vol. 159, no. 5 the american naturalist may 2002A Theory of Stochastic Harvesting in Stochastic EnvironmentsNiclas Jonze´n,*Jo¨rgen Ripa,†and Per Lundberg‡Department of Theoretical Ecology, Ecology Building, LundUniversity, SE-223 62 Lund, SwedenSubmitted June 27, 2001; Accepted November 12, 2001abstract: We investigate how model populations respond to sto-chastic harvesting in a stochastic environment. In particular, we showthat the effects of variable harvesting on the variance in populationdensity and yield depend critically on the autocorrelation of envi-ronmental noise and on whether the endogenous dynamics of thepopulation display over- or undercompensation to density. Thesefactors interact in complicated ways; harvesting shifts the slope ofthe renewal function, and the net effect of this shift will depend onthe sign and magnitude of the other influences. For example, whenenvironmental noise exhibits a positive autocorrelation, the relativeimportance of a variable harvest to the variance in density increaseswith overcompensation but decreases with undercompensation. Fora fixed harvesting level, an increasing level of autocorrelation inenvironmental noise will decrease the relative variation in populationdensity when overcompensation would otherwise occur. These andother intricate interactions have important ramifications for the in-terpretation of time series data when no prior knowledge of demo-graphic or environmental details exists. These effects are importantwhenever the harvesting rate is sufficiently high or variable, condi-tions likely to occur in many systems, whether the harvesting iscaused by commercial exploitation or by any other strong agent ofdensity-independent mortality.Keywords: population dynamics, environmental stochasticity, colorednoise, harvesting.The temporal fluctuation of natural populations is one ofthe most well-studied problems in population ecology. Theunderlying processes causing the observed patterns havebeen attributed to both intrinsic demographic (density-dependent) mechanisms and to stochastic or more regularvariations in the environment (Higgins et al. 1997; Leirset al. 1997; Forchhammer et al. 1998; Grenfell et al. 1998;* Corresponding author; e-mail: [email protected].†E-mail: [email protected].‡E-mail: [email protected]. Nat. 2002. Vol. 159, pp. 427–437. 䉷 2002 by The University of Chicago.0003-0147/2002/15905-0001$15.00. All rights reserved.Myers et al. 1998; Bjørnstad et al. 1999; Stenseth 1999;Stenseth et al. 1999a, 1999b). A major challenge has been,and still is, to disentangle the relative contribution of en-dogenous and exogenous factors determining changes inabundance in time. It is generally agreed that observedpatterns necessarily are a combination of them both (e.g.,Turchin 1999; Lundberg et al. 2000; Fromentin et al. 2001).In this article, we show how a stochastic populationtheory can be used to partition different sources of vari-ation in population density. The basic components of thistheory apply to any population that can be reasonablywell described and analyzed by a nonstructured single-population model in a nonspatial context.Although the variance partitioning is general, we chooseas a model system a population that is harvested. A con-siderable part of our current understanding of populationdynamics is based on data collected or motivated by har-vesting (Kendall et al. 1998), for example, the famous furtrade records of the Canada lynx (Lynx canadensis) col-lected by the Hudson Bay Company and Statistics Canada(Stenseth et al. 1999a). Also, exploited populations are veryclearly affected by two major processes determining theirdynamics: recruitment and harvesting. One could equallywell view the problem as one of a population in a seasonalenvironment where reproduction (or, rather, recruitment)is the major source of density dependence, whereas thereis strong, but density-independent, mortality during thenonbreeding season. Taking the harvesting example, weadd two important aspects to the existing harvesting the-ory: stochastic harvesting and temporally autocorrelatedenvironmental noise.Harvesting is often considered to be nothing but anextra source of mortality, decreasing average populationdensity unless completely compensated for (Kokko andLindstro¨m 1998; Boyce et al. 1999; Jonze´n and Lundberg1999). Independent of whether a constant effort or adensity-dependent harvesting strategy (e.g., a threshold orfixed-stock policy) is implemented, the fraction annuallyremoved from a population is best described as a stochasticprocess (Lauck et al. 1998; Patterson 1999; Mangel 2000).In this article, the annual harvest fraction is a stochasticvariable independent of density, and we note that possiblemechanisms of a time-variant harvest fraction are obser-428 The American Naturalistvation error on which the target is based as well as im-perfect control.Contemporary harvesting theory (e.g., May et al. 1978;Shepherd and Horwood 1979; Horwood and Shepherd1981; Getz and Haight 1989; Lande et al. 1995, 1997) isbuilt on the assumption of uncorrelated environmentalstochasticity, so-called white noise. There are, however,good reasons to believe that positively autocorrelated noiseis a better null model for environmental variability (Steele1985; Pimm and Redfearn 1988; Halley 1996). Recent workhas shown that autocorrelation per se influences popu-lation models in terms of expected extinction risk (Ripaand Lundberg 1996; Petchey et al. 1997; Morales 1999;Heino et al. 2000) and also that it may be significant indetermining how exploited populations should be man-aged (Koslow 1989; Spencer 1993; Walters and Parma1996).Using an exploited population as a model system doesnot imply that we are interested here in the managementproblem per se but, rather, in how two important sto-chastic processes work in concert to produce observedpopulation dynamics. Then, of course, understanding towhat extent population variance can be explained by dif-ferent processes does have both a general theoretical in-terest as well as potential far-reaching implications forpopulation management and conservation.ModelConsider a general population model where harvesting (or,strictly speaking, any density-independent mortality fac-tor) takes place after reproduction such that˜N p f(N , u ), (reproduction) (1a)ttt˜Y p H(N , w ), (harvest) (1b)ttt˜N p N ⫺ Y ,(1c)t⫹1 ttwhere and Ntare the population densities before