A comparative study of models for predation and parasitism

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74 However, the similarity in their shapes can be compared, if desired, by parallel translation of the relative position of the coordinate systems between the observed and hypothetical relationships, since the curves are drawn on the ln-ln scale. Then, the shapes of curves (a) and (a') in Fig. lla are comparable to a certain part of curve (1) in Fig. 12b. The scattergram (e) in Fig. llb resembles curve (3) in Fig. 12a, and so on. A strict comparison will not be attempted here for the above reason, however. It should be mentioned finally that fitting a straight line to these observed relationships may be justified only for the purpose of showing the declining tendency of the value of ~ with increasing parasite density. In other words, the only conclusion that one can draw from such linear regression analysis is restricted to the suggestion that social interference is involved among parasites. However, there is no justified basis for adopting the hypothesis that the relationship is linear. Also, the assumption of the 'quest constant' by HASSELL and VARLEY (1969) (see w 4h) is justified only in the linear regression analysis of those data in which host density is known to be constant: the assumption is, however, hardly justified for speculating about the stability of host-parasite oscillations in which host density is changing all the time. The possibility of stable oscillations induced by social interference among parasites is yet to be demonstrated on a more reasonable basis; until it is, the suggestion by HASSELL and VARLEY is only a possibility. Appendix to w 4i. Is the concept of 'area of discovery' useful in studies of predation and parasitism ? It has been shown in this paper that the concept of 'area of discovery', originally introduced by NICHOLSON (1933), cannot be used as a geometric attribute of the hunting process, since this simple, but highly hypothetical, concept involves a contradiction from the energetics point of view. But the concept has been shifted, as one way of expressing the hunting efficiency of predators or parasites, and has been widely used in the literature of population dynamics. The shifted concept is now defined as d in eq. (4b. 7) for predation and in eq. (4b. 8) for parasitism. The definition, however, is not a straightforward expression of hunting efficiency, as it is the logarithm of the reciprocal value of the survival rate, for a specified value of the initial density of the hunted species per hunter. My question here is whether this concept of 'area of discovery' is altogether useful in the study of predation and parasitism. Of course, the concept has played a significant role in its original context as a species specific constant under a given condition. But, once the original meaning of this index as a species specific constant is lost, what does the shifted concept signify ? Is there any particular advantage in using this index in shifted, and more general, situations ? In order to answer these questions, the index d will be evaluated in various models reviewed in this paper and will be compared with the instantaneous hunting function on which each model
75 is based. As the evaluation of the//is in general different between models for predation and parasitism, I shall use a symbol //, for predation and i/2 for parasitism, so that: //1 : 1 x0 In _r Xo--z ii2=l_ln X X-z" Also, the expression f(x, Y) will be used as a general form of the instantaneous hunting function; x should be replaced by X for parasitism. 1. The LOTKA-VOLTERRA model The definition of //1 does not fit here, since the model takes into account changes in the densities of both predator and prey populations during the hunting period, t. In other words, first, the value of Y, defined as a fixed predator density during t, does not exist, and secondly the value of z is influenced by mortality in the prey population due to factors other than predation. Under these general circumstances, the redefined concept of 'area of discovery' just does not exist. If, however, it is as- sumed as a specific case that mortality in the prey population does not occur except by predation, and that predator density is fixed during t, the model converges to the NICHOLSON-BAILEY model. 2. The NICHOLSON-BAILEY model f(x, Y)=ax //1 =at. 3. HOLLING'S disc model f(x, Y)=ax/(l+ahx) //1 :at-ahz/Y //2-at/(l+ahX), under the THOMPSONIAN assumption. 4. IVLEV'S model f(x, Y) = b (1 - e -~') //1 = - (1/Y) In [-1 + (1/axo) In { (1 - e- axo) e- abYt + e- aXo } ] 5. WATT's model Same as IVLEV'S, but a~ayl-~. iiz-bt(1-e-~x)/x, under the THOMPSONIAN assumption. 6. ROYAMA'S model in w 4e f(x, Y) =a (x) x Yt/{1+ a (x) hx} where a (x) is defined in p. 46, w 4e. 7. THOMPSON'S model ~i2=n/XY, //1: may be evaluated from eq. (4e. 9), but since its analytical solution with respect to z is difficult, the evaluation will not be attempted here. 62 =a (X) t~ {1 +a (X) hX}, under the THOMPSONIAN assumption.
Page 1 and 2:
A COMPARATIVE STUDY OF MODELS FOR P
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the model. The mathematical model r
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etween the assumption (CHAPMAN's (1
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observed system, So : the structure
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the notions 'realistic' and 'unreal
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11 situation is easily seen from Fi
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As the prey density is reduced from
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15 assumption that f(x) is a linear
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17 of social interactions among pre
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19 and unparasitized hosts and on t
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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 and 52: 5T dency towards a number of miniat
Page 53 and 54: .... -o ~ 300 O 1.1.1 N I.-- 9 0BS.
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: 73 the curve must be decreasing for
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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