POPULATION FACTORS AND SELECTED POPULATION PROBLEMS 



359 



A highly stimulating paper, and one of 

 considerable interpretative value in terms 

 of the Robertson eflfect, was published by 

 Ludwig and Boost (1939). In brief, these 

 investigators showed that when the data of 

 some nineteen authors, studying bacteria, 

 yeasts, and protozoa and concerned in one 

 form or another with the phenomenon of 

 allelocatalysis, were analyzed in a new 

 way, it could be shown that a stimulation 

 of fission rate by certain degrees of crowd- 

 ing was the rule rather than the exception. 

 This held true even for several cases in 

 which an author himself claimed that his 

 investigations ran counter to a Robertson 

 type of effect. To put it diflFerently, Lud- 

 wig and Boost concluded that no conclu- 

 sive case against the fact of allelocatalysis 

 had been recorded, while some actually 

 positive evidence had been incorrectly in- 

 terpreted as negative. 



Ludwig and Boost pointed out that most 

 of the earher data were plotted in the 

 same form as that used in graphing a lo- 

 gistic curve, i.e., numbers plotted against 

 time (see p. 301). They restudied the 

 problem by graphing the data on a differ- 

 ent coordinate system based on equations 

 of their own derivation. 



If available space is unlimited and if the 

 environment is held reasonably constant, 

 then it can be shown that 



1 dN „ 



where N = population size, t = time and 

 K = a constant. If the space is limited so 

 that iV = the number of individuals that 

 can exist in it, the logistic equation holds. 

 In the notation of Ludwig and Boost this 

 becomes 



dX 

 Ndt 



= K(N* - N), (where K = >0). 



This equation upon integration yields a lo- 

 gistic curve. 



Since it is hard to distinguish "allelo- 

 catalytic" or other effects from the popula- 

 tion growth curve because of random fluc- 

 tuations and individual variabiHty, Ludwig 

 and Boost proposed that a more accurate 

 and interpretative curve is afforded by 

 plotting specific growth rate, or for these 



cases fission rate, - *-{-, on the ordinate, 



A dt ' 



against total population growth on the 



abscissa. When this is done, the type of 

 effect can be adduced directly by inspec- 

 tion of the graph. Thus a Unear relation- 

 ship indicates equiUbrium; a convexity 

 (i.e., where the curve is convex to the zero 

 point or point of joining of the coordi- 

 nates) indicates deceleration or inhibition 

 of population growth; and a concavity in- 

 dicates acceleration. 



In the "Empirischer Teil" of their paper 

 Ludwig and Boost apply this analysis to 

 various pubhshed studies and, barring the 

 initial or lag phase, reach the afore-men- 

 tioned conclusion about the usualness of 

 stimulation of reproduction by population 

 size. 



It is suggestive that, despite their sub- 

 stantiation of the Robertson effect, Ludwig 

 and Boost cast doubt upon his explanation 

 and suggest a "more harmless" one. In 

 general terms this runs somewhat as fol- 

 lows: Since, naturally, protozoans exist as 

 populations of variable density, they are 

 best adjusted to this density range or at 

 least to part of it. Therefore, the optimum 

 conditions of pH, oxidation-reduction po- 

 tentials, and so forth that, to some extent 

 at least, are dependent on metaboHc prod- 

 ucts, would be present in cultures of such 

 densities, and the division rate or growth 

 rate would not depend on any such highly 

 specific mechanism as that postulated by 

 Robertson.* 



For a recent and suggestive contribution 

 to this general problem the work of Kid- 

 der (1941) on the ciUate Tetrahymena 

 geleii merits attention. Kidder showed that 

 a conditioned medium, one which has sup- 

 ported populations of this species for as 

 much as sixty hours, has both accelerating 

 and inhibiting properties for growth. If 

 the conditioning population is removed 

 from the medium by centrifugation, ac- 

 celeration of the assay culture results. If 

 the conditioning population is removed by 

 filtration, using a Seitz or Berkfeld filter, 

 inhibition results. Kidder advances the in- 

 teresting suggestion that Tetrahymena cul- 

 tures elaborate two types of substances. 

 One is an accelerator that "acts freely dur- 

 ing the early growth phases but which is 

 removed either by filtration or by adsorp- 

 tion. The other substance produces inhibi- 

 tion of growth, and when the accelerator 



*• We have discussed the paper of Ludwig 

 and Boost with Dr. Benson Ginsburg and are 

 indebted to him for his comments. 



