The Marginal Value Theorem

The Marginal Value Theorem (MVT) is an analytical tool for optimizing Benefit to Cost Ratios. In Behavioral and Evolutionary Ecology, the MVT has been used to model such diverse phenomena as optimizing the tradeoff between egg size and egg number (Smith & Fretwell 1974),  optimal courtship persistence and mate guarding (Parker 1974, Parker & Stuart 1976), and optimal foraging (Charnov 1976).

Here is a general form of the MVT, which can be applied to a variety of situations. Assume that Benefit, B(C), is an increasing sigmoid function of Cost, C, and that Fitness, W, is defined as...

1.     W = B(C)/C

If B(C) represents offspring survival as a function of egg size, then W represents the product of offspring survival times offspring number (where ovary mass, a constant, divided by C, egg size, the decision variable, equals egg number). If B(C) represents net energy gain as a function of time spent foraging in a patch (or eggs fertilized as a function of time spent mate guarding), then W represents net benefit divided by total time, time foraging (or mate guarding) plus time spent searching for a new patch (or mate). 

The task is for the organism to choose a level of cost, C*, so as to maximize fitness. To solve for the optimal cost, we differentiate 1 with respect to C and set it equal to zero.

2.     dW/dC = (dB/dC) x (1/C) - B/C²

Which simplifies to...

3.     dB/dC = B/C

at C*. The left hand side of 3 is the instantaneous rate of return on investing cost, C, into benefit, B. 

Benefit - B

Cost - C

Straight lines through the origin represent fitness isoclines of the form, B = W x C, where fitness is the slope of the isocline; W = B/C.  Because fitness is the slope of the isocline, that isocline that intersects the the benefit curve, B(C), with maximum slope, W, indicates maximum fitness at the point of intersection, C*. Thus, the right hand side of equation 3 represents the fitness isocline.

The second derivative of 1, evaluated at C*, simplifies to...

4.     d²W/dC² = (1/C) x d²B/dC²,

which is negative if d²B/dC² is negative at C*. Thus C* is a maximum.

An organism is under selection to keep increasing C until dB/dC equals B/C, to then abandon its current benefit (egg, patch, mate), and to start over by investing in its next benefit.

References:

Parker, G. A. 1974. Courtship persistence and female-guarding as male time investment strategies. Behavior 48: 157-184.
Smith, CC & Fretwell, SD 1974. The optimal balance between size and number of offspring. American Naturalist 108:499-506. (link)
Charnov, EL 1976. Optimal foraging: the marginal value theorem. Theoretical Population Biology 9:129-136. (link)
Parker, GA & Stuart, RA 1976. Animal behavior as a strategy optimizer: evolution of resource assessment strategies and optimal emigration thresholds. American Naturalist 110: 1055-1076. (link)
Sargent, RC, Taylor, PD, & Gross, PD 1987. Parental care and the evolution of egg size in fishes. American Naturalist 129:32-46. (link)