A comparative study of models for predation and parasitism

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52 in Fig. 9, in which the total number of hosts parasitized, z, is plotted on the vertical axis against the total number of eggs laid, n, on the horizontal axis. It should be noticed, however, that this method of presentation is comparable to that in Fig. 8 for the predation model, showing the effect of diminishing return3. Here again, a wavy tendency is discernible, which shows that the occurrence of superparasitism is periodically increased in accordance with changes in the number of eggs laid. One important difference between the predation model and the present parasitism one is that the number of hosts freshly found hardly increased after 130 eggs had been laid in this example. This is because the parasite could not get out of the area already searched because no host individuals were removed from the area and the parasite repeatedly re-parasitized them. Also, it is noticeable that the observed number of hosts attacked was always lower than that expected in a random encounter, i.e. a binomial series (SToY's formula; see w 4g), because of excessive superparasitism. Although the indiscriminate parasite in this model is unable to avoid superparasitism, to continue searching in the area already searched is obviously inefficient. If, however, the parasite periodically moved to start a new search elsewhere, it would raise its hunting efficiency, and so would be favoured by natural selection. If the parasite is unable to know whether the area where it starts a new search has already been searched by itself or by other parasites, the shift of hunting area can be only at random. Of course, a random shift of hunting area certainly involves the risk of hitting an area which has already been searched by itself or by other parasites; but the parasite at least avoids the disadvantage of staying in an area which has just been searched by itself. In the second series of 'experiments', the parasite left the area after each five eggs had been laid and travelled for a certain distance in a randomly determined direction (Fig. 10 a). Again, a wavy trend is seen, but the observed number of hosts v : ~- OBS, 80 ....... THEOR. /o".-'" / __N nr ff~ 4O l.-- ffl 0 212 ~ 20 o 6 Z o 40 ~ 1oo 12o ~o No. OF EGGS LAiD (n) Fig. 10a. A result from the second series of Monte Carlo simulations of parasitism similar to the first series, except that a parasite, after every five eggs have been laid, moves 7. 5 unit lengths in a randomly determined direction.
.... -o ~ 300 O 1.1.1 N I.-- 9 0BS. o- ..... ~ THEOR. 20O nr" n U~ U') 0 "r 100 b- o Z / I I I I i i i i i J SO0 1000 No. OF EGGS LAID (n) Fig. 10 b. A relationship between z and n actually observed with a parasite species, Encarsia formosa G^~AN, laying eggs on its host species, Trialeurodes vaporariorum (W~sTW.), in BURNETT'S experiment. The solid line with black dots is the observed relationship and the broken line with open circles is the one expected from the binomial distribution (adapted from BURI~ET1" 1958; table IV). attacked is much closer to that expected by THOMPSON'S formula for a random distribution of eggs. Thus the efficiency was on the whole raised in the second experiment as compared with the first; it should be noticed that for the lower numbers of eggs laid, less superparasitism occurred than expected in a random distribution. Only two published sets of data are available to compare with my simulation model; in other published data, changes in the distribution pattern were not observed in accordance with the number of eggs laid. The first is BURNETT'S (1958) study of the distribution of the eggs laid by Encarsia formosa GAHAN on Trialeurodes vaporariorum (WESTW.), in which periodic deviations from a random distribution in accordance with the number of parasites searching is clearly shown. For comparative purposes, the number of hosts attacked is plotted against the number of eggs laid by all parasites (Fig. 10b, adapted from BURNETT'S table IV). A striking similarity, in the way that the observed curve deviates from the expected, is immediately apparent in Fig. 10aand b. BURNETT (Op. cit.) also showed a periodic deviation from a random distribution in accordance with changes in host density, rather than parasite density. Although I did not attempt to make any simulation experiment to compare with this experiment by BURNETT, the result may be deduced from Fig. 10a and b. Obviously, the curves shown in Fig. 10a and b are cross-sections of the hunting surface (as in Fig. 3a) parallel to the z-Yt plane, whereas the figure for BURNETT'S second experiment is a cross-section parallel to the z-xo plane. Now, the periodicity of waves (or wave length) is likely to be subject to change according to the initial prey density x0. That is, it is likely that the wave length becomes longer as x0 increases, since the
Page 1 and 2: A COMPARATIVE STUDY OF MODELS FOR P
Page 3 and 4: the model. The mathematical model r
Page 5 and 6: etween the assumption (CHAPMAN's (1
Page 7 and 8: observed system, So : the structure
Page 9 and 10: the notions 'realistic' and 'unreal
Page 11 and 12: 11 situation is easily seen from Fi
Page 13 and 14: As the prey density is reduced from
Page 15 and 16: 15 assumption that f(x) is a linear
Page 17 and 18: 17 of social interactions among pre
Page 19 and 20: 19 and unparasitized hosts and on t
Page 21 and 22: 21 HOLLING (1966). In order to main
Page 23 and 24: course an instantaneous hunting fun
Page 25 and 26: number of hosts attacked in terms o
Page 27 and 28: Now, if the prey density in the pre
Page 29 and 30: 29 the attack period is now possibl
Page 31 and 32: 31 In this scheme, the NICHOLSON-BA
Page 33 and 34: 33 appear. This is perhaps because
Page 35 and 36: 35 d). The IVLEV-GAusE equation IVL
Page 37 and 38: 37 big the container Was or if the
Page 39 and 40: 39 particles. As the average number
Page 41 and 42: 41 (dy/dt)/y = C (1- e-~) (4d. 6) w
Page 43 and 44: and BAILEY but redefined in w 43 as
Page 45 and 46: 45 Fig. 5. ROYAMA'S first geometric
Page 47 and 48: 47 mately true. Consequently, HOLLI
Page 49 and 50: 49 -- -- rr R~x Xo + (l/V2) [2fO(V'
Page 51: 5T dency towards a number of miniat
Page 55 and 56: 55 for a sufficiently long period o
Page 57 and 58: 57 to z=F (xo, Y, t). Also one must
Page 59 and 60: 59 a-ring model, m winch the area o
Page 61 and 62: 61 7; or 1968, figure ii. 2). It is
Page 63 and 64: and so zM=MX {1- (1-1/MX) n~} z =X
Page 65 and 66: Az/An = (X-z)/X from which we get d
Page 67 and 68: 67 0.200 Iol 0,100 0.050 0,020 0,0]
Page 69 and 70: 9 of this fact, it is curious that
Page 71 and 72: 71 Now, if interference is involved
Page 73 and 74: 73 the curve must be decreasing for
Page 75 and 76: 75 is based. As the evaluation of t
Page 77 and 78: 77 of Hierodula crassa GIGLIO-TOs.,
Page 79 and 80: 79 population caused by factors oth
Page 81 and 82: 81 'without experience'. If one use
Page 83 and 84: 83 justification of its use, see RO
Page 85 and 86: 85 which has been and Still is to a
Page 87 and 88: 87 TINBERGEN, L. and H. KLOMP (1960
Page 89 and 90: (1) assumption of a PoxssoN distrib
Page 91: 91 4. LOTKA, VOLTERRA, NICHOLSON ~

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