S

A

JK

a

ARRAA

KDTB

1

iiabtauATS

tb(

P

Tcm

0d

Ecological Modelling 220 (2009) 3472–3474

Contents lists available at ScienceDirect

Ecological Modelling

journa l homepage: www.e lsev ier .com/ locate /eco lmodel

hort communication

note on density dependence in population models

.V. Ross ∗

ing’s College, University of Cambridge, Cambridge CB2 1ST, UK

r t i c l e i n f o

rticle history:eceived 29 June 2009eceived in revised form 25 August 2009ccepted 27 August 2009vailable online 26 September 2009

eywords:ensity dependenceheta-logistic modelasic-Savageau model

a b s t r a c t

One of the most studied phenomena in ecology is density dependent regulation. The model most fre-quently used to study this behaviour is the theta-logistic model. However, disagreement has developedwithin the ecology community pertaining to the interpretation of this model’s parameters, and thus as toappropriate values for the parameters to assume. In particular, the parameter � has been allowed to takenegative values, resulting in the ‘growth rate parameter’ estimated to be negative for species which areextant and exhibit no signs of becoming extinct in the short-term. Here we explain this phenomenon byformulating the theta-logistic model in the manner in which the original logistic model was formulatedby Verhulst (1838), in doing so providing a simple interpretation of model parameters and thus restric-tions on values the parameters may assume. We conclude that � should (almost always) be restricted to

values greater than −1. This has implications for studies assessing the form of density dependence fromdata. Additionally, another model appearing in the literature is presented which provides a more flexible

ence

model of density depend

. Introduction

As a population grows competition for resources amongst itsndividuals typically increases. This competition slows the growthn population abundance, until the population is reduced to a levelt which resources are more readily accessible. The relationshipetween abundance and growth that emerges is one of the cen-ral issues of ecology. Studies of such density dependent regulationbound, typically highlighting its fundamental importance on pop-lation dynamics (Verhulst, 1838; Pearl and Reed, 1920; Gilpin andyala, 1973; Getz, 1996; Turchin, 2003; Sibly et al., 2005; Stacey andaper, 1992; Clutton-Brock et al., 1997; Sæther et al., 2000, 2002;inclair, 2003; Gerber et al., 2004; Chamaillé-Jammes et al., 2008).

The classical model of population dynamics, the logistic equa-ion (Verhulst, 1838), incorporates density dependent regulation,ut in a restrictive way, as it implies that the per-capita growth ratePGR) of the population declines linearly with population size:

GR = 1n

dn

dt= r

[1 −

(n

K

)]. (1)

he parameter r is a growth rate parameter and K is the carrying

apacity of the population (i.e. the stable equilibrium point of theodel in the case r > 0).

∗ Tel.: +44 77806 85323; fax: +44 12233 31315.E-mail address: jvr25@cam.ac.uk.

304-3800/$ – see front matter © 2009 Elsevier B.V. All rights reserved.oi:10.1016/j.ecolmodel.2009.08.024

at the expense of only one additional parameter.© 2009 Elsevier B.V. All rights reserved.

A model developed to allow more freedom in the form of densitydependence is the theta-logistic equation (Gilpin and Ayala, 1973)

dn

dt= rn

[1 −

(n

K

)�]

, (2)

where � is a parameter describing the curvature of the relationship.When � < 1 there is a concave relationship between abundance andPGR and when � > 1 a convex relationship exists (see Fig. 1).

The theta-logistic model has been applied to several speciesincluding acorn woodpecker (inter alia Melanerpes formicivorous)(Stacey and Taper, 1992) and several other populations of birds(Sæther et al., 2000, 2002, 2008), sea otters (Enhydra lutris L.) offthe coast of Washington State, U.S.A. (Gerber et al., 2004), and ele-phant (Loxodonta africana Blumenbach) in Hwange National Park,Zimbabwe (Chamaillé-Jammes et al., 2008). In fact, it has been usedin the largest study of density dependence undertaken to date, inwhich it was fitted to a large number of population time series fromthe Global Population Dynamics Database (GPDD; NERC, 1999) todetermine if any general patterns of density dependence could befound for mammals, birds, fish and insects (Sibly et al., 2005).

In a number of these studies the parameter � has been allowedto take negative values. Consequently the growth rate parameterr has been estimated to be negative for species which are extantand exhibit no signs of becoming extinct in the short-term. These

studies have created disagreement in the ecology community as towhether such values for parameters are realistic and confusion overthe true interpretation of parameters. The purpose of this note is toaddress these issues and thus hopefully settle any disagreement asto the interpretation of parameters in the theta-logistic model and

J.V. Ross / Ecological Modelling 220 (2009) 3472–3474 3473

F lation

ta

ugrrsd

2

he

wfiT1

s

b

warEh

so

tdi

ig. 1. (a) Per capita growth rate (PGR), and (b) population growth rate, versus popu

he values appropriate for parameters to assume. Additionally anlternative more general model of density dependence is discussed.

For future reference, we note here that dn/dt denotes the pop-lation growth rate, whilst (1/n)(dn/dt) denotes the per-capitarowth rate. Additionally, we will consider the population growthate to consist of a population birth rate minus a population deathate; each of these components, when divided by the populationize n, will be referred to as the per-capita birth rate and per-capitaeath rate, respectively.

. Density dependence

The logistic model was first formulated in the literature by Ver-ulst in 1838 (Verhulst, 1838; Gabriel et al., 2005). He proposed thequation

dn

dt= an − bn2, (3)

here a is the per-capita birth rate and b is a positive ‘friction’ coef-cient which influences the rate of density dependent regulation.his equation may be expressed in its now more common form (cf.) as

dn

dt= an

[1 −

(n

a/b

)], (4)

o that r = a and K = a/b.Following this original formulation, the theta-logistic model can

e written

dn

dt= an − bn� , (5)

here � allows more flexibility in the form of density dependence,nd it must be larger than 1 to serve its purpose of populationegulation if a and b are to retain their original interpretations.xpressing this equation in its more ubiquitous form (cf. 2), weave

dn

dt= an

[1 −

(n

(a/b)(1/(�−1))

)�−1]

, (6)

o that r = a, K = (a/b)(1/(�−1)) and � = � − 1. Thus, from this linef reasoning, the parameter � must be greater than 0.

The confusion that arises in the literature, and amongst some ofhe ecology community, is that Eq. (5) (unlike Eq. (3)) can capture

ensity dependent regulation with parameters having different

nterpretations. In particular, the model

dn

dt= cn� − dn (7)

size for various values of � for the theta-logistic model (2) with r = 1 and K = 100.

also displays density dependence, in the sense that the populationgrowth rate is positive for small population sizes and negative forlarge population sizes, when � < 1. This can be incorporated in theprevious framework (5) by setting a = −d and b = −c. Thus if � < 1,then a and b can both be chosen to be negative and the resultingmodel displays density dependent regulation:

dn

dt= |b|n� − |a|n. (8)

Note that PGR approaches a as the population size n → ∞. Thus,the parameter � may assume negative values, but in such a casethe parameter r is no longer the per-capita birth rate, but is theper-capita growth rate as the population size n tends to infinity.

Closer inspection shows that in (5) the dynamics are composedof a constant per-capita birth rate a (and thus linear populationbirth rate), and a population death rate which increases with pop-ulation size like n� , whereas in (8) the dynamics are composedof a constant per-capita death rate |a| (and thus linear populationdeath rate), and a population birth rate which changes with pop-ulation size like n� . In the former we have both the rate of birthand death increasing with population size n, but death increasingat a faster rate than birth (super-linear population death rate com-pared to linear population birth rate). In the latter, for 0 < � < 1we have, once again, both birth and death increasing with popu-lation size, and death increasing at a faster rate than birth; this isdue to population birth rate increasing at a sub-linear rate (withdeath rate remaining linear). However, when � < 0, we no longerhave the population birth rate increasing with population size—it isalways decreasing with increasing population size n. To be emphatic,if � < 0 a population with n = 4 individuals always has a smallerpopulation birth rate than one with n = 2 individuals. It can beseen from this argument that the assumptions required for use ofthe model with � < 0 will rarely (if ever) be satisfied in reality.For this reason care needs to be taken when fitting the model todata—the model with � estimated to be less than 0 may providea ‘best’ fit to the particular data available (typically observationsof the population close to the carrying capacity) but the physicalproperties required for use of such a model will almost never besatisfied for the species from which the data was collected; in sucha case estimation should be performed with � restricted to positivevalues.

Translating the above arguments into restrictions on the param-eter �, we can see that it will typically be necessary to restrict� > −1. We note that this is the parameter region considered by

Sæther et al. (2008), and is less restrictive than that suggested inearlier comments on this subject (Ross, 2006). Sæther et al. (2008)comment (page 1203) that this restriction on � results in estimatesof r that can be considered biologically plausible; the current studyexplains their finding. A summary of the form of the population

3474 J.V. Ross / Ecological Modelling

Table 1Form of the population birth and death rates (with respect to population size)and interpretation of r for different values of � in the theta-logistic model. PGRn

represents the per-capita growth rate at population size n.

� r Population birth rate Population death rate

> 0 Per-capitabirth rate

Linear increasing Super-linear increasing

= 0 PGR Linear increasing Linear increasing(−1, 0) PGR{n→∞} Sub-linear increasing Linear increasing

b�

torwpi(itdoi

gmtt

w�mSi1a

w

iapfotp

3

oha

Tsoularis, A., Wallace, J., 2002. Analysis of logistic growth models. Mathematical

= −1 PGR{n→∞} Constant Linear increasing(−2, −1) PGR{n→∞} Sub-linear decreasing Linear increasing= −2 PGR{n→∞} Linear decreasing Linear increasing< −2 PGR{n→∞} Super-linear decreasing Linear increasing

irth and death rates, and interpretation of r, for different values ofis provided in Table 1.

With respect to the theta-logistic model, finally we note thathe case � = 1 (equivalently � = 0) results in exponential growthr decay depending upon whether a − b is positive or negative,espectively, and thus implies no density dependence. Whereasith formulation (2) the case � = 0 appears to result in no change toopulation size (dn/dt = 0), using formulation (5) we can see that

n this case the dynamics are governed by the equation dn/dt =a − b)n. This is because (2) ignores the dependence of the carry-ng capacity K on the parameters a, b and � . Thus, with respecto estimation of model parameters, the crux of assessing densityependence, it may be useful to estimate these parameters, aspposed to r, K and �; in any case, estimates of K tell us importantnformation about the species characteristics (b).

As discussed above, the theta-logistic model imposes linearrowth in either the population birth or population death rate. Aore flexible model would allow for non-linear growth in both of

hese rates. This may be achieved with a model of the same ilk ashose previously considered:

dn

dt= an�1 − bn� , (9)

here a and b are both positive and �1 < �; for the same reasons as> 0 is almost always a requirement for use of the theta-logisticodel, we will have �1 > 0. This is the so-called ‘basic’ model of

avageau, derived from similar considerations as here in formulat-ng models from their underlying determinants (Savageau, 1979,980; Tsoularis and Wallace, 2002). Reformulating the model intoform more analogous to the theta-logistic model, we have

dn

dt= rn�1

[1 −

(n

K

)�2]

, (10)

here r = a, K = (a/b)(1/�2) and �2 = � − �1.The basic-Savageau model provides a more flexible and real-

stic framework in which to study density dependent regulation,llowing for simultaneous non-linear effects of population size onopulation birth and death rates. Whilst this model certainly allowsor a wider range of behaviours than the theta-logistic model, it willbviously still remain inadequate for many populations where fac-ors such as habitat destruction, the Allee effect, and harvesting areresent.

. Summary

Through formulating the theta-logistic model in a manner anal-gous to Verhulst’s original formulation of the logistic model weave clearly determined the interpretation of model parametersnd thus have determined appropriate values to allow the param-

220 (2009) 3472–3474

eters of the theta-logistic model to assume. It appears necessary torequire � > −1. In the case � > 0 the parameter r is the per-capitabirth rate, whilst when � < 0 the parameter r is the per-capitagrowth rate of the population as the population size tends toinfinity. Formulating the theta-logistic model in this manner alsoelucidates the behaviour of the model in the case � = 0 (see Table 1).Of course one may ignore the formulation provided herein and usethe theta-logistic model without contstraint; however, the rela-tionship between the model and the population it is proposing tomodel may be lost. Finally, a more flexible model of density depen-dence was discussed, the basic-Savageau model (see (9) or (10)),which will hopefully aid in gaining a better understanding of thedensity dependent regulation of populations.

Acknowledgments

Thanks to an anonymous reviewer, the Editor, and David Sirl forcomments that improved this note, and King’s College for financialsupport (Zukerman Research Fellowship).

References

Chamaillé-Jammes, S., Fritz, H., Valeix, M., Murindagomo, F., Clobert, J., 2008.Resource variability, aggregation and direct density dependence in an open con-text: the local regulation of an African elephant population. Journal of AnimalEcology 77, 135–144.

Clutton-Brock, T.H., Illius, A.W., Wilson, K., Grenfell, B.T., MacColl, A.D.C., Albon, S.D.,1997. Stability and instability in ungulate populations: an empirical analysis.American Naturalist 149, 195–219.

Gabriel, J.-P., Saucy, F., Bersier, L.-F., 2005. Paradoxes in the logistic equation? Eco-logical Modelling 185, 147–151.

Gerber, L.R., Buenau, K.E., Vanblaricom, G., 2004. Density dependence and risk ofextinction in a small population of sea otters. Biodiversity and Conservation 13,2741–2757.

Getz, W.M., 1996. A hypothesis regarding the abruptness of density dependence andthe growth rate of populations. Ecology 77, 2014–2026.

Gilpin, M.E., Ayala, F.J., 1973. Global models of growth and competition. Proceed-ings of the National Academy of Sciences of the United States of America 70,3590–3593.

National Environment Research Council (NERC), 1999. Centre for Population Biol-ogy, Imperial College, Global Population Dynamics Database, available atwww.sw.ic.ac.uk/cpb/cpb/gpdd.html.

Pearl, R., Reed, L., 1920. On the rate of growth of the population of the United Statessince 1790 and its mathematical representation. Proceedings of the NationalAcademy of Sciences of the United States of America 6, 275–288.

Ross, J.V., 2006. Comment on the regulation of populations of mammals, birds, fish,and insects II. Science 311, 1100.

Sæther, B.-E., Engen, S., Lande, R., Arcese, P., Smith, J.N.M., 2000. Estimating the timeto extinction in an island population of song sparrows. Proceedings of the RoyalSociety of London Series B 267, 621–626.

Sæther, B.-E., Engen, S., Matthysen, E., 2002. Demographic characteristicsand population dynamical patterns of solitary birds. Science 295, 2070–2073.

Sæther, 2008. Forms for density regulation and (quasi-) stationary distributions ofpopulation sizes in birds. Oikos 117, 1197–1208.

Savageau, M.A., 1979. Growth of complex systems can be related to the properties oftheir underlying determinants. Proceedings of the National Academy of Scienceof the United States of America 76, 5413–5417.

Savageau, M.A., 1980. Growth equations: a general equation and a survey of specialcases. Mathematical Biosciences 48, 267–278.

Sibly, R.M., Barker, D., Denham, M.C., Hone, J., Pagel, M., 2005. On the regulation ofpopulations of mammals, birds, fish, and insects. Science 309, 607–610.

Sinclair, A.R.E., 2003. Mammal population regulation, keystone processes andecosystem dynamics. Philosophical Transactions of the Royal Society of LondonSeries B 358, 1729–1740.

Stacey, P.B., Taper, M., 1992. Environmental variation and the persistence of smallpopulations. Ecological Applications 2, 18–29.

Biosciences 179, 21–55.Turchin, P., 2003. Complex Population Dynamics. Princeton University Press, Prince-

ton, NJ.Verhulst, P.F., 1838. Notice sur la loi que la population suit dans son accroisement.

Correspondence Mathematique et Physique 10, 113–121.