Evolutionarily stable strategies in food selection models with fitness sets

Evolutionarily stable strategies in food selection models with fitness sets

J. theor. Biol. (1984) 109, 489--499 Evolutionarily Stable Strategies In Food Selection Models with Fitness Sets NILS CHR. STENSETH Department of Bi...

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J. theor. Biol. (1984) 109, 489--499

Evolutionarily Stable Strategies In Food Selection Models with Fitness Sets NILS CHR. STENSETH

Department of Biology, Division of Zoology, University of Oslo, PO Box 1050, Blindern, N-0316 Oslo 3, Norway (Received 25 February 1983, and in revised form 26 January 1984) Most current models for optimal food selection apply to ecological and behavioural optimization. I n this paper optimal food selection theory is extended to apply to evolutionary optimization. A general evolutionary model for optimal food selection must incorporate the concept of fitness sets--or that variables, changing as a result of natural selection in evolutionary time, cannot, in general, vary independently of each other. A "Charnov type" optimal food selection model with a fitness set is investigated, and evolutionarily stable strategy (ESS) solutions of the evolutionary variables (i.e., the efficiencies of using available food types) are found. From this analysis it follows that the relative frequency of various food types in the environment may, under specified conditions, influence the evolutionarily optimal diet. Secondly, the analysis demonstrates that a food type not in the optimal diet may, in evolutionary time, be added to this by becoming more abundant. Thirdly, it follows from the analysis that the ecological result of MacArthur and Pianka, that food types are worth eating even if there is competition for them, is not generally applicable when referring to an evolutionary time scale. Finally, it is pointed out that for the diet to be an ESS, it is necessary that the consumer's density is stable and that the consumer's population dynamics are subjected to some density-dependent factor.

1. Introduction Since the pioneering works of MacArthur & Pianka (1966) and Emlen (1966), a large n u m b e r o f optimal food selection models have been published (see reviews in Schoener, 1971; Pyke, Pulliam & Charnov, 1977; Hassell & Southwood, 1978). However, these models almost exclusively treat ecological optimization (referring to an ecological or behavioural time scale); evolutionary changes (referring to an evolutionary time scale) are generally not treated (but see Schoener, 1973, 1974a). The often-cited model developed by Charnov (1976) refers explicitly to the ecological or behavioural time scale. Unfortunately, many authors assume that the ecological conclusions from this and similar models are valid in evolutionary time too. 489 0022--5193/84/160489 + 1I $03.00/0

© 1984 Academic Press Inc. (London) Ltd.




In this paper I extend the "Charnov type" model (e.g. Charnov, 1976) to refer to evolutionary time, by applying the concept of evolutionarily stable strategies (or ESS) developed by Maynard Smith & Price (1973). The specific formulation developed by Lawlor & Maynard Smith (1976) (see also Reed & Stenseth, 1984) is used. I apply Maynard Smith's (1974) formulation of the "Charnov type" model, in which the connection between populations dynamics and food selection is made explicit (see Fig. 1).

Oplirnol foroging

demogrophic schedules /

ei, 19, ond T,

hi ond Et

r(x) ond y



d.x dtx

Moximizofion of resource OCCluisilion

l ESS- orgument

FIG. 1. Diagrammatic representation of the model components and how they relate to various parts of the consumer's biology. See main text for definition of the various quantities. In this diagram, I have indicated at what level the resource maximization argument (referring to ecological or behavioural time) for finding the optimal diet applies; this is the common approach in ecological food selection optimization. I have also indicated at what level the ESS-argument (referring to evolutionary time) for finding the optimal solution of evolutionary variables (in our case, the e*-values) applies: this is the approach applied in this paper. See text for further discussion. The figure is modified from Pianka (1976) and Stenseth (1977).

In evolutionary arguments of this kind it is necessary to incorporate the fact that, in general, various evolutionary variables cannot be optimized independently (e.g., Rosenzweig, 1974, 1979): they are constrained by some kind of fitness set (see Levins, 1968; Maynard Smith, 1978). That is, when a species in evolutionary time improves one aspect of its phenotype, it must, in general, do so at the expense of some other features. 2. The Model (A) DEFINITIONS Let x be the density of the species whose diet is studied. Its population dynamics are described by the model d x / d t = xr(x). Besides depending on



the population density, r depends on the evolutionary and behavioural strategy being chosen. For a particular evolutionary strategy, the ecologically stable equilibrium density giving d x / d t = 0 is denoted x*. Let hi be the density of food type i (one of several); ni is the number of attacks (per time unit) on such food items; E~ is the expected total amount of resources (e.g. energy) contained in one item; T~ is the expected proportion of available time used per food item of the ith type which is attempted taken (successfully or not); i.e. T~ is the length of time from the moment an animal decides to pursue an item of type i until it starts searching for another food item (of any type) ; p~ is the probability of pursuing an ith food item when such an item is encountered; and e~ is the efficiency of utilizing (i.e. capturing and digesting) the item, given that the consumer decides to pursue it when detected. The ei-quantities may take on all values between (and including) 0 and 1 ; e~ is the evolutionary variable to be studied in this paper. To simplify, I consider in the following only two food types. This is done without any loss of generality. Ecological (or behavioural) optimizations models usually aim at finding those p-values (called/~) optimizing net energy gain given that everything else is constant (for review, see Pyke et al., 1977). (Similarly, T~ may be optimized in ecological time; see, e.g. Maynard Smith, 1974.) Ecological optimization of this kind assumes constant ei-values; hence, it was sufficient for Charnov's (1976) purpose to consider the product e~E~ as such. Evolutionary optimization, on the other hand, consists of finding those values of the evolutionary variables, e, optimizing individual fitness (x -~ d x / d t ) , given that everything else is constant and in particular that the p-values (and the T-values) are at their current ecological optimum. (B) THE ESS-DIET

Natural selection operates on variations in the evolutionary variables, e~. For simplicity, I assume that no evolutionary changes occur on the food resources (i.e. no co-evolution); hence, El are treated as constants. The ESS-value of ei is denoted e*. The ESS-diet will then correspond to e* and /~i-/~(e*, e * , . . . ) ; the corresponding food densities are denoted A*. (C) THE F I T N E S S SET

Assuming that the evolutionary variables are constrained by a fitness set (Levins, 1968) implies that a consumer cannot, in general, specialize on the ith food type without losing some of its ability to utilize other food types.




Only those combinations o f el and e2 within the first quadrant below a fitness set function, g, defined as e 2 = g(e~) are possible combinations (Fig. 2); combinations of e~ and e 2 outside this region are biologically impossible. Only combinations o f e~ and e2 on the fitness set function, g, are o f interest in the present context as they are the ones which result in the highest possible net rate o f resource gain.

e2m , ox e2 ~ _ _ el

elm , ax

FIG. 2. A fitness set for a consumer species being exposed to two food types (1 and 2) is defined as all combinations o f e 1 and e2 falling below the g-function (the fitness set function) in the first quadrant; ej and e 2 combinations outside the fitness set are n o t biologically feasible combinations. Notice in particular that e~ and e 2 combinations in the hatched region are n o t possible combinations even though both e I and e 2 are less than their maximal possible overall values, e~,m~x and e2,raax, respectively.

Several authors (e.g., MacArthur & Levins, 1967) have correctly pointed out that if the consumer is exposed to dissimilar food types, the fitness set function (g) is concave (i.e. d2g/de 2 > 0); it is difficult for animals to have teeth capable for simultaneously using both meat and vegetation at a high rate. Otherwise g is convex. (D) T H E POPULATION D Y N A M I C S M O D E L

The rate of growth of the consumer density may be given as dx dt

x (K ~ einiEi-y)



(Maynard Smith, 1974, p. 121), where K is a constant conversion factor converting obtained energy into recruited adults; and y is the resourceindependent component of mortality. The term K ~ ejniE~ represents the resource-dependent net rate o f increase (birth minus death). Any o f the






quantities e~, n~ and 3, may incorporate density-dependent terms; see below. Maynard Smith (1974) demonstrated that equation (1) could be rewritten as

dx ( Kl ~" piAieiE~ --=x g i dt


1 + K1Y'. p,A,T~ 3'



where K~ is another conversion factor; K~-~ is in fact the half-saturation constant in Holling's (1959) Type II functional response curve. We need equation (2), rather than equation (1), in the subsequent argument since we need p~ explicitly incorporated; equation (1) is, however, biologically easier to understand. Equation (2) may be simplified as dx - x[f(e) - 3'] dt


where e = {el} and

Ki ~ piAieiEi f(e) = K


1 + gl ~ piAiTi



Charnov (1976, p. 142) (and, e.g. Engen & Stenseth, 1984) derived the quantity f(e) by a slightly different argument (see also Pulliam (1974) for an earlier derivation of an almost identical model). I shall call all food selection models based on an expression conceptually equivalent to equation (4), "Charnov type" models. 3. Results (A) TIlE


For an asexually reproducing population without age-structure, the ESSsolution, e*, is the one maximizing x-' d x / d t constrained by (x -~ dx/dt)* = 0 where (x -j dx/dt)* is evaluated at the ESS-value (see Fig. 3; see Lawlor & Maynard Smith, 1976; Read & Stenseth, 1984). Thus, for e* to be an ESS, it follows that f(e*) is the maximum o f f and that this maximum must be equal to y. (B) T W O



(a) Abundant food supply In the case of abundant food supply, the consumer spends most of its time (devoted to feeding activities) handling food items and little time


N, C, S T E N S E T H


r=r(x'(e'); e)

FIG, 3. An ESS-vatue e = e* is found by maximizing the specific growth rate x - ' dx/dt, of the evolving population with respect to the evolutionary variable subject to the constraint o f an ecological equilibrium density x = x*(e*).

searching for food [i.e. a consumer similar to MacArthur's (1972) "purs u e r " - - a species with a food item always in sight, so that ~ piA~T~>>1]. If an individual feeder only has to decide between "'taking a particular food item, i" or "'nothing" (see Engen & Stenseth, 1984) only one food type (the one with maximal e~E~/T~-value) is included in the optimal diet (e.g. Maynard Smith, 1974; Charnov, 1976). In this case, e* = ei. . . . and e* = 0 for all j % i. The population dynamics for the ESS-phenotype are then given by

dx dt=X

(e'E,) K2---~-i - 7


where K2 = KK~. The population consisting of only ESS-individuais will exhibit a stable population density if one or more of the following inequalities are satisfied: dT,./dx > O, d3~/dx > 0, and deJdx < O. Hence, for this case of abundant food supply, the theory predicts food specialization and an all-or-none strategy to be evolutionarily optimal. This is consistent with an extension of MacArthur's (1972) ecological conclusion.

( b ) Sparse food supply In the case of sparse food supply, the consumer spends most of its time searching for food items which take little time to handle and digest [i.e. a "searcher" feeding on easy-to-handle food types (MacArthur, 1972); p~A~L<< 1]. Under such circumstances natural selection will maximize

f,(e) = K ~ etA,E,. i


(The index, 1, on the f-function in equation (6) only indicates that it is a special case of the general expression given in equation (4).) Maximizing



f~ requires that the c o r r e s p o n d i n g e * - v a l u e s on the fitness set function, g, are such that de_.__~2I g'(e*)=del


, (e*l,e*z)




(see Fig. 4, a n d see also L a w l o r & M a y n a r d Smith (1976) for h o w e q u a t i o n (7) is derived f r o m e q u a t i o n (6)). E q u a t i o n (7) applies only to solutions where b o t h e* # 0 ; i.e. only for the o p e n interval o f possible ei-values. The possibility o f a c o n s u m e r specializing on one or the other f o o d type (i.e. e* = 0 for i = 1 or i = 2) m u s t be treated separately (Reed & Stenseth, 1984); this has b e e n d o n e in o r d e r to reach the c o n c l u s i o n s b e l o w (see Fig. 4(b)).


~ ~

eXr'e2lonXve~ ~}~






relaliveto k~


SmolI el, mal/~

\ (o)

k~ el


FiG. 4. Fitness set function, g, for a consumer being exposed to two food types (1 and 2) requiring (a) somewhat similar adaptations to be utilized, and (b) greatly different adaptations to be utilized. The parallel straight lines with negative slope given by -(A*EffA*E2) represent

adaptive functions (Levins, 1968) for a given consumer in a particular environment. Two such situations are depicted in both figures; one with relatively high abundance of food type 1, another with relatively high abundance of food type 2. The optimal feeding strategies are, in (a), given by (e*, e2*) and depicted for each environmental situation; this optimal strategy is, in (a), found by the point of the fitness set function, g, touching the adaptive function with the largest M =f~(e) (defined in the main text). In (b) the optimal solution (e/*, e*) is also found by the point on the fitness set function, g, touching the adaptive function with the largest M=f~(e); in this case the optimal e* is, contrary to the case depicted in (a), found as the most extreme e-values. See the main text and Fig. 3 for definition of other symbols. In this case o f sparse f o o d supply, the p o p u l a t i o n d y n a m i c s o f the E S S - p h e n o t y p e are given b y dx

- - = x ( K 2 E k * e * E , - 3')




which m a y satisfy the requirement o f a stable p o p u l a t i o n density if either



d3,/dx > 0 and/or dei/dx < 0 [see (a) above]. However, in this case of low food supply, the found e*-values may correspond to an. ESS-diet even though e* and 3' both are independent of the consumer's density, x; it is sufficient here to require that the consumer affects its food supply; i.e. dAJdx < 0. Thus, if a consumer is exposed to relatively similar (but not identical) food types so that a convex fitness set function applies (Fig. 4(a)), a generalized diet will result; MacArthur (1972) suggested such a feeding strategy to be optimal for "searchers" in general. However, if the consumer is exposed to more different food types so that a concave fitness set function applies (Fig. 4(b)), a specialized diet is predicted to be optimal. This is contrary to what is expected on the basis of MacArthur's ecological conclusion. 4. Conclusion and Discussion (A) C H A R A C T E R I Z A T I O N O F T H E E S S - D I E T

Predictions concerning the evolutionarily optimal set of utilization efficiencies, or the ESS-diet, may now be summarized. (1) High food abundance of all food types, all of which are relatively difficult to handle, leads to food specialization; anatomical features specializing on efficiently using one (or a few) food type(s) are favoured by natural selection. The relative density of the different food types plays no role in determining the evolutionarily optimal utilization efficiencies. Furthermore, in this case of abundant food supply a food type not in the optimal diet will not be added to this diet even if the food type becomes, in evolutionary time, more abundant. (2) For easy-to-handle food types, all of which occur in low density, generalization is predicted whenever the available food types are fairly similar (i.e. need similar adaptations in order to be used). Specialization is, however, predicted whenever the available food types differ greatly. Dietary changes over evolutionary time will occur gradually in the first case, but abruptly (relatively speaking) in the latter. In this case of low food supply, the relative densities of the food types play an essential role in determining the evolutionarily optimal utilization efficiencies. A food type which at present is not in the optimal diet may, in evolutionary time, be added to this diet if it becomes more abundant. (B) T H E I M P O R T A N C E OF T H E ESS A R G U M E N T

The strategy of maximizing the rate of net resource intake may not necessarily be the evolutionarily optimal diet (or ESS-diet). Or, "the diet



which maximizes fitness [or the ESS diet] may not be the diet that maximizes the rate of caloric intake" (Pulliam, 1974, p. 70). This is illustrated in Fig. 5. As pointed out above, the essential feature necessary for an ESS-diet to exist is density-dependence of the rates entering equation (l). Lack (1954) and Orians (1962), among others, have earlier pointed out the importance of density-dependence in evolutionary arguments. Even though such density-dependence has not been treated with sufficient care in the literature on optimal food selection theories, density-dependencies have often been considered implicitly in earlier works; and, to a certain degree Emlen (1968) does this explicitly.

r(e) I










FIG. 5. Differences between the approach ofsearching for strategies, eo, maximizing resource intake (a), and for strategies, e*, being an ESS-solution (b); the latter approach requires, in addition to representing a maximum for the f(e)-function (defined in the main text), that its maximum is equal to the death rate, y. There is no reason to assume e*= eo. In figure (a), the strategy eo cannot be invaded by mutants, e', if e ' < ea; the strategy eo can, however, be invaded by mutants, e', if ea < e' < eb. Similar comments apply for the strategy eb.


The effect of a competitor is often to reduce the density of the food types differentially. It follows from equation (7) that if a competitor takes, e.g., a great amount of food type l, thus reducing h*, an evolutionarily optimal diet for the considered consumer may in the presence of its competitor consist mainly (Fig. 4(a)) or exclusively (Fig. 4(b)) of food type 2. However, in the absence of its competitor, A* may on a relative scale be small, rendering food type 1 (and not 2) the preferred one. However, MacArthur & Pianka (1966) concluded that food types worth exploiting in the absence of competition are always worth eating even if there is competition for the




food types. It is not clear whether they referred to ecological or evolutionary optimization. [MacArthur & Wilson (1967), however, explicitly refer to ecological time.] If the conclusion derived by MacArthur & Pianka (1966) is restricted to ecological optimization, they are correct. If, however, their conclusion is related to evolutionary optimization (as some assume, e.g. Rapport, 1980), their conclusion is not generally valid; see sections 3(a) and 4(A). My conclusions are a supplement to Schoener's (1974a, b) who treated the evolutionary problem. To say that dietary changes may occur in evolutionary time as a result of competition does not preclude habitat shift (as Schoener concluded). Hence, our results are compatible; Schoener's treatment is less detailed on the feeding part, but says something about habitat selection; my treatment considers only the feeding strategy when the consumer is constrained to remain in a particular habitat. Great stimulus for the work reported here was given by participants in two parallel graduate courses on Evolutionarily Stable Strategies at the Department of Animal Ecology in Lurid and at the Department of Biology in Oslo, This work was in part funded by the Science Foundations NAVF (Norway) and NFR (Sweden). Among the many who have commented on the work reported here, I want to single out Steinar Engen; he is certainly the one who has influenced my thinking on "food selection" the most. REFERENCES CHARNOV, E. L. (1976). Am. Nat. 110, 141. EMLEN, J. M. (1966). Am. Nat. 100, 611. EMLEN, J. M. (1968). Am. Nat. 102, 385. ENGEN, S. & STENSETH, N. C. (1984). Theor. pop. Biol. (in press). HASSELL, M. P, & SOUTHWOOD,T. R. E. (1978). A. Rev. Ecol. Syst. 9, 75. HOLLING, C. S. (1959). Can. Ent. 91, 293. LACK, D. (1954). The Natural Regulation of Animal Numbers. Oxford: Oxford University Press. LAWLOR, L. R. & MAYNARDSMITH, J. (1976). Am. Nat. 110, 79. LEVINS, R. (1968). Evolution in Changing Environments. Princeton, New Jersey: Princeton University Press. MACARTHUR, R. H. (1972). Geographical Ecology. New York: Harper and Row. MACARTHUR, R. H. & LEVINS, R. (IX)67). Am. Nat. 101, 377. MACARTHUR, R. n. & PIANKA, E. R. (1966). Am. Nat. 100, 603. MACARTHUR, R. H. & WILSON, E. O. (1967). The Theory o f Island Biogeography. Princeton, New Jersey: Princeton University Press. MAYNARD SMITH, J. (1974). Models in Ecology. Cambridge: Cambridge University Press. MAYNARD SMITH, J. (1978). ,4. Rev. Ecol. Syst. 9, 31. MAYNARD SMITH, J. & PRICE, G. (1973). Nature 246, 15. ORIANS, G. H. (1962). Am. Nat. 96, 257. PXANKA, E. R. (1976). Am. Zool. 16, 775. PULLIAM, H. R. (1974). Am. Nat. 108, 59. PYKE, G. H., PULLIAM, H. R. & CHARNOV, E. L. (1977). Quart. Rev. Biol. 52, 137. RAPPORT, D. J. (1980). Am. Nat. 116, 324.



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