the bioenergetic model
The model assumes an individual anole is an energy-maximizing sit-and-wait predator, the foraging radius of which is limited by lizard density. Details of the basic model are provided by Roughgarden (1997). The model is extended to include temperature dependence by Buckley & Roughgarden (2005). The model is an animal counterpart for the neighbourhood model, in which plants interact with their adjacent neighbours (Pacala & Silander 1985). Essentially, the model balances energetic input from foraging with energetic costs associated with metabolism and reproduction. Lizards are assumed to forage on a linear interval, which simplifies the spatial dynamics and produces model predictions that are comparable to empirical transect counts. The assumption is biologically reasonable because lizards tend to maintain perch positions that limit prey sighting to a linear band.
We model lizards as optimal foragers that maximize the energetic yield per unit time. The foraging energetic yield, E(d), of foraging within a radius, d, is derived as the energetic input less the energetic cost divided by the total foraging time:
( eqn 1)
where ei is the energy per insect; ew and ep are energy per unit time expended waiting and pursuing, respectively; and tw and tp are time expended waiting and pursuing, respectively. The pursuit and waiting times are a function of prey density, a (insects m−1 s−2), and lizard velocity, v (m s−1) (tw = 1/ad and tp = d/v; Roughgarden 1997; Buckley & Roughgarden 2005). The handling time is assumed to be minimal and included in the pursuit time. The energetic cost of handling is accounted for by discounting the energetic content of each insect by the assimilation efficiency (Buckley & Roughgarden 2005). We convert the insect catch (m−2 s−1) to number of insects (m−1 s−1) by assuming that each lizard forages within 0·5 m to each side of the linear transect.
At low densities, the solitary foraging radius, ds, is that which optimizes E(d). Density dependence is introduced when crowding forces the territory size to be less than the energetically optimal d for solitary anoles, and thus reduces the energetic yield from foraging for each lizard. A specified transect length, L, is partitioned between N foragers (Roughgarden 1997). The model of individual foraging energetics is extended to population dynamics by calculating the change in population per unit time (production function) as the product of the population growth rate, based simply on birth minus death, and the population size, N, as follows:
ΔN = [bE(d) − λ]N (eqn 2)
where λ represents mortality and the reproductive cost of metabolism while not foraging, and b is the reproductive rate per unit net energetic yield. All density dependence is included in the expression for E(d), which can be substituted into the production function. As the foraging energetic yield is dependent on population size, N, we can explicitly solve for equilibrium population size (carrying capacity, K, where the population growth rate equals 0, bE(d) − λ = 0 and the initial rate of population growth (intrinsic rate of population increase, r0):
( eqn 3) ( eqn 4)
where tf is the duration of foraging; L = 1000 m; a = 0·037 ± 0·13 insects m−1 s−1 (mean ± 95% CI); b = mtf, λ = µ + m(24 × 60 × 60 − tf)ew. For A. aeneus, v = 1·33 m s−1; ep = 0·15 J s−1; ew = 0·008 J s−1; ei = 3·53 J. For A. richardi, v = 1·51 m s−1; ep = 0·15 J s−1; ew = 0·02 J s−1; ei = 9·78 J. The daily mortality rate, µ, is assumed to be 1/365 day−1. The parameter m is the quantity of eggs produced per joule × the probability of surviving to adulthood, and is assumed to be 1/e × 0·0001 eggs J−1.
Model parameterizations are detailed by Buckley & Roughgarden (2005). All parameters except tf are assumed to remain constant with respect to elevation. Briefly, from lizard length (Schoener 1970), we use empirically well established relations to derive lizard mass (Pough 1980); resting and maximum metabolic rate (Bennett & Dawson 1976; Bennett 1982); maximum sprint speed (Huey & Hertz 1982; Losos 1990; Irschick & Losos 1998); and prey size (Schoener & Gorman 1968). The energetic yield of each prey item, ei, is discounted by the capture rate. 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 (f = 0·5, Roughgarden 1995; Buckley & Roughgarden 2005). We assume that the probability of the insect remaining stationary is distributed exponentially to solve for the capture rate. The influence of small-scale microclimate variation and thermal physiological influences on processes such as digestion were omitted to maintain model simplicity.
The qualitative predictions of the bioenergetic model are robust to parameterizations other than body size, thermal constraints and prey abundance (Buckley & Roughgarden 2005). The most uncertain model parameter is insect abundance, which we measured empirically (Buckley & Roughgarden 2005). We use the 95% CIs of insect abundance to depict the sensitivity of the model outcomes on Grenada. For the model fitting, we parameterize the model with the lower 95% CI for insect abundance on Grenada, as it yields more empirically realistic abundances. This parameterization does not influence the shape of the abundance trend or relative species abundances. In addition to the insect abundance data we collected for the islands of Petit Bateau and Carriacou, we use data from Roughgarden & Fuentes (1977). The estimation methods were equivalent, with the exception of manufacturer changes in the sticky collecting substance. We determine the abundance of a specified prey size by multiplying the overall insect abundance by the probability density function using an abundance–size relationship for a wet tropical forest (Schoener & Gorman 1968).
Temperature dependence is incorporated by calculating the duration of foraging, tf, as the period during which the environmental temperature falls within each lizard's functional temperature range. We used hourly sea-level environmental temperature data during the sampling season (NOAA National Weather Service) and the wet adiabatic lapse rate (0·65 °C/100 m) to derive a temperature trend as a function of elevation and hour. The trend was fitted to 2 years’ hourly temperature data collected at sea level on St Lucia, which is just north of the Grenadines. The extremes of the critical temperate range were measured empirically as the temperature at which the lizard is too cold to roll over and the temperature at which the lizard begins to pant (A. aeneus = [22·6 °C, 39·2 °C]; A. richardi= [22·6 °C, 37·1 °C], Buckley & Roughgarden 2005). This critical temperature range was measured on the island of Grenada and applied to the Grenadines. From the critical temperature range, we derive the temperature at which lizards can run at maximum velocity and the temperature at which it is sufficiently warm to initiate foraging (VanBerkum 1988).
Previous research on Caribbean anoles suggests that lizard habitat choice provides at least partial thermoregulation along elevation gradients (Grant & Dunham 1990; Hertz & Huey 1992; Huey, Hertz & Sinervo 2003). Accordingly we assume that, once the environmental temperature falls within the lizard's function temperature range, the lizard behaviourally thermoregulates to the temperature at which it reaches maximum velocity (VanBerkum 1986; Bennett 1990; Irschick & Losos 1998). Resting and maximal metabolic rates are calculated as a function of temperature and mass at this optimal performance temperature (Buckley & Roughgarden 2005). We assume the scale of dispersal relative to island size prevents intraspecific differences in thermal physiology along the elevation gradient (VanBerkum 1986; Sultan & Spencer 2002).
incorporating interactions in the bioenergetic model
We use Lotka–Volterra approximations to the discrete time-logistic growth equations to simulate competition. The equilibrium abundances, N1 and N2, are found from simultaneously solving two linear equations:
N1 + β12N2 = K1; β21N1 + N2 = K2( eqn 5)
where Ki is the carrying capacity of species i and βij is the competition coefficient for the effect of species j against species i. The Kis are obtained as equilibrium solutions of the single-species production function, parameterized separately for each species. The βijs are treated as phenomenological coefficients, the values of which are not (yet) known empirically. The implications of various hypothetical values of βij are explored in the simulations. We solve for equilibrium population sizes (N1 and N2) in the presence of competition. As the carrying capacity and intrinsic rate of increase varies spatially in our model, we are unable to analyse the dynamics of the competition model to make coexistence predictions for the Grenadines. Rather, we calculate the equilibrium abundance for each pixel given a competition coefficient and sum up the predicted abundance over the pixels.
We examine how varying the form and strength of competition influences the predicted temperature-based abundance patterns. We explore the following forms of competition and combinations thereof: no competition; constant competition; linearly varying competition along the elevation gradient; and competition varying exponentially along the elevation gradient. For each form of competition, we solve for the values of β12 and β21 that minimize the sum of squares summed over both species along the elevation gradient. For linear competition, we include the approximate maximum elevation predicted by the bioenergetic model (800 m) in the competition coefficient (β12 = c1 − (c2/800)x, where c1 and c2 are constants). The fit of the bioenergetic model with competition to empirical lizard abundance on Grenada was compared with that of the null model (no effects) by anova. The correlation coefficient, F statistic, and two-tailed P value are reported for each model. The strengths of the fits are approximate, as several high-elevation sites occur in regions where model outcomes are undefined and could not be included in the sum of squares. These regions are where density dependence ceases but the potential for population growth is >0. Where lizard performance is low at high elevation, energetic yield, E(d) is not sufficiently high for the population to reach the equilibrium carrying capacity. All analyses were performed using r (R Foundation for Statistical Computing). We spatially projected the theoretically predicted and empirical lizard abundances onto the Grenadine Islands in gis using a 90-m resolution digital elevation model (NASA Shuttle Radar Topography Mission).
Methods for lizard censuses on Grenada are described in Buckley & Roughgarden (2005) and are analogous to those described below for the Grenadines. We surveyed all Grenadines islands that contained a substantial proportion of undisturbed habitat. On the larger islands, sites were distributed along both windward and leeward elevation gradients. On the smaller islands, lizard abundance was surveyed in lizard habitats near sea level. Observations were conducted in July–August 2004, which is within the wet season. We surveyed during hours of peak anole activity (between 10 : 00 and 16 : 00 h). Anole abundance was estimated by the first author while pacing a linear transect for 2 h. This method allows greater geographical coverage than mark-and-recapture techniques (Diaz 1997). Repeat censuses of sites on other islands in a subsequent year confirmed the robustness (both relative and absolute abundance) of the censusing technique (Buckley & Roughgarden 2005). The ≈100-m transect was often along a low-use trail and was chosen to be passable, representative, and to have little or no elevation change. If a substantial distance (>100 m) was traversed in <2 h, an additional transect was surveyed adjacent to the initial transect to maintain constant habitat.
Vegetation was scanned for anoles from the forest floor to canopy within 2 m on each side of the transect. Estimates of abundance differences are conservative, as more time was spent collecting lizard data where lizards were more abundant. Elevation was estimated as the mean of GPS measurements at each end of the transect, and confirmed with digital elevation models. A total of 36 sites were surveyed on 13 islands. The empirically projected lizard densities are linear regressions for islands on which there was a significant decline in abundance with increasing elevation, and the regression accounted for at least 60% of the variation. For the remaining islands, data are means. We spatially projected the theoretical and empirical lizard abundances onto the Grenadine Islands in arcgis using a 90-m resolution digital elevation model (NASA Shuttle Radar Topography Mission). Island-wide abundance estimates are derived by summing projected abundance across pixels.