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presumably due to an error in estimating either it or rumen ca pacity The original discussion of the moose s problem Belovsky 1978 was more Complicated than the version presented here Two kinds of terrestrial food were distinguished herbs and the leaves of de ciduous trees Herbs cannot be eaten as quickly as the leaves of trees but they yield a little more energy per unit mass Also ac count was taken of the danger of overheating in the midday sun Moose seem to have to spend part of the day on land and part in water to control body temperature in summer Since three kinds of food were being distinguished it was not possible to solve the problem by drawing a twodimensional graph like Fig 42 The equivalent graph was threedimensional with planes instead of lines representing the constraints Such graphs are inconvenient to draw but linear programming problems in three or more dimensions can be solved by algebraic methods see for instance K00 1977 in twodimensional problems the solution always occurs at the intersection of two of the constraint lines In three dimensional problems it occurs at the intersection of three constraint planes and in multidimensional problems at an intersection of the hyperplanes that represent the constraints The standard method of solution examines the intersections sys tematically avoiding any that are obviously not the optimum The three dimensional analysis indicated that in the optimum diet about 90 of the terrestrial food should be leaves of trees and about 10 should be herbs Moose were observed to eat these foods in almost exactly these proportions Belovsky 1978 also considered what the optimum diet would be if the aim were not to maximize energy intake but to minimize time spent feeding In that case the optimum diet would include no herbs 43 When to give up The larvae of ladybirds coccinellid beetles feed on aphids eat ing out the soft tissues and leaving the hard exoskeleton In the 72 l TlMUM BEHAVIOUR early stages feeding is quick and easy but as the meal continues the remaining food becomes more dif cult to extract How long should a larva persist with one aphid before leaving it to look for another It will be assumed as in section 41 that it is advanta geous to make the mean rate of intake of food as large as possible Larvae which feed fast will probably grow and develop faster than ones which feed slowly and produce offspring sooner Hence nat ural selection should favour behaviour which increases the rate of intake of food Figure 4 3ta shows the evidence that during a meal the remain ing food becomes progressively more difficult to extract Cook and Cockrell 1978 Ladybird larvae which had previously been starved for 24 h were given an aphid each to feed on Their meals were interrupted after various times and the remains of the aphids weighed Thus the graph for feeding on a single aphid was ob tained Notice that its gradient becomes smaller the rate of extrac tion of food falls as the meal proceeds The other graph shows that this was not due to diminishing appetite when fresh aphids were supplied at ten minute intervals feeding continued at an almost constant rate for fifty minutes Let a ladybird larva spend time t feeding on each aphid and let the mass of food it extracts in this time be mt The letter t in parentheses indicates that m is a function of t Let the mean time required to find a new aphid after leaving a partiallyeaten one be T The mean rate of intake of food Q is given by QmtTt 47 The optimum value of t is the one which makes Q a maximum and could be found by calculus if we had an algebraic expression giving m t as a function oft Such an expression could be chosen to fit the graph in Fig 43a that shows the time course of a typical meal on a single aphid Figure 4m3b shows a different easier method of nding the Optimum The curve is a graph of mt against t copied from Fig 4 3a Straight lines have been drawn from the point 7quot 0 to intersect the curve at various values of t The gradient of each Food 0 intake Succesmon A of aphids A A lt7 AA Single aphid A Time min Feeding timeimm 4O 20 O 20 40 Mean searching time mm c Figure 43 Graphs describing ladybird larvae feeding on aphids a Graphs of the dry mass of food extracted against time Graphs for a single meal and a series of ten minute meals on successive aphids are shown b A diagram based on the graph for a single meal showing how the optimum duration of the meal tnpll can be determined T is the time required to find an aphid c A graph of feeding lime 1 against search time T The points Show observed values and the line shows the theoretical optimum behaviour The points in a and C have been taken from Cook 81 Cockrell 1978 In c they are Shown one standard error of these lines is mit 39T t so the steepest possible line for any given value of T indicates the optimum value of t This line is the one which is a tangent to the curve Award 74 I TlMllM BEHAVIOUR Optimum values of 139 obtained in this way are shown in Fig 4 3c These values are only rough estimates different values would have been obtained if the smooth curve through the points in Fig 43a had been drawn slightly differently Figure 4 3c also shows the mean feeding times I actually used by ladybird larvae Between 2 and 32 aphids were distributed even ly on a tray 05 m square A ladybird larva was put on the tray and watched continuously for 4 h Any aphid it ate was replaced imme diately The time spent feeding on each aphid and the time spent searching for the next one were recorded Hence mean values of t and T were obtained The more aphids there were on the tray the shorter the searching time T and also the time i spent feeding on each aphid The observed values of t are close to the graph of theoretical optimum values in Fig 43c The ladybird larvae behaved approximately as the theory suggested they should A similar analysis has been applied to the mating of dungflies Sarcophaga Parker amp Stuart 1976 This is another situation in which persistence brings diminishing returns Females arriving at cowpats to lay eggs are intercepted by males who copulate with them although most of the females have sperm from previous matings in their spermathecae and could lay fertile eggs without further copulation The longer the male prolongs copulation the more of the sperm from previous matings are displaced and the larger the proportion of the offspring that are fathered by him This proportion increases at a diminishing rate just as the rate of feeding of a ladybird larva diminishes in the course of a meal To maximize his number of offspring a male should copulate for a particular optimum time and then go in search of another mate It was estimated from field observations using a graph like Fig 4 3th that the optimum duration for copulation in the particular circumstances was 41 minutes The observed mean duration was only a little different it was 36 minutes The theories for the ladybird larvae and dungflies are particular cases of the Marginal Value Theorem Charnov 1976b This the orem refers to situations in which food or some other resource is patchily distributed and exploitation of a patch gives diminishv 7quot 13 ing returns In the case of the ladybird larvae the patches were individual aphids and the resource was their food content In the case of the dungflies the patches were individual females and the resource was unfertilized eggs The theorem also assumes that when one patch is abandoned appreciable time is needed to find or travel to the next The theorem states that the rate of benefit is maximized by exploiting each patch until the rate of bene t from it falls to the maximum mean rate that can be sustained over a long period The theorem has also been applied to animals searching for hid den food Cowie 1977 studied great tits Pants major looking for mealworms hidden in small jars of sawdust The tits found the first few mealworms in each jar faster than they could find the re mainder the fewer the mealworms that remained the longer on average it took to find each one The tits moved on from each jar to the next at about the optimal times The theorem could also be applied to natural patches of hidden food For a bird feeding on caterpillars each plant on which caterpillars might be found would be a separate patch The theorem applies only if patches are searched at random If the prey are stationary and the predator searches each patch sys tematically never going over the same ground twice the number of prey removed has no effect on the rate of nding the remainder and the theorem does not apply 44 Ideal free ducks Imagine two people feeding the ducks in a pond One on the north bank throws them bread at a rate pieces per minute m The other on the south throws bread at a rate Vs The ducks divide into two groups nN of them going to the north and rig to the south The ducks in the north group get bread at an average rate rNnN and those in the south group at a rate VgVlg If mnN lt rgng a duck in the north group can hope to improve its rate of feeding by moving to the south and vice versa Therefore if each duck is trying to maximize its rate of feeding we should expect them