R2

In the state space of resource abundances 1 and 2, this equation describes a straight line that has a negative slope when ei/hj > k. Combinations of R1 and R2 that lie outside this isocline yield harvest rates greater than k; combinations inside the isocline yield feeding rates less than k. When k represents the subsistence level of resource consumption by the forager, the corresponding consumption isocline is the zero net growth isocline, ZNGI, at which the forager's per capita growth rate is zero (Vincent et al. 1996).

Gut modulation may take the form of adjustments in the rate of nutrient assimilation, which we model by allowing e1 and e2 to increase or decrease in relation to a changing diet composition. Similarly, gut modulation may take the form ofvariation in the rate offood transport through the gut (gut retention time), modeled by assuming that h1 and h2 implicitly include both pre- and postconsumptive handling of food, and thus change in response to changing diet. We will restrict our development here to the case of active nutrient transport involving the e terms; the h modulation case is very similar (Whelan et al. 2000).

This model allows the e terms to vary between two gut modulation modes, which we will designate A and B, respectively. Each mode has its own consumption isocline (see below). In some circumstances, the isoclines will intersect so that one mode is more efficient at certain resource abundances, while the other is more efficient at other resource abundances. We assume that the modulation mode is chosen to maximize the forager's fitness, written as G = max{fA, Jb}. This objective function applies for a family of fitness functions (Brown 1992).

Assume that gut modulation strategy A increases the rate of assimilation of resource 1 via an increase of active 1 transporters, e1, coupled with a decreased rate of assimilation of resource 2 via a decrease of active 2 transporters, e2. Let the opposite be true for gut modulation strategy B. The variables e1A and e1B represent the assimilation rates for resource 1 under modulation strategies A and B, respectively. For a given constant feeding rate k, we now have two consumption isoclines, one for each gut modulation strategy:

Each equation is a straight line with negative slope. When assimilation of resource 1 is greater for modulation mode A than it is for modulation mode B (e2A > e2B) and assimilation of resource 2 is less for modulation mode A than it is for modulation mode B (e2A < e2B), then the two lines must cross at positive values for resource abundances. This indicates that each digestive strategy yields a higher feeding rate (in terms of assimilated energy per unit time) at some combinations of resource abundances. At the point of intersection, both gut modulation strategies yield the same feeding rate.

Resource abundance combinations for which the two gut modulation strategies yield the same feeding rate define the modulation isoleg (sensu Rosenzweig 1981). Some algebra shows that this is

Figure 5.1.1. Graphical representation of the effect of modulation ofdigestive processing on consumption isoclines. Families of paired equal consumption rate isoclines forthree levels of harvest rate or fitness, k, when the forager modulates between strategies A and B (labeled for harvest level k3). Each isocline represents the relative combinations of resources 1 and 2 that result in a constant harvest rate. Note that the modulation isoleg (indicated by MI), that combination of resources 1 and 2 that results in an equal harvest rate for both modulation strategies, cuts through the intersection of each pair of consumption isoclines.

Abundance of Resource 1

Figure 5.1.1. Graphical representation of the effect of modulation ofdigestive processing on consumption isoclines. Families of paired equal consumption rate isoclines forthree levels of harvest rate or fitness, k, when the forager modulates between strategies A and B (labeled for harvest level k3). Each isocline represents the relative combinations of resources 1 and 2 that result in a constant harvest rate. Note that the modulation isoleg (indicated by MI), that combination of resources 1 and 2 that results in an equal harvest rate for both modulation strategies, cuts through the intersection of each pair of consumption isoclines.

a straight line with positive slope. Points on the isoleg further from the origin represent higher feeding rates, k (fig. 5.1.1). Above the isoleg, gut modulation strategy B (fa< fB) yields a higher feeding rate, and below the isoleg, modulation strategy A (fA< fB) yields a higher feeding rate. When resource abundances lie above the isoleg, the forager should modulate nutrient transport to become more efficient on resource 2. Similarly, below the isoleg, the species should modulate nutrient transport to become more efficient on resource 1. The net effect of modulation results in an "effective" consumption isocline that is piece-wise linear and is composed of the part of each component isocline [equations (5.1.3A) and (5.1.3B)] that lies within that of the other. This effective consumption isocline approximates that for antagonistic resources (see Tilman 1980, 1982), despite the fact that the model specifically treats resources as perfectly substitutable (fig. 5.1.2).

Figure 5.1.2. Graphical representation of the effect of modulation of digestive processing on consumption isoclines. Following completion of modulation to digestive strategy A and B, respectively, the piece-wise linear "effective" consumption isocline approximates that for antagonistic resources. Note thatthis "effective" consumption isocline bows out from the simple line connecting the intercepts of the abscissa (a) and ordinate (b), which would be the expected consumption isocline for two perfectly substitutable resources.

Abundance of Resource T

Figure 5.1.2. Graphical representation of the effect of modulation of digestive processing on consumption isoclines. Following completion of modulation to digestive strategy A and B, respectively, the piece-wise linear "effective" consumption isocline approximates that for antagonistic resources. Note thatthis "effective" consumption isocline bows out from the simple line connecting the intercepts of the abscissa (a) and ordinate (b), which would be the expected consumption isocline for two perfectly substitutable resources.

Whelan et al. (2000) analyzed the consequences of their functional response equations in consumer-resource models that allowed analysis of gut modulation modes and diet selection under three ecological scenarios. First, when the consumer does not deplete its resources, the resource standing crop determines the optimal modulation strategy. Second, when a consumer population of a fixed size depletes its resources, and the standing crop of resources results from a dynamic equilibrium between resource renewal and resource consumption, the equilibrium between renewal and consumption determines the optimal gut modulation strategy. Finally, when resource renewal, depletion, and consumer population size all equilibrate, the intersection of the consumer's depletion trajectory with the modulation isoleg at the consumer's zero net growth isoclines (ZNGIs) determines the optimal gut modulation strategy (Whelan et al. 2000).

These analyses show that we cannot fully understand the consequences of modulating gut physiology independently of an organism's ecological circumstances. They also hint at reasons why some foragers modulate digestive processes while others do not. Foragers that exploit nondepletable resources should show rapid modulation in response to changes in the standing crop of food. The situation is more complex and nonintuitive for foragers that exploit depletable resources. To illustrate, consider a scenario in which resource renewal, depletion, and consumer population size equilibrate. In this circumstance, the relation of the carrying capacity of the resources and the depletion vector (the trajectory of resource consumption) that intercepts the "elbow" of the modulation isoclines (fig. 5.2) determines the optimal modulation mode. When the resource supply points lie above this special depletion vector, the forager should modulate its physiology appropriately for resource 2 in figure 5.2, even though it may consume mostly resource 1 (a surprising result!). When the resource supply points lie below this special depletion vector, the forager should modulate its physiology appropriately for resource 1, the resource it is consuming predominantly (a much more intuitive result).

Nutrient Transfer Functions

Raubenheimer and Simpson (1998) present a graphical framework that views the digestive process as nutrient transfer between serially connected processing compartments. The nutrient transfer functions that apply at each junction are key points ofintegration between the behavioral and physiological components of input regulation. Raubenheimer and Simpson's framework focuses on two nutritional variables, the rate ("power") and efficiency of nutrient processing, and the transfer from one processing compartment to the next.

Raubenheimer and Simpson (1998) plot the processing time for a given quantity of food at stage Si (where i = 1,2,..., n serial stages of processing,

Figure 5.2. Graphical representation of consumer-resource model when resource renewal, depletion, and consumer population sizes equilibrate. K1 and K2 represent the carrying capacity of resources 1 and 2, respectively, forthree resource supply points, A, B, and C. In this case, the optimal digestive physiology modulation mode is determined by the depletion vector connecting the resource supply point to the intersection of the ZNGI (resource supply point A). When below this depletion vector, the consumer should always modulate to digestive mode A (resource supply point C), and when above, the consumer should always modulate to digestive mode B (resource supply point B).

Figure 5.2. Graphical representation of consumer-resource model when resource renewal, depletion, and consumer population sizes equilibrate. K1 and K2 represent the carrying capacity of resources 1 and 2, respectively, forthree resource supply points, A, B, and C. In this case, the optimal digestive physiology modulation mode is determined by the depletion vector connecting the resource supply point to the intersection of the ZNGI (resource supply point A). When below this depletion vector, the consumer should always modulate to digestive mode A (resource supply point C), and when above, the consumer should always modulate to digestive mode B (resource supply point B).

and may represent foraging, ingestion, digestion, absorption, etc.) against the cumulative release (or transfer) of the product of processing at stage Si to the following stage (stage Si+i). Following Sibly (1981), they assume that a sigmoidal curve represents this nutrient transfer relationship (fig. 5.3). Given this sigmoidal relationship between time of processing in compartment Si and transfer of the product to the next serial compartment, Si +i, the model finds the maximal rate of transfer using tangent construction techniques, as in graphical solutions of the marginal value theorem. If natural selection maximizes efficiency, rather than rate, then processing in compartment Si should proceed until the transfer curve reaches its asymptote (Raubenheimer 1995; Raubenheimer and Simpson 1994, 1995, 1997; Simpson and Raubenheimer 1993b, 1995, 2001).

A potential flaw in Raubenheimer and Simpson's graphical approach may be that sigmoidal enzyme reaction kinetics pertain to allosteric enzymes, but many digestive enzymes and carrier-mediated (saturable) transport mechanisms follow Michaelis-Menten kinetics, which are monotonically increasing with decelerating slope [as in the type II functional response of equation (5.1.1)]. The logic of the marginal value theorem may still apply, however. For instance, if one considers a nutrient's "travel time" (say, from oral cavity to reaction chamber), the marginal value theorem approach can still be applied in the manner of Raubenheimer and Simpson (1998; see, for instance, fig. 3b in Penry andJumars 1986; see also Cochran 1987).

Figure 5.3. An example of nutrient transfer relationships hypothesized by Raubenheimer and Simpson. Note the general similarity to the graphical model of Sibly (fig. 5.1). Thex-axis represents the amount of time that digesta (or nutrient) is processed at one stage in a serial nutritional pathway, and the y-axis represents the cumulative release to the next stage in the pathway. The slope of the smooth linear line (labeled R) connecting the origin to the nutrient transfer function (heavy solid line) represents the maximal rate of transfer from stage i to stage i + 1. Dropping a vertical line from this point on the transfer curve to the x-axis indicates the associated processing time in compartment Si. The slope of the dashed linear line (labeled E) represents the rate of transfer from stage i to stage i + 1 when processing at stage i is allowed to proceed to completion (maximum efficiency). Maximum rate of transfer from stage i to stage i + 1 is accomplished at time ft. Maximum efficiency is accomplished at time t2. (After Raubenheimer and Simpson 1998.)

Time at Si

Figure 5.3. An example of nutrient transfer relationships hypothesized by Raubenheimer and Simpson. Note the general similarity to the graphical model of Sibly (fig. 5.1). Thex-axis represents the amount of time that digesta (or nutrient) is processed at one stage in a serial nutritional pathway, and the y-axis represents the cumulative release to the next stage in the pathway. The slope of the smooth linear line (labeled R) connecting the origin to the nutrient transfer function (heavy solid line) represents the maximal rate of transfer from stage i to stage i + 1. Dropping a vertical line from this point on the transfer curve to the x-axis indicates the associated processing time in compartment Si. The slope of the dashed linear line (labeled E) represents the rate of transfer from stage i to stage i + 1 when processing at stage i is allowed to proceed to completion (maximum efficiency). Maximum rate of transfer from stage i to stage i + 1 is accomplished at time ft. Maximum efficiency is accomplished at time t2. (After Raubenheimer and Simpson 1998.)

Incorporating Digestive Processing into the Functional Response

Recently, a number of investigators have considered the influence of digestion and food quality on the functional response (Verlinden and Wiley 1989, 1997; Hirakawa 1997a, 1997b; Farnsworth and Illius 1998). We refer to their closely related models as digestive rate models (DRM), after Verlinden and Wiley (1989). In these works, digestive capacity is modeled as an on/off inequality constraint. These studies suggest that under some circumstances (e.g., high food abundance, low food quality), digestive quality (energy gain per throughput time) determines diet selection when digestive rate is limiting. Under these circumstances, the diet is composed of a smaller number of food types of higher quality, partial preference is expected for one food type, and all other food types are either always accepted or always rejected (the zero-one rule; Hirakawa 1997a). A critical conclusion of the digestive rate model is that the digestive properties of foods, which we refer to as their bulk properties, can play a major role in diet selection.

In a recent review of the functional response, Jeschke et al. (2002) suggested that most predators (in the broad sense, including carnivores, herbivores, parasites, and parasitoids) are, in fact, digestion limited. They proposed the steady-state satiation model, which incorporates both the handling and digestion of prey. Digestion influences a predator's hunger level, and this in turn determines the likelihood that the predator will search for prey. While digestion is a background process, gut fullness influences feeding rate as a sliding motivational state that can result in the forager choosing to cease foraging—it does not forage when satiated. This model has similarities to the "digestive pause" of Holling (1965). Because Jeschke et al. (2002) consider foraging on only a single food type, their model does not suggest how this digestive pause should influence diet selection.

Whelan and Brown (2005) developed an extension of Holling's (1965, 1966) disc equation that incorporates the passage rate of food through the gut (referred to as postconsumptive handling) as an integral component of total food handling time. In a manner similar to Jeschke et al. (2002), they modeled the extent of the "digestive pause" on a sliding scale, but one that reflects gut fullness (rather than satiation). In contrast to Jeschke et al. (2002), they developed their model for a forager that consumes two or more food types, and thus their model considers the effect of digestive processing on both harvest rates and diet selection. Postconsumptive handling time may be partially exclusive of time spent searching for and handling additional food items (preconsump-tive activities). In contrast to the DRM, in which the effect of internal gut passage on harvest rate is a step function (operable or inoperable), it is continuous in Whelan and Brown's model. However, in a manner similar to the DRM, the bulk properties of foods, via their effects on postconsumptive handling, can also have strong effects on harvest rates and diet selection.

Whelan and Brown (2005) begin with a modification of the type II functional response (Holling 1965, 1966), in which they include terms for external (preconsumptive) and internal (postconsumptive) handling of food:

External handling, h, is identical to that in the original disc equation. Internal handling consists of two variables. The first, g, represents the actual processing of food within the gut; the second, m(B), represents the proportion of gut handling time that is exclusive of alternative foraging activities, and can take any functional form with a monotonically positive slope. External handling, h, and internal food processing, g, have units of (time/item). Internal food processing, g, is determined by the quotient of food bulk per item, b (ml/item), and the volumetric flow rate of food through the gut, V0 (ml/ time): g = b/V0. But V0 = gut capacity, k (ml), divided by retention or throughput time, T(time) (seeJumars and Martinez del Rio 1999; McWhorter and Martinez del Rio 2000). Thus, passage time per item is given by g = (bT)/k. For simplicity, let m(B) = B (a linear function), the proportion of gut volume occupied by food. Gut fullness, B, is given by the bulk rate of intake (bulk of the resource, b, multiplied by its ingestion or harvest rate, H) and the retention time of food in the gut (the quotient of throughput time, T, and gut volume, k): B = (bHT)/k. This definition of m(B) allows the exclusivity of internal handling to be a continuous sliding scale that reflects the extent to which gut volume is filled from food consumption. Substituting g and B into equation (5.2) and simplifying yields

This model can be solved explicitly for H(see Whelan and Brown 2005), but this explicit expression obscures the way in which external and internal food handling influence the forager's consumption rate. Equation (5.3) has three interesting consequences, which we explore graphically in figure 5.4. First, we now see the intimate connection between harvest rate and gut processing: we need the harvest rate to specify the gut processing rate, and we need the gut processing rate to specify the harvest rate. Harvest rate and gut processing rate mutually feed into each other. Second, equation (5.3) shows transparently that pre- and postconsumptive food handling jointly limit harvest rate. Third, we see that external handling and internal handling are qualitatively different phenomena. External handling, h, has a fixed cost per item consumed that is paid in time—it operates qualitatively like a batch reactor (Martinez del Rio et al. 1994) that is full (on) or empty (off). Internal handling,gB (= g(bHT)/k), in contrast, has a variable cost paid in time because one component, harvest rate, H, is continuous (see also Jumars and Martinez del Rio 1999). In other words, internal handling operates like a continuous reactor, such as a plug-flow reactor (Martinez del Rio et al. 1994).

An analogous expression can be written for consumption oftwo (or more) food types:

T {1 + a1 R1 [h + h(T2/k2)(b1 H1 + b2 H2)] + a2 R2[h2 + b^T1/k2)^H1 + b2 H2)]}

The behavior of equation (5.4) is qualitatively similar to that of equation (5.3) and is illustrated by plotting Ht as a function of R1 and R2 (fig. 5.5). In all cases, increasing R increases Hi and decreases Hj, where i = j. This occurs because resource 1 (or resource 2) reduces the forager's consumption of resource 2 (or resource 1) through both external and internal handling times. By handling an item of resource 1, the forager spends less time looking for food. The external handling time is independent of the forager's overall harvest rates on resources 1 and 2. However, the internal handling time increases with harvest rates and

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