© 2000 by The Society for Integrative and Comparative Biology
Calculating Climate Effects on Birds and Mammals: Impacts on Biodiversity, Conservation, Population Parameters, and Global Community Structure1
1 Department of Zoology, University of Wisconsin, 250 N. Mills St., Madison, Wisconsin 53706
2 Department of Chemical Engineering, University of Wisconsin, 1415 Johnson Drive, Madison, Wisconsin 53706
3 Institute for Environmental Studies, 1225 West Dayton Street, Madison, Wisconsin 53706
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This paper describes how climate variation in time and space can constrain community structure on a global scale. We explore body size scaling and the energetic consequences in terms of absorbed mass and energy and expended mass and energy. We explain how morphology, specific physiological properties, and temperature dependent behaviors are key variables that link individual energetics to population dynamics and community structure.
This paper describes an integrated basic principles model for mammal energetics and extends the model to bird energetics. The model additions include molar balance models for the lungs and gut. The gut model couples food ingested to respiratory gas exchanges and evaporative water loss from the respiratory system. We incorporate a novel thermoregulatory model that yields metabolic calculations as a function of temperature. The calculations mimic empirical data without regression. We explore the differences in the quality of insulation between hair and feathers with our porous media model for insulation.
For mammals ranging in size from mice to elephants we show that calculated metabolic costs are in agreement with experimental data. We also demonstrate how we can do the same for birds ranging in size from hummingbirds to ostriches. We show the impact of changing posture and changing air temperatures on energetic costs for birds and mammals. We demonstrate how optimal body size that maximizes the potential for growth and reproduction changes with changing climatic conditions and with diet quality. Climate and diet may play important roles in constraining community structure (collection of functional types of different body sizes) at local and global scales. Thus, multiple functional types may coexist in a locality in part because of the temporal and spatial variation in climate and seasonal food variation. We illustrate how the models can be applied in a conservation and biodiversity context to a rare and endangered species of parrot, the Orange-bellied Parrot of Australia and Tasmania.
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INTRODUCTION |
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SYNOPSISINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONCONCLUSIONSAppendixReferences |
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A brief history
Ever since the era of Charles Darwin biologists have been intrigued by how and why animals live where they do and what is it about their properties that makes them appear where they do, and appear in the species associations that they form. Hutchinson (1959)
defined the concept of the niche. MacArthur et al. (1966), Roughgarden (1974) and many others explored aspects of how size and habitat may influence community structure. Norris (1967) and Bartlett and Gates (1967) were the first to calculate explicitly how climate affects animal heat and mass balance and the consequences for body temperature in outdoor environments. The climate space concept emerged from steady state heat and mass balance calculations and was used to explore how climates might constrain animal survival outdoors (Porter and Gates, 1969).
Those early animal models of the 1960s were limited by the lack of models for distributed heat generation internally, distributed evaporative water loss internally, and a first principles model of gut function. Batch reactor, plug flow and other models were already in existence in the chemical engineering literature (Bird et al., 1960) and it would take time for the biological community to rediscover them. Also missing were a first principles model of porous insulation for fur or feathers, an appendage model, and a general microclimate model that could use local macroclimate data to calculate the range of local microenvironments above and below ground. It became possible to estimate convection heat transfer properties knowing only the volume of an animal (Mitchell, 1976). Another useful development was the appearance of a countercurrent heat exchange model for appendages (Mitchell and Myers, 1968) and the measurement of heat transfer characteristics from animal appendage shapes (Wathen et al., 1971, 1974). It also became possible to deal with outdoor turbulence effects on convective heat transport (Kowalski and Mitchell, 1976). A general-purpose microclimate model emerged in the early 1970s (Beckman et al., 1971; Porter et al., 1973; Mitchell et al., 1975) that calculated above and below ground microclimates. The ability to deal with local environmental heterogeneity and calculate percent of thermally available habitat came later (Grant and Porter, 1992). Over time general-purpose conduction–radiation porous media models for fur appeared in the biological literature (Kowalski, 1978) and it became possible to refine and test them in a variety of habitats and on many species (Porter et al., 1994). The extension of the models to radial instead of Cartesian coordinates and the implementation of first principles fluid mechanics in the porous media (Stewart et al., 1993; Budaraju et al., 1994, 1997) added important new dimensions to the models, which could now calculate temperature and velocity profiles and therefore heat and mass transfer within the fur from basic principles. A test of the ectotherm and microclimate models to estimate a species' survivorship, growth and reproduction at a continental scale appeared in the mid 1990s (Adolph and Porter, 1993, 1996).
Thanks to these developments and the ones reported in this paper, such as the temperature dependent behavior linked to the new thermoregulatory model, it is now possible to ask: "How does climate affect individual animals' temperature dependent behavior and physiology and what role(s) does it play in population dynamics and community structure?" This paper attempts to address some of these questions.
We approach the problem from the perspective of a combination of heat and mass transfer engineering and specific aspects of morphology, physiology and temperature dependent behavior of individuals. We show how this interactive combination is essential to calculate preferred activity time that minimizes size specific heat/water stress.
Preferred activity time is a key link between individual energetics and population level variables of survivorship, growth and reproduction, since it impacts all three population variables. Both individual and population level effects may place constraints on community structure. At the individual level, climate at any given time and food type and quality affect the optimal body size that maximizes discretionary mass and energy, the resources needed for growth and reproduction. Climate also affects community structure by affecting individual survivorship directly (heat balance/metabolic costs) and indirectly (activity time overlap of predator and prey). Climate affects seasonal food availability, distribution of food in space and time, and the cost of foraging for that food at different times during a day. Survivorship is affected by temperature dependent behavior changes that allow animals to move to less costly microenvironments at any time. For small mammals, underground burrows or under snow tunnels provide temperatures that never stay below 0°C due to local heating effects of the animal's metabolic heat production.
At the population level, climate plays a very important role in population numbers. Each species interacts in its own way with climate, affecting its abundance, and community structure. As Ives et al. (1999 p. 546) have pointed out
Our main result is that interspecific competition and species number have little influence on community-level variances; the variance in total community biomass depends only on how species respond to environmental fluctuations. This contrasts with arguments (Tilman and Downing, 1994Consequently, assessing the effect of biodiversity on community variability should emphasize species-environment interactions and differences in species' sensitivities to environmental fluctuations (for example, drought-tolerant species and phosphorus-limited species) (McNaughton, 1977
Exactly how climate variation, vegetation differences, animal morphology, and foraging behavior all interact to constrain multiple functional types' existence as a community is still largely unknown. Very little is known about temperature dependent foraging in mammals, although this has been well studied in reptiles and insects. Quantitative consequences of functional morphology on encounter probability and food handling time also are relatively unexplored as yet in mammals.
Temporal climate variation in a locality creates the opportunity for multiple optimal body sizes over annual cycles. The spatial local variation in topography and vegetation creates multiple local climates. Thus temporal and spatial variation in climate creates opportunities for multiple functional types (sizes) to coexist as communities, because as we shall see below, different body sizes interact differently with climate. Qualitatively, this idea is not new. However with likely major shifts in global climates and the rapid global changes in land use, there is urgent need to move these qualitative ideas to a quantitative framework for protection of biodiversity, conservation biology, and a number of other applications. We focus in this paper on applications to mammals and birds.
An overview of this paper
The structure of the paper begins with an overview of how macroclimate drives microclimates, which in turn impact individual animal properties. We then show how key individual properties determine population level parameters that can be used to calculate population dynamics variables. We then illustrate how individual properties also impact on community structure, that in turn feed back to temperature dependent animal properties of individuals.
The initial overview provides a context for an analysis of the model components and their interactions in hierarchical contexts. We start with the model components from the core to the skin, then from the skin through the insulation to the environment. We demonstrate how these components collectively can define the metabolic cost to mammals ranging in size from mice to elephants. We show how the empirical mouse-to-elephant metabolic regression line for animals of different sizes changes depending upon the animal's climate and posture.
Then we explore how changing mammal body size affects discretionary energy across all climates. Once the mammal model is explored, we repeat the process for the bird model. We demonstrate how we can estimate metabolic cost across bird sizes ranging from hummingbirds to ostriches. We show how postural changes and air temperature can alter metabolic cost estimates for birds.
Once sensitivity analyses are completed, we explore how temporal and spatial variation in global climate impact body size dependent discretionary energy assuming no food limitation and thereby place constraints on the potential combinations of body sizes (community structure) of mammals at the global scale.
Finally, we show how these models can be applied to estimate for the first time from basic principles the metabolic costs and food requirements of an endangered species of bird, the Orange-bellied Parrot of Tasmania and Australia. We show these results for body sizes ranging from hatchling to fully mature adult for a wide range of environmental conditions.
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MATERIALS AND METHODS |
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SYNOPSISINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONCONCLUSIONSAppendixReferences |
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The models
Overview
Figure 1 is a flow diagram that shows qualitatively how we connect macroclimate, microclimates, individual properties, population level effects, and community attributes. The macroclimate–microclimate connection is achieved in part by general climate data available through the National Oceanic and Atmospheric Administration (NOAA). The microclimate model has been described for a variety of habitats that range from southwestern deserts (Mitchell et al., 1975
; Porter et al., 1973) to Santa Fe Island in the Galapagos (Christian et al., 1983) to Michigan bogs (Kingsolver, 1979). It is a one-dimensional finite difference model that simultaneously solves the heat and mass balance equations for the ground surface and below. It also calculates wind speed and temperature profiles from the ground surface to two meter reference height, where meteorological data are typically measured. Clear sky solar radiation is calculated from basic principles (McCullough and Porter, 1971).
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Microclimate calculations for heterogeneous environments can determine percent of thermally available habitat and temperature dependent feeding frequency (Grant and Porter, 1992
). Grant and Porter showed that item feeding frequency was a linear function of the thermally available percent of the habitat (the percentage that allows the animal to stay within its preferred temperature range, thereby avoiding significant thermoregulatory heat stress costs). A summation of a day's preferred activity times over a month and over the year yields total annual activity time.
Total annual activity time is a key variable linking individual energetics with population and community level phenomena. Annual activity time for a terrestrial vertebrate was first calculated from basic principles in 1973 (Porter et al., 1973). By "basic principles" we mean equations derived from thermodynamic principles that do not involve regression equations. Total annual activity time can be used to calculate key life history variables, such as survivorship, growth, and reproductive potential (Adolph and Porter, 1993; Adolph and Porter, 1996), that are used to calculate population dynamics.
Survivorship (mortality) probability/hour is affected by activity time, which is affected by temperature dependent habitat selection. Climate change may affect survivorship, partly by modifying predation probabilities that change with seasonal changes in overlap of predator and prey preferred activity time (Porter et al., 1973; Porter and James, 1979) and partly due to climate stress (Porter and Gates, 1969).
Growth and reproduction potential depend on mass and energy intake and expenditures. The difference between intake and expenditure is the capital available for growth or reproduction. We are in a strong position to calculate mass/energy expenditures. Intake of mass and energy is more challenging. Intake depends on item feeding frequency and handling time. Handling time depends on the size of food "packages" and morphology of the feeding apparatus. Calculations in this paper assumed no shortage of food and that the mass flow through the gut scales with mass (Calder, 1984) and meets the body size/climate imposed metabolic demand. The mass flow absorbed over a day is assumed sufficient to meet basic thermoregulatory requirements for the day plus a user defined multiplier (up to 7) above the minimal metabolism needed to maintain core temperature in the current climate. This was done to try to establish an upper bound for absorbed mass for different sizes of animals.
Different sizes of animals may represent different trophic levels in the community Only some of the connections between a species' individual energetics, population dynamics and community attributes are shown in Figure 1. Other species within the habitat may influence temperature dependent behavior by competing with the arbitrarily chosen animal species represented here, thereby affecting their numbers (Ives et al., 1999). The reader may imagine multiple layers of this graph for individual species interconnected vertically to allow for explicit multiple species descriptions.
Model cross section
Figure 2 represents a diagrammatic cross section through an arbitrarily chosen part of an animal. This could represent a torso whose geometry may be approximated by a cylinder, sphere, or ellipsoid, or even a cross section through an appendage, if the heat loss by respiration is removed. There may or may not be a porous insulation beyond the skin. Figure 2 shows what would be needed for heat and mass transfer calculations. Data needed are the mean length of the fibers (hair or hair-like elements in feathers), fiber density as a function of depth, fiber diameter, and the depth of the insulation. Length and depth of the fibers are usually different unless the fibers extend outward normal to the skin. Solar reflectivity and transmissivity of the fibers also must be known if the animal is diurnal and exposed to sunlight. The environmental conditions that specify the climate boundary conditions for an individual include solar radiation, infrared fluxes from the sky and ground, air temperature, wind speed, and relative humidity of the air passing over the animal. These values are calculated based on the animal's average height above ground and the microclimate calculations for environment conditions above ground. The microclimate equations have been described (Mitchell et al., 1975; Porter et al., 1973).
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Most of the equations describing porous media heat flux without convection through the fur are described (Conley and Porter, 1986
; Porter et al., 1994). Heat and mass flux equations describing flow through fur are complex (Stewart et al., 1993; Budaraju et al., 1994, 1997). Solar radiation was incorporated in the model used here by assuming that solar radiation is absorbed very close to the fur/feather–air interface, which is usually the case for bird feathers and dark, dense fur (Porter, unpublished data). Absorbed solar radiation heats the fiber elements, which then emit infrared radiation outward toward the sky and inward through the porous insulation. The watts of absorbed solar radiation were treated as an additional source of thermal radiation from the sky for the half of the animal exposed to the sky. Thus, the diffuse infrared radiation equations already in model were also used for incorporating absorbed solar radiation in the model.
The porous media model is only part of the animal model used to calculate metabolic heat production that will maintain core temperature given the internal and external morphology of the animal, including its insulation (Porter et al., 1994). The radial dimension of an animal is calculated from its weight and geometry. An iterative searching routine named Zbrent guesses the metabolic heat production needed to maintain any specified core temperature (Press et al., 1986). Zbrent finds the unique metabolic heat production that satisfies the heat and mass balance equations (Appendix, Porter et al., 1994) given the body allometry, dimensions, specified core temperature, insulation properties, and environmental conditions. Because the equations are interconnected, relatively few variables determine these solutions (Porter et al., 1994).
Inside the body
The type of food in the gut determines the relative proportions of carbohydrates, proteins and lipids that are absorbed by the body. A healthy body will utilize these absorbed molecules as substrates. The demand for energy and the substrates being oxidized determine the amount of oxygen needed. The oxygen consumption is associated with heat generation. The proportion of the substrates oxidized determines the amount of carbon dioxide produced and hence the respiratory quotient. The oxygen demand specifies the moles of air that must pass through the respiratory system to meet the demand. Thus, the type of food in the gut affects indirectly the amount of incoming respiratory air, which in turn affects the water balance in the respiratory system in the heat generation-ventilation-gut coupled model described below.
Heat generation models
Figure 3 shows how the current model of distributed heat generation throughout the body creates a parabolic temperature profile from the body core to skin. The equations describing uniform heat generation for rectangular (slab), cylindrical, spherical, and ellipsoid geometry (Porter et al., 1994) all show that the internal heat generation and the temperature gradient from core to skin are functions of the body radius squared. The model solves the heat and mass balance equations (Porter et al., 1994) for heat generation needed to maintain core temperature by iterative guessing the solution for each hour of simulation throughout a 24 hr daily cycle. The coupled equations of heat and mass transfer simultaneously yield solutions for water balance, gut absorbed food requirements, hours of activity time and discretionary mass and energy available for growth or reproduction or fat deposition as described below.
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Earlier metabolic heat generation models, such as a slab approximation, assumed a heat source only at the center of the animal (Porter and Gates, 1969
; Porter et al., 1973). This assumption creates a simple linear temperature profile from core to skin (Fig. 3, Porter et al., 1994), but not shown here. This type of construct frequently uses the term "thermal conductance," the reciprocal of" thermal resistance." Thermal conductance is a linear model of heat transfer commonly used in many biological publications referring to animal heat transfer. Unfortunately it is only relevant in the context of non-heat generating materials.
A cylindrical geometry with a heat source only at the center (axis) does not mathematically allow for the heat source only at the axis, since it is undefined there (Bird et al., 1960). A central heated region is required. Simple conduction (but not added heat generation by the conducting tissues) of heat radially from the perimeter of the core region yields a logarithmic temperature profile. This logarithmic profile has different heat generation requirements to maintain a specified core temperature in the center region than a model using distributed heat production from the core to the skin.
Respiration
An important addition to the current model is the distributed respiratory water loss, which represents lungs that span most of the body cavity. This innovation gives much better agreement of predicted metabolic rates with measured values.
Figure 4 shows the system diagram for the lung molar balance model. A dashed line labeled 1 represents the entrance surface to the respiratory system. The dashed line labeled 2 represents the exit surface from the respiratory system. Moles of nitrogen, oxygen, water, and carbon dioxide enter the respiratory system. The moles of air entering are calculated from the product of the moles of oxygen needed for the current guess for heat generation requirements times the sum of the percent composition of the components of air divided by the percent of oxygen in the air, which may change in burrows. Thus, the current iterative guess for metabolic heat production specifies how many moles of oxygen are needed to meet the metabolic demand from the respiratory system. The type of diet (carbohydrate/protein/lipid) specifies the joules of heat produced from the oxidation of a mole of oxygen (Schmidt-Nielsen, 1979). The oxygen extraction efficiency of the respiratory system and the properties of air determine how many moles of air are needed per unit time by the respiratory system. The amount of carbon dioxide added to the respiratory system air is calculated from the respiratory quotient, RQ, which is the ratio of moles of carbon dioxide produced per mole of oxygen consumed (Schmidt-Nielsen, 1979).
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The RQ changes with different substrates oxidized. The respiration model uses the RQ for carbohydrates, proteins, or lipids, or a combination of the three, to calculate the amount of carbon dioxide that flows out of the respiratory surfaces. The user-specified proportions of carbohydrate, protein, and lipid in the food consumed thus ultimately determine the RQ. Thus, the metabolic oxygen demand to maintain core temperature and the current properties of air specify the volume of airflow and the amount of water added to saturate the respiratory system air. At expiration, the user specified temperature difference between the air in contact with nasal surfaces as air exits at surface 2 and the free stream external air (1–3°C) is used to calculate the amount of water recovered by condensation on the nasal surfaces. The calculated skin temperature of the body would not be relevant for estimating nasal air temperature at exit because of the different convective environment inside the nares vs. the outer skin covered with fur or feathers. Since we were trying to estimate maximum recovery rates as an upper bound, we used experimental data summarized from the literature (Welch, 1980
) for the calculations and used a 3°C difference between exit air temperature and local external (free stream) air temperature.
Temperature regulation model
Another important addition to the model was temperature regulation responses. Sensitivity analyses of the model done by increasing air and radiation temperatures revealed that the calculated skin temperature, which is a function of the specified core temperature, must not exceed core temperature. If it does exceed the core temperature, metabolic heat production must be dissipated by evaporation of respiratory water to achieve steady state. The molar balance model for the lungs just described clearly showed a limited capacity for heat dissipation by water vaporization in the lungs, which is consistent with experimental data (Welch and Tracy, 1977; Welch, 1980). A user specified minimum core–skin temperature difference was added to the model. The value used in our calculations was 0.5°C. If an iterative solution for heat generation given the specified core temperature produced a skin temperature less than the minimum core–skin difference, a three-level hierarchy of physiological responses was invoked.
First, flesh thermal conductivity increases to the maximum value measured in the literature. That was never sufficient to increase the core-skin temperature gradient, since it only serves to increase skin temperature.
Second, the percentage of the skin surface assumed covered with tiny water drops increases up to 100 percent of the skin surface area to cool the skin. The amount of cooling is constrained by air temperature, wind speed, relative humidity, and the boundary layer thickness at the skin. The latter is a function of body characteristic dimension, insulation properties, and wind properties defined in Nusselt and Reynolds numbers (Bird et al., 1960). The Nusselt number is simply a nondimensional ratio of the heat transfer coefficient times a characteristic dimension (often defined as the distance a fluid such as air travels when passing over the object of interest) divided by the thermal conductivity of the fluid. The Reynolds number is also a nondimensional ratio. It is the product of the fluid density, velocity, and the characteristic dimension divided by the dynamic viscosity of the fluid. The Nusselt number is often plotted against the Reynolds number. The regression of the data plotted is a relationship that allows for the calculation of the heat transfer coefficient (used to calculate convective heat loss) for any value of Reynolds numbers variables, such as changing characteristic dimension (body size).
Third, failing all else, the core temperature is allowed to rise in 0.1°C increments until a stable solution of the equation is found that allows a 0.5°C temperature difference between core and skin. This approach causes a rise in metabolic rate at high temperatures that is observed experimentally (Schmidt-Nielsen, 1979). It also mimics the rise in core temperature that is observed experimentally (Schmidt-Nielsen, 1979). No regressions are needed to emulate the experimental data.
The gut
Figure 4 also shows the system diagram for the molar balance gut model. It is related to the well-known batch reactor and plug flow model originally developed in chemical engineering and subsequently applied to animal digestive systems (Penry and Jumars, 1987). The model used here allows for any type of ingested food consisting of user specified proportions of carbohydrates, lipids, proteins and water content. The food can enter the gut any time during activity time in any amount, subject to the constraint that the volume of food ingested per day may not exceed the wet mass of the animal. The energy value of absorbed carbohydrates, lipids, and proteins is well known (Schmidt-Nielsen, 1979). Details of the model are in the Appendix.
Temperature dependent feeding
Figure 5 shows how these animal models respond to different temperatures. The metabolic rate of an endotherm changes with increasing environmental temperature in a distorted U-shaped curve (Bucher et al., 1986; Kleiber, 1975; Morris and Kendeigh, 1981; Schmidt-Nielsen, 1979; Scholander, 1940). It is commonly assumed from a physiological perspective that the capacity to absorb food is independent of environmental temperature because of the relatively constant body temperatures that endotherms usually maintain. This is in contrast to the temperature dependent digestion of ectotherms (Waldschmidt et al., 1987).
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However, the temperature dependent foraging behavior and appetite levels of endotherms are frequently ignored, although they have been considered with respect to domestic animals (Kleiber, 1975
). Recent seed tray experiments under natural foraging conditions show that desert rodents are extremely sensitive to substrate temperatures that affect willingness to forage (Mitchell et al., ms), and similar results have been reported for free ranging raccoons (Berris, 1998). Predation risk and competition also influence foraging costs. Birds and mammals may compete for the same resource (Brown et al., 1997). Predation risk and competition can be expressed in terms of energetic cost (Brown et al., 1994). Thus, the difference between temperature dependent foraging (mass and chemical energy intake) and temperature dependent metabolic costs (mass and chemical energy expenditure) yields temperature dependent discretionary mass and energy intake. Discretionary mass and energy intake is the oval area in Figure 5 bordered by intake and expenditure rates. Climate and type of food available are important constraints on fitness that can now be calculated from basic principles. As we shall soon see, body size and diet are additional important constraints on fitness in different climates.
Porous insulation
Fur vs. feathers
Figure 6 shows schematically the difference between fur and plumage density as a function of distance above the skin. The densities of hair elements are greatest near the skin. The density of feather elements, in contrast, is lowest near the skin and greatest at the feather–air interface. The consequence of this difference in density profiles is that air can penetrate at least the outer parts of fur more easily. Feathers, however, seal off the plumage–air interface, creating a conduction–radiation environment that is easier to analyze.
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This is true, irrespective of whether the fur or plumage elements are normal to the skin or at an angle relative to the skin. Changing angle relative to the skin modifies density profiles for either type of porous insulation, but does not change the general shape of the density profile with depth. In reality, parts of the skin of birds do have plumaceous elements near the skin. Individual plumages will vary between taxa and between different locations on the body. The density will also vary with degree of elevation of contour feathers, degree of density at the same level of the rachis/vane junction, and the presence and density of down feathers. However, vertical serial sections of crow feathers embedded in plastic under vacuum have shown that the greatest density of elements is at the plumage–air interface (Porter, unpublished data).
Fortunately, in low wind environments, changes in individual element density with height do not have a significant impact on porous insulation heat loss, unless the fibrous elements are either very sparse or extremely dense across a wide range of body sizes (Fig. 25 in Porter et al., 1994). At very low individual element density, the porous insulation becomes very open, allowing substantial convective and radiant heat transfer from the skin. In contrast, at very high individual element density, the effective thermal conductivity of the porous insulation approaches that of keratin, rather than air. This amounts to an increase in thermal conductivity by a factor of about eight, thus increasing heat loss. Sensitivity of heat loss due to density changes with depth in fur in a conduction-radiation heat exchange is very small (Kowalski, 1978; McClure and Porter, 1983). Kowalski used a measured density of fur as a function of depth for cow fur, which is described by a hyperbolic tangent function (Fig. 6).
If fur density has an optical thickness (Porter et al., 1994) less than 0.001, the fur is so sparse that it is assumed to be transparent to infrared radiation, and conduction heat transfer along the fibers is negligible. Under these conditions, the model assumes the functional equivalent of bare skin. For example, a user of the model can explore the consequences of changing insulation or removing insulation merely by altering the input data file, the depth and density of fur or plumage. If it is set to zero, or very low density, the program automatically checks to be sure that a porous model is appropriate and changes to a bare skin model, if necessary.
Finite elements and flow through the fur
Moderate and high wind environments can force penetration of air through fur. Thus, it was important to develop a basic principles model that would permit calculation of velocity and temperature profiles in a porous medium with nonlinear coordinates on a round body. Integration of the profiles allows for calculations of heat energy and mass transfer from basic principles, a task first accomplished only recently (Stewart et al., 1993; Budaraju et al., 1994, 1997). We will briefly review the basic features of this sophisticated model.
Figure 7 shows the finite element model in cylindrical (radial and angular) coordinates from an end view. These curvilinear boxes are used to compute the mass and heat flows in the radial and angular directions relative to the skin and to the direction of incoming air for variable densities of keratin elements projecting from the skin of an endotherm. This concept is also appropriate for birds, under conditions where air penetrates the feathers. Examples include resting birds with fluffed feathers or flightless birds like kiwis, whose plumage strongly resembles mammalian fur. In principle, this model would also be useful for the pulsing (changing angular orientation relative to the skin) feather conditions of active bird flight.
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Appendages
Figure 8 shows an appendage model for birds developed for three large ratites, the rhea, Rhea americana, the cassowary, Casuarius casuarius, and the ostrich, Struthio camelus. These appendages are largely bare and constitute a significant percentage of the surface area of the standing bird. Appendage dimensions were measured relative to torso dimensions using photographs of the birds of known height and weight from the side, and the front. We measured only the exposed areas of appendages. The portions of the legs covered by torso feathers were assumed to be part of the volume of the torso for heat transfer calculations.
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The same cross-section model shown in Figure 2, but without porous insulation and respiratory water loss, was used to calculate heat loss in the radial dimension from these appendages. Heat loss from the bottom of the appendages in contact with the ground was assumed to be negligible. Total calculated heat loss was the sum computed from the torso plus the heat losses from the appendages.
The regression equations that were fitted to the appendage dimensions, areas, and volumes are listed in the Appendix. The appendage dimensions were computed from regressions based on body weight. Appendage volumes were then computed, added and the total subtracted from the total volume of the bird based upon its weight. The difference was the torso volume. The torso length, width, and height were calculated from the dimension ratios of the feathered torso. The calculated torso dimensions then were used to compute the effective torso skin area and plumage depths. Plumage depths agreed well with the few data available.
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RESULTS |
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SYNOPSISINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONCONCLUSIONSAppendixReferences |
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Modeling an individual
Internal body temperature profiles
Figure 9 demonstrates the temperature profile from core to skin to the insulation–air interface. The temperature profiles in the porous insulation (left side) represent no absorbed solar radiation vs. (right side) absorbed solar radiation. When sunlight is absorbed by a porous insulation, it will act as a distributed heat source in the medium (Rs – Rf). The color of the insulation and its transparency affect the depth over which the radiation is absorbed. The temperature profile from center to skin (0 – Rs) is parabolic in a slab, cylinder, sphere or ellipsoid with uniform heat generation per unit volume (Porter et al., 1994
: Appendix, equations 1–3). The temperature difference from core to skin is equal to the heat generation per unit volume multiplied by the radius squared divided by the product of a geometry constant and the thermal conductivity. The value of the geometry constant may be 2, 4, or 6, depending on whether the geometry is a slab/ellipsoid, a cylinder, or a sphere (Bird et al., 1960; Porter et al., 1994).
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The calculation of distributed respiratory water loss had to take this parabolic temperature profile into account. The average temperature for a nonlinear profile can be calculated easily using an integration procedure (Bird et al., 1960
, p. 270). The volume average temperature was used as the average lung temperature for the purpose of calculating saturation vapor pressure. The amount of water lost in respiration was equal to the water added internally to saturate the air in the lungs at average temperature less the water recovered at saturation at the exit temperature at surface 2 in Figure 4.
The insulation
The flow of air through fur or plumage is influenced by the diameter and density of the fibrous elements the air encounters as well as the pressure generated in the flow field around the animal. We illustrate the combined effect of these variables on flow through fur in a red deer (Cervus elaphus elaphus) in Figure 10a, b, and c modified from Budaraju et al. (1997). Red deer fur is well characterized (Steudel et al., 1994).
Flow at very low wind
Figure 10a shows calculated flow through the fur of red deer at 0.01 m/sec using the finite element model (Budaraju et al., 1997). The streamlines shown are trajectories of airflow through the fur layer. Free convection is the dominant airflow pattern. Nearly still air in the vicinity of the animal enters at the bottom (ventral surface), flows along the sides of the animal and exits at the top (dorsal surface). The velocity is greatest near the skin along the sides of the animal, and least near the ventral and dorsal surfaces.
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Flow at 0.5 and 3 m/sec
Figure 10b and c show calculated flow through the fur of red deer in a 0.5 and 3 m/sec wind using the finite element model (Budaraju et al., 1997
). Now the wind is assumed to be blowing horizontally from left to right. The long axis of the body is assumed to be normal to the direction of wind flow. The streamlines diverge on the windward side, enter the fur, and then move toward the dorsal and ventral parts of the animal. Air recirculates near the top and bottom because of pressure differences that are a consequence of the geometry and the properties of air. A recirculating eddy then enters the fur, moving toward the leeward side of the animal and finally exits the fur. We might expect similar types of air movement if birds fluffed their feathers at rest in a moderately strong wind, where air could enter the plumage from the windward side. However, birds typically only fluff their feathers in very low wind environments. Figure 10a suggests that extending the insulation further laterally and normal to the flow might slow free convection around the body.
Scaling across mammal body sizes
Mouse to elephant metabolic rate
Skin temperature is a consequence of the solution of heat flux equation (the correct guess of heat generation to maintain core temperature) from the core to the skin. It is simultaneously the basis for heat exchange from the skin through the porous insulation to the environment. The total heat generation must satisfy the coupled body and insulation equations whose boundary conditions are core temperature and environmental conditions.
Figure 11 shows these heat generation calculations assuming an ellipsoid geometry for animals ranging in size from mice to elephants. The line with filled circles assumes a bare skin. Alternatively, if we put deer mouse (Peromyscus maniculatus) fur on all body sizes instead, we get the line with open circles on it. The free-floating diamonds represent the empirical data.
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An appendage model was also developed for mammals using the same principles as just described for birds. However, the torso, head, and neck were assumed to be cylindrical in shape with an elliptical cross section. The ratio of the elliptical cross-section based on photographic measurements of elephants and cattle was 1.27 for the ratio of height to width of the torso. The appendage model was not used for animals smaller than 100 kg.
It is particularly interesting to note that for the standing mammal model using no fur vs. fur of red deer (a form of elk), there is no observable difference in metabolic rate from 100 kg to 3,833 kg. This is reasonable, since the low wind speed used here would probably not disrupt the substantial boundary layer of these large mammals. Fur would be of little help at this low wind speed in an environment where the air and radiant temperatures are assumed to be the same.
It is important to note that the experimental data in the literature used to test this model were all collected in a metabolic chamber at an air and radiant temperature of 28°C and a core temperature of 36°C. Benedict measured the core temperature of all of his animals in his classic publication (Benedict, 1938). A sensitivity analysis of the conduction–radiation fur model shows that core, air and radiant temperatures are the most sensitive of all variables affecting heat loss from an animal with porous insulation (McClure and Porter, 1983).
Figure 12 illustrates the impact of changing air temperature on the mouse to elephant curve. The frame of reference is the experimental data shown as free-floating diamonds. A simulation for 28°C is the dashed line with open circles. Metabolic rates at a lower temperature of –25°C and an upper temperature of 40°C are also calculated. The increase in metabolism at small body size for 40°C air temperature is part of a consequence of the thermoregulatory model used in these calculations. It is an emergent property of the model.
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It is also important to realize that the postures of Benedict's animals were largely unknown. In fact, posture is rarely monitored in metabolic rate measurements. Metabolic costs can vary considerably due to simple changes in posture (Porter et al., 1994
). Ellipsoid geometry is the best intermediate approximation for estimating metabolic costs in mammals (Porter et al., 1994).
The term "thermal conductance" is an amalgam of all the variables associated with specific morphological, physiological and climate variables. Those state variables have been explicitly defined in an appendix (Porter et al., 1994), where the detailed equations for the model reside. The metabolic heat lost by an animal is a consequence of (1) body morphology and insulation properties, (2) core temperature and thermal conductivity of body tissues, and (3) environmental conditions, such as air temperature, sky radiant temperature, ground radiant temperature, and wind speed. Solar radiation and relative humidity were not explicitly included in the earlier endotherm model (Porter et al., 1994), but they are added in the present model.
Figure 12 demonstrates the limitations of the regression assumptions in the standard mouse to elephant metabolism regression line when applied to natural environments. The slope and intercept of the mouse to elephant curve changes with air/radiant temperature and body size.
Mouse to elephant discretionary energy uptake
Figure 13a and b show explicitly the calculated discretionary energy uptake for mammals of different sizes represented qualitatively in Figure 5. Selection for maximum discretionary energy uptake represents selection for growth and reproduction potential, key elements of fitness. The three dimensional and the contour map of body size and temperature effects on discretionary energy uptake assume no food limitation. The animals are simply allowed to fill their gut, digest and absorb the food. At temperatures where sweating must be initiated to cool the animal, it is assumed that animals no longer forage. These figures are an upper estimate of discretionary mass intake as a function of body size. Air temperature is assumed to be the same as the radiant temperature of the sky and ground. Thus, this set of calculations represents metabolic chamber environments. The effects of changing wind, sunlight and humidity are not included in these figures.
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A very interesting feature of the three dimensional and contour surfaces is the suggestion of discontinuous optimal body sizes in nature, where temperature varies in time and space. There is a "ridge" of maximum discretionary energy (fitness) that is irregular in elevation as it curves upward and to the left in Figure 13b from its starting point at about 28°C where the body size is 0.01 kg body size. As body size increases, the optimal environmental temperature that maximizes discretionary energy for body size decreases. The discretionary energy maximum traces a jagged ridge with peaks and valleys going to lower and lower temperatures. This result occurs because of differences in the temperature at the lowest metabolic rate for animals of different sizes and the maximum limit on gut mass flow per day at each body size. The temperature associated with maximum distance between those two curves changes with body size. Thus, the greater payoff in discretionary energy (fitness) at some temperature/body size combinations relative to others of lower value at nearby temperatures suggests there should be a selection for the size that has the highest value in the neighborhood of a small temperature range. This would create one or more "gaps" in the distribution of animal sizes in nature. The largest gap shown here is between –5 and 0°C and a body size of approximately 100 kg.
Diet effects on optimal body size
Figure 13a and b were calculated assuming a diet of seeds for all body sizes. The proportions of carbohydrate, fats and protein in the dry matter are similar to those of alfalfa hay and grass hay forage (see Appendix). However, the water content is very different. When the diet is assumed to be alfalfa hay (reference number 100056; U.S.–Canadian nutrient composition tables, 1994) with the same digestive efficiency of 57% as for seeds, Figure 13c shows a similar optimal body size landscape, but the "ridge" is shifted to larger body size for the same temperature. The magnitude of the "peaks" is much reduced. The landscape is also much flatter. The body size where the animal is in negative mass balance is much larger for the same temperature.
Figure 13d shows the landscape for an assumed diet of grass hay (reference number 102250; U.S.–Canadian nutrient composition tables, 1994) for all animals ranging in size from mice to elephants. The digestive efficiency is 61%, to reflect the overall digestive ability of ruminants. This makes for slightly higher peaks but the same general landscape. The change in digestive efficiency more than compensates for the change in quality of diet. Thus, when examining all three contour plots, the higher the quality of diet, the smaller the optimal body size at any given temperature.
Figure 13e compares four different sizes of mammals all assumed to be consuming seeds as their diet. The ten gram mammal has a net throughput capacity that matches metabolic cost slightly below 0°C. Temperatures lower than this we calculated will cause the animal to be in negative mass balance without an insulating nest or change in posture (Porter et al., 1994). A 30 g mammal has the same intersection at approximately –18°C. A1 kg mammal has a body size large enough to assure adequate gut mass flow until approximately –40°C, assuming deer mouse fur and appendages that are extended. If a 4,000 kg mammal were able to process seeds of the same nutritional properties as the smaller mammals, it would have sufficient gut throughput to allow it to have no mass flow constraints, even with only deer mouse fur at this very low wind speed of 0.1 m/sec. A quick examination of the maximum distance between the two curves of the four sizes of animals shows different temperature optima where maximum discretionary energy is available assuming unlimited seeds. As we saw from Figure 13c and d, a change in diet from seeds to green forage shifts the optimum body size at any given temperature to higher values. These results ignore coprophagy and other subtle, but significant modifications of digestive systems. Nonetheless, it provides some useful guidelines for identifying future modifications of the gut model and provides some understanding of multiple constraints on optimal body size of animals.
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Bergmann's Rule
These results are reminiscent of Bergmann's rule, an empirical observation that as climates get colder, animal sizes tend to get larger. Body size increases with decreasing temperature provide the greatest advantage at small size (Steudel et al., 1994
). At larger body sizes, changes in fur insulation confer a greater advantage Steudel et al., 1994). Experimental data from different types of fur on a flat plate (Scholander et al., 1950) suggested this, but animals of larger size also have thicker boundary layers. A thicker boundary layer reduces convective heat loss and simultaneously enhances radiation temperature effects (Porter and Gates, 1969). Larger animals are taller, which means exposure to greater wind speeds higher above the ground. Higher wind speed reduces boundary layer thickness and may engender greater wind penetration of the fur. A first principles fur model can separate boundary layer effects due to size and wind from fur properties effects and provide better estimates of combined effects.
Assessment of consequences of Bergmann's rule have pointed out that larger animals have the advantage of longer fasting ability under conditions of climate or food availability stress (Morrison, 1960). However, smaller animals have the advantage of lowering body temperature and seeking much more favorable microclimates, especially underground habits in severe cold. Careful transient modeling analyses of these two strategies in the animals' microclimates would yield a testable hypothesis of the relative benefits of these different solutions to the same problem of dealing with cold.
Of course, survival in extreme temperature events is also important in affecting community structure. However, extreme temperature survival may be overrated in terms of its effects on community structure, at least for mammals. Temperature dependent behavior and selection of microhabitats by both small and large animals can greatly reduce cold or heat stress. For example, moving under or into trees and modifying the solar and infrared radiation and wind protection they provide can change equivalent local microenvironment temperatures by 20°C or more. Underground burrows or tunneling beneath the snow can provide habitats that typically do not drop below 0°C in winter when an animal is present, due to local heat from metabolism. Photoperiod-induced temperature dependent physiology, such as hibernation or estivation is another way that mammals can persist in habitats during periods of extreme heat or cold stress and thereby maintain community structure. Birds typically opt to migrate from extremely cold habitats in winter that they occupy in the summer. By exercising temperature dependent behavioral selection of microclimates through migration, the scale of their selection movements is simply larger due to the short time and lower costs of long distance bird transport.
Scaling across bird body size
Hummingbird to ostrich metabolic rate—Appendage effect
Figure 14 shows the impact of appendages on heat loss in large birds by comparing data on bird metabolism with calculations of bird metabolism using two different models. The filled circles represent bird metabolism vs. body size in the absence of sunlight (Schmidt-Nielsen, 1979). The sitting bird model calculations use an ellipsoid approximation for a bird with legs and head tucked into the feathers. The standing bird model represents the torso as an ellipsoid, and the appendages as cylinders with diameters averaged over the appendage length. The solid line marked with solid triangles is the sitting bird model. We assumed an air temperature and radiant temperature of 30°C. The dashed line with open triangles represents the standing bird model calculations only for the ratite birds; rheas, cassowaries and ostriches. All smaller birds are assumed to be sitting. We assume that the birds are maintaining a core temperature of 39°C at the center of the torso and at the centers of the appendages.
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