A comparative study of models for predation and parasitism

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7O i). A geometric model for social interaction among parasites (this study) In w 3, I introduced a function S, by which the effect of social interaction among attacking species upon the instantaneous hunting efficiency is indicated. Thus, if social interaction is involved, the instantaneous hunting equation for predation is given by eq. (3. 14) rather than eq. (3. 4). As already explained, however, the instantaneous equation for parasitism does not take the form of a differential equation as in eq. (3. 14), but, for indiscriminate parasites, it is expressed in terms of the number of eggs laid per unit area, i.e. n, as in eq. (3. 1). Thus, the equation for indiscriminate parasitism, equivalent to eq. (3. 14) for predation, will be written as: n=S(Y, X)f(X) Yt (4i. 1), and from eq. (3.22), we have an overall hunting equation for indiscriminate parasites as below: z=X{1-r (Y, X)f(X) Yt/X, V)} (4i. 2). In order to study some fundamental characteristics of the function S, I shall again use a geometric model similar to those used previously. Suppose, firstly, that a given parasite individual encounters, within an area 8 around itself, other parasite individuals in the course of hunting. these other parasites encountered within the area 8 is 0, 1, 2 ....... If the number of or i, the instantaneous hunting efficiency of the given parasite, i.e. f(X), is changed by factors 2o, 21, ~2 ...... or 2~ respectively. It is conceivable, as a more general case, that 2 is influenced not only by the number of parasites in the 8, but also by the number of hosts. This is because, as already pointed out in w 3, the effect of, say, interference might be strengthened or weakened if a lesser or greater number of hosts, respectively, is available within the 8. Thus, it is more appropriate to indicate the number of hosts too. Thus, 1~ is the index of the degree of social interaction when there are i parasites and j hosts within the area 8; it should be noted that both i and j take discrete values, 0, 1, 2 ..... , independently of each other. Secondly, let m (j) be the probability of finding j hosts within the 6. Then the average partial realization of the potential efficiency for a fixed value of i will be oo 2~:~o(j). Similarly, let o(i) be the probability of finding i parasites within the 3. j-0 Then the overall degree of changes in the instantaneous hunting efficiency for all j's and i's, i.e. S(Y, X), will be S(Y, X)= ~{ ~ 2~:~o(j)} o(i) (4i. 3). i=o j=0 Now, ~o is the probability-distribution function of j (and can be determined when both the average number of hosts within the area 8 and its variance are known). So the oo value of ~2~jco(j) can be determined for a given value of X and for each value of i. j-0 Therefore, if the value of X is fixed in the following argument, the expression co ~,~o(j) can be indicated simply by 2,(X). Then, eq. (4i. 3) will be written as j=o co S(Y, X) = ~ ,~,(X) o(i) (4i.4). i~0
71 Now, if interference is involved among parasites, the value of ~l~ must decrease as i increases while j is fixed, i.e. 2~j~2~+lj, and facilitation is indicated conversely by ,~ 2~+I(X), this indicates that the effect of interference outweighs that of facilitation, and vice versa. If all 2's are equally unity, this indicates that there is no social interaction, since from eq. (4i. 3), c~ co S(Y, X)=32,~o(j) ~o(i)=1. j=0 i=0 In order to make further investigations of the nature of the function S and its influence on eq. (4i. 2), it may be more convenient to assume a certain concrete form of the function 0. For this purpose, let us assume that 0 is a PomsoN distribution function, i.e. p (i) : e -~r' (8 Y') '/i ! (4i. 5), where 6Y' is the mean number of parasites within the area 6 around a given parasite individual (excluding the given individual), and 6Y' = 8Y/(1-e -~r) -1 (4i. 6). (For the derivation of eq. (4i. 6), see Appendix 3.) If we adopt at this stage the THOMPSONIAN model, i.e. eq. (4g. 8), as a concrete form for eq. (4i. 2): z :X(1 -e- {f(X) Yt/X} e -~Y' X (2~ (X) (5II')*/i !} ) (4i. 7). The evaluation of the // in eq. (4i. 7) is, from eq. (4b. 8), d= {f (X)t/X} e -at' X {,is(X) (6Y')'/i !} (4i. 8). Equation (4i. 8) is compared to eq. (4h. 1'), and if we take the logarithm of both sides of eq. (4i. 8), i.e. In ii =In {f (X) t/X} + ln[e-Sr'X {~, (X) (6 Y') '/i !} ] (4i. 9), and this equation is directly comparable with eq. (4h. 1) or with the curves in Fig. 11. The following are comparisons between eqs. (4h. 1) and (4i. 9), or between eqs. (4h. 1') and (4i. 8). First, while the value of In Q in eq. (4h. 1) is constant, the equivalent term (i. e. the first term of the right-hand side) in eq. (4i. 9) is a function of X; this term in eq. (4i. 9) can be treated as constant when the value of X is fixed, since the term is independent of Y. Secondly, while the second term of the right- hand side of eq. (4h. 1) is a linear function of In Y, and independent of X, the equivalent term in eq. (4i. 9) is not a linear function of In Y, and at the same time it is generally a function of X too; the term becomes independent of X only when 2,5 is independent of X for a given value of i. Thirdly, while the value of // in eq. (4h. 1) will increase without limit as Y decreases, the d in eq. (4i. 9) will converge to a finite value for a given fixed value of X, i.e. lim ?i =20 (X) f (X) t/X (4i. 10). Y~0 Now, I shall examine the shape of curves that are generated by eq. (4i. 9), and compare them with the observed data in Fig. 11. For the purpose of maintaining the generality of this model, the examination will be made analytically (i. e. mathemati- cally), and some concrete examples will be shown later. In order to find conditions
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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
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observed system, So : the structure
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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
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15 assumption that f(x) is a linear
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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 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: 9 of this fact, it is curious that
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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