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Abundance of Resource 2, Rj

Figure 11.2. Functional response of an optimal forager exploiting two prey species that contain different mixes of two essential nutrients. (A) Consumption of resource (prey type) 1 with a fixed abundance of resource 2. (B) Consumption of resource 1 with a fixed abundance of resource 1. Abrams (1987) argues that consumer fitness should be an increasing function of the following quantity: minimum of {k1C1 R1 + feC2R2, f (1 - WC1R1 + (1 - k2)C2R2}, where Ri is population density of resource i, Ci is the attack rate on resource i, ki is the proportion of nutrient a in resource i, (1 - kO is the proportion of nutrient b in resource i, and f is the ratio of nutrients a and b required in the diet for the consumer to survive. Because the consumer needs both nutrients a and b, fitness can be assigned to the consumer only by determining which nutrient is limiting. To do this, the amount of each nutrient being consumed must first be compared, taking into account the ratio necessary for survival. The first term of the equation, (k1C1R1 + k2C2 R2), is the amount of nutrient a that the forager is consuming. The second term of the equation, f (1 - k1)C1R1 + (1 - k2)C2R2, is the amount of nutrient b the forager is consuming, but multiplied by f, which uses the ratio of nutrients necessary for survival to convert nutrient b into the equivalent units of nutrient a. Whichever amount is smaller is the nutrient limiting the consumer; therefore, the fitness of the consumer is the minimum of the first or second term in the equation. (After Abrams 1987.)

show considerable variation in encounter rates with prey, and this variation may influence overall population dynamics. Moreover, by focusing on individuals, one can explore the implications for population dynamics of features of foraging behavior such as sampling, learning, and state dependence (e.g., dependence of foraging decisions on hunger). Focusing on the behaviors of discrete individuals as a basis for developing population models is a "bottom-up" approach to modeling. The development of high-powered computers has allowed the ready exploration of models that incorporate the rich detail

Figure 11.2. Functional response of an optimal forager exploiting two prey species that contain different mixes of two essential nutrients. (A) Consumption of resource (prey type) 1 with a fixed abundance of resource 2. (B) Consumption of resource 1 with a fixed abundance of resource 1. Abrams (1987) argues that consumer fitness should be an increasing function of the following quantity: minimum of {k1C1 R1 + feC2R2, f (1 - WC1R1 + (1 - k2)C2R2}, where Ri is population density of resource i, Ci is the attack rate on resource i, ki is the proportion of nutrient a in resource i, (1 - kO is the proportion of nutrient b in resource i, and f is the ratio of nutrients a and b required in the diet for the consumer to survive. Because the consumer needs both nutrients a and b, fitness can be assigned to the consumer only by determining which nutrient is limiting. To do this, the amount of each nutrient being consumed must first be compared, taking into account the ratio necessary for survival. The first term of the equation, (k1C1R1 + k2C2 R2), is the amount of nutrient a that the forager is consuming. The second term of the equation, f (1 - k1)C1R1 + (1 - k2)C2R2, is the amount of nutrient b the forager is consuming, but multiplied by f, which uses the ratio of nutrients necessary for survival to convert nutrient b into the equivalent units of nutrient a. Whichever amount is smaller is the nutrient limiting the consumer; therefore, the fitness of the consumer is the minimum of the first or second term in the equation. (After Abrams 1987.)

of individual foraging behaviors. Individual-based models have burgeoned in popularity (Grimm and Railsback 2005; Schmitz 2001). For example, Turner et al. (1994) modeled individual elk (Cervus elaphus) and bison (Bison bison) foraging in Yellowstone National Park. The landscape of the model was a grid with features matching spatially explicit data describing the Yellowstone landscape. The model tracked individual elk and bison as they foraged across the landscape under different winter conditions and fire patterns. These authors concluded that the proportion of elk and bison that could survive a severe winter depended on the spatial pattern of fire in the landscape, a conclusion that gives crucial guidance to park managers. Individual-based models made it easier to incorporate realistic spatial information about the landscape and details of individual foraging behavior. Individual-based models also permit investigators to explore the implications of alternative scenarios.

As with the bison and elk foraging model, most individual-based models begin with the investigator giving each individual a set of rules that define its behavior, position in space, and fate through time. These models typically represent space explicitly because each individual occupies a specific position. The computer takes these rules and applies them, individual by individual, to project the state of the system through time. Individual-based models commonly use probabilistic rules ofindividual behavior, which build stochasticity into the system automatically. We will describe an individual-based model for predator switching below.

Individual-based models do have disadvantages, however. To draw inferences from individual-based models, one must compute averages over large numbers of simulation runs; in complicated models, this makes it hard to survey the available parameter space thoroughly. In addition, the complexity of individual-based models makes it difficult to deduce which features of the system account for a particular observed outcome. Individual-based models can become so complex that they become a world unto themselves, requiring so much effort to understand that they distract from the model's original goals. It can be very useful to use a hybrid approach that combines "bottom-up" and "top-down" approaches. Several studies illustrate the benefits of such hybrid approaches (e.g., Keeling et al. 2000; Illius and Gordon 1997).

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