DENSITYDEPENDENCE AND THE STABILIZATION OFANIMAL NUMBERS 2.THE PINE LOOPER* by P.J. DENBOER (BiologicalStationof the Agricultural University, Kampsweg 27,9418 PD Wijster, The Netherlands) SUMMARY Totest the hypothesis thatthe density-dependent mortalityof advanced larvae ofthe pinelooper kept density within limits (i.e. "regulated"), thefluctuations of numbers inthe field population studied byKlomp arecompared withthose inthe null model inwhich, onaverage, thismortality ofolder larvae iskept at the field level, but the densitydependence isremoved. Itis concluded thatthe density-dependent mortality ofadvanced larvae, whether ornot in concert with the following density-dependent reductionoffecundity, didnot regulate density, butrather had asomewhat destabiliz- inginfluence byincreasing thechance that very low densities arereached. Inan earlierpaper the same effect was shown tooccur in the winter moth population studiedbyVarley and Gradwell. Itis suggested thatthis destabilizing tendency may resultfrom the fact that the action ofa significantlydensity-dependent factorcan be frustratedbycorrelations withother, more independent andmutually non-randomly interrelated,mortality factors, bywhich thepotentially stabilizing effectcan get lost, andmay even turn into the reverse. INTRODUCTION Thisarticle is published inmemory of HermanKlomp with whom in thecourse of abouttwenty years, I frequentlyand thoroughly dis- cussedproblems concerning the persistence ofpopulations. We were bothconvinced that significantevolution results from continuing selectionprocesses, which we agreed will occur only in long-persisting populations.Klomp thought that populations could only persist long enoughfor this to happen, ifdensity was kept-` `regulated"-within safelimits by powerful density-dependent processes. Otherwise, den- sity wouldrapidly "random-walk", either to extinctionor to catastrophicallyhighvalues (KLOMP, 1962). In particular,I did not agreewith the latter statement. In myopinion, the trend of density willdepend on theparameters of thefrequency distribution of the multiplicationfactors (R = netreproduction), which determine the *Communication no.315 of the Biological Station. 221 randomwalk of densities. Also, I couldimagine random walks that wouldlet density fluctuate within restricted bounds, for hundreds of years.Such favourable distributions of R-values might result, for instance,from a largenumber of factors, each independently affecting density,because there is, then, a goodchance that the influence ofone factorwill be levelled, or evennullified, by the effects of others (DEN BOER,1966). Klomp, however, objected that it can hardly be expected thatthe effectsof such"non-reactive" factors would balance one another,so that meanlnR wouldbe keptvery close to zero,a necessarycondition to avoida significanttrend of numbers.My answerwas that populationswhich do not conformto this condition-whichin my opinion is lesscritical than supposed by Klomp-wouldhave a lowchance of beingobserved (especially if meanIn R < 0),because they would rapidly become extinct. In other words,the populations westudy form a biasedsample with respect to distributionsofR-values. Thiscontroversy dominated the meetings ofthe "Discussion group forpopulation dynamics" (founded in 1962by Klomp and Kluyver) forsome years, and led me to formulate the concept of "spreadingof risk"(DEN BOER, 1968). Although it could be shown (REDDINGIUS & DENBOER, 1970) that spreading ofrisk may significantly contribute to thestabilization ofanimal numbers, and thus could be an alternative to regulationofnumbers, it was not possible to provewhich of these two(or possibly other) principles was generally responsible for the supposedpersistence ofnatural populations. REDDINGIUS (1971),who thoroughlyexamined this question, concluded that the regulation hypothesisloses much of its deterministic rigidity, and probably also muchof its appealto manypopulation ecologists, when it is generalizedtoa morerealistic, i. e. probabilistic, model. On the other hand,however, "spreading of risk"is modelledin probabilistic terms,quite naturally. Therefore, he advisedthat population hypothesesshould be formulated inprobabilistic terms, and he recog- nizedthe potential role of spreadingof riskin thestabilization of numbers.However, this is not to say that regulation ofnumbers may notplay a partin thepersistence ofnatural populations. Therefore, Klomphad good reasons to continuehis search for the "regulating mechanism"ofthe pine looper (Bupalus piniarius) population, which hestudied for many years. Inthe following I will try to evaluate the stabilizing influence ofthe density-dependentprocesses found by Klomp, and which he expected tomake some contribution tothe "regulation ofnumbers" ofthis pine looperpopulation. 222 THEPINE I,OOPER In hismonograph on thedynamics ofthe pine looper population of "HogeVeluwe" (near Otterlo, The Netherlands)KLOMP (1966) carefullydiscussed all aspects of hisinvestigation. We will only be engagedhere with his key-factor analysis, as describedby VARLEY & GRADWELL(1960), which he adoptedto evaluatethe influenceof density-dependentfactors on the pattern of density fluctuations. The relevantdata on densitiesin successivestages of thelife cycle are broughttogether in tableI. I coulduse only the data from 1953 onwards,because only in thatyear did Klomp discriminate between themortality of olderlarvae in Septemberand October (k4 and k5 respectively),which appears to bedistinctly density-dependent, and the mortalityof nymphsin November(k6), which is notdensity- dependent.In table II thedensities offour stages, viz. pupae in April, reproducingfemales, eggs, and olderlarvae in September,are presentedasloglo-values, together with the values (k) of the different mortalityfactors, which are the differences between the loglo-values oftwo succeeding stages in table I (e.g.k8 + k9 in 1956= log 10(2.58) - logI0(1.52)= 0.4116 -0.1818 = 0.230). Todiscover the key factor (MORRIS, 1963), i.e. the mortality factor withthe greatest variation in effectbetween years, and which there- foreis expectedto governthe pattern of densityfluctuations, the changesintime of each mortality factor, distinguished byKlomp, are comparedwith changes in thetotal generation mortality (K): fig. 1. It willbe obvious that eggmortality + juvenilemortality of larvaeI (k2) and of larvae II/III (k3), can be considered the key fac- tor, becausethe variationin total generationmortality (K = ki +k2 + .....+ kf2) mainly results from that in kl_3 (correlation(Pear- son)K/kf-3: r = + 0.78,P =0.006); compare table III, fromwhich willbe clear that also k, +k2 can be considered the key factor. Figure 1 nextshows that k4 + k5 has some tendency (though not significantly so:r = +0.13) to be low when the values of Kare highest. This results fromthe density-dependent relationship between the magnitude of thismortality and the density of the larvae in September,onwhich it acts:p (Spearman) + = 0.77 (P0.01). = Curiously enough, high den- sitiesof larvae in Septembernot only coincide with a highmortality ofthese larvae (k4 + k5), but also with a relativelyhigh reduction of fecundity(kI2) of the moths that develop from the surviving larvae: p = + 0. 76(P =0.016). See also table III. Onaverage k4 + k5 + kl2 isresponsible for18.3% of generation mortality (K), but the variation in thismortality from year to year is high (table IV: between 1.4 and 34.0 %The ). correlation oflarvae Sept. to k4 + k5 +is kI2p = + 0.78 223 224 225

ofthe at Veluwe" Fig. 1.Key-factor Modifiedanalysisfrom KLOMP pinelooper Forpopulationfurther "Hoge see the tables (TheI Netherlands). (1966).andII. expanation

(P =0.014). No other k-factors showed significant correlations with thedensities on which they act. Weare thusleft with two interesting questions concerning the regulationofthe numbers of the pine looper: (1)What are the causes of the density-dependent relationship between larvalmortality in September(k4 + k5) and the subsequent fecundity (kl2)? (2)Can the population be "regulated"by only18% of generation mortality,in spiteof the fact that the variance of k4+ k5 (table II: 0.0325)is smallerthan that of the key factor (table II: 0.071)? 226 TABLEIII Product-moment(Pearson)correlations (r)between totalmortality (K)and compos- ingmortality factors (k), as well as between thedifferent mortality factors mutually; 100r29lo ? variation "explained" bythat variable.

Note:Correlations between mortality factors aretaken inthe normal time order only, e.g.k¡ + k2 with the k8 + kg-values following intime (next year). IfI r I >0.602 the correlationissignificant at5 % .

TABLEIV Proportionalshare( % of )the different mortality factors intotal mortality (K)in dif- ferentyears.

Note:K= = k¡ + k2 + k'$ + k4of' year(t) + k5 + k6 + k7 of ycar(t). + 1). + ks + kg + klO + k¡t + k12 227 Thefirst question was studied by GRUYS (1970, 1971). He found that advancedlarvae which make contact during the nightwill regurgitategut fluid, and transmit it to eachother. Because of this, growthis reduced,and this results in smallermoths with lowered fecundity.However, larval and pupal mortality, longevity of the adults,and viability ofthe progeny are all unaffected bythese noctur- nalencounters ofolder larvae. It hadalready been shown by KLOMP (1958)that the density-related reduction of fecundity(k12) cannot regulatepopulation density. He rightly concluded that the range of reductionismuch too small for that (KLOMP, 1966); see also table II (Var =0.0018) and table IV. This is confirmedby fig.2: overthe entirerange of densitiesa higherdensity of reproducingfemales resultsin a nearlyproportional increase inthe density of eggs (i. e.the regressioncoefficient hardly differs from unity). GRUYS (1971) sug- gestedthat the smaller moths, which result from the density-related growthreduction of advancedlarvae, might have an increased likelihoodof dispersalfrom the population(because of a more favourablewing-loading), andthat this would restrict density in the nextgeneration. However, BOTTERWEG (1978)showed that this hypothesisalso has to be rejected, because dispersal islow, and neither density-relatednordependent on moth size. KLOMP (1966) supposed that,because of the nearly perfect concealment of the older larvae, mutualinterference among them might result in spacing them out, so thatthey would be more difficult to findby bird predators, that hunt bysight. However, if this supposition is right the spacing behaviour cannotbe veryeffective, because kq + k5,which results from bird predationand from disease, is stillhighly density-dependent. More- over,GRUYS (1971) did not findindications that grouped larvae showedany increased tendency tospace out. What then might be the biologicalfunction-if any-of the remarkable"territorial behaviour"ofthe advanced larvae? Concerningthe second questinn. (KLOMP (1c)66) had serious doubts himself(p. 292), and did not reach any conclusion about the possible regulatingeffects of the density-dependent mortality ofadvanced lar- vae(k4 + k5), whether or notin concertwith the following reduction of fecundity(k12)' It, therefore,seems useful to apply the test that was developedfor the winter moth in Wytham Wood (DEN BOER, 1986a) to thedata for the pine looper at "HogeVeluwe", to investigate the likelyeffects of the density-dependent factors. METHODSANDRESULTS VARLEYetal (1973: 19) defined "a regulatedpopulation" tobe: "one whichtends to return to an equilibrium density following any depar- 228

2. between of = Xand Fig. Relationship= Yto log,o(meanshow the effect densityof thereproducing females)reduction of log¡o(meanineggthe density) at Veluwe".Productmomentdensity-dependent + 0.95 fecundity pinelooper= "Hoge of least corr.(r)Y a + + bX= (n= 12),p(Spearman) 0.97 2.23366 (P= 0.001); +line 0.9587X. squares turefrom this level". This is thought to be brought about by density- dependentprocesses: "Density dependent mortality serves to regulate thepopulation density and keeps it withinlimits" (VARLEY etal., 1973:112). Therefore, I will compare the limits of densityas LR (LogarithmicRange: DEN BOER, 1971) = logl0(highestdensity)- log¡o(lowestdensity), with and without the density-dependent effects (ceterisparibus). To be sure that the density-dependent effectof k4 + k5 is notmade ineffective by factors operating on stageswhich follow, andis not lost before the next generation, I will mainly apply the test to thedensity of advanced larvae in September,i.e. just before the operationofk4 + k5. Beforeapplying the test I willfirst consider separate generations, withand without density dependence. Starting from the actual larval 229 densityin yeart (Nt),larval density in yeart +1 is computedwith k4+ k5 fixed at itsmean value (0.3444; table II). For each generation, thiscalculated value Nit+ 1-from which density dependence is thus removed-isplotted against the observed value Nt +1 fig.3A. This figureshows whether or notdensity dependence resulted in a direct andgeneral limitation of highdensities (Nit 1+ > Not , z.1,e. black dotsabove vertical bar = mean)or low densities (Nt 1+ > Nlc+ 1,1'. e. blackdots below the mean). It appearsfrom these data that the density dependenceofk4 + k5 has indeed some effect on the density of larvae inthe next generation: high densities (except for the highest one!) are somewhatrestricted, and low densities (except for the lowest one!) are increased. Ifwe fix all k-values, except k4 + k5, at their mean values (table II), comparisonofthe Nit + 1's,then computed, with the observed Nt +1's willgive some idea of themaximum possible power of thedensity dependenceof kq + ink5 separate generations (fig. 3B), i. e.if Bupalus existsin an otherwise(demographically) constant environment. Figure3B shows that potentially the density dependence ofk4 + k5 is notpowerless: onaverage, high densities are reduced by about 19'/C, andlow densities increased byas much as 68 %Although ! such effects cannotbe expected tooperate each year (not in 3 out of 11 years: open circles),they most probably would keep density within limits. How- ever,in eachgeneration, the density of advanced larvae of the pine looperis alsoaffected, ofcourse, by other powerful variables, e.g.- apartfrom the keyfactor-by k8 + k9 (pupalmortality between Decemberand April from predators, parasitoids and disease). This is responsibleforan average mortality of 15 %(table IV), and shows a highervariance (table II: 0.062)than kq + k5.Nevertheless, even underfield conditions, the density dependence ofk4 + k5 might still havesome effect in the next generation: onaverage high densities are restrictedby about 6.7 %and low densities increased by 24.8 %(fig. 3A). In spiteof thisencouraging result, we still do not know whether k4+ k5 indeed kept the density of advanced larvae of the pine looper withinlimits. To showthat, we have to demonstratethat the net effectsof k4 + k5 in separategenerations accumulated in such a way that,during the eleven years of the study, density was kept within nar- rowerlimits with, than it wouldhave been without, density dependence.Therefore we have to considerthe entire time series. Startingfrom the density of larvae in September 1953, i. e.4.5 (table I; loglo= 0.653,table II), all other larval densities were calculated anew,in a singlerun from the data in table II, butnow with k4 + k5 fixedat itsmean value ( = 0.3444)in eachgeneration. To avoid the 230

Fig.3. Possibleeffects ofthe density-dependent mortalityof advanced larvae (k4+k5) on larval density in the next generation ofthe pine looper at "Hoge Veluwe".A.The effect ofk4 + k5 in the field: for each generation actuallarval den- sity(N, + 1)isplotted against the value calculated afterhaving fixed k4+ k5 at its meanvalue (Nit + 1)'Black dots: observed densities lessextreme than calculated (k4+k5 had a stabilizingeffect); open circles: observed densities more extreme than calculated(k4+k5 had adestabilizing effect).B. Maximally possible effect ofk4 + k5: foreach generation thefield value oflarval density + is1)plotted against thevalue computedafterhaving fixed all k-values-except k4+ k5-at their respective mean values(Ni, +Black 1)'dots: calculated values more stabilizing thanfield data; open circles:calculated values less stabilizing thanobserved densities. Verticalline: mean value oflogIO(N, + 1). 231 accumulationofsmall inaccuracies inthis long computation, it isbest firstto calculatefor eachgeneration the newcoefficient of net reproductionR' according to 10glOR't =1og10N' 1-1°?lONt t + if loglpN' t + 1logl0Nt+l = I + [(kq + k5)t-0.3444],and R' t = antilog(logl0R't). For instance, log10N' S4 = 1.06+ (0.248-0.3444) = 0.9636,by which loglpR' 53= 0.9636-0.653 = 0.3106, and R' S3 = 2.045.By multiplying thesuccessive R' -values (starting from 4.5), thesuccessive densities in thenew time series can be simulated: fig. 4. Comparisonofthe "Logarithmic Range" (LR) of the new time serieswith that of thefield data will tell us whetherthe density dependenceofk4 + k5 kept density within narrower limits than with- outdensity dependence. It isclear from fig. 4 thatthe density dependence ofk4 + k5 did not keeplarval density in September within limits. The density range was evenwider (LR =1.51) with, than without (LR =1.06), density dependence(table V). Although the density dependence of k4 + us restricteddensity in 1956,it wasat thesame time responsible forthe dangerouslylow density in 1957,and it alsoresulted in a veryhigh densityin 1961.Apparently, positive effects in separategenerations (suchas in fig.3) do not necessarily predict the course of density in thewhole time series, as was pointed out previously byMILNE (1962) andby DEN BOER (1968). It maybe suggested that this contradiction couldhave something to do withthe imperfection of the density- dependentrelationship under field conditions. Ifwe replace the values ofk4 + k5 by 0.381164.loglo(larval density in Sept.)-fromthe least squaresline of kq + k5 on 1og10(larval density in Sept.)-weremove this imperfectionby makingk4 + k5"deterministically" density- dependent,without changing the average mortality. Figure 4 shows thatthis does not solve the above problem: the density in 1956is more extremeand that in 1957somewhat less extreme than in the field, but neitheris any better than in the case without any density dependence: LR =1.32, i.e. intermediate between cases (11 and (2) of table V. Hence,even if thedensity dependence of k4 + ks exertedits full power,it wouldnot have kept larval density in September within nar- rowerlimits than without any density dependence in this mortality, i.e.kq + k5could not regulatedensity. This becomes even more obviousif we apply our test to otherstages of the life cycle: whether k4+ k5 is density-dependent ornot does not appreciably influence the limitsof density of either reproducing females, or eggs,or pupaein April,table V, (1)and (2). This may partly be caused by k6, being somewhatantagonistic tok4 + k5 (r = -0.46,table III), see also fig. 1. In someyears k6 will immediately nullify the effect of k4 + k5; com- paretable V, (2) and(4). Alsotaking into account the density 232

Fig.4. Pattern ofdensity fluctuations ofadvanced larvae ofthe pine looper intime series.The fluctuations ofnumbers inthe field data (LR = 1. 5are 1)compared with thoseofthe series with k4 + ks deterministically density-dependent (LR= 1.32), and withthe fluctuations inthe series with the density dependence removed, i.e. k4 +k5 fixedatits mean value (LR= 1.06). See table III, (1) and (2). dependenceofthe reduction of fecundity(k12) does not change the resultsof our test: table V, (3). Objectionmay still be made, that fixing k4 + k5 at itsmean value takesaway part of the variation of mortality, which may reduce the rangeof variation in density.However, the likelihood that fixing a variable,which operates in themidst of other variables with the same orgreater variances, would reduce overall variability, will not be very high.Nevertheless, it seems useful to allocateto k4+ k5 random valuesin time,either according to a fittednormal distribution, or- whichis much better-according to the frequency distribution ofthe actualk4 + k5-values, in such a waythat the sum of thesimulated k4+ k5-values does not deviate more than 2 °/o(say) from the sum of thefield values (table II). If werepeatedly allocate random values to k4+ k5 according to a normaldistribution fitted to the field data, LR ofadvanced larvae in Septemberisgreater than in thefield (1.51) in only 11 cases,and smalleror equalin 87 cases(X2 =58.9, P<<0.001 mean; LR= 1.24).When simulating a frequency distributionthat accurately fits that of the field data for k4 + k5 (i. e. with6 classesof 0.10 between 0.03 and 0.63, and with frequencies 2,0,3,2,2,2respectively, table II) we get only 5 caseswith LR > 1.511 and79 cases with LR < 1.51(Xz = 65.2,P < < 0.001;mean LR = 1.24),and in anothersimulation mean LR = 1.17(in all cases LR < 1.51). 233 TABLEV Influenceofthe different mortality factors onthe fluctuation pattern ofnumbers in differentstages ofthe pine looper asexpressed bythe limits ofdensity, i.e. LR [ = logjo(highest dcnsity)-loglo (lowestdensity)] ofthe pertinent time series.

The abovesimulations tell us the sameas before:the density dependenceofk4 + k5 did not contribute to regulation ofdensity-in thesense of "keepingthepopulation within limits". On the contrary, in allcases, the limits of density were wider-or more often wider- withthan without density dependence. Wehave already seen (table V) thatsomething similar can be said of other stages of the life cycle. In this respect,k4 + k5 doesnot differfrom other, non-density- dependent,mortality factors: table V, (4),(5), (6). This remarkable "insensitivity"ofthe density range to the removal ofthe variance of oneor othermortality factors probably results from these variances (and/oraverages) not beingvery differentfrom each other: Var(k4-6)(l 044 = (mean ll 571,Var(ko +kg) = 0 n6?(mpan (1 336), Var(kl_3)0.12 = (mean 0.67)-see also table II-by whichnone of thesefactors, not eventhe "keyfactor" kl-3, will conspicuously dominatetotal mortality K (Var =0.228, mean =2.26; table II). Otherwiseisit notclear why the density range of larvae in September "reacts"most to fixingjust one of themortality factors. It seems unlikelythat this is because this density isreached just after the action ofk1 _3(the key factor), and just before that of k4 + k5, since kl +k2 is notcorrelated with k4 + k5 (table III), as it appearedtobe the case in thewinter moth (DEN BOER, 1986a). Note that k3, which shows an interestingcorrelation with k4 + k5 (table III) has a toosmall effect (tableIV) to haveany appreciable influence. 234 Theconclusion from these tests is, that if a density-dependentpro- cessacts on the same numbers-though not in thesame stage of the lifecycle-as other variables with similar, or greater,variances, the resultingpattern of density fluctuations will be unpredictable, or at leastnot more predictable than the effect of these other variables on density(compare the problems concerning weather-forecasts formore thanone or two days ahead). In thepresent case of the pine looper at "HogeVeluwe" this means that the doubts expressed by KLOMP (1966)were justified: the density dependent processes found operating wereinsufficient to keep density within limits. DISCUSSION Justas in thewinter moth (DEN BOER, 1986a), in thepine looper the density-dependentmortality did not keep density within limits, but ratherhas a destabilizinginfluence on thedensity of larvae.This destabilizingeffect resulted in a higherVar(logI0R) = 0.363 of field data than in a serieswith k4 + ksfixed at its meanvalue (Var(logloR')= 0.211). The same was found in thewinter moth: Var(log,OR)= 0.226, Var(logI0R') = 0.107 (DEN BOER, 1986a; table 3).Because of this,the density ranges of (500) time series with per- mutatedR-values (from the field data) are generally wider than those oftime series with permutated R' -values (from the series with k4 + k5 fixed).The chance of the density of a permutatedtime series staying withinthe range of the field data (LR = 1.51)in thefirst case is only 0.082(or 0.080 for reproducing females), in the latter case 0.59. In thewinter moth these chances were 0.052 (or 0.046) and 0.37 respec- tively.This means that, in both species, the density dependence ofpart ofmortality increased the chance of reaching extreme (especially low; seefig. 4, andDEN BOER, 1986a: fig. 2) densities, instead of "keeping densitywithin limits" (compare: DEN BOER, 1968: 184). As in the wintermoth, the density-dependentpupal mortality makes up a greaterpart of totalmortality (35 %on average)than mortality of advancedlarvae in the pine looper (15% on average), it is understand- ablewhy in thewinter moth the destabilizing effect of thedensity dependenceextends over other stages of the life cycle as well (see table 3, (1),(2) and (3) in DENBOER, 1986a), whereas in thepine looper it doesnot: table V, (1)and (2). Whydoes the density dependence of larval(pupal) mortality have a destabilizing,rather than a stabilizingeffect in bothspecies? In my opinion,this results from the factthat density is influencedby a numberof different mortality factors that are both mutually correlated todifferent extents (table III) and have quantitative effects that are not 235 lessthan those of k4 + k5 in mostyears (table IV). This also means, thatif thesemortality factors would have been differently-especially less-correlatedmutually, and/or the more independent factors would haveformed a lower part of totalmortality, the density dependence of larval(pupal)mortality might not necessarily have had a destabiliz- ingbut possibly even a favourableeffect (see also: DEN BOER, inprep. ). However,such a morefavourable effect of density-dependent mor- talitydoes not imply that it wouldhave "regulated" the population. Moreover,such speculations do not apply to thereality under con- siderationthe :pine looper at "HogeVeluwe", or the winter moth at WythamWood. Each case will have to be considered separately. More generally,we may expect that only if the density-dependent factoralso is thekey factor may it governthe complex ofother factors and thus alsothe range of densities, although even then this is notnecessarily so. Doesthis mean that in most cases the persistence ofpopulations is onlya matterof good luck? I don'tknow, because much depends on whatis meant by "population". For instance, if a speciesoccupies a largeand demographically heterogeneous area, the risk of extinction willbe spreadover the different population units (subpopulations), andwill thus be diminished. DEN BOER (1981) showed that, because ofthis risk spreading, the carabid beetle Pterostichus aersicolor canbe expectedto surviveat theHeath of Kralo and Dwingeloo (1200 ha) formany thousands ofyears (ceteris paribus). Isthat merely a matter of goodluck? Not more and not less than adaptation asa resultof natural selectionisonly a matterof "goodluck" (see e.g. EHRLICH &HOLM, 1963:120-122). Bethat as it may,it at leastmakes the persistence of naturalpopulations and species comprehensible; seealso: DEN BOER (1986b). Butcan it beexpected that a numberof "non-reactive"factors will balanceone another, without any feedback, such that mean In R will keepclose to zero? This was the main objection of Klomptomy ideas. Asfar as I cansee, we indeed cannot expect this generally to occur. In populationunits (interaction groups: DEN BOER, 1981) of the carabidspecies studied, mean In R variedwidely, on average between + 0.1and -0.1in only50 %of 62 species, between + 0.2and -0.2in 84 %and , between +0.3 and -0.3in 90 %of the species (DEN BOER, 1985,fig. 5). At the same time, this means that most population units willnot survive for very long: on average less than 10 years in 31 % of65 carabid species, in 54 %< 20years, in 77 %< 50years, and in 88%< 100years (DEN BOER, 1985, fig. 3). However,if suchpopulation units are partsof a multipartite population(ANDREWARTHA & BIRCH, 1984), with exchangeof 236 individualsbetween subpopulations, such short-lived units will not usuallyendanger the persistence ofthe (natural) whole population, becausenot all units will die out in the same generation, and refound- ingof subpopulations willoccur with relative ease. With the help of stochasticmodelling, under very broad conditions I could show (unpublished)that multipartite populations, consisting of abouta hundredunits, which die out and are refounded randomly with on averagesimilar frequency, will survive for some geological periods (ceterisparibus). It is expectedthat the same will also apply to many multipartitepopulations, that consist of lessthan a hundredunits, especiallyifthe mean survival times of subpopulations arein the order of decadesinstead of years (compare: DEN BOER, 1981). In fact,in my opinion,this solves the problem of the persistence ofnatural popula- tions,because these generally will be multipartite: ANDREWARTHA & BIRCH(1984). However, this also means that if sucha multipartite populationis considered as a singleand uniform population, i. e. if samplesfrom different subpopulations areconsidered similar and thus puttogether (as often will be done), density will appear to be surpris- inglystable, and the mean In R willoften be very close to zero(see DENBOER, 1986b). However, usually this stability will not have much todo with "regulation",because the latter rests on significant interac- tionsbetween individuals and therefore by definition operates within populationunits (interaction groups). This stability will result from spreadingthe risk of extinctionover population units (DEN BOER, 1968,1981). This means that the controversiesabout population "regulation"partly result from an inappropriate useof the concept "population".The study of each problem asks for the use of adequate units.

ACKNOWLEDGEMENTS In thefirst place I takethis opportunity to gratefully recall the many stimulatingdiscussions with Herman Klomp, which significantly con- tributedto thedevelopment ofmy thoughts about the persistence of populations. Further,I especially like to thankJack Dempster (Abbots Ripton, U.K.)for valuable comments, and for carefully correcting the English of thispaper. Also the comments of Hans Reddingius (Groningen, Netherlands)and of BobKowalski (Plymouth, U.K.) were very useful.The drawings were made by Reindert de Fluiter (Wageningen, Netherlands). 237 REFERENCES ANDREWARTHA,H.G. & L. C. BIRCH, 1984.The ecological web.More onthe distribution andabundance ofanimals. Chicago Univ. Press, Chicago. BOER,P.J. DEN,1966. Risk-distribution andstabilization ofanimal numbers. Preliminarymsfor restricted circulation. Comm. 127 of theBiological Station Wijster. BOER,P. DEN, 1968. of riskand stabilization of animal numbers. Acta Biotheor.J. 18:165-194. Spreading BOER,P.J. DEN, 1971. Stabilization ofanimal numbers andthe heterogeneity ofthe environment:Theproblem ofthe persistence of sparse populations. In:P. J. DEN BOER&G. R. GRADWELL(Eds.):Dynamics ofpopulations; 77-97.PUDOC Wageningen. BOER,P.J. DEN, 1981. On the survival of populations ina heterogeneous andvariable environment.Oecologia (Berl.) 50: 39-53. BOER,P.J. DEN,1985. Fluctuations ofdensity and survival of carabid populations. Gecologia(Berl.)67:322-330. BOER,P.J. DEN, 1986a. Density dependence andthe stabilization ofanimal numbers. 1.The winter moth. Oecologia (Berl.) 69: 507-512. BOER,P.J. DEN,1986b. Environmental heterogeneity andthe survival ofnatural populations.Proc.3rd Eur. Congress Entomol., Amsterdam: 345-356. BOTTERWEG,P.F., 1978. Moth behaviour anddispersal ofthe pine looper, Bufialus piniarius(L.)(Lepidoptera, Geometridae). Neth.J. Zool. 28: 341-464. EHRLICH,P.R. & R. W. HOLM, 1963. The McGrawHillBook Co., NewYork. processof evolution. GRUYS,P.,1970. Growth inBupalus piniarius (Lepidoptera, Geometridae) inrelation to larvalpopulation density. Agric. Res. Rep. 742: 1-127. PUDOC, Wageningen. GRUYS,P., 1971. Mutual intereference in Bupalus piniarius (Lepidoptera, Geometridae).In:P. J. DEN BOER &G. R. GRADWELL (Eds.):Dynamics of popula- tions:199-207. PUDOC, Wageningen. KLOMP,H.,1958. Larval density and adult fecundity ina natural population ofthe pinelooper (Bupalus piniarius L.).Arch. néerl. Zool. 13: 319-334. KLOMP,H.,1962. The influence ofclimate andweather onthe mean density level, the andthe ofanimal numbers. Arch. néerl. Zool. 15: 68-109.fluctuations, regulation KLOMP,H.,1966. The dynamics ofa field population ofthe pine looper, Bupalus piniariusL.(Lep., Geom.). Adv. Ecol. Res. 3: 207-305. MILNE,A.,1962. On the ofnatural control of insect Theoret. Biul.3. 19-50. theory populations.J. MORRIS,R.F., 1963. The of epidemicsprucebudworm populations. Mem. ent.Soc. Can. 31: 1-322. dynamics REDDINGIUS,J., 1971.Gambling forexistence. Adiscussion ofsome theoretical problems in animalpopulation ecology.Acta Biotheor. 20(suppl.): 1-208. REDDINGIUS,&P. DEN BOER, 1970. Simulation stabiliza- tionof animalJ. J.numbers ofrisk. experiments illustrating5: 240-284. G.C. & G. R. GRADWELL byspreading 1960. Oecologiain (Berl.)studies. Anim. VARLEY,Ecol.20: 251-273. Key-factorspopulation J. VARLEY,G.C., G. R. GRADWELL &M. P. HASSELL, 1973.Insect population ecology,an analyticalapproach. Blackwell. Oxford, London.