it is still present, and Hc has the same form, for 4 > n > 2 where there is no condensation.

Production of highly excited neutral atoms for injection into plasma devices: II.

We derive an expression for the charge-exchange cross section σ(N) for the formation of excited final states with principal quantum number N in the limit N > 1. This provides a rigorous proof of the results made plausible by Butler and Johnston in the preceding paper.We also derive a closed expression for the fractional contribution to σ(N) made by the various angular momentum states (l = 0, 1, 2, ... , N − 1). For high energies only S states contribute: in the neighbourhood of the resonance in σ(N) both P and D states are also formed in significant amounts. (At the resonance σ(N) comprises 29% S states, 54% P states and 15% D states.)

Theory of low energy scattering by velocity dependent potentials

Abstract The low energy scattering by pure velocity dependent potentials is investigated. It is found that the potential V 1 =(2m) −1 p · U(r) p gives a phase shift of order k2l+1 except in S states where the phase shift is of order k5, while the potential V2=(4m)−1 [p2W(r)+W(r)p2] gives a phase shift of order k2l+3. A non vanishing scattering length may be restored by including a static potential, but to simulate a hard core exactly this potential must have properties which are not usually assumed in attempts to fit the scattering data with velocity dependent potentials. The difference between V1 and V2 is also a static potential, but again this is not of the form usually assumed. The analysis yields a linear integral equation for the coefficient of k2l+3 in scattering by V2, but in general a non-linear equation for the leading coefficient in scattering by V1. For this potential however the coefficient of k5 in S wave scattering is given by a closed expression.

Statistical mechanics of velocity-dependent potentials

Abstract We consider the statistical mechanics of a system in which the particles interact through a velocity-dependent potential, of the form p ij V(| r ij |)· p ij ( p ij is the relative momentum of the i th and j th particles ) . The aim of the investigation is to gain further understanding of the formal relationship between velocity-dependent potentials and hard-core potentials in an explicity many-body context. In the limit of classical statistics, we find that the above interaction is indistinguishable from a hard-sphere potential; in particular the specific heat is just that for an ideal gas, a property which for velocity-independent potentials is specific to the hard-sphere case. In the limit of quantum statistics, the two potentials (velocity-dependent and hard-sphere) are no longer equivalent: it is no clear whether addition of a static potential to the velocity-dependent one could restore the exact equivalence.

Collision Damping of Two-Stream Instabilities in a Plasma

Doubt exists as to whether the Gross-Krook collision model is adequate to describe the collisional damping of two-stream instabilities. It is shown here that the model is indeed adequate—the instability ceases to grow after a time of the order of one collision time.