Scaling up from functional response to numerical response in vertebrate herbivores. A.W. ILLIUS AND I.J. GORDON* Institute of Cell, Animal and Population Biology, University of Edinburgh, West Mains Rd, Edinburgh, EH9 3JT, Scotland and *Macaulay Land Use Research Institute, Craigiebuckler, Aberdeen, AB15 8QH, Scotland.
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SUMMARY Considerable progress has been made in the last decade towards understanding the relations between vertebrate herbivores and their food supply. Mechanistic approaches to analysing the constraints on food intake, and the consequences for population dynamics, are replacing the classical theoretical descriptions of predator-prey dynamics. The challenge of the former approach is to discover what our mechanistic understanding can reveal about process and pattern in plant-herbivore relations. The paper describes the modelling of the processes of food intake and diet selection, from the level of the individual bite, up to daily nutrient intake, metabolism, energy balance, reproduction and mortality, thus integrating the mechanisms underlying population dynamics. Two examples, of a temperate and a savanna grazing system, are used to show how far mechanistic modelling can be used to explain: the relationship between vegetation and herbivore abundance; the physiological basis of overcompensatory population dynamics; and the size-related divergence in herbivores' diets and in their consequent impact on vegetation.
INTRODUCTION In analysing trophic interactions and their population consequences, classical mathematical ecology aims to achieve a tractable analytical description of the phenomena of interest. This approach requires the introduction of the minimum number of parameters consistent with capturing the main properties of the system's behaviour, in order that the parameter space can be investigated analytically or graphically (eg Rosenzweig & McArthur 1963; Noy-Meir 1975). Simplified representations of ecological interactions are justified, not only by the clarity with which they may reveal each parameter's effects, but also in the absence of sufficient knowledge of underlying mechanisms to build more detailed and, hence, more realistic models. However, considerable progress has been made in the last few decades towards a mechanistic understanding of the relations between vertebrate herbivores and their food supply. Mechanistic approaches, addressing the physiological processes which relate food intake to its consequences for population dynamics, are now set to augment the classical theoretical descriptions of predator-prey dynamics. Instead of treating population dynamics explicitly, as for example in setting a parameter governing density dependence, mechanistic modelling defines the nature of plant-herbivore relations implicilty, allowing quantitative hypotheses to be made and tested. The challenges for the mechanistic approach are to discover what further can be revealed about processes and patterns in plant-herbivore relations; to show how far our mechanistic understanding can account for population and community phenomena, and to produce quantitative and prescriptive solutions to practical ecological problems. This paper sets out an approach to mechanistic modelling of mammalian herbivore population ecology, to show how the functional response can be scaled up to the numerical response. In order to substantiate the claim that such an approach is feasible, a detailed description of the modelling of the component processes is presented, using the St Kilda grazing system as an example. A second example, of a savanna system, is presented in less detail, to show how the approach can be generalised to other systems. The unmanaged population of feral Soay sheep on the island of Hirta, in the St Kilda archipelago, off NW Scotland, shows erratic population dynamics in a temperate
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environment. The sheep and their grazing system have been the subject of long-term study (Jewell, Milner & Morton Boyd 1974; Clutton-Brock et al 1991). Although climatic effects on sheep population dynamics may yet be revealed by longer-term study, the over-riding influence appears to be the low or absent density-dependence of fecundity, with the consequence that the sheep population periodically exceeds the island's winter carrying capacity. This results in over-compensatory mortality (Grenfell et al 1992): a sharplyfocussed decline in sheep numbers from around 1500 to 700 in late winter. The purpose of modelling this grazing system in terms of the physiological processes governing the flow of energy and N from the vegetation and through the sheep population is to test whether the system can be adequately represented in this way, and to examine the causes of overcompensatory mortality. In contrast to the relatively simple and temperate system of grass and sheep on Hirta, our second example is of tropical savanna grazing systems. Semi-arid savanna ecosystems are complex, in consisting of a range of vegetation and herbivore types, and are dominated by annual and seasonal variability in rainfall, which are the main source of variation in herbivore population size. A model of herbivore and vegetation dynamics is used to test whether secondary production can be predicted mechanistically from rainfall, and to examine whether our assumptions about the body-mass allometry of animal physiological properties can explain the observed patterns of animal and plant community dynamics. COMPONENT PROCESSES We are concerned here with describing the processes of food intake and diet selection, from the level of the individual bite, up to daily nutrient intake, metabolism, energy balance, reproduction and mortality, thus integrating the mechanisms underlying population dynamics of mammalian herbivores. These descriptions are then combined in a model which, in brief, calculates the daily flux of material and energy through compartments representing the vegetation and the sheep population. Further details of the model structure and of the vegetation sub-model are given in the Appendix.
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Herbage intake and nutrient supply. Daily nutrient intake is the product of intake rate while feeding, the nutrient content of the diet and the time spent feeding. Short-term nutrient intake rate is largely determined by the animal's physical responses to vegetation structure and abundance. The degree of selectivity exerted represents a trade-off between the nutrient content of the diet and the rate at which it can be eaten. The time spent feeding is driven by the animal's appetite, and is constrained by environmental factors such as daylength and weather, and is constrained by fatigue, the ability to utilize or store mutrients, and by digestive constraints (see Illius & Jessop 1996; Illius 1997). Intake rate. Both experimental and theoretical analyses of the functional response of mammalian herbivores show that bite mass is the variable exerting the greatest effect on intake rate (Hodgson 1985; Spalinger & Hobbs 1992). Grass swards consist of a threedimensional array of plant tissues, and bite mass varies with the volume of the sward which the animal can enclose in each bite and with the bulk density of the grazed horizon (Black & Kenney 1984; Illius & Gordon 1987; Burlison et al. 1991). Bite volume is determined by the horizontal area of the sward covered by each bite and the depth to which the incisors penetrate into the sward. Bite mass can be estimated by the model of Illius & Gordon (1987) from the animal's incisor arcade breadth and bite depth (governing bite volume) and from the height and biomass of the vegetation (governing the bulk density of the herbage enclosed in a bite). Incisor arcade breadth in Soay sheep is given by the allometric relationship: IB = aW0.245 mm, where a is 12.28 for females and 12.03 and males (r2=64; residual sd=1.6mm; Illius et al. 1995). Bite depth of Soays grazing Agrostis-Fescue swards was found to be closely related to sward surface height, H, by the following expression: BD=4.45+0.4H mm (r2=0.94, residual sd=0.4 mm; A.W. Illius et al., unpublished observations). The relationships between sward surface height and sward biomass for the three vegetation types were taken from Armstrong et al. (in press). The determinants of the rate of biting were greatly clarified by the work of Spalinger & Hobbs (1992), who recognised that bite rate could be limited either by the animal's need to search for its next bite or by its need to chew ingested herbage. They derived equations for the
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functional response under conditions where intake rate is limited by one of three processes: rate of encounter with cryptic food items (Process 1); rate of encounter with apparent items (Process 2); and rate at which food can be chewed and swallowed (Process 3). Given an animal with maximum foraging velocity Vmax, maximum eating rate Rmax, minimum handling time per bite in the absence of chewing h and width of search path w, foraging on plants offering bites at density D (per m2), and offering a bite of mass S, then the rate of biting B can, adapting Spalinger & Hobbs (1992) slightly, be written as: Process 1:
Vmax WD B1 = 1 + nVmax WD
(1)
Process 2:
Vmax D B2 = (1 + nVmax D )
(2)
Process 3:
Rmax B3 = ( S + Rmax h)
(3)
(
)
Intake rate is simply the product of B and S. In (1) and (2), n is a coefficient describing the time lost to future encounters whilst pausing to take a bite, and can probably be approximated by h. The first two equations describe rate of biting as limited by encounter rate, and distinguish cases where (1) potential bites can only be detected at close range, perhaps because obscured by dead herbage, from (2) those where bites can be detected at a distance. Process 1 applies where the average distance between plants is greater than the detection distance, ie where 1/√D > W. Equation (3) describes the case where encounters with large S are sufficiently frequent to cause bite rate to be limited by chewing rate. Thus, as bite mass becomes restricted, bite rate can increase up to a maximum, h. The actual rate of biting will be governed by whether finding or chewing bites is slower. If bite mass and biomass density are known, then intake rate and distance moved during foraging can be calculated. Vmax is an allometric funtion of animal weight W (kg) and is 0.5W0.13 m/s (Pennycuik 1979). Search path width, w can be assumed to scale with animal stature, and so with limb length, or with W0.25 (Alexander 1977). Shipley et al. (1994) determined Rmax to be 12.3W0.69 mg/s and h=0.9 s.
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Plant community selection and nutrient content of the diet. On each day, the vegetation type selected to be grazed was assumed to be that offering the highest energy intake rate. Within a community, diet composition is determined by the vertical distribution of biomass in each leaf compartment, and by the effect of bite depth on the relative amounts removed from each leaf age (Fig 1; see also Hodgson 1985). It was assumed that seasonal variation in nutrient content of the vegetation is primarily accounted for by the changing proportions of live and dead leaf. The organic matter digestibility of live (L1 - L3) and dead (D) Agrostis-Fescue leaf was estimated from the data of Armstrong et al. (1986) to be 0.77 and 0.44 respectively, with values of 0.68 and 0.41 for live and dead Molinia. The metabolizable energy (ME) content of digested organic matter is 15.6 MJ kg-1 (ARC 1980). The N contents of Agrostis-Fescue live and dead leaf were taken to be 24.5 and 10.6 g kg-1 with 17.5 and 8.0 g kg-1 for Molinia (Milner & Gwynne 1974; Armstrong et al. 1986). Quantification of the metabolism and utilization of energy and N follows ARC (1980) and IDPW (1993). Grazing time. Given the short-term rate of nutrient intake, daily intake is determined by the time spend grazing, which, assuming that animals seek to maximize their daily nutrient intake, will be the maximum allowed by environmental and other constraints. Additionally, digestive constraints may limit daily intake to the maximum daily turnover of digestive tract contents. Preliminary analysis of the diet quality selected by Soays suggested that digestive constraints are unlikely to apply, and this factor is therefore not included in the model. The setting of digestive constraints in cases where they might apply, such as in savanna ecosystems, is given in the next section. The settings of constraints on grazing time which are thought to be operative on Hirta are as follows: Soay sheep rarely graze at night (Grubb & Jewell 1974; Stevenson 1994); ruminants appear to have an upper limit on the number of bites they will take in a day (Hodgson 1985); and intake will ultimately be constrained by the animal's ability to utilize or dispose of nutrients (see Illius & Jessop 1996). It was thus assumed that the grazing time of Soays is constrained to the minimum of: (a) the hours of daylight; (b) the time taken for 40 000 bites (estimated as the upper limit in a small-bodied ruminant: see Illius and Gordon 1987); (c) the time required to meet the animal's daily energy
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requirements (which are described below). Rutting also reduces grazing time in rams, and this effect of male reproductive investment is discussed later. The resulting predictions of grazing time are compared with observations of Grubb & Jewell (1974) in Fig. 2. Digestive constraints. Daily intake of foods of low digestibility is constrained by the low rates of ruminal digestion and passage (eg Laredo and Minson 1973). The model of Illius & Gordon (1991; 1992) quantified the relationship between food and animal characteristics and allows prediction of the maximal intake of a food in relation to its digestibility and the animal's size. Model output from a range of these inputs was summarised by regression to give daily intake (kg DM): I = 0.1D1.1 M 0.81Ug (grass)
( browse)
I = 0.15 D1.3 M 0.78Ug
(4)
( )
where Ug = M A
0.75
The term Ug scales gut capacity to body mass in immature animals, accounting for the relatively larger digesta load of immature (but weaned) animals (unpublished data of B.G. Lowman). The prediction equations agree well with published observations at low digestibility, when digestive constraints would be expected to apply, but overestimates intake by up to 25% on higher digestibility diets, which experimental animals may not choose to eat to fill (see Illius 1997). Energy expenditure. The first call on the animal's energy intake is to supply its requirements for basal metabolism, activity and thermoregulation. Maintenance and activity. Thermoneutral resting metabolism is 0.3WA-0.27 MJ d-1 (Taylor, Turner & Young 1981), where W is body mass and A is the mature mass of 32 and 24 kg for males and female Soay sheep (McClelland, Bonaiti & Taylor 1976). Energy costs of foraging arise from maintenance of posture (4.22 W0.735 Js-1 of foraging), from travel (15.8W0.589 Jm-1; Taylor, Heglund & Maloiy 1982; Table 2) and from eating (1.54 times the resting
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metabolic rate; Graham 1964). Sheep were assumed to move 500 m during foraging and 500 m commuting between feeding and sheltering sites (I.R. Stevenson, pers. comm.). Thermoregulation. Sheep, being warmer than their environment on St Kilda, continuously lose heat. To maintain body temperature, heat generated by maintenance energy expenditure, activity, and the waste heat of metabolism from growth and reproduction must either exceed the heat loss to the environment, or additional energy expenditure must be incurred to make up the deficit. Heat production during metabolism is calculated according to ARC (1980). Heat loss to the environment was modelled according to the electrical analogue model (McArthur 1987) in which heat flows through the series of resistances posed by skin, coat and boundary layer. Radiative exchanges are considered separately. Parameter values were taken from Campbell, McArthur & Monteith (1980), McArthur and Monteith (1980a, b), McArthur (1987), Monteith & Unsworth (1990) and Stevenson (1994). Environmental temperature and windspeed were taken to be the daytime mean temperature and 5 ms-1 while foraging, and 2o C and 1.25 ms-1 while sheltering (Campbell 1974; Stevenson 1994). Webb & King (1984) showed that wetting an animal's coat reduces its resistance to sensible heat flux by a factor of 2, so the effect of the high rainfall on St Kilda was modelled by assuming that half of the coat of sheep is wet during half the period foraging. The surface area and trunk diameter of Soay sheep ranging in size from juveniles to adults was described by the allometric functions: 0.14W0.565 m2 and 0.085W0.376 m, respectively (n=14, r2=0.98 for both, residual cv=6.0, 4.0%; A.W. Illius & G.A. Lincoln, unpublished data). Energy deficits are met by metabolism of fat and protein, yielding respectively 39.3 and 13.5 MJ kg-1 (Blaxter 1989), after allowing for energy lost in urea and 4.6 MJ lost as heat of synthesis of urea. The protein proportion of energy loss was estimated from the data collated by Reeds and Fuller (1983) as (0.12-0.19M)/(1-M) for M
