A comparative study of models for predation and parasitism

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34 the effect of Y. In order to show the point, let us assume that a and h in eq. (4c. 6) are the true estimates of the factors originally defined, whereas those in eq. (4c. 3) are superficial and are written a' and h' respectively. Then the fit of eq. (4c. 3) to data, which should be correctly fitted by eq. (4c. 6), means that a'Xt/ (1 + a'h'X) = aXYt/ (1 + ahX) and transposing we get h'=h/Y+ (a'-aY)/aa'XY or a'=aY/ (l +aX(h- h'Y) ). Thus, estimates a' and h' not only change as the predator density changes but also they are a decreasing function of the prey density, unless a'=aY and h'=h/Y. Fur- ther, they are not independent of the units in which the densities of both populations are measured. Hence, such estimates have no universal value. Secondly, the depletion of prey or, in the case of parasitism, superparasitism, often occurred in the observations. If eq. (4c. 3) is applied to such data, it is the same as fitting the instantaneous equation to one of the cross-sections parallel to the Z-Xo plane of the surface generated by eq. (4c. 9) in the case of predation, or eq. (4c. 10) in the case of parasitism. Obviously, the curve of a cross-section parallel to the Z-Xo plane, as in Fig. 3a, changes as Y or t changes, and therefore the estimates for factors a and h should change accordingly. Again such estimates have no universal value. The third point is concerned with a precaution that should be taken when prey density is changed by changing the size of the experimental universe, a method often adopted to obtain an extremely high prey density with relatively few individuals (e. g. MORRIS 1963; HAYNES and SISOJEVIC 1966). If this is done, however, it should be noticed that the predator density changes too. To assess z in this method, we measure a cross-section of the surface which goes diagonally across the xo-Yt plane. As far as I know, all the published literature in which HOLLING'S disc equation is applied, involves at least one of the above three misapplications. Yet, the goodness of fit is, in many cases, remarkably high. This is because the equation involves two factors that are normally estimated from the observed relationships between z and x0. That is to say, the nature of the estimation ensures that the resultant curve fits the observation well. Hence, a good fit does not constitute verification of the theory. In my opinion, the significance of HOLLING'S simulation experiment with discs and tapping is to show the importance of the factor h, and this simple model is sufficient to show that "the area of discovery", i.e. /i, cannot be independent of prey or host density, as clearly deduced from eqs. (4c. 9) or (4c. 10). However, an application of the original model as in eq. (4c. 3) to a situation in which the effect of diminishing returns is obviously involved, is incorrect.
35 d). The IVLEV-GAusE equation IVLEV (1955), in his study of the feeding ecology of fish, proposed an equation and used it rather extensively for the analysis of predation processes. One of IVLEV'S fundamental ideas which led him to the study of feeding ecology appears to have emerged from his dissatisfaction with one of VOLTERRA'S assumptions that the number of prey taken by predators is a linear function of the prey density. his book (pp. 20-21, English edition, 1961): IVLEV wrote in "... , according to VOLTERRA'S position, in the case of an unlimited increase in the concentration of the food material, there must also be an unlimited increase in the amount of the food taken. This 'unlimited' increase is a biological absurdity, since each individual is only capable of consuming a strictly limited quantity of food in each unit time." This criticism led IVLEV to propose a new hunting equation in the following quota- tion: "... the actual ration of food eaten by the predator over a certain period of time will, under favorable feeding conditions, tend to approach a certain definite size, above which it cannot under any circumstances increase and which also corresponds to the physiological condition of full saturation. Hence the mathematical interpret- ation of the given law takes a form which has been used fairly widely in quanti- tative biology and physical chemistry. If the amount of the maximal ration is taken as b, then the relation between the size of the actual ration u and the density of the prey population v must be proportional to the difference between the actual and maximal rations and can be expressed by du/dv =a (b- u), (4d. 1) where a represents the coefficient of proportionality. Integrating this equation, we get u =b (1 - e -~') ." (4d. 2) (The symbols are mine). Although IVLEV'S criticism of VOLTERRA is a correct one, there appear to be some ambiguities and confusions in the second statement quoted above. It is not clear what IVLEV was aiming at in these equations, because of inadequate definition of the symbols in the equations. As far as I know, it is not explicitly mentioned anywhere in his book, whether eq. (4d. 2) represents an overall hunting equation or an instantaneous relationship. IVLEV'S treatise covers a wide range of problems in feeding ecology, which includes problems in natural population and competition between species feeding on the same food resources. Needless to say, these problems cannot be solved unless the depletion of the resources is considered ; the notion of 'competition' in the VOL- TERRA-GAusE (as well as in the NICHOLSON-BA1LEY) line of thought would not have emerged without this fundamental phenomenon, the depletion of some essential requi-
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: 33 appear. This is perhaps because
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 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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