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Population growth rates: Determining factors and role in population regulation

P Bayliss1 & D Choquenot2

1 Ecological Risk Assessment Environmental Research Institute of the Supervising Scientist

GPO Box 461, Darwin NT 0801

2 Arthur Rylah Institute for Environmental Research Natural Resources and Environment

PO Box 137 Heidelberg, VIC 3084, Australia

June 2003

Registry File SG2003/0098

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Contents

Population growth rates: Determining factors and role in population regulation (Abstract) 1

The numerical response: rate of increase and food limitation in herbivores and predators 3

(Paper by P Bayliss & D Choquenot published in Philosophical Transactions of the Royal Society (London B)(2002) issue 357)

Population growth rate: determining factors and role in population regulation 19

(Proceedings compiled and edited by RM Sibly, J Hone & TH Clutton-Brock, published 29 September 2002)

Powerpoint presentation 21

1

Population growth rates: Determining factors and role in population regulation

The numerical response function: rate of increase and food limitation in herbivores and predators

Peter Bayliss

Abstract Animal populations vary in abundance over time. Some populations have declined towards extinction and others have increased dramatically. The patterns of, and reasons for, such variation have been topics of active research for decades. Recent developments in the field, such as more detailed case studies and refined mathematical analysis, allow greater exploration of why populations vary in abundance. The Royal Society held a discussion meeting in London between 6–7 February 2002 on ‘Population Growth Rates: Determining Factors and Role in Population Regulation’ which examined the recent developments for animal populations around the world and provided directions for future research and wildlife management. I presented a seminar on one theme: ‘The numerical response function: rate of increase and food limitation in herbivores and predators’ with colleague Dr David Choquenot. All papers at this conference were published in the Philosophical Transactions of the Royal Society (London B)(2002) issue 357. Subsequently all papers were published by the Royal Society separately in a book. Details of the conference, the paper, book and seminar are outlined in this Internal Report.

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48 The numerical response: rate of increase and food49 limitation in herbivores and predators50

51 Peter Bayliss1* and David Choquenot2

521Environment Australia, Environmental Research Institute of the Supervising Scientist (eriss) and National Centre

53 for Tropical Wetlands Research, GPO Box 461, Darwin, NT 0801, Australia54

2Arthur Rylah Institute for Environmental Research, Natural Resources and Environment, PO Box 137 Heidelberg,55 VIC 3084, Australia

56 Two types of numerical response functions have evolved since Solomon first introduced the term to57 generalize features of Lotka–Volterra predator–prey models: (i) the demographic numerical response,58 which links change in consumer demographic rates to food availability; and (ii) the isocline numerical59 response, which links consumer abundance per se to food availability. These numerical responses are60 interchangeable because both recognize negative feedback loops between consumer and food abundance61 resulting in population regulation. We review how demographic and isocline numerical responses have62 been used to enhance our understanding of population regulation of kangaroos and possums, and argue63 that their utility may be increased by explicitly accounting for non-equilibrium dynamics (due to environ-64 mental variability and/or biological interactions) and the existence of multiple limiting factors. Interferen-65 tial numerical response functions may help bridge three major historical dichotomies in population ecology66 (equilibrium versus non-equilibrium dynamics, extrinsic versus intrinsic regulation and demographic ver-67 sus isocline numerical responses).

68 Keywords: numerical response; population growth rate; regulation; herbivores; predators;69 non-equilibrium

7071

72 1. INTRODUCTION

73 (a) Herbivores and predators: types of consumer–74 resource systems75 The resources used by animal populations are either non-76 consumable or consumable (Caughley & Sinclair 1994).77 While the absolute level of non-consumable resources is78 generally not influenced through its use (e.g. shelter), the79 level of consumable resources is (e.g. food). The most80 comprehensive classification of the relationship between81 resources and animals is that developed for grazing sys-82 tems by Caughley & Lawton (1981). They accounted for83 the degree to which herbivores interact with their food84 resources and interfere with each others capacity to access85 those resources. Interactive grazing systems are those in86 which herbivore consumption influences the rate of87 renewal of food plants, which in turn influences the88 dynamics of the herbivore population itself. Interactive89 grazing systems are further differentiated into interferen-90 tial systems in which herbivores can affect each others91 capacity to assimilate food plants, and laissez-faire systems92 in which they do not. Non-interactive grazing systems are93 those in which herbivore feeding has no influence on the94 rate of renewal of food plants and, hence, no reciprocal95 influence on the dynamics of the herbivore population.96 Non-interactive grazing systems are differentiated between1

27 * Author for correspondence ([email protected]).28

29 One contribution of 15 to a Discussion Meeting Issue ‘Population growth30 rate determining factors and role in population regulation’.

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2 Phil. Trans. R. Soc. Lond. B 02TB006D.1 2002 The Royal Society3 DOI 10.1098/rstb.2002.11246

1 TRANSB: philosophical transactions of the royal society2 02-07-02 14:35:23 Rev 16.03x TRANSB114P

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97reactive systems in which rate of change in herbivore98abundance is a function of food plants, and non-reactive99systems in which herbivore population dynamics are larg-100ely independent of food availability. We argue that this101classification encompasses the range of mechanisms that102link most animal consumer systems to their food resources103and so is applicable to both herbivores and predators. Any104food resource available to an animal population has the105potential to elevate average reproduction and/or survival.106The availability of food resources to an animal population107will be potentially reduced through the use of those108resources by the animal population itself (i.e. the negative109feedback loop).

110(b) Food availability and consumer abundance111(a short history)112Solomon (1949) recognized that an increase in food113availability would generally elicit two responses in a con-114sumer population limited by those food resources; a ‘func-115tional response’ which elevates the per capita rate of food116intake, and a consequent ‘numerical response’ which117increases consumer abundance through enhanced repro-118duction, survival or both. By directly linking food avail-119ability and consumer population demography and120abundance through the numerical response, Solomon121(1949) was generalizing features of more specific models122of trophic interaction (primarily Lotka–Volterra predator–123prey models) to his central theme of animal population124regulation. These models assume that both prey (food)125mortality due to predation and predator (consumer) sur-126vival are proportional to the product of food (H) and con-

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1 02TB006D.2 P. Bayliss and D. Choquenot Numerical response functions in herbivores and predators��1��2

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1396 Figure 1. A diagram describing a Lotka–Volterra model of1397 interaction between a predator (consumer) and its prey1398 (food). Symbols are those used in equations (2.3) and (2.4):1399 food availability (V ); food intake (CH); consumer1400 demographic rates (annual rate of increase1401 rH = births � deaths or b � d ); and consumer abundance1402 (H).

127 sumer (P) abundance (i.e. bHP and cHP respectively). In128 effect, this implies that both the functional response of129 consumers to variation in food availability and the conse-130 quent change in consumer demographic rates are linear,131 indicating that the transfer of biomass from the food to132 consumer populations is conserved. Perhaps more133 importantly, the structure of the model drives changes in134 consumer abundance according to the direct effect that135 food intake rate has on consumer demography (figure 1).136 Since Solomon’s original definition, two types of137 numerical response have been defined and used to help138 elaborate the broad interactive dynamics between con-139 sumer populations and their food. These are: (i) a ‘demo-140 graphic’ numerical response that links rate of change in141 consumer abundance to food availability (Caughley 1976;142 May 1981a); and (ii) an ‘isocline’ numerical response that143 links consumer abundance per se to food availability (see144 Holling (1965, 1966) for total predator responses).145 In this paper, we review how both approaches to the146 numerical response have been used to enhance under-147 standing of herbivore and predator population regulation,148 and attempt to increase their realism and utility by149 explicitly accounting for the existence of non-equilibrium150 dynamics due to environmental variability, biological151 interactions and situations where multiple factors simul-152 taneously limit rates of change in population abundance.153 The different approaches to describing numerical154 responses have also been recently reviewed by Sibly &155 Hone (2002).

156 2. DEMOGRAPHIC NUMERICAL RESPONSES

157 (a) Single-species logistic models of population158 growth159 The dominant paradigm in large herbivore ecology pro-160 posed that density-dependent mortality regulates popu-161 lation density through food shortage (i.e. the so-called162 ‘food hypothesis’ (Sinclair et al. 1985)). Most tests of the163 relevance of this hypothesis to large herbivores have either164 reduced herbivore population density (or allowed a natu-165 ral catastrophe to do so), and assessed whether the popu-166 lation returns to its pre-reduction level (Houston 1982;167 Sinclair et al. 1985), or looked for density dependence in168 r or some valid demographic correlate of r (i.e. growth,169 body condition, fecundity or survival) (O’Roke & Ham-1

2 Phil. Trans. R. Soc. Lond. B3


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170merston 1948; Woodgerd 1963; Boyd & Jewell 1974;171Sinclair 1977; Sauer & Boyce 1983; Skogland 1983, 1985;172Messier & Crete 1984; Clutton-Brock et al. 1985; Eber-173hardt 1987; Fryxell 1987; Choquenot 1991; Messier1741991). Both of these approaches focus on the dynamics175of the herbivore population, interpreting any decline in r176or its index as the population moves towards its hypotheti-177cal equilibrium as the effect of declining per capita food178availability. Because these tests do not consider food179explicitly, they are either implicitly or directly under-180pinned by single-species models of interaction between181herbivores and their food resources (Caughley 1976). The182simplest model that is generally applied to herbivore popu-183lations is the generalized logistic which has the form:

184r = rm�1 �NK�Z, (2.1)

185

186where rm is the maximum rate of increase, K is the density187of the herbivore population where the rate of renewal in188food resources is just sufficient to balance reproduction189and survival (where r = 0), N is prevailing population size190and z is a coefficient describing the degree to which the191density-dependent decline in r with N is delayed until192higher levels of N are attained (Fowler 1981, 1987; figure1932a). The value of z reflects the degree to which the194amount of food currently available to herbivores is195determined by the number of herbivores currently con-196suming that food (z = 1), or the number that have fed on197the food in the past (z � 1). Eberhardt (1987) used a fairly198high value of z = 11 in fitting equation (2.2)��2�� to199population census data for elk (Cervus elaphus) in Yel-200lowstone National Park in the western United States,201implying that current food availability was heavily depen-202dent on past elk density. Delayed effects of density on r203mean that most density dependence is observed at den-204sities near K (figure 2b).205The most pressing limitation of single-species models206for large herbivores (and hence on tests of the food207hypothesis based on single-species models), is that Kmust208be assumed to be relatively constant if the relationship209between population density and r is to be consistent (and210hence detectable) (Caughley 1976; Choquenot 1998).211The importance of this assumption can be illustrated by212contrasting the growth trajectories for elk projected from213Eberhardt’s (1987) model, where K is alternatively stable214(1% year-to-year variation in K; figure 3a) or unstable215(5% year-to-year variation in K; figure 3b). While growth216towards equilibrium follows a clearly density-dependent217trajectory where K is relatively stable, density-dependent218population growth is not evident where K is less stable.

219(b) Interactive consumer–resource models220The more explicit demographic numerical response221links change in consumer demographic rates to food avail-222ability. In contrast to the single-species density-dependent223approach above, consumer–resource models are by nature224multi-species models, but only in the sense that separate225predator and prey components are explicitly modelled and226linked. Additionally, whilst only one predator species is227usually modelled, prey may involve all food species228lumped on one axis or a subset of most important food229species. As a step in formulating an interactive plant–her-230bivore system, Caughley (1976) described a demographic

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1 Numerical response functions in herbivores and predators��1�� P. Bayliss and D. Choquenot 02TB006D.32

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1408 Figure 2. Hypothetical relationships between (a) rate of1409 population increase (r) and population size predicted from1410 the generalized logistic model in which z is varied from 1 to1411 3 (see lines on graph), and (b) the generalized logistic model1412 for Yellowstone elk estimated from population census data1413 (Eberhardt 1987). The parameter values estimated for elk1414 are rm = 0.2 p.a., carrying capacity K = 12 000 and z = 11.

231 numerical response (after May 1981a) that linked vari-232 ation in herbivore demographic rates (summarized by the233 instantaneous rate of population increase, rH), to the234 biomass of available food (V ):

235 rH = �a� c1(1 � e�Vd1), (2.2)236

237 where a is the maximum rate at which the population238 declines in the absence of food, c1 is a constant describing239 the difference between the maximum rate at which the240 population can increase (rmH) and a (i.e. rm = c1 � a), and241 d1 is the demographic efficiency of the population indexing242 how quickly r changes from being negative to positive as243 vegetation biomass increases. The general form of the244 response and a diagram of the full interactive model are245 shown in figure 4a,b.246 The other components of Caughley’s interactive model247 were the growth of ungrazed vegetation and the herbivore248 functional response. Vegetation growth in the absence of249 grazing was modelled using a simple density-dependent250 logistic function to link the instantaneous rate of change251 in vegetation biomass (rV ) to standing biomass (V):1



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1420Figure 3. Trajectories of growth for elk populations1421predicted from a generalized logistic model estimated by1422Eberhardt (1987), with stochastic variation in K equivalent1423to (a) 1% of the mean, and (b) 5% of the mean value of K.

252rV = rmV�1 �VK�, (2.3)

253

254where rmV is the maximum rate of increase in vegetation255biomass and K is vegetation biomass where shading or256competition for water or nutrients limits further plant257growth. The herbivore functional response, which258describes the increase in per capita vegetation offtake by259herbivores (CH) with increasing vegetation biomass, was260modelled using the same exponential form as the numeri-261cal response:

262CH = c2(1 � e�(V�Vg)d2), (2.4) 263

264where c2 is the maximum rate of vegetation intake by each265herbivore, Vg is the vegetation biomass where intake by266herbivores falls to 0 (i.e. the value of Vg determines267whether or not the curve goes through the origin (May2681981a, table 5)), and d2 is the efficiency of the functional269response describing how rapidly vegetation intake270increases to its maximum rate with increasing vegetation271biomass.272The important differences in the model developed by273Caughley and the Lotka–Volterra model described above274are: (i) the curvilinear functional and numerical responses275(equations (2.1) and (2.3)); and (ii) the fact that Caugh-

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1429 Figure 4. (a) The general form of a demographic numerical1430 response described by Caughley (1976), and (b) the1431 structure of the interactive model within which the response1432 was used. The dashed line in (b) indicates a relationship that1433 is explicit in Lotka–Volterra models but is subsumed by the1434 demographic numerical response in Caughley’s interactive1435 model.

276 ley’s numerical response links consumer demography277 directly to food availability rather than food intake rate.278 The more complex form of the functional response used279 in Caughley’s model accommodates more sophisticated280 ideas on how food availability and other environmental281 factors influence animal foraging behaviour (e.g. Watt282 1959; Ivlev 1961; Allden 1962; Holling 1966). In applying283 the same general form to the demographic numerical284 response, Caughley’s model simply allows the possibility285 that transfer of biomass between adjacent trophic levels is286 conserved in the same way as is assumed in Lotka–Vol-287 terra models (i.e. maximum reproduction and survival is288 dependent entirely on the rate of food acquisition). Under289 these conditions the functional and numerical responses290 can be parameterized so that maximum rates of increase291 (rm) are approached at levels of food availability that pro-292 duce maximum rates of food intake (C). This would293 reproduce the linear relationship between the rate of food294 intake and rate of change in predator abundance used in295 Lotka–Volterra models. Of course, other forms of this296 relationship are possible. Crawley (1983) argued that the297 relationship between food intake rate and r would be298 curvilinear where: (i) maximum reproduction or survival299 was limited by factors other than food intake; or (ii) a300 threshold rate of food intake was required before repro-301 duction was possible. Different forms for the numerical302 response would need to be considered if these alternatives303 were to be accommodated.304 While the demographic numerical response used in the305 interactive model subsumes the direct link between food306 intake and animal demography, it provides a very powerful307 summary of the indirect effect that food availability, as a1



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308limiting factor, has on an animal population. Demo-309graphic numerical responses to food availability have been310estimated for a range of herbivores including kangaroos311(Bayliss 1987), brush-tailed possums (Bayliss & Cho-312quenot 1998), wild pigs (Choquenot 1998) and wild313house mice (Pech et al. 1999). The primary use to which314these numerical responses have been put is in the develop-315ment of simulation models that explore dynamic interac-316tions between herbivore populations and their limiting317food resources (Caughley 1976, 1987; Caughley & Gunn3181993; Bayliss & Choquenot 1998; Choquenot 1998).319Perhaps one the best examples of how demographic320numerical responses can be applied to help understand321interactions between animal populations and their food322resources is the work of Caughley (1987) and his co-work-323ers. They estimated the components of the interactive324model described by equations (2.2), (2.3) and (2.4) for325the grazing system comprising red kangaroos and native326pastures in Australia’s eastern rangelands. This grazing327system is highly stochastic, being driven by the vagaries of328rainfall which varies up to 47% from year to year, with329low correlation between years and between seasons within330years. This highly stochastic variation leads to wide, seem-331ingly random fluctuations in the abundance of kangaroos332and the pastures they feed on. Between 1977 and 1985,333Caughley and his co-workers exploited these natural fluc-334tuations to estimate the form of density-dependent pasture335responses to rainfall (Robertson 1987a,b), and the336numerical response of kangaroos to pasture biomass337(Bayliss 1985a,b, 1987). During that time, the functional338response describing pasture intake by kangaroos to339changes in pasture biomass was also estimated in a series340of graze-down trials using captive kangaroos held in semi-341natural enclosures (Short 1985, 1987).342The vegetation response obtained was modified from343that described in equation (2.3) to account for empirically344estimated effects of variation in rainfall on pasture growth345and die-back, over and above the density-dependent346effects of pasture biomass. The modelled vegetation347response was:

348�V = �55.12 � 0.01535V� 0.00056V2

349� 3.946R, (2.5) 350

351where �V is the pasture growth increment over three352months in the absence of grazing, V is pasture biomass at353the start of those three months and R is the rainfall in mm354over that period. The pasture growth increment was taken355as a random draw from a normal distribution with mean356equal to the solution of equation (2.5) and a standard357deviation of 52 kg ha�1, equivalent to the variation in pas-358ture growth not accounted for by rainfall and standing359biomass (Robertson 1987b). The functional response of360red kangaroos (Short 1985) was estimated as:

361C = 86(1 � e�V/34), (2.6) 362

363(assuming an average body weight of 35 kg), and their364demographic numerical response as:

365rH = �1.6 � 2(1 � e�0.007V). (2.7) 366

367The dynamics of the grazing system were simulated over368100 years, with seasonal rainfall drawn from normal distri-369butions with means and standard deviations estimated370from long-term records. Successive pasture growth

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1441 Figure 5. Temporal variation in (a) pasture biomass and1442 (b) kangaroo density, predicted from a model of interaction1443 between kangaroos and pasture, developed by Caughley1444 (1987).

371 increments were estimated from equation (2.5), and372 changes in the per capita rate of pasture consumption and373 kangaroo density from equations (2.6) and (2.7). Changes374 in pasture biomass and kangaroo density were accounted375 weekly. Figure 5a,b shows changes in pasture biomass and376 kangaroo density, respectively, from a typical run of the377 model.378 Caughley (1987) found that despite high season to sea-379 son variation in pasture biomass, and year to year variation380 in kangaroo density, kangaroos persisted indefinitely in the381 modelled grazing system, neither crashing to extinction382 nor increasing without limit. Stochastic rainfall variation383 led to dramatic fluctuations in pasture biomass that were384 largely independent of kangaroo density. These fluctu-385 ations constantly buffeted the grazing system away from386 its potential equilibrium, creating the rapid oscillation in387 pasture biomass and large swings in kangaroo density evi-388 dent in figure 5a,b. However, despite this constant buf-389 feting, the reciprocal influence kangaroos and pasture390 exerted over each others abundance, imparted a suf-391 ficiently strong tendency towards equilibrium (i.e.392 ‘centripetality’), that kangaroos persisted indefinitely. The393 tendency that the grazing system has towards equilibrium394 is driven essentially by density-dependent competition395 amongst kangaroos for available pasture. The fact that396 kangaroos compete for pasture is evident in the 43%397 increase in average predicted pasture biomass that occurs398 when kangaroos are removed from the model. This indi-399 cates that the grazing system achieves centripetality400 through the same trophic processes that are represented401 implicitly by the density dependence of single-species402 models of other herbivore populations (Sinclair 1989).403 For example, in the absence of density-independent fluc-404 tuations in pasture biomass, the pattern of variation in r1



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1450Figure 6. Relationship between density and rate of increase1451for kangaroos in the absence of stochastic variation in1452rainfall and pasture growth, predicted from a model of1453interaction between kangaroos and pasture developed by1454Caughley (1987).

405for kangaroos with density conforms to that of a gen-406eralized logistic model (figure 6). Hence, the interactive407model is the general case for vegetation–herbivore sys-408tems, single-species models being a ‘short-hand’ or ‘con-409tracted’ version that represents the statistical association of410herbivore density and rate of increase that emerges when411density-independent perturbation of these systems is low412or uncommon.

413(c) Consonance with observation414(i) Non-equilibrium dynamics (including multiple equilibria)415Environmental variability and kangaroos416Bayliss (1987) developed numerical response models417for red and western grey kangaroos in two locations (a418national park and a sheep station). Caughley (1987) used419a slightly modified version of these functions to simulate420overall grazing system dynamics. An Ivlev (1961) function421was fitted (figure 7a) to the rate of increase versus food422availability data for red kangaroos using maximum likeli-423hood estimation. Results here are for red kangaroos on a424national park. Similar patterns were found for both kanga-425roo species in all locations. The a priori model assumes426that food is the major proximate factor that regulates427kangaroo population dynamics. Two post-drought outliers428were hence excluded from the original analysis because429they did not fit the a priori assumption (figure 7a). How-430ever, a time-trace of the rate of increase data (figure 7b)431show that the population dynamics of kangaroos entering432a drought from conditions of high food abundance is quite433different to that for populations recovering from a drought434from conditions of low food abundance. For populations435recovering from drought, rates of increase remain negative436despite high levels of food (pasture biomass). This ‘hyster-437esis’ or ‘lens’ pattern (resulting in two equilibria where438r = 0) may reflect the existence of two alternate system439states over a drought cycle, where the transition between440each is across a threshold or break point.441A density-dependent (isocline) numerical response442model demonstrates the non-equilibrium system proper-443ties more clearly (figure 7c). Results are for western grey444kangaroos on Kinchega National Park, however similar445patterns were found for both species in all locations. A446time-trace of rate of increase with a six month time-lag

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1460 Figure 7. (a) Demographic numerical response (r p.a.) for1461 red kangaroos and their food availability (pasture biomass V,1462 kg.ha�1��12��). The fitted function is r = �0.8 � 1.141463 (1 � e�0.007V) which excludes two post-drought data (square1464 symbols, Bayliss (1987)); (b) time-trace of the same rate of1465 increase data including previously discarded outliers (solid1466 line, high density populations entering a drought; dotted1467 line, low density populations leaving a drought��11��); and1468 (c) time-trace of western grey kangaroo rate of increase1469 versus lagged density (km�2, six month time-lag), showing1470 the break point transition between two domains of attraction1471 (drought and non-drought conditions).

447 shows clearly the break point or hysteresis between two448 postulated ‘domains of attraction’, reflecting drought and449 non-drought conditions. Although lagged density may450 confound both extrinsic (pasture food) and intrinsic451 (spacing behaviour) regulation processes, P. Bayliss and452 D. Choquenot (unpublished data) argue that the two par-453 allel and negative linear correlations may simply reflect the454 high and low phases of a stable limit cycle (i.e. an open-455 ended ellipse). Once again, this graphical analysis points456 to the existence of two equilibria (where r = 0), one at high1



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1477Figure 8. Phase plane trajectory plotting isoclines of pasture1478food (V, kg.ha�1��12��) and kangaroo density (D,1479nos.ha�1��3��) using a demographic numerical response1480model. Note the apparent existence of two domains of1481attraction at high (H) and low (L) densities. Each domain is1482locally stable (centripetal) but globally unstable because of1483unpredictable changes in rainfall-driven pasture biomass.

457density and high pasture biomass, the other at low density458and low pasture biomass. Both equilibria may be locally459stable but globally unstable because pasture biomass is460driven largely by stochastic rainfall events (Robertson4611987a,b).462Hence, globally, the kangaroo grazing system is a non-463equilibrium system but with two postulated local ‘domains464of attraction’ towards stability. The trajectories and pos-465itions within this binary system depend critically on initial466conditions of pasture biomass (food) and kangaroo den-467sity. Populations at high density entering a drought exhibit468different population dynamics to low density populations469emerging from a drought (e.g. different sex and age struc-470tures, response time-lags and reproductive condition (see471Bayliss 1980; Cairns & Grigg 1993)). The phase plane472trajectory (figure 8), or time-trace of zero isoclines of473kangaroo density (H, nos.ha�1��3��) and food abun-474dance (V, kg.ha�1��3��), clearly illustrate the dynamics475between alternate periods of very high and low to medium476kangaroo densities.477This result is surprising given that the numerical478response model is in fact an equilibrium model applied to479a stochastic environment. A probable cause may be the480asymmetrical relationship between rate of increase and the481availability of food which is in itself driven largely by482stochastic rainfall events; for the same amount of rainfall483about the annual mean (where r = 0), a much greater rate484of decrease occurs than a rate of increase. Hence, rainfall485variance reduces long-term mean density (Caughley4861987), but may also create a ‘two-state’ system (a run of487high density periods followed by a run of low density per-488iods; see figure 5b).489Although not incorporated into the Caughley (1987)490‘structural’ interactive grazing model, the composition of491pastures is also likely to differ between pre- and post492drought domains, which should add yet another dimen-493sion of complexity and stability. Pastures of course494respond in two ways to grazing (see McNaughton (1979)495for grassland–herbivore dynamics in the Serengeti), either496through changes in biomass and productivity (modelled497here), or shifts in pasture composition (not modelled498here). Surprisingly, even eastern grey kangaroo popu-

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1489 Figure 9. Annual trends in an index of abundance (numbers1490 observed km�1 ± s.e.) of eastern grey kangaroos on1491 Tidbinbilla Nature Reserve (ACT, Australia), showing1492 apparent 4–5 year stable limit ‘cycles’ between 1975–19911493 (n = 17 years). Kangaroos were counted at night by spotlight1494 along fixed transects across the reserve.

499 lations living in more stable seasonal, temperate environ-500 ments of eastern Australia exhibit non-equilibrium501 behaviour. Figure 9 shows an apparent 4–5 year stable502 limit cycle for eastern grey kangaroos on Tidbinbilla Nat-503 ure Reserve, ACT, Australia (P. Bayliss, unpublished504 data). Although we do not know what causes these appar-505 ent cycles (fox predation, competition with rabbits and/or506 other macropods, disease, el nino, seasonality effects or a507 combination of causes), or if in fact they are cycles508 (coincidence), the system is definitely not an equilibrium509 system as we would predict from the Caughley (1976,510 1987) interactive kangaroo grazing model applied to more511 stable environments. Hence, kangaroos appear to be very512 good examples of non-equilibrium systems because their513 numbers over time are characteristically unstable; the514 abundance of all censused populations of kangaroos var-515 ies widely.

516 Biological interactions and possums in New Zealand517 Caughley & Krebs (1983) identified two categories of518 regulation in order to explain the apparent dichotomy519 studies of small and large mammal population dynamics.520 One is intrinsic (self) regulation, where rate of increase521 (r) is suppressed by some form of spacing behaviour (or522 physiological process) as density increases. This type of523 regulation is generally expressed in terms of the negative524 prediction between r and instantaneous density (e.g. sin-525 gle-species logistic models). The other is extrinsic regu-526 lation, where r is governed by the relationship between the527 consumer and an external factor (food availability, pre-528 dation, disease, weather or a combination of factors).529 However, Erb et al. (2001) found that patterns of popu-530 lation dynamics in small versus large mammals contradict531 those predicted by the Caughley & Krebs (1983) hypoth-532 esis. Nevertheless, their distinction between intrinsic and533 extrinsic regulatory mechanisms remains an important534 distinction and is retained here. The term ‘density depen-535 dent’ generally refers to a prediction between r and den-536 sity, and density independent a lack thereof. However,537 density per head of population may index a limiting538 resource (extrinsic regulation) and/or spacing behaviour539 (intrinsic regulation). Because the population regulatory540 mechanisms are not explicitly defined, single-species ‘den-1



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541sity-dependent’ models often subsume or hide biological542process rather than expose them.543Although extrinsic and intrinsic regulation are not544mutually exclusive they are generally modelled as such.545However, some populations may be regulated by both546mechanisms (e.g. grazing interference or facilitation may547attenuate the numerical response to food availability (see548Vessey-Fitzgerald 1968)). A theoretical framework for549such combined regulatory influences on a species popu-550lation dynamics already exists (e.g. Caughley & Lawton5511981; Caughley & Krebs 1983), and is encapsulated in552a class of numerical response models called interferential553models (Caughley & Lawton 1981; Caughley & Krebs5541983; Barlow 1985). Caughley & Lawton (1981) exam-555ined a common form of the interferential numerical556response, such that:

557rH = rm�1 �JHV �, (2.8)

558

559where rH and V are as defined previously, and J is a pro-560portionality constant related to the availability of food561needed to sustain consumer H at equilibrium. However,562Barlow (1985) argued that, despite its widespread use, this563particular type of interferential numerical response model564is biologically meaningless. Nevertheless, the addition of565an intrinsic density dependent factor which is unrelated566to the extrinsic availability of food is easily expressed by567including a density term in laisez-faire numerical response568models exemplified by equation (2.2). A good example is569that provided by Tanner (1975) and explored by Caugh-570ley & Krebs (1983), such that:

571rH = a � c1(1 � e�Vd1) � gD, (2.9) 572

573where rH, a, c1, V and d1 are as previously defined, with574D being instantaneous density and g a coefficient575depending on the magnitude of the effect. Ginzburg576(1998) suggests that it is preferable to keep separate any577terms in the numerical response function which reflect578unrelated biological phenomena, and this is the approach579adopted by Caughley & Krebs (1983), Bayliss & Cho-580quenot (1998) and Pech et al. (1999). This is a more581realistic and explicit model of interference or aggregation582effects than the model proposed by Caughley & Lawton583(1981), as it separates the food intake and self-regulation584terms in the consumer numerical response whilst leaving585the nature of the self-regulation unspecified, as highlighted586by Barlow (1985).587Bayliss & Choquenot (1998) suggested that interferen-588tial numerical response models of equation (2.9) may pro-589vide a more useful framework for understanding590population dynamics because the combination of extrinsic591(consumer–resource) and intrinsic (animal–density) regu-592lation processes may embody the much broader spectrum593of population regulation mechanisms that most probably594exist in species. They examined this proposition for the595introduced possum in New Zealand forests, which is sum-596marized below.597Population models developed for possums per se have598curiously been entirely single-species logistic models (e.g.599Barlow & Clout 1983) and, hence, ignore plant–herbivore600interactions despite the marked impact that possums have601on native forests (although Barlow (1991) and Barlow et

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602 al. (1997) developed comprehensive multi-species pos-603 sum-disease models as extensions of the earlier single-604 species logistic model). Hinau (Elaeocarpus dentatus) is an605 endemic New Zealand hardwood tree that lives up to 400606 years and is known to mast; fruit production alternates607 between periods of low and super abundance in cycles of608 two plus years. Possums are primarily folivorous, but also609 feed extensively on fruits, flowers and buds of many native610 trees such as hinau. Hinau fruit is a critical winter food611 source and may even index all winter food sources (Bell612 1981). Bell found that the birth date of possums, the per-613 centage of females with pouch young and body weight all614 were positively correlated to the annual crop of hinau fruit.615 Cowan & Waddington (1990) found that hinau fruit pro-616 duction increased dramatically when possums were eradi-617 cated, and that fruit production was suppressed again with618 subsequent recolonization. The ecological relationships619 between possums and food availability were examined in620 greater detail in two stages, using hinau fruit to index food621 availability. First, the physiological relationships between622 possums and hinau abundance were examined and used623 to underpin the population-level analyses. Second, an624 interferential numerical response model was then625 developed using the availability of hinau fruit as an index626 of food supply overall and ‘instantaneous density’ to index627 possible spacing behaviour effects (e.g. feeding inter-628 ference and/or competition for nest sites) which may be629 independent of the effects of food. We argue that behav-630 ioural density-dependent effects are likely to be instan-631 taneous (i.e. no time-lag) and, by contrast, additional632 extrinsic effects unrelated to food (e.g. predation or633 disease) are more likely to be indexed by lagged density634 effects. Nevertheless, the exact cause of any instantaneous635 density-dependent effect is unknown and, therefore,636 additional extrinsic effects (in interaction or in637 combination) cannot be ruled out.638 Possum reproductive and body condition (weights and639 fat storage) were collected during a long-term trap–kill640 study of possums in the Pararaki Valley (n = 32 years;641 1965–1997), New Zealand (M. Thomas and J. Coleman,642 unpublished data.). Female body fat increased with643 increasing availability of hinau fruit at Orongorongo Valley644 (20 km distance from Pararaki Valley; figure 10a). The645 trend is mostly linear although overall significantly nonlin-646 ear (quadratic, concave down) due to one point. The pro-647 portion of female possums with pouch young increased648 with increasing body fat condition up to a maximum level649 (figure 10b). An Ivlev curve was fitted by maximum likeli-650 hood estimation, explaining a high proportion of variance651 (r2 = 89%). The proportion of females with pouch young652 increased with increasing food availability up to a653 maximum level (figure 10c). A logistic curve was fitted by654 maximum likelihood estimation explaining a high pro-655 portion of variance (r2 = 99.5%). There was a negative lin-656 ear correlation between birth date (arbitrary estimated as657 days since the 1st January on a logarithmic scale) and658 hinau food availability (figure 10d). More possums were659 born early when food levels were high compared with660 more possums being born late when food levels were low.661 However, only a low proportion of variance was explained662 (r2 = 15%) by this relationship.663 Long-term (n = 31 years; 1966–1997 (Efford 1998,664 2000)) population level data were collected at Orongo-1



3

665rongo Valley by mark–recapture and contemporaneously666with estimates of the annual crop of hinau fruit (via seed-667fall traps (Cowan & Waddington 1991; P. E. Cowan et al.668unpublished data��4��). A classic Ivlev curve was fitted a669priori by maximum likelihood estimation to the numerical670response between rate of increase and food (figure 11a),671explaining 40% of observed values (rposs = �0.60672� 0.85(1.0 – e�0.037V). The estimate of rm is 0.25 p.a.673which compares favourably with the range of estimates674(0.22–0.25 p.a.) derived by Hickling & Pekelharing675(1989). There was a significant and negative linear corre-676lation between rate of increase and instantaneous density677(figure 11b), explaining 35% of observed values678(rposs = 0.65–0.084D). Backwards extrapolation to the y-679axis (D = 0) predicts a rm value of 0.65 p.a., significantly680higher than that estimated by the demographic numerical681response above. However, the linear extrapolation is well682outside the range of observed data, and may be an over-683estimate if the relationship between r and density is non-684linear as expected (because of the interaction between685limiting food resources).686The dynamics of hinau fruit production as impacted on687by possums, and of possums as influenced by intrinsic and688extrinsic regulatory mechanisms, are characterized by a689family of curves or relationships and, hence, multiple equi-690libria (figure 12a,b). Bayliss & Choquenot (1998)691developed an interferential numerical response function692for possums by statistically combining both extrinsic (rposs

693versus hinau, V ) and intrinsic (rposs versus density,694Dposs ha�1) numerical responses (figure 12b) into a joint695multiple regression equation (rposs = {[�0.60 � 0.85696(1 � e�0.037V )] � 0.01Dposs}). Both variables were statisti-697cally significant entries into the overall regression equ-698ation, which explained 69% of observed values. These two699numerical response models are a good example of the con-700trasting paradigms described by Sibly & Hone (2002).701The dynamics of hinau fruit production in the presence702of consumption by possums is best described by a logistic703model with an independent term for the negative impact704of possum density (figure 12a). The high intrinsic rate of705increase of annual hinau fruit production (rhin(m) � 2.0)706produces stable limit cycles with a periodicity of 2.0 years707(see years 1–20; figure 12c). Without the impact of pos-708sums hinau would mast every two years (i.e. low one year709and high the next), although environmental variability (P.710E. Cowan et al. unpublished data��4��) may mask detec-711tion of any regular cycles. Figure 12c shows a simulated712trend in hinau and possum abundance after a liberation713of 0.1 possums ha�1. The modelled stable limit cycle of714hinau fruit is flattened out after 20 years because the715increasing abundance of possums and associated absolute716consumption of hinau has an equilibriating effect. The717abundance of hinau fruit eventually stabilizes at a level of71844 (cf. an observed mean value of 47). Similarly, possum719densities are predicted to equilibriate at 8.4 ha�1 in con-720trast to mean observed densities of 8.2 ha�1 (and mean721rposs = 0, as predicted by the joint regression model).722Although this close fit is not an independent test of the723model, the outcomes at least concord with observed data.724Without the density effect in the possum numerical725response function, the model predicts an equilibrium pos-726sum density of 10.7 ha�1 , 30% above the observed mean,

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1501 Figure 10. Physiological responses of possums in the Pararaki Valley (New Zealand) to winter food supply as indexed by the1502 availability of hinau fruit in the Orongorongo Valley 20 km away (trap–kill data, 1965–1997) for: (a) female body fat index1503 versus food (hinau fruit) availability (quadratic polynomial regression: r2 = 48%, d.f. = 2/13, p � 0.01); (b) proportion of1504 females with PY versus female body fat index (r2 = 89%, n = 17, p � 0.01); (c) proportion of females with PY versus food1505 (hinau fruit) availability (logistic model: r2 = 99.5%, n = 20, p � 0.001); and (d) timing of births (natural logarithm of arbitrary1506 number days since 1 January) versus food (hinau fruit) availability (negative linear correlation, r2 = 15%, n = 17, p � 0.05). All1507 nonlinear functions were fitted using maximum likelihood estimation.

727 whilst hinau fruit abundance decreases to 33% less than728 the observed mean.

729 3. ISOCLINE NUMERICAL RESPONSES

730 (a) Underlying assumptions731 An isocline numerical response links changes in the732 abundance of a consumer population per se directly to the733 availability of its food resources (figure 13a). Isocline734 numerical responses are generally formulated as an735 asymptotic increase in consumer abundance, indicating736 that the upper limit to consumer abundance is imposed737 by some factor which is independent of food availability738 (e.g. available territories, nest sites or some socially739 mediated crowding effect). Isocline numerical responses740 subsume both the effect that food availability has on the1



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741rate of food intake through the functional response, and742the influence food intake has on demographic rates of the743consumer population (figure 13b). While the isocline744approach greatly simplifies the way in which interaction745between food and consumer abundance can be rep-746resented, it also assumes that territorial behaviour or inter-747ference competition regulates populations at high density,748and the accessibility or availability of limiting resources at749low density (Choquenot & Parkes 2000��13��).

750(b) Stability properties of the wolf–moose system:751graphical analyses752Probably the best known application of isocline numeri-753cal responses is in the graphical analysis of the stability754properties of consumer–resource systems (Rosenzweig &755MacArthur 1963; Noy-Meir 1975; Messier 1994; Caugh-

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1513 Figure 11. The (a) extrinsic numerical response (r p.a.1514 versus index of food availability V: Ivlev curve fitted a priori1515 is rposs = �0.60 � 0.85(1.0 � e�0.037V) where r2 = 40%,1516 n = 50, p � 0.001) and (b) intrinsic numerical response (r1517 p.a. versus instantaneous density D, nos.ha�1:��3��1518 rposs = 0.65–0.084D where r 2 = 35%, n = 31, p � 0.001) of1519 possums in Orongorongo Valley, New Zealand (see1520 Bayliss & Choquenot (1998) for methods). Data are winter1521 and summer annual exponential rates of increase (r p.a.),1522 and the index of food availability (hinau seedfall) has a six1523 month time-lag. The parameters of the Ivlev curve fitted to1524 (a) were estimated by maximum likelihood estimation; rm1525 was estimated at 0.25 p.a.

756 ley & Sinclair 1994). In these analyses, proportional gains757 and losses to the food population are contrasted over the758 full range of its potential abundance in order to identify759 levels of food availability which are relatively stable.760 Potential increases in the abundance of the food popu-761 lation are usually represented by a generalized-logistic762 model similar to that shown in figure 2a. This model763 assumes that in the absence of the consumer population,764 the upper limit to food population abundance is imposed765 by interspecific competition for some limiting resource or766 through social regulation. Potential losses from the food767 population are the product of consumer population’s1



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1531Figure 12. Plant–herbivore model for possums. The1532numerical response (rhin p.a.) of (a) hinau fruit production as1533combined negative functions of the previous years1534production V (via a logistics model) and possum density1535(Dposs, nos.ha�1)��3��, rhin = 2.2(1 � V/100) � 0.15Dposs,1536(r2 = 68%, p � 0.001, d.f. = 1/47); and (b) possums1537(rposs p.a.) as functions of hinau (food) availability (V ) and1538possum density (Dposs, nos.ha�1)��3��,1539rposs = {[�0.60 � 0.85(1 � e�0.037V )] � 0.01Dposs}, (r2 = 69%,1540d.f. = 1/47). A family of numerical response curves exist for1541both hinau and possums depending on possum density, and1542are here illustrated with 2 and 10 possums ha�1. (c)1543Simulated equilibriation between hinau fruit production1544(solid line) and possums (dashed line) 50 years after1545liberation predicting extinction of the hinau ‘masting’ cycle.

768abundance (estimated from its isocline numerical response769to food availability; figure 14a) and per capita food intake770(estimated from its functional response). When potential771loss is expressed as a proportion of the food population it

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1551 Figure 13. (a) The general form of an isocline numerical1552 response derived by Rosenzweig & MacArthur (1963), and1553 (b) the structure of the predator–prey model within which1554 the response was used. The dashed lines in (b) indicate1555 relationships that are explicit in Lotka–Volterra models but1556 subsumed by the isocline numerical response used in1557 Rosenzweig & MacArthur’s model.

772 is generally termed the consumer total response. Figure773 14b shows an example of a graphical analysis of the stab-774 ility properties for a wolf–moose system in North America775 using predator–prey population parameters estimated by776 Messier (1994). Our graphical derivation of an equilib-777 rium point between wolves and moose agree with the con-778 clusion by Eberhardt & Petersen (1999) that a two-state779 system need not apply. The isocline numerical response780 underpins calculation of the proportion of the moose781 population that would be consumed by wolves at given782 moose densities. These isocline curves are essentially the783 same as those shown by Sinclair et al. (1998) for predation784 in general.

785 4. DISCUSSION

786 (a) Density dependence and population regulation787 While density-dependent, density-independent and788 inversely density-dependent factors can all limit popu-789 lation density, only density-dependent factors impart a790 tendency towards an equilibrium. Hence, while any pro-791 cess that affects population density will be a limiting fac-792 tor, only density-dependent factors are also regulating793 factors (Sinclair 1989).794 Density dependence can arise from both intrinsic and795 extrinsic factors operating on a population (Krebs 1985).796 Changes in the exponential rate of increase (r) of an intrin-797 sically regulated population slows at high density through798 the effect of some type of spacing behaviour on mortality,799 fecundity or migration as population density increases.800 Populations regulated in this way can be thought of as1



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1563Figure 14. The wolf–moose system in North America1564showing (a) an example of an isocline numerical response1565linking the density of consumer populations ( y, wolves) to1566the availability of their food resources (x, moose) (Messier15671994) with the fitted function y = 58.7(x� 0.03)/0.76 � x1568and, (b) a graphical stability analysis representing the1569interaction between moose and wolves based on empirically1570derived functions in Messier (1994). The lower curve,1571describing the percentage loss of moose to wolves as a1572function of prevailing moose density, was calculated from1573the functional and isocline numerical responses of wolves to1574moose density (see (a)). The upper curve, describing the1575gain in moose abundance in the absence of wolves, was1576derived by fitting a generalized-logistic model to data1577(rm = 0.22 p.a., K = 2.0 moose km�2 and z = 4.0) presented in1578Messier (1994). The dashed line indicates the equilibrium1579moose density predicted by the model.

801‘self-regulating’, with rate of change in their abundance802determined instantaneously by their prevailing density803(Caughley & Krebs 1983). By contrast, r for an extrinsi-804cally regulated population is determined by the availability805of some environmental resource such as food or nesting806sites, or by the effect of some limiting environmental agent807such as predators or disease (Caughley & Krebs 1983).808Rate of change in the abundance of an extrinsically regu-809lated population is determined instantaneously or cumu-810latively by the availability of the critical resource or the811effect of the critical agent.

812(b) Population regulation: single-species models813The abundance of large herbivore populations is widely814held to be limited by extrinsic factors, most commonly815food supply (Caughley 1970, 1987; Laws et al. 1975;816Sinclair 1977; Houston 1982; Skogland 1983; Sinclair et

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817 al. 1985; Fryxell 1987; Choquenot 1991, 1998), pre-818 dation (Bergerud 1980; Gasaway et al. 1983; Messier &819 Crete 1984; Messier 1991, 1994), or both (Caughley820 1976, 1977). Factors that limit the size of large herbivore821 populations may or may not also regulate them,822 depending on whether they operate in a density-depen-823 dent fashion. Sinclair (1989) reviewed studies of regu-824 lation in large terrestrial mammals and concluded that the825 majority (including all ungulates) were regulated by den-826 sity-dependent mortality related to food shortage. Similar827 conclusions were reached by Fowler (1987) in a review828 based on many of the same studies. Sinclair (1989) also829 found that while predator removal experiments have830 shown predation to be an important limiting factor for831 large herbivore populations, there was no empirical evi-832 dence that predation could also be a regulating factor for833 large herbivores. Skogland (1991) and Boutin (1992) con-834 curred with Sinclair (1989), finding no consistent evi-835 dence for regulation of ungulate populations by predation.836 However, Messier (1994) inferred from a comparative837 study of interaction between moose (Alces alces) and838 wolves (Canis lupus) across North America, that moose839 populations could be regulated by wolf predation. If pre-840 dation does not commonly regulate the abundance of large841 herbivore populations, processes that could impart density842 dependence to their dynamics reduce to intrinsic (socially843 mediated) mechanisms, the debilitating effects of disease844 or parasites, or food availability.845 Several recent reviews (e.g. Gaillard et al. 1998) of846 empirical evidence for density dependence in large herbi-847 vore demography indicate that the role of density-inde-848 pendent variation in the abundance of herbivores and their849 key food resources has not been fully recognized in tests850 of the food hypothesis. Caughley & Gunn (1993) argued851 that in areas with highly variable environments, factors852 such as unpredictable precipitation could introduce sig-853 nificant degrees of density-independent variation in food854 availability and herbivore abundance. When combined855 with lags and overcompensation in vegetation and herbi-856 vore responses, density-independent variation can obscure857 any tendency that herbivore density may have towards858 equilibrium. Similarly, Putman et al. (1996), and Saether859 (1997) considered that even in temperate grazing systems,860 stable equilibria between large herbivores and their food861 resources were unlikely because of the direct effects of862 environmental variation on herbivore demographic rates863 and overcompensation in herbivore responses to variation864 in food availability. If stable K cannot be assumed for large865 herbivore populations, tests of the food hypothesis which866 fail to detect density-dependent variation in r (or its867 correlates) cannot differentiate between perturbation of868 the system by density-independent limiting factors, and869 the absence of regulation through density-dependent food870 shortage. To account for the potential effects density-871 independent variation in food availability and herbivore872 abundance have on the tendency of a herbivore population873 towards equilibrium, interaction between herbivores and874 their food resources must be considered in a more explicit875 framework than can be provided by single-species density-876 dependent models (Choquenot 1998).

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877(c) Population regulation: interactive models878Explicit models of vegetation–herbivore interaction879were developed by Noy-Meir (1975) and Caughley880(1976), who derived them from the predator–prey models881of Rosenzweig & MacArthur (1963) and May (1973). The882interactive model described by Caughley (1976) for a883deterministic environment is described in detail here as884applied to kangaroos in the Australian rangelands, a stoch-885astic environment driven by unpredictable rainfall886(Caughley 1987). The three components of the interactive887model (growth of ungrazed plants, functional and numeri-888cal responses) operate collectively as two negative feed-889back loops governing the influence that vegetation and890herbivores exert over each others abundance. Density-891dependent growth of ungrazed vegetation forms a veg-892etation biomass feedback loop, reducing vegetation893growth at high biomass and keeping vegetation in check894regardless of how good seasonal conditions may be for895plant growth, or how low vegetation offtake by herbivores896is. The functional and numerical responses of the herbi-897vore form a vegetation–herbivore feedback loop, increas-898ing the number of herbivores and how much vegetation899each consumes at high vegetation biomass, and reducing900herbivore abundance and their per capita consumption of901vegetation at low vegetation biomass. Caughley (1976)902combined these feedback loops in two linked differential903equations which predict coincident variation in the abun-904dance of herbivores and the vegetation they feed on.905Herbivores reach a stable equilibrium point that is906qualitatively similar to that produced by the generalized907logistic model described for elk. However, in the inter-908active model, equilibrium is achieved through the recipro-909cal influence vegetation and herbivores exert over each910others abundance, while in the generalized logistic model,911equilibrium reflects an algebraic limit to herbivore popu-912lation growth imposed by the existence of K. This does913not mean that the herbivore population is not regulated914through essentially density-dependent processes.915Regardless of the density from where the herbivore916population starts, its density moves back towards equilib-917rium through the same series of dampening oscillations.918While the processes producing the tendency towards equi-919librium are essentially density dependent (i.e. the vector920of vegetation and herbivores at any point in time is a direct921consequence of past grazing activity which is a conse-922quence of past herbivore density), the tendency itself is of923more importance to the stability of the grazing system than924is the attainment of any specific point equilibrium. To925reflect this, Caughley (1987) coined the useful term ‘cen-926tripetality’ to describe the tendency that a vector in veg-927etation and herbivore abundance has towards equilibrium.928Centripetality de-emphasises the importance of an equi-929librium in the dynamics of vegetation–herbivore systems,930focusing instead on the stabilizing properties that the931potential existence of an equilibrium imparts.

932(i) Stochastic rangelands grazing systems: kangaroos933Caughley (1987) demonstrated that stochastic rainfall934variation in the rangelands of Australia produced high fre-935quency and high amplitude fluctuations in pasture936biomass with wearying monotony and largely independent937of kangaroo density. Hence, the kangaroo grazing system938is constantly buffeted away from its potential equilibrium.

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939 McLeod (1997) argued that, in this environment, the con-940 cept of equilibrium carrying capacity density has no mean-941 ing. Nevertheless, the reciprocal influence that kangaroos942 and pasture exerted over each others dynamics and, ulti-943 mately abundance, imparted a strong tendency towards944 equilibrium (i.e. centripetality). Hence, in this model eco-945 system kangaroos were able to persist indefinitely. Inter-946 active models may be the general case for vegetation–947 herbivore systems (and by extension all consumer–948 resource systems such as predator–prey systems). By con-949 trast, contracted single-species models are, however, a950 ‘short-hand’ proxy, often represented by a statistical nega-951 tive correlation between herbivore rate of increase and952 density that manifests when density-independent pertur-953 bation of these systems is low or uncommon. By contrast,954 McCarthy (1996) used variable rainfall as a surrogate for955 pasture biomass (see Bayliss 1985a,b) to examine the956 combined statistical relationships between red kangaroo957 rate of increase, food availability and past kangaroo den-958 sity. This approach falls neatly between the single-species959 and interactive consumer–resource approach, and could960 be better modelled with an interferential numerical961 response model.

962 (ii) Habitat effects on food limitation963 Like most animals, large herbivores balance their forag-964 ing efficiency with exposure to direct sources of mortality965 or debilitation in order to maximize the rate at which they966 can utilize their food resources to increase individual fit-967 ness (Belovsky 1981, 1984; Stephens & Krebs 1986).968 Hence, the quality of the various habitats in which large969 herbivores may elect to spend time will reflect both the970 availability and quality of food found there, and the degree971 to which habitat-related constraints effect the rate at972 which this food can be assimilated to enhance their repro-973 duction or survival. Habitat-related constraints can influ-974 ence the rate at which herbivores can find and ingest food975 relative to its availability (foraging constraints), or the976 degree to which ingested food can be used to enhance977 reproduction or survival (demographic constraints). For978 example, increased predation risk in open areas reduces979 the foraging efficiency of snowshoe hares (Lepus980 americanus) by limiting their access to the food available981 in these areas (Hik 1995). Alternatively, the demographic982 efficiency of caribou (Rangifer tarandus) is limited by the983 availability of habitat that affords their calves protection984 from predators, over and above any effects of food avail-985 ability (Skogland 1991).986 Habitat-related constraints on foraging or demographic987 efficiency inhibit the potential a herbivore population has988 to respond demographically to variation in its food989 resources. Hence, models of how habitat-related foraging990 and demographic constraints influence large herbivore991 population dynamics need to be formulated within a992 framework which explicitly represents interaction between993 the herbivores and their food resources. Caughley’s994 (1976) interactive model links herbivore foraging995 efficiency to food availability through a functional996 response, and herbivore demographic efficiency to food997 availability through a numerical response. These two998 responses collectively form a vegetation–herbivore feed-999 back loop that controls the interdependent effects veg-1000 etation and herbivores exert over each others abundance.1



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1001However, the vegetation–herbivore feedback loop implies1002that variation in food availability effects herbivore1003demography (summarized as r) independently of its effect1004on their rate of food intake. While this simplification is of1005little consequence where constraints on foraging or demo-1006graphic efficiency are constant, it compromises the useful-1007ness of the interactive model where these constraints vary1008between habitats. Hence, where habitat quality is in part1009determined by constraints on herbivore foraging or demo-1010graphic efficiency, the general form of the interactive1011model cannot be used to consider how habitat quality1012influences herbivore population dynamics. For example,1013Choquenot & Dexter (1996) hypothesized that wild pigs1014in the rangelands thermoregulate when radiant heat loads1015are high by seeking refuge under the more or less continu-1016ous cover afforded by riverine woodlands. Behavioural1017thermoregulation when ambient temperatures are high is1018obligatory for wild pigs inhabiting other arid and semi-arid1019environments (Van Vuren 1984; Baber & Coblentz 1986).1020Choquenot & Dexter (1996) suggested that the thermore-1021gulatory needs of wild pigs when temperatures were high1022could link the spatial accessibility of riverine woodlands1023to either: (i) their foraging efficiency by restricting the area1024over which they could forage; or (ii) their demographic1025efficiency by restricting the area within which they could1026survive and reproduce. In either case, the quality of any1027particular location to wild pigs will be determined by both1028habitat composition of the immediate area around the1029location (i.e. the accessibility of riverine woodlands), and1030the availability of food in that area.

1031(d) Conclusions1032The two main reasons why we construct ecological1033models are to predict and to aid understanding of the sys-1034tem (Caughley 1981). Both modelling functions are essen-1035tial to the development and implementation of population1036management goals for whatever objective (conservation,1037control or harvesting). Simulation of population manage-1038ment scenarios to assess their efficacy is becoming increas-1039ingly popular if not necessary because of the general lack1040of experimentation. However, the success of this approach1041depends entirely on the ability of the model to capture1042real ecological processes, but progress in understanding1043ecological systems has been less than spectacular. A little1044scrutiny shows that most wildlife populations are still1045managed by trial and error rather than by scientific knowl-1046edge, and that most managers still lack tight criteria for1047the success or failure of their actions (including doing1048nothing). Nevertheless, the contributions of consumer–1049resource models to the research and management of over-1050abundant kangaroos and possums have been substantial.1051Curiously though, the modelling approaches adopted in1052Australia and New Zealand have followed two inde-1053pendent paradigms (or multi-species versus single-species1054models; or mechanistic versus density (Sibly & Hone10552002)). Our re-examination of current kangaroo and pos-1056sum models, however, indicate that a more useful frame-1057work for understanding how marsupial populations work1058may be obtained by combining the two modelling1059approaches. A marriage between extreme extrinsic1060(animal–resource) and intrinsic (animal–density) regu-1061lation models could embody the much broader spectrum1062of population mechanisms that most likely exist within

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1063 species. Although more complex population interactions1064 may be exposed, the trade-off may be increased predictive1065 power and, hence, utility. This approach increases the1066 realism and predictive power of existing population mod-1067 els for kangaroos and possums at manageable levels of1068 complexity.1069 We conclude by reinforcing the axiom that in order to1070 manage populations effectively we need to understand1071 their dynamics. However, research costs can be substantial1072 both in terms of time and money. For example, the pos-1073 sum model was developed a posteriori (no model in mind)1074 with data collected after 32 years of intensive study of a1075 population more or less in equilibrium. By contrast, the1076 kangaroo model was developed a priori (a model in mind)1077 with experimental data collected after 5 years of intensive1078 study of a non-equilibrium grazing system. Hence, one1079 impediment to more widespread use of more useful inter-1080 active ecological models is the daunting and costly task of1081 ‘parameterizing’ such models, especially for populations1082 that exhibit little dynamics within time-scales dictated by1083 funding and career cycles. An adaptive management strat-1084 egy (Walters 1997), however, may allow a new breed of1085 population models to be developed and tested cost-effec-1086 tively by integrating focused population-scale manage-1087 ment experiments with the modelling process. One such1088 model is the interferential numerical response function,1089 because it may help bridge three major historical dichot-1090 omies in population ecology (equilibrium versus non-equi-1091 librium dynamics, extrinsic versus intrinsic regulation and1092 demographic versus isocline numerical responses).

1093 We thank Malcome Thomas, Jim Coleman, Murray Efford,1094 Phil Cowan (New Zealand Landcare Research) and Environ-1095 ment ACT (Lyn Nelson) for access to unpublished data,1096 Environment Australia-eriss (Max Finlayson), all staff of the1097 NT University Key Centre for Tropical Wildlife Management1098 (in particular Barry Brook, Peter Whitehead & David1099 Bowman) and staff of the Arthur Rylah Institute (Victorian1100 DNRE) for support. We also thank Jim Hone, Richard Sibly1101 and Suzi White for organizing the conference and workshop,1102 and The Royal Society and Novartis Foundation for financial1103 support to attend.

1104 REFERENCES

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1388GLOSSARY

1389PY: pouch young 1390

Population growth rate: determining factors and role in population regulation Compiled and edited by RM Sibly, J Hone and TH Clutton-Brock Published 29 September 2002

Wildlife populations vary in abundance over time. Some populations have declined towards extinction and others have increased dramatically. The patterns of, and reasons for, such variation have been topics of active research for decades. Recent developments in the field, such as more detailed case studies and refined mathematical analysis, allow greater exploration of why populations vary in abundance.

This volume derives from a Royal Society Discussion Meeting held on 6 and 7 February 2002, which examined the recent developments for wildlife populations around the world and provided directions for future research and wildlife management.

For a limited period this special issue of Philosophical Transactions is available at a discounted price of £45/US$70 (normal price £85/US$135). To purchase the complete volume complete the order form below and return it to: The Royal Society, (TB 1425Q), PO Box 20, Wetherby, West Yorkshire LS23 7EB, UK ___________________________________________________________________

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Contents _________________________________________________________ Introduction RM Sibly, J Hone and TH Clutton-Brock Population growth rate and its determinants: an overview RM Sibly and J Hone Demographic, mechanistic and density-dependent determinants of population growth rate: a case study in an avian predator J Hone and RM Sibly Estimating density dependence in time-series of age-structured populations R Lande, S Engen and B-E Sæther Pattern of variation in avian population growth rates B-E Sæther and S Engen Determinants of human population growth W Lutz and R Qiang Two complementary paradigms for analysing population dynamics CJ Krebs Complex numerical responses to top-down and bottom-up processes in vertebrate populations ARE Sinclair and CJ Krebs The numerical response: rate of increase and food limitation in herbivores and predators P Bayliss and D Choquenot Populations in variable environments: the effect of variability in a species’ primary resource SA Davis, RP Pech and EA Catchpole Trophic interactions and population growth rates: describing patterns and identifying mechanisms PJ Hudson, AP Dobson, IM Cattadori, D Newborn, DT Haydon, DJ Shaw, TG Benton and BT Grenfell Behavioural models of population growth rates: implications for conservation and prediction WJ Sutherland and K Norris Comparative ungulate dynamics: the devil is in the detail TH Clutton-Brock and T Coulson Population growth rate as a basis for ecological risk assessment of toxic chemicals VE Forbes and P Calow Population growth rates: issues and an application HCJ Godfray and M Rees

21

Powerpoint presentation

Population growth rate: Population growth rate: determining factors & role in determining factors & role in

population regulationpopulation regulationThe numerical response: rate

of increase & food limitation in herbivores & predators

Peter B aylis s & David Choquenot

Seminar OutlineSeminar Outline•Framework

– Regulation

– Consumers & resources

– Demographic & isocline numerical responses

•Kangaroos•Possums•Pigs

•Implications

22

Types of populationTypes of population regulationregulation• What regulates population dynamics (or rate of increase)?

r = (b - d) + (i – e)

• Caughley & Krebs (1983) identified two categories of regulation which are not mutually exclus ive:

2. Intrins ic (=self regulation) – spacing behaviour &/or phys iological constraints ; additional to extrins ic factors

r = - g (dens ity)

1. Extrins ic – external, often multiple factors

r = f (resources , predators , weather)

Consumers & food resources Consumers & food resources –– types of systemstypes of systems(after Caughley & Lawton 1981)

Grazing sys tems(Plant-herbivore)

InteractiveNon-interactive

Reactive Non-reactive Lais sez-faire Interferential

Class ification s ys tem generic to mos t consumer-resource s ys tems : e.g . predator-prey & paras ite-hos t

23

Earlies t interactive consumer-resource model Lotka-Volterra predator-prey model

Consumption rate of prey

Rat

e of

incr

ease

pr

edat

or

Time

Num

bers

PredatorPrey

Population trends of predator & prey - coupled s table limit cyc les

numerical response is food - dependent

B iomass convers ion principle

PREYFood

availability

PREDATORConsumerabundance

Consumer demographicrates

Foodintake rate

Functional response

Links between consumers & food in Lotka-Volterra predator- prey model

Numerical response

24

Food availability

Ani

mal

abu

ndan

ce

Isocline numerical response

Food availability

Inst

anta

neou

s ra

te o

f in

crea

se (r

)

+

_

Demographic numerical response

Two types of numerical response functions s ince Solomon (1949)

Foodavailability

Consumerabundance

Consumer demographicrates (summarised as r)

Foodintake rate

Functional responseDemographic

numerical response

Explic it in Lotka Volterra modelsSubsumed in demographic numerical response

L inks between consumers & food in demographic numerical responses

25

Foodavailability

Consumerabundance

Consumer demographicrates (summarised as r)

Foodintake rate

Functional responseIsocline

numerical response

Explic it in Lotka Volterra modelsSubsumed in isoc line numerical response

L inks between consumer & food in isocline numerical responses

Examples : numerical responses & foodExamples : numerical responses & food

Demographic numerical response for pigs-pas ture (Choquenot 1994)

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

0 500 1000 1500

Pasture biomass (kg.ha-1)

Rat

e of

incr

ease

(r p

.a.)

0

10

20

30

40

50

60

0.0 0.5 1.0 1.5 2.0 2.5

Moose km-2

Wol

ves

103 k

m-2

Isocline numerical response for wolf-moose sys tem. After Mess ier (1994)

26

Kangaroo grazing s tudy in the sheep Kangaroo grazing s tudy in the sheep rangelands of NSW (rangelands of NSW (19801980-- 1984)1984)

Caughley et al. (1987)

Aim: to parameterise an a prioriinteractive plant - herbivore model for

kangaroos

InteractiveInteractive plant plant -- herbivoreherbivore model: deterministicmodel: deterministic

Explic it functions model

• Plant growth function

• Functional response herbivores

• Numerical response herbivores

( )Vd

VmV eHc

KVVr

dtdV

111 1−−−

−=

( )[ ]VdecaHdtdH

212−−+−=

Phase plane trajectory H & V in absence predators

0

2

4

6

8

10

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0

Plant density

Her

bivo

re d

ensi

ty

Interactive plant-herbivore model

0

2

4

6

8

10

0 10 20 30 40 50 60

Time (year)

Plan

t & h

erbi

vore

den

sity

27

BIOENERGETICS MODELDeterministic environment 3 explicit nonlinear energy

transfer functions

0

20

40

60

80

100

0 200 400 600 800 1000

Food (vegetation, kg/ha)

Food

inta

ke (g

/day

)

2. Functional response

-2.0

-1.5

-1.0

-0.5

0.0

0.5

0 200 400 600 800 1000

Food (vegetation, kg/ha)R

ate

of in

crea

se (r

p.a

.) 3. Numerical response

-400

0

400

800

0 200 400 600 800 1000

Vegetation (kg/ha)

Gro

wth

(kg/

ha/3

mo)

1. Plant growth response

Unpredictable rainfall is the driving variable Unpredictable rainfall is the driving variable in rangeland ecosystemsin rangeland ecosystems

BOOM BUST

28

Pasture biomass Kangaroo dens ity

Boom-bust trend in pasture biomass &kangaroo density during a drought period

Kinchega National Park (1980 – 1984)

0

200

400

600

800

1000

1200

1980 1981 1982 1983 1984 1985

Time (year)K

anga

roo

dens

ity in

dex DROUGHT RAINSRAINS

W. greys

Reds

0

200

400

600

800

1000

1200

1980 1981 1982 1983 1984 1985

Time (year)

Past

ure

biom

ass

(kg/

ha) RAINS RAINSDROUGHT

0

20

40

60

80

100

0 200 400 600 800 1000


Food

inta

ke (g

/day

)

2. Functional response

-2.0

-1.5

-1.0

-0.5

0.0

0.5

0 200 400 600 800 1000


Rat

e of

incr

ease

(r p

.a.) 3. Numerical response

BIOENERGETICS MODELStochastic, rainfall driven

environment 3 explicit nonlinear energy

transfer functions -1200

-800

-400

0

400

800

0 200 400 600 800 1000

Vegetation (kg/ha)

Gro

wth

(kg/

ha/3

mo)

1. Plant growth response

increas ing ra infa ll

29

The The a prioria priori interactiveinteractive kangaroo grazing kangaroo grazing model Feedback loops & regulationmodel Feedback loops & regulation

Pasture(food)

kangaroos

Pasture intakerate

3. Numerical3. Numericalresponseresponse

Instantaneous rate ofkangaroo increase

2. Functional2. Functionalresponseresponse

1. Density- dependentpasture growth

PasturePasture-- herbivoreherbivorefeedbackfeedback

looploop

PasturePasturebiomassbiomassfeedback feedback

looploop

RAINFALL

100–year stochastic simulation The interactive kangaroo grazing model

PASTURE BIOMASS

0

200

400

600

800

BIO

MA

SS (k

g.ha

-1)

RAINFALL

0

50

100

150

200

250

RAIN

(mm

)

KANGAROO DENSITY

0.0

0.5

1.0

1.5

100-YEARS

DEN

SITY

(ha-

1)

Phase plane trajectory for kangaroos & food (vegetation)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 200 400 600 800

Pasture Biomass (kg.ha-1)

Dens

ity (n

os.h

a-1)

High

Low

30

•• PostPost--hochoc s imulations indicate two s table s tates . s imulations indicate two s table s tates .

Unstable equilibrium dynamics?

•• Demands reDemands re--examination of numerical response models .examination of numerical response models .

•• A prioriA priori hypothes is hypothes is -- “centripetality”“centripetality” -- the tendency the tendency towards a s ingle s table equilibrium point. towards a s ingle s table equilibrium point.

High dens ity entering drought

Low dens ity leaving drought

post-drought data excluded in 1987 analysis

Time trace: sugges ts two equilibria where r = 0

Hysteresis pattern of red kangaroo numerical response over a drought period

-1.0

-0.8

-0.5

-0.3

0.0

0.3

0.5

0 200 400 600 800 1000


Rat

e of

incr

ease

(r p

.a.)

-1.0

-0.8

-0.5

-0.3

0.0

0.3

0.5

0 200 400 600 800 1000


Rat

e of

incr

ease

(r p

.a.)

>

>

?

A priori model sugges ts one equlibrium point where r = 0 at V*

31

A two state system: a limit cycle in a variable environment?

Time trace of w. grey kangaroo rate of increase vs density(T- 6 months) over a drought period, 2 points where r =0

Non-drought to start of drought

Drought to break of drought

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

0 5 10 15 20 25 30 35

Density t - 6 months (nos.km-2)

Rat

e of

incr

ease

(r p

.a.)

>

>

>

S table limit cycle: rate of increase & dens ity

Numbers with T-1 time lag Numbers with no time lag

Time

Num

bers

Lagged Numbers (Nt-1)

Rat

e of

incr

ease

(r p

.a.)

>

<

Numbers (Nt)

Rat

e of

incr

ease

(r p

.a.)

>

>

32

0

200

400

600

800

1000

0 200 400 600 800 1000 1200

Vegatation (kg.ha-1)

Den

sity

inde

x

Isoc line numerical response – kangaroos vs food (pas ture biomass) 1980-1984

Time trace – s table limit cycle?

Red kangarooW. grey kangaroo

>

<

<

1980

1984

Trends in kangaroo density 1973 - 1984

0

10

20

30

40

1972 1974 1976 1978 1980 1982 1984 1986

Time (year)

kang

aroo

s / k

m2

Tidbinbilla Nature Reserve ACT, eas tern Aus tralia (1975 – 1991; quarterly counts = 17 yrs )

Eastern grey kangaroos in seasonal, temperate environments – stable limit cycle?

periodicity 4 - 6 years?

Mis s ing s eries300

400

500

600

700

800

900

1974 1979 1984 1989 1994

Time (year)

Abu

ndan

ce in

dex

33

KangarooKangaroo dynamics dynamics rere--vis itedvis ited

• Rangelands: binary states – droughts & non - droughts.

• Kangaroo dynamics different in each state.

• An “unstable” limit cycle in arid rangeleands.

• States locally stable but globally unstable.

• But stable limit cycle in temperate zone.

The numerical response & The numerical response & multiple limiting factorsmultiple limiting factors

• Introduced possums in New Zealand fores ts

Food + spacing behaviour indexed by ins tantaneous dens ity

34

Population sizeRa

te o

f inc

reas

e (r)

z = 1

z = 7

z = 3

Generalised logistic model

r = rm (1 – N/K) Z

Introduced possums in New ZealandIntroduced possums in New Zealand• New Zealand's number 1 vertebrate pes t

• Kiwis use logis tic population growth model;

• where DD recruitment &/or mortality is as sumed – no evidence

Time

Poss

um d

ensi

ty

Hinau fruit production (T)

B ody condition (fat & weight)

Winter survivalIn s itu yearling recruitment•Proportion females with young

•Timing births

Rate of increase

Possum dens ity

Possum Possum –– plant modelplant model

3 negative3 negativefeedback loopsfeedback loops

+/- Hinau fruit (T-1)

Other frugivores

35

Orongorongo Valley: trends in possum density & hinau seedfall (1966-1997; n = 31 yrs)

02468

101214161820

1965 1975 1985 1995

Time (year)

Poss

um d

ensi

ty (h

a-1) &

hi

nau

abun

danc

e

Hinau fruitPossums

r = food (extrins ic regulation)

r = dens ity(intrins ic regulation?)

Possum numerical response models

rposs = - 0.60 + 0.85 (1.0 - e 0.037FOOD)

(n = 50, P < 0.001, R2 = 40%)

rposs = 0.65 - 0.084 (DENSITY)(n= 31, P < 0.001, R2 = 35%)

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0 20 40 60 80 100 120

Food availability (hinau fruit index)

Rat

e in

crea

se (r

p.a

.)

rm = 0.25 p.a.

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0 2 4 6 8 10 12 14

Instantaneous density (nos.ha-1)

Rat

e in

crea

se (r

p.a

.)

36

-0.8

-0.6

-0.4

-0.2

0

0.2

0 25 50 75 100

Food a va ila b ility (h ina u)

Rate

of i

ncre

ase

(r p.

a.)

20 10

2

DENSIT Y

2. Possum numerical response: combined regulation model

1. Numerical response of hinau fruit: logis tic growth - possum dens ity

+

= 3. Possum - hinau fruit interaction

(B aylis s & Choquenot 1997)0 .0

5 .0

1 0 .0

1 5 .0

0 1 0 2 0 3 0 4 0 5 0

Y e a r

Poss

um d

ensi

ty (h

a-1)

&

hina

u in

dex

x 10

POSSUM

HINAU

-2

-1

0

1

2

0 25 50 75 100

P re vious ye a rs crop hina u

Rate

of i

ncre

ase

(r p.

a.)

2

4

10

DENSITY

(R2 = 68%)(R2 =69%)

rhin = logis tic growth – g(Dposs)

Possums & food – take home messages

Logistic model:

• OK for possums & bovine Tb

• Inappropriate for complex ecological processes

37

Pigs Pigs -- habitathabitat quality quality & food availability& food availability

• Take landscape ecology approach

• Multiple habitat related limiting factors

• Implications for habitat quality & food resources

How does “habitat quality” affect interaction between animal populations and their food resources?

FOOD FOOD AVAILABILITYAVAILABILITY

FORAGING FORAGING EFFICIENCY (dEFFICIENCY (d11))

FOOD OFFTAKEFOOD OFFTAKE

DEMOGRAPHYDEMOGRAPHY

DEMOGRAPHIC DEMOGRAPHIC EFFICIENCY (dEFFICIENCY (d22))

HABITAT QUALITYHABITAT QUALITY

( )Vd1

1e1ccintake food −−=

( )Vd2

2e1ca rincrease ofrate −−+−=

38

LandscapeLandscape quality influences “foraging efficiency” via quality influences “foraging efficiency” via little dlittle d1 1 in the interactive model in the interactive model

Pasture biomass

Functional response Observed relationshipObserved relationship

Numerical response

dd11

( )Vd1

1e1cc −−=

( )Vd2

2e1ca r −−+−=

Biomass conversion principle

Rate of food intake

Rat

e of

incr

ease

15 10 5 -5

Strip Distance (km)

NSW

Main channel of the Paroo RiverMain channel of the Paroo River

Pigs & habitat quality

39

ResultResult

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

-5 5 10 15

Kilometers west of the Paroo main channel

Obs

erve

d r

- pr

ojec

ted

r(+

95%

CI)

HypothesisHypothesis• Does foraging dis tance influence pig rate of increase

because of reduced foraging effic iency?

Summary of pig case study Summary of pig case study • Key importance of biomass convers ion principle & hence

Lotka-Volterra framework

• Landscape approach yields key ins ights into pig-resource interactions & population dynamics

• Specifically, behavioural tradeoffs & habitat quality (e.g. thermoregulation & foraging)

• Implications for metapopulation analys is & linking population processes to landscape scales .

• Multiple limiting factors supported.

40

The upshot The upshot

• Unders tanding population dynamics & managing wildlife are tightly coupled.

• Lotka-Volterra model & demographic numerical responses : bas ic building block for multi-factorial consumer-resource interactions .

• Need to experimentally tes t a priori-multiple-working-hypotheses .

• Great global effort required.

• Critica l for conservation, control & harves ting of wildlife.

THE ENDTHE END

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