Optimal Diet Model

Optimal Diet Model—originally formulated in 1966 by the American ecologists Robert H. MacArthur and Eric R. Pianka in The American Naturalist and cast into its canonical algebraic form (the prey model or contingency model) by Eric L. Charnov in 1976—is the foundational mathematical theory of behavioral ecology and optimal foraging theory that predicts diet breadth and prey selectivity in predators, herbivores, and parasitoids. Assuming that natural selection shapes foraging behavior to maximize the long-term net rate of energy intake per unit foraging time while partitioning a predator's activity budget into mutually exclusive searching time (T_s) and handling time (T_h), the model ranks all potential prey types i = 1, 2, ..., n within an ecosystem by their individual profitability (e_i/h_i), defined as net assimilable energy gain e_i divided by handling time h_i. From C. S. Holling's (1959) multi-species disk equation, the MacArthur–Pianka–Charnov theory derives three famous, empirically testable predictions: (1) the Zero–One Rule (a prey type should either be attacked on 100% of encounters or ignored on 100% of encounters—never taken with partial preference), (2) the Independence of Low-Value Prey Abundance (whether a lower-ranked prey type is included in the optimal diet depends only on the encounter rates of more profitable prey types and is independent of its own abundance), and (3) Diet Specialization During Resource Booms (foragers become specialists when high-value prey are abundant and expand their diet into generalists when preferred prey become scarce).

Evolutionary Economics of Foraging Time Budgets

Prior to 1966, dietary analyses in community ecology largely treated a predator's diet as a passive reflection of the relative abundances of prey species in the environment, modified only by morphological constraints such as gape size or bill shape. In twin landmark papers published back-to-back in the November–December 1966 issue of The American Naturalist, Robert H. MacArthur and Eric R. Pianka ("On Optimal Use of a Patchy Environment") and J. Merritt Emlen ("The Role of Time and Energy in Food Preference") introduced microeconomic optimization theory into evolutionary biology, launching the field of optimal foraging theory (OFT).

MacArthur and Pianka partitioned the total foraging time per item captured into two components that respond in opposite directions when a predator expands its diet breadth from the k most profitable prey types to k + 1 prey types:

1. Mean Search Time per Item Eaten (ΔS < 0): Adding a new acceptable prey type k + 1 increases the total density of acceptable food items in the environment, which reduces the average time spent searching between consecutive captures.
2. Mean Pursuit and Handling Time per Item Eaten (ΔH > 0): Because prey types are added in descending order of profitability, adding a less profitable prey type k + 1 increases the average pursuit, capture, and consumption time per unit of energy gained.

MacArthur and Pianka reasoned that the optimal diet breadth k* is reached at the threshold where the marginal decrease in search time (−ΔS) is outweighed by the marginal increase in handling time (+ΔH). Between 1971 and 1976, Thomas W. Schoener, H. Ronald Pulliam, John R. Krebs, and Eric L. Charnov translated MacArthur and Pianka's graphical marginal analysis into algebraic inequalities using Holling's (1959) functional response disk equation.

Partitioning Searching vs. Handling Time and Prey Profitability

Consider a predator foraging in a fine-grained habitat patch containing n distinct prey types i ∈ {1, 2, ..., n}. The predator's total foraging time T = T_s + T_h is divided into two mutually exclusive behavioral states:

• Searching Time (T_s): Time spent scanning or moving through the habitat searching simultaneously for all prey types. While searching, the predator cannot handle prey, and encounters prey of type i as a Poisson process at constant encounter rate λ_i (measured in encounters per unit of search time T_s).
• Handling Time (T_h): Time spent pursuing, subduing, consuming, and digesting an encountered prey item after deciding to attack it. While handling a prey item, the predator cannot search for other items.

Each prey type i is characterized by three parameters:

1. Encounter Rate (λ_i): Number of type-i prey encountered per unit of search time T_s (proportional to prey density D_i and perceptual detection width vw_i).
2. Expected Handling Time (h_i): Expected total time required to pursue, capture, and consume one encountered item of type i (normalized by attack success probability).
3. Expected Net Energy Gain (e_i): Assimilable caloric energy content of one item of type i minus the metabolic energy expended during handling h_i (assessed relative to the baseline metabolic rate of searching).

The profitability of prey type i is defined as the net rate of energy return during the handling process itself:

Profitability of prey i ≡ e_i/h_i (in J s⁻¹)

Without loss of generality, all n prey types in the environment are indexed in non-increasing order of their profitability:

e₁/h₁ ≥ e₂/h₂ ≥ e₃/h₃ ≥ ··· ≥ e_n/h_n

Multi-Species Holling Disk Equation for Rate of Energy Intake

Suppose the predator adopts a foraging strategy defined by a vector of attack probabilities p = (p₁, p₂, ..., p_n), where 0 ≤ p_i ≤ 1 is the probability that the predator attacks an item of prey type i given that it has just encountered one.

During a total search duration of T_s seconds:

• The number of type-i prey encountered is λ_iT_s, and the number attacked and consumed is N_i = p_iλ_iT_s.
• The total net energy gained across all prey types is E(T_s) = T_s∑_i=1ⁿ p_iλ_ie_i.
• The total time spent handling all captured prey is T_h = T_s∑_i=1ⁿ p_iλ_ih_i.
• Consequently, the total foraging time elapsed (search time plus handling time) is T = T_s + T_h = T_s(1 + ∑_i=1ⁿ p_iλ_ih_i).

Dividing total net energy gained E by total foraging time T cancels T_s and yields the multi-species Holling disk equation for the long-term average rate of net energy intake R(p):

R(p₁, p₂, ..., p_n) = (∑_i=1ⁿ p_iλ_ie_i)/(1 + ∑_i=1ⁿ p_iλ_ih_i)

(If the metabolic cost of searching c_s per unit time has not already been subtracted from e_i, it appears as −c_s in the numerator with e_i = E_gross,i − (c_h,i − c_s)h_i.)

Proof of the Zero–One Rule and the Optimal Diet Criterion

To determine the optimal attack probability p_j ∈ [0, 1] for any specific prey type j, Charnov (1976) differentiated the long-term energy intake rate R(p) with respect to p_j using the quotient rule. Let A_−j = ∑_i≠j p_iλ_ie_i and B_−j = 1 + ∑_i≠j p_iλ_ih_i denote the numerator and denominator contributions from all other prey types i ≠ j. Then:

R(p_j) = (A_−j + p_jλ_je_j)/(B_−j + p_jλ_jh_j)

Taking the partial derivative ∂R/∂p_j gives:

∂R/∂p_j = (λ_jh_jB_−j/(B_−j + p_jλ_jh_j)²) [(e_j/h_j) − (A_−j/B_−j)]

Because the sign of ∂R/∂p_j is independent of p_j itself, R(p_j) is monotonic in p_j on the interval [0, 1] and achieves its maximum at one of the two boundary endpoints (p_j = 1 or p_j = 0):

• Include Prey j Always (p_j = 1) if ∂R/∂p_j > 0, which occurs if and only if its profitability exceeds the average intake rate of the diet without j: e_j/h_j > A_−j/B_−j.
• Exclude Prey j Always (p_j = 0) if ∂R/∂p_j < 0, which occurs if and only if its profitability falls below the average intake rate of the diet without j: e_j/h_j < A_−j/B_−j.

Because prey types are ordered by decreasing profitability (e₁/h₁ ≥ e₂/h₂ ≥ ··· ≥ e_n/h_n), the optimal diet always consists of a contiguous block of the top k* most profitable prey types {1, 2, ..., k*}, where k* is the largest integer k such that the profitability of prey k exceeds the long-term intake rate R(k − 1) achieved on the top k − 1 prey types. Equivalently, prey type k + 1 is excluded from the optimal diet if and only if:

e_k+1/h_k+1 < R(k) = (∑_i=1^k λ_ie_i)/(1 + ∑_i=1^k λ_ih_i)

For the simplest two-prey system (most profitable Type 1 prey with e₁/h₁ > e₂/h₂ and less profitable Type 2 prey), Type 1 is always attacked (p₁ = 1), and Type 2 is dropped from the diet (p₂ = 0) whenever:

e₂/h₂ < (λ₁e₁)/(1 + λ₁h₁) ⇔ λ₁ > e₂/(e₁h₂ − e₂h₁)

This threshold encounter rate λ_1,crit = e₂/(e₁h₂ − e₂h₁) is the specialization threshold of the two-prey Optimal Diet Model.

Why Opportunity Cost Explains the Independence of λ_k+1

The second prediction—that the inclusion of a less profitable prey item (Type 2) is independent of its own abundance λ₂—was initially viewed by field ecologists as paradoxical, since animals in nature often consume common items. Charnov (1976) clarified why this follows directly from opportunity cost: when the predator encounters a single Type 2 item right in front of it, the decision is whether to spend the next h₂ seconds handling that item (which yields net energy e₂) or to ignore it and resume searching the environment for h₂ seconds (which yields expected energy R(1)h₂ from capturing and handling Type 1 prey). Whether there are 10 other Type 2 items or 10,000 other Type 2 items elsewhere in the habitat has no bearing on whether e₂ > R(1)h₂ during those h₂ seconds.

Classic Laboratory and Field Tests

The quantitative predictions of the MacArthur–Pianka–Charnov Optimal Diet Model have been verified in controlled laboratory and intertidal field experiments:

1. Great Tits (Parus major) on Conveyor-Belt Prey (Krebs, Erichsen, Webber & Charnov 1977): John R. Krebs and colleagues at the University of Oxford presented captive great tits with large (8-segment, higher profitability e₁/h₁) and small (4-segment, lower profitability e₂/h₂) mealworm pieces (Tenebrio molitor) moving past a perch on a motorized conveyor belt, allowing the experimenters to manipulate the encounter rates λ₁ and λ₂ independently. When the encounter rate of large prey λ₁ was low, birds ate both large and small prey non-selectively; as soon as λ₁ exceeded the theoretical threshold λ_1,crit, the birds selectively dropped small prey from their diet, and varying the encounter rate of small prey λ₂ by fourfold had zero effect on their rejection of small prey.
2. Bluegill Sunfish (Lepomis macrochirus) Foraging on Daphnia (Werner & Hall 1974): Earl E. Werner and Donald J. Hall fed bluegill sunfish three size classes of water fleas (Daphnia magna) at low, medium, and high total prey densities. At low prey density (50 per class), fish consumed all three size classes in proportion to encounter rate; at intermediate density, the smallest size class was dropped; and at high density (300–350 per class), fish specialized on the largest Daphnia size class.
3. Shorebirds (Tringa totanus and Haematopus ostralegus) on Intertidal Mudflats (Goss-Custard 1977): John D. Goss-Custard showed that foraging redshank (Tringa totanus) feeding on estuarine polychaete worms (Nereis diversicolor and Nephtys hombergi) ignored small worms as a decreasing function of the biomass density of large worms in the mudflat, regardless of the density of small worms themselves.

Why Real Predators Exhibit Sigmoidal Partial Preferences

While empirical tests confirm the ranking of prey by e_i/h_i and the dependence of specialization on λ₁ rather than λ₂, real predators rarely display an instantaneous mathematical step function from p₂ = 1 to p₂ = 0 at λ_1,crit; instead, they exhibit a steep sigmoidal transition (partial preferences, 0 < p₂ < 1). Subsequent extensions of the Optimal Diet Model identified four biological mechanisms that smooth the step function into a sigmoid:

• Cryptic Prey and Search Image Formation (Hughes 1979; Erichsen et al. 1980): When prey are camouflaged, a predator must spend recognition time t_r inspecting each encountered item to distinguish profitable prey from unprofitable mimics or twigs, introducing λ₂ into the recognition-time budget.
• Sampling and Imperfect Information in Changing Environments (McNamara & Houston 1985): Because real encounter rates λ₁ fluctuate over space and time, a Bayesian forager must occasionally sample lower-ranked prey or update its estimate of λ₁, producing apparent partial preferences over finite observation windows.
• Nutrient Complementarity and Toxin Constraints (Pulliam 1975; Westoby 1978): Herbivores and omnivores often require essential amino acids, sodium, or trace minerals absent from the single highest-calorie food item, or must limit intake of plant secondary metabolites from a single host species, shifting the optimal strategy from a pure corner solution to a mixed diet (analogous to niche complementarity and Tilman's R* resource ratio rule).
• Variance-Sensitive Foraging and Starvation Risk (Stephens & Charnov 1982): Under bet-hedging and risk-sensitive foraging theory, a predator facing an overnight starvation threshold optimizes its probability of surviving the night rather than its long-term mean intake rate R, accepting lower-profitability prey if they have lower handling-time variance.

Why is prey profitability defined as energy divided by handling time (e_i/h_i) instead of energy divided by search plus handling time?

Because the decision to attack or ignore an item is made at the instant of encounter, the search time required to find that specific item is already a sunk cost in the past. Once the item is right in front of the predator, the only remaining time investment required to obtain its energy e_i is the handling time h_i, so the marginal rate of return from attacking the encountered item is e_i/h_i.

Why does the Optimal Diet Model predict that abundance of a low-value prey item never affects whether it should be eaten?

Whether attacking an encountered low-value prey item (Type 2) increases or decreases the predator's long-term energy intake rate depends on whether the return rate during handling (e₂/h₂) exceeds the opportunity cost of foraging on high-value Type 1 prey (R(1) = λ₁e₁/(1 + λ₁h₁)). Because R(1) depends only on Type 1 parameters (λ₁, e₁, h₁), making unprofitable Type 2 items more abundant (λ₂ larger) never raises e₂/h₂ above R(1); eating them would still waste time that could be spent earning a higher rate on Type 1 prey.

When do real animals violate the Zero–One Rule and show partial preferences?

Real animals exhibit partial preferences (0 < p_i < 1) when one or more idealizations of the basic contingency model are relaxed: (1) when cryptic prey require finite recognition time t_r to identify after detection, (2) when encounter rates λ_i vary across space or during a bout as prey are depleted, (3) when physiological digestive bottlenecks or essential micronutrient requirements force dietary mixing (Pulliam 1975), or (4) when individual prey items within "Type 2" vary continuously in size or escape distance.

1. MacArthur, R. H., & Pianka, E. R. (1966). "On Optimal Use of a Patchy Environment." The American Naturalist, 100(916), 603–609.
2. Charnov, E. L. (1976). "Optimal Foraging: Attack Strategy of a Mantid." The American Naturalist, 110(971), 141–151.
3. Werner, E. E., & Hall, D. J. (1974). "Optimal Foraging and the Size Selection of Prey by the Bluegill Sunfish (Lepomis macrochirus)." Ecology, 55(5), 1042–1052.
4. Krebs, J. R., Erichsen, J. T., Webber, M. I., & Charnov, E. L. (1977). "Optimal Prey Selection in the Great Tit (Parus major)90064-1)." Animal Behaviour, 25, 30–38.
5. Pyke, G. H., Pulliam, H. R., & Charnov, E. L. (1977). "Optimal Foraging: A Selective Review of Theory and Tests." The Quarterly Review of Biology, 52(2), 137–154.