## Introduction

Environmental temperatures and topographic diversity are primary determinants of a solitary species’ distribution. Interactions of two or more species along an environmental gradient modify responses to these factors (Heller & Gates 1971; Heller 1971; MacArthur 1972; Lawton 1993; Gaston 2003). Geographic range boundaries often coincide with identifiable climatic conditions (Muth 1980; Rogers & Randolph 1986; Caughley *et al*. 1987; Root 1988; Taulman & Robbins 1996). However, establishing whether climate causes observed correlations is difficult (Huntley 1994; Sykes, Prentice & Cramer 1996; Hill, Thomas & Huntley 1999; Gioia & Pigott 2000). Models that use environmental temperatures to predict species distributions of birds (Root 1988), butterflies (Kingsolver & Watt 1983; Bryant, Thomas & Bale 2002), mammals (Porter *et al*. 1994), and reptiles and amphibians (Tracy 1982; Porter & Tracy 1983; Porter 1989; Huey 1991; Grant & Porter 1992) are becoming increasingly mechanistic. However, this study is one of the first to couple energetic and population dynamic models to project species abundances on to a landscape.

Temperature-dependent metabolism has gained attention as a means to predict population growth rates (Savage *et al*. 2004) and species richness or distributions (Porter *et al*. 2000; Allen, Gillooly & Brown 2002). We use simple *Anolis* lizard communities on the Lesser Antilles islands to investigate how thermal energetics influence species distributions. Integrating individual energetic and population dynamic models is desirable but often prevented by community complexity and insufficient knowledge of life histories (Lawton 1991; Chown, Gaston & Robinson 2004).

How species interactions affect elevation trends in abundance has not been demonstrated (although the influence on distributions has been well demonstrated; Heyer 1967; Sullivan 1981; Conroy 1999). Environmental temperature interacts with species interactions and dispersal to determine species distributions, adding difficulty to isolating how each factor influences distributions in complex communities (Pearson & Dawson 2003). Our bioenergetic null-model provides a spatially explicit demonstration of how species interactions alter landscape-scale abundance patterns.

Our model of temperature-dependent foraging energetics predicts lizard abundance trends along an elevation gradient. We compare model outcomes with empirical abundance patterns on Lesser Antilles islands with and without species interactions and with differential topography. At the landscape scale, we hypothesize that abundance will decline with elevation due to thermal constraints on locomotion and metabolism. Further, we hypothesize that competing species partition the available energetic resources along the elevation gradient to enable coexistence.

### lesser antillean *anolis*model system

The simple anole communities on the Lesser Antilles islands enable understanding how competition and environmental temperatures interact to determine abundance distributions along elevation gradients. Each of the 27 Lesser Antilles islands contains either one anole species or a size-dimorphic pair of species (Creer *et al*. 2001; Schneider, Losos & deQueiroz 2001). Most species on the sympatric islands are separated in size by a factor consistent with Hutchinson's Rule, while most of those on one-species islands are of intermediate size (Schoener 1970). The degree to which sympatric anoles compete for resources is related to niche overlap in body size and perch position (Pacala & Roughgarden 1982) due to overlapping insectivorous diets (Schoener & Gorman 1968; Rummel & Roughgarden 1985).

Lizard survival varies inversely with island size and the number of bird species on small Bahamian islands (Schoener & Schoener 1978). As the species richness of birds, including predators, tends to increase with island size and with proximity to the equator (Mclaughlin & Roughgarden 1989; Ricklefs & Lovette 1999), we would expect increasing predation with decreasing latitude. However, we observe a greater abundance of insects as well as lizards on the southern islands. This suggests that environmental temperature and prey abundance exert a greater influence on landscape-scale lizard abundance than predation on the study islands.

The northern and southern Lesser Antilles Islands are topographically distinct and their anole communities differ in phylogenetic origin. Northern anole species are related to anoles in Puerto Rico and western North America, whereas southern species have western South American affinities (Poe 2004). While the northern Lesser Antilles islands are predominately low elevation with steep mountains in the centre of the island, the southern islands tend to be more uniformly mountainous (Roughgarden 1995). By comparing altitudinal abundance patterns between the island groups, we can address whether the evenness of the distribution of area into elevation classes influences abundance patterns. The Lesser Antilles enable isolating the influence of species interactions and topography on habitat partitioning.

We examine these two factors – species interactions and topography – by comparing model outcomes to empirical abundance trends on four study islands. The northern one- and two-species study islands are Montserrat and St Kitts, respectively. The southern one- and two-species study islands are St Lucia and Grenada, respectively. On St Kitts, the smaller and larger species are *A. schwartzi*[mean snout vent length, SVL = 53·6 mm] and *A. bimaculatus* (78·5 mm), respectively. On Grenada, the smaller and larger species are *A. aeneus* (66 mm) and *A. richardi* (101 mm), respectively. On Montserrat and St Lucia, the solitary anole species are *A. lividus* (61 mm) and *A. luciae* (77 mm), respectively (Schoener 1970). Phylogenetic relatedness (Poe 2004; but see Losos *et al*. 2003) and similar habitat use (i.e. trunk-ground ecomorphs, Williams 1972) enable comparing habitat use by species on the one- and two-species islands. The island pairs have comparable topographic reliefs, vegetation types, and areas (with the two-species islands being somewhat larger than their one-species counterparts).

Anoles in the Lesser Antilles partition habitat at both local (within habitats) and landscape (between habitats along the elevation gradient) scales (Ricklefs & Schluter 1993). We focus here on landscape-scale habitat partitioning. A greater degree of broad scale habitat partitioning occurs in the southern islands (Roughgarden, Heckel & Fuentes 1983).

Among northern two-species islands, the smaller species numerically dominates everywhere on islands with size dimorphism and in higher-elevation forests on islands with little size dimorphism. A contrasting distribution pattern occurs on the southern islands, where the smaller lizard is restricted to open sites near sea level and the large anole is dominant in higher-elevation forests. Roughgarden *et al*. (1983) hypothesize that the more mountainous topography of the southern islands has resulted in a greater degree of landscape-scale habitat partitioning. We evaluate this hypothesis.

### the model

Anole distributions are predicted using a behaviour-based model of anole population dynamics that predicts equilibrium abundance from foraging energetics. Foraging energetics provides a good basis for predicting lizard distributions as ectothermic lizards are strongly influenced by ambient temperatures (Spotila & Standora 1985) and foraging accounts for a large proportion of daily energy expenditures (Bennett & Gorman 1979). At the least, foraging energetics provides a sensible null model for addressing the influence of other factors such as species interactions and resource requirements (Muth 1980). The basic model, which we modify to incorporate thermal physiology by making the parameters temperature-dependent, is presented by Roughgarden (1997).

Briefly, the model assumes an individual anole is an energy maximizing sit-and-wait predator whose foraging radius is limited by lizard density. The model is an animal counterpart for the neighbourhood model, where plants interact with their adjacent neighbours (Pacala & Silander 1985). The model assumes that the anoles forage on a linear interval (*r* to *r* + *dr*). This simplifies spatial dynamics and produces model predictions that are comparable with empirical transect counts. This approach ignores the vertical distance an anole travels from its tree perch to its prey. The approximation is reasonable because anoles have similar intraspecific perch-height preferences (Schoener & Gorman 1968) and it is relatively seldom that multiple lizards are observed in the same tree at different heights.

The number of prey encountered in an interval is *p*(*r*)*dr* = *adr*, where *a* is prey density (insects per metre per second). If the cut-off distance, the maximum distance an anole is willing to run to obtain prey, is *r*_{s}, the average waiting time to encounter an insect is

and the average pursuit time, to the insect and return, is

where *v* in the forager's sprint velocity. The foraging energetic yield per unit time, *E*(*r*), of foraging within a radius, *r*, is derived as the energetic input less the energetic cost divided by the total foraging time:

where *e*_{i} is the energy per insect; *e*_{w} and *e*_{p} is the energy per unit time expended waiting and pursuing, respectively, and *t*_{w} and *t*_{p} is time expended waiting and pursuing, respectively.

The handling time is assumed to be minimal and included in the pursuit time. The energetic cost of handling is included in the assimilation efficiency, as outlined below. At low lizard densities, the optimal cut-off distance for a solitary forager is the *r*_{s} that maximizes *E*(*r*_{s}):

Density dependence is introduced when crowding forces the territory size to be less than the *r*_{s} for solitary anoles. The model assumes that pairs of foragers are distributed facing each other along a circle of length *L* and that foragers equally partition foraging space (see Roughgarden 1997) for assumption rationale). The interindividual distance, *d*, for *N* foragers is then *L/*(*N*/2), where *L* is the standard unit of length of a transect (1000 m).

To extend this model of individual foraging energetics to the scale of population dynamics, the change in population per unit time (production function) is calculated as the product of the population growth rate, based simply on birth minus death, and the population size, *N*, as follows:

*N*= (

*bE*(

*r*) −

*v*)

*N*( eqn 5 )

where µ represents mortality and the reproductive cost of metabolism while not foraging and *b* is the reproductive rate per unit net energetic yield. *v* and *b* are lumped parameters,

*v*= µ +

*m*(τ − τ

*)*

_{f}*e*( eqn 6 )

_{r}*b*=

*m*τ

_{f}( eqn 7 )

where µ is the daily mortality, τ is the day length (24 * 60 * 60 s), τ_{f} is the average daily foraging duration (seconds), *e*_{r} is the average metabolic rate while not foraging, and *m* is the quantity of eggs produced per joule of foraging yield times the probability of surviving to adulthood.

When the average distance between lizards on the transect is less than the energetically optimal foraging radius, *r*_{s}, each crowded lizard intakes less energy from foraging. This form of density dependence enables solving for the equilibrium population size. Explicit forms of the production function are used to solve for equilibrium population size (carrying capacity, *K*, where *r* = 0) and the initial rate of population growth (the intrinsic rate of population increase, *r*_{0}):

### incorporating temperature dependence

We make the model temperature-dependent by scaling running velocity and metabolic rates with temperature. Changes in lizard pursuit speed influence the energetically optimal foraging radius, the proportion of escaping insect prey, and the duration of foraging. Model predictions for population dynamics along the elevation gradient are contingent on whether the anoles are able to thermoregulate behaviourally along the elevation gradient.

Previous research on Caribbean anoles suggests that lizard habitat choice provides thermoregulation along elevation gradients (Huey & Webster 1976; Hertz 1979; Huey, Hertz & Sinervo 2003). While body temperature does drop somewhat with elevation, the drop is less than the corresponding decline in environmental temperature (Hertz, Huey & Stevenson 1993; Huey *et al*. 2003). With increasing altitude, anoles perch in more open habitats (Rand 1964; Adolph 1990) and shift their activity times (Hertz 1981; Hertz & Huey 1981; Grant & Dunham 1990; Hertz 1992). Hertz & Huey (1981) show that, among anole populations along an altitudinal gradient, the degree of thermoregulation increases with increasing altitude. At the highest elevations, the slope of the regression of body temperature on ambient temperature is substantially less than 1 (Hertz & Huey 1981).

If we assume that active lizards are able to choose habitats to maintain their preferred body temperature, then metabolic rate and pursuit velocity are constant over the elevation gradient. In our model, a primary determinant of population dynamics is the time period during which a lizard is able to forage. When parameterizing the model with empirically observed environmental temperatures, the temperatures are never sufficiently warm to prevent midday foraging. Low temperatures at high elevation are the exclusive limitation to foraging duration in our model. Hence, populations of behaviourally thermoregulating lizards decline with elevation as their foraging duration declines. Conversely, metabolic rates and running speeds will vary along the elevation gradient and will influence abundance predictions if we do not assume that lizard habitat choice buffers changes in environmental temperature.

We examined the implications of behavioural thermoregulation along the altitudinal gradient by examining model outcomes under assumptions of both behavioural thermoregulation and lack thereof. We only present predictions derived by assuming that lizards behaviourally thermoregulate to the temperature at which they reach maximum velocity, as empirical studies suggest that behavioural thermoregulation is quite effective and the model yields more realistic altitudinal patterns. The assumption may be approximate as lizards might thermoregulate to a temperature slightly below their optimal performance temperature (R. Huey, personal communication). We assume that the scale of dispersal relative to island size prevents intraspecific differences in thermal physiology along the elevation gradient (VanBerkum 1986; Sultan & Spencer 2002).

During periods of inactivity, we make the simplifying assumption that lizards are at their optimal body temperature. This assumption has a limited influence on energetics as the active metabolic rate is an order of magnitude greater than the inactive metabolic rate. We do not account for the thermal dependence of other metabolic parameters such digestive rate (Vandamme, Bauwens & Verheyen 1991). Rates of digestion drop substantially as the body temperatures of inactive lizards declines with elevation (Angilletta 2001; McConnachie & Alexander 2004).

### model parameterizations

#### Lizard parameters

Relations used to parameterize the bioenergetic model and their sources are presented in Table 1. Lizard mass is a power-law function of lizard SVL (Pough 1980; Stamps, Losos & Andrews 1997). Lizard metabolic rate has been summarized at different body temperatures as a power-law function of mass, with the power-law exponent varying linearly with temperature (Bennett & Dawson 1976; Bennett 1982). Lizard maximum sprint speed is estimated as a power-law function of length (Huey & Hertz 1982; Losos 1990). We estimate lizard pursuit velocity as 70% of maximum sprint speed, corresponding to field observations by Irschick & Losos (1998).

Relation | Source | |
---|---|---|

Lizard parameters | ||

Fresh mass, M(g) from snout-vent length, SVL (mm) | M = 3·1 * 10^{−5}SVL^{2·98} | Pough (1980) |

Resting metabolic rate from mass, M, and temperature, T (°C) (O_{2} consumption, mL O_{2} h^{−1}) | M R_{resting} = 0·28M^{0·023T−0·94}, r^{2} = 0·57 | Bennett & Dawson (1976); Bennett (1982) |

Maximum metabolic rate from mass, M, and temperature, T (°C) (O_{2} consumption, mL O_{2} h^{−1}) | M R_{max} = 3·75M^{0·015T−0·89}, r^{2} = 0·72 | Bennett & Dawson (1976); Bennett (1982) |

Metabolic rate conversion (mL O_{2} h^{−1} to J s^{−1}) | 20·11 | Bennett & Gorman (1979) |

Maximal sprint speed (m s^{−1}) from SVL (mm) | v_{max} = 1·54SVL^{0·30}, r^{2} = 0·537 | Huey & Hertz (1982); Losos (1990) |

Ratio of pursuit speed to maximum sprint speed | 0·7 | Irschick & Losos (1998) |

Prey size (mm) from lizard SVL (mm) | L_{insect} = 0·042SVL + 0·15 | Schoener & Gorman (1968) |

Proportion insect that can be assimilated | 0·75 (approximate) | Kitchell & Windell (1972) |

Assimilation efficiency (varied insectivorous diets) | 0·75 (approximate) | Kitchell & Windell (1972) |

Foraging window parameters | ||

Temperature, T (°C), from elevation, × (m), and hour, h | T = 19·4 − 0·0065x + 1·58h − 0·06h^{2} | Derived from NOAA NWS data |

Temperature, T_{maxv} (°C), of maximum velocity from panting temperature, T_{pant} (°C) | T_{maxv} = 0·8T_{pant} | Huey (1983); VanBerkum (1988) |

Temperature, T_{b80} (°C) of 80% of maximum sprint speed from T_{maxv} (°C) | T_{b80} = T_{maxv} − 4·6 | VanBerkum (1988) |

Proportion of activity window that an individual lizard is active | 0·75 (approximate) | Pacala (1982) |

Proportion of activity time spent foraging | 0·75 (approximate) | Bennett & Gorman (1979) |

Prey parameters | ||

Insect abundance, a(insects/(ms)) | 0·016 ± 0·003 | Unpublished data |

Prey dry mass, M (mg) from length, L (mm) | Schoener (1977) | |

Prey energy content J/mg_{drymass} | 23·85 | Reichle (1971); Andrews & Asato (1977) |

#### Foraging window parameters

We calculate the foraging window using empirically derived diurnal trends for air temperature (NOAA NWS). The diurnal temperature trend was fit, using 2 years of hourly temperature data collected at sea level on St Lucia, with a parabola. Ocean buffering maintains fairly warm night-time temperatures, resulting in a parabola fitting the diurnal temperature trend better than a sine wave. The shape of the diurnal temperature trend at higher elevations is assumed to approximate that at sea level. Higher altitudes tend to experience larger diurnal temperature fluctuations (Rundel 1994). However, this trend is likely moderated by ocean buffering. Assuming that lizards maintain optimal body temperatures while inactive eliminates the need to account for the altitudinal trend in night-time temperature fluctuation. We use the wet adiabatic lapse rate (0·65°C per 100 m) to shift the diurnal temperate trend with elevation (Table 1). The environmental temperature (measured by a grey body or biophysical calculation; Roughgarden, Porter & Heckel 1981; Shine & Kearney 2001), is a better predictor of lizard body temperature than air temperature. However, we use air temperature data in this study as extensive time series data are available.

Once a site becomes sufficiently warm, lizard velocity increases linearly as a function of temperature to the lizard's maximum velocity. The velocity then remains near maximum before declining rapidly when approaching the upper critical temperature (VanBerkum 1986; Hertz, Huey & Garland 1988; Bennett 1990; Irschick & Losos 1998). We assume that lizards thermoregulate to the temperature at which they reach maximum velocity, which consistently occurs at 80% of their critical upper temperature. The proportional temperature at which maximum velocity occurs is consistent for Costa Rican and Puerto Rican anoles (mean ± SE = 0·80 ± 0·010; *n* = 7 species; Huey 1983; VanBerkum 1986, 1988).

We empirically estimated the upper critical temperature – the temperature at which lizards pant and are too warm for activity – by holding the lizards, caught at sea level, in the sun until they begin to pant (Hertz & Nevo 1981; Roughgarden 1995; Fig. 1). On the northern two-species island, the smaller species reaches its thermal maximum at a lower temperature, consistent with its lesser thermal inertia. In contrast, on the southern two-species island, the smaller species has a significantly greater panting temperature.

Foraging is restricted to periods of sunlight when the environmental temperature is sufficiently high for lizards to forage effectively. At what temperature can a lizard begin foraging? We conducted morning counts of active lizards. We confirmed that foraging begins at temperatures above the lowest temperature at which lizards are able to move (unpublished data). However, variability prevented our deriving a conclusive relation describing how temperature and time of day determine when lizards begin foraging. We assume that lizards forage when they can run at least 80% of their maximum sprint speed. The temperature breadth within which Costa Rican anoles can sprint at least 80% of capacity is consistent (mean ± SE = 9·2 ± 0·48; *n* = 7; VanBerkum 1988). We thus assume that lizards initiate foraging at temperatures 4·6°C below the temperate of maximum velocity.

We assume that lizards are active for three-quarters of the activity window (Pacala 1982) and that three-quarters of the activity time is spent foraging (Bennett & Gorman 1979). We account for the metabolic cost (assuming the resting metabolic rate) for the portion of the activity window during which the lizard is not foraging.

#### Prey parameters

We account for decreasing proportional prey capture with decreasing lizard sprint speed by defining *f* as the probability per second that an insect moves from its initial location (Roughgarden 1995). We assume that the probability of the insect remaining stationary for *t* or more seconds is distributed exponentially with parameter *f*, *e*^{−ft} (adapted from Roughgarden 1995). The successful fraction, *F* of all pursuits within the cut-off distance, *r*_{s}, where *r* is the distance from a given insect to the lizard and *v* is lizard velocity, is

We reduce the energy input per insect by the proportion of pursued insect that are not caught. *F* is a free parameter as empirical data are unavailable. We estimate *f* to be 0·5 by solving for when the lizard foraging cut-off radius is 2 m, an empirically realistic value.

We estimate prey length as a function of lizard SVL using data from the intermediate sized anole, *A. roquet* on Martinique (Schoener & Gorman 1968). Although the data choice does not qualitatively change model outcomes, we chose their Martinique data over data for Grenada. The Grenada data predict that large lizards eat larger prey than characteristically available in some areas of the Lesser Antilles (Rummel & Roughgarden 1985). Conversions from insect head length to dry mass (Schoener 1977), energy content (Reichle 1971; Andrews & Asato 1977) and ultimately assimilated energy (Kitchell & Windell 1972) are provided in Table 1.

#### Insect abundance

We assume that insect abundance is log-normally distributed with mean 1·15 mm and variance 2 mm, corresponding to insect data for a tropical wet forest (Schoener & Gorman 1968). We determine the abundance of a specified prey size by multiplying the overall insect abundance by the probability density function for the specified prey size. As we lack data on the breadth of prey sizes consumed, we assume that the lizards are picky and calculate abundance based on a single prey size.

We empirically measured insect abundance along the elevation gradient on each island (methods in Appendix I). Elevation is not a significant determinant of insect abundance on any of the study islands (maximum likelihood mixed effect model with site as a random effect, *F*-test, *P* > 0·1 for each island). Among the northern and southern two-species islands, the southern island has significantly more insects (*F*_{[8,231]} = 25·67, *P <* 0·001, ancova with elevation as the covariate). Although the difference is not significant, the southern one-species also tends to have more insects than the northern island (*F*_{[7,180]} = 0·67, *P <* 0·5, ancova with elevation as the covariate). The model was parameterized with island-specific insect abundances [St Kitts: 0·009 ± 0·0006, *n* = 96; Grenada: 0·037 ± 0·006, *n* = 80; Montserrat: 0·008 ± 0·0008, *n* = 80; St Lucia: 0·011 ± 0·0007, *n* = 80 *insects*/(*m***s*) (mean ± SE)].

#### Model sensitivity

The most uncertain model parameter is insect abundance. Hence, we use the 95% confidence intervals of empirically measured insect abundance to depict the sensitivity of the model outcomes. This may underestimate the actual variance as insect measurements were conducted in a single day on each island. However, it does reveal how the relative abundance patterns of lizards respond to insect abundance. We also examine the model sensitivity to other less-certain parameters. Varying the proportion of maximum velocity at which a lizard begins foraging does not alter the predicted relative abundance patterns. Decreasing the proportion of maximum velocity at which foraging initiates primarily increases the maximum elevation at which the lizards are able to persist. Insect flightiness, *f*, is a free parameter. Varying *f* does not influence relative abundance patterns but does shift the absolute abundance trends. Increasing the insect flightiness decrements the energetic yield of foraging. Decreased energetic yield decreases both the predicted abundance and the maximum elevation at which lizards are able to persist. All other parameters are derived from empirically well established lizard morphology and physiology.