A comparative study of models for predation and parasitism

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58 where ci and ce are integral constants. Hence, the following relationship should hold: f ~f,dN~-cl= f r~dP-c~. That is, the functions rl and r2 are not independent of each other. (2), an ecological condition that the function rl should reflect: since rl is a partial derivative ON~/ONo, it should reflect the limited capacity of each predator according to its maximum number of attacks, i.e.K. (3), an ecological condition that the function r2 should reflect: since r2 is a partial derivative ONA/~P, it should reflect the effect of both types of competition, i.e. the effect of diminishing returns and the effect of social interference. In the derivation of eq. (4f. 7), WATT evaluated the factor A arbitrarily, neg- lecting the first condition and part of the third condition, i.e. the effect of diminishing returns. Although it seems as though WATT was considering the effect of competition, i.e. condition (3), his consideration in terms of the factor A was concerned only with the effect of social interference. This is obvious because eq. (4f. 4) does not involve No, whereas the effect of diminishing returns should be a function of No. The neglect of the effect of diminishing returns in WATT'8 mathematics, and hence the failure to determine the function r2 in relation to r~, resulted in the contra- diction when eq. (4f. 7) was regarded as an overall hunting equation. My alternative interpretation of eq. (4f. 7) is therefore that it is an instantaneous hunting equation equivalent to eq. (3. 1); i.e. Na is not z, but n. It automatically follows that No is X, and this means that the density of the attacked species is kept constant during the attack period. Then P is Y. Now, K was defined by WATT as 'the maximum number of attacks made per P during an attack period' (though the expression 'per P' is not clear, this is perhaps 'per individual predator'). Then K corresponds to the expression bt used in my toss-a-ring experiment, i.e. b was the frequency of tosses per unit time so that the maximum possible number of attacks for time t per individual ring was bt (see w 4d). mine as above, except A, eq. (4f. 3) is rewritten as On/OX= A Y(Ybt - n) So, changing WATT'S notations to and eliminating A from the above equation, using the relationship A--aY-a (cf. eq. (4f. 5) in which P~ Y, and a-=a), we have n=b(1-e -aYI-pX) Yt (4f. 8). If we set ~=1, then eq. (4f. 8) becomes n=b(1-e -~) Yt, and this equation is identical to eq. (4d. 3), i.e. IVLEV'S equation with the addition of the attack period t and the density, Y, of the attacking species. While WATT stated that the structure of eq. (4f. 3) was intuitively obvious, it appears that intuition was a poor guide in this case, and the structure has become intelligible in the light of the toss-a-ring experiment. That is, eq. (4f. 7), which is equivalent to eq. (4f. 8), represents a generalized instantaneous equation of the toss-
59 a-ring model, m winch the area of the ring (t. e. a=a) diminishes, or increases, by the factor Y:-P, as the number of rings per unit area, i.e. Y, increases. Therefore the equation is considered to imitate the effect of social interaction incorporated into the effective area of each attacking individual; the effective area is now aY a-~ rather than simply a. Clearly, (1) if 0
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 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: 57 to z=F (xo, Y, t). Also one must
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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A comparative study of models for predation and parasitism

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