A comparative study of models for predation and parasitism

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44 vicinity of any one of the marked molecules must have higher chances of being encountered by the marked ones than those located remotely. However, if the number of marked molecules is high, all the unmarked molecules have an equal chance of being encountered by any of the marked one~. LAING (1937) and ULLYETT (1947) were aware of a part of the above point and stated, in their criticism of the NICHOLSON-BAILEY theory based on random searching, that a predator or a parasite can perceive the location of prey or host at a certain distance away from it ; the search therefore can only be at random (a free path or undirected path would be better words for random searching : my comment) as long as the predator or parasite remains outside this zone of perception;as soon as it enters this zone, its actions cease to be random and become, instead, directed towards the prey or host. THOMPSON (1939) thus wrote that a theory that equates animal action and random action covers at least only a fraction of the field, and that any theory based on the assumption that search is random cannot be accepted as a valid general theory. Although these criticisms sound reasonable, no attempt has been made, as far as I am aware, to find what and how much bias might be involved in a theory based on random searching as against the more likely situation raised by the above authors. I investigated this problem to some extent in a previous paper (RoYAMA 1966), but because it was written in a language not normally accessible to English-speaking readers, and also because I have since found some mathematical errors in it, all the points will be revised here. The investigation will be made with a situation in which a predator recognizes a prey from some distance. For the moment, it is assumed that only one predator is searching and that the prey density is kept constant. Suppose a number of particles of kind P (prey) are scattered at random over a sufficiently large plane (a two-dimensional space), each particle being given a circle of radius R as the recognition radius. One particle of another kind Q (a predator) can recognize P only within the area covered by the circles, so that Q's path is undirected when Q is outside the recognition area, but it is directed towards the nearest P within the recognition area. For simplicity, it is supposed that all P's remain stationary, but that Q moves around in the manner described. Also the radius R is the same for each P (although this assumption is not quite essential). At the moment, the time spent handling P's after they are caught by Q is not considered. Now, the starting point of Q will be determined on the plane at random. Then one or other of the following two cases wilt occur (see Fig. 5). Case 1: the starting point happens to be outside any of the circles of P's. Then Q determines the direction of its movement at random and moves along a (straight) free path until it contacts a P's circle; then goes to the centre of the circle to capture the P. Case 2: the starting point happens to be within one of the circles. Then Q immediately goes to the nearest centre, where the P is located, along the shortest path. A situation is first
45 Fig. 5. ROYAMA'S first geometric model in which a predator (Q) recognizes its prey (P's, black dots) within a circle around each P, and searching by Q is discontinued each time a P is captured. A starting ponit (cross) is determined at random for each new search. For further explanation see the text. considered in which Q discontinues searching each time it captures a P, and so the next starting point is determined again at random over the plane. This is perhaps comparable to a bird collecting food for its young, and each time a food item is found it is taken to the nest, and the next hunting starts independently of the previous search. Suppose n of P's were caught by Q during the total time spent in searching, ts, (note that the number caught can be smaller than the number seen). Let n~ be the number of occurrences of case 1, L1 the average distance that Q travelled between the starting point outside any circle and the periphery of the first circle that Q hap- pened to encounter, and G the average speed of movement while Q was on an undirected path. Similarly, let n2 be the number of occurrences of case 2, L2 the average distance between the starting point inside the circles and the nearest centre of the circles, and Ve the average speed of movement along a directed path. Then we have the following formula ts = ( L1/ VI +R/V2) nl q- ( L2/ V2) ne (4e. 1). Now, let Pr{nff and Pr{ne} be the probability that cases 1 and 2 occur respectively, then but since nl+n2=n, m=nPr{n,} n2 = nPr {n2}, n2 =n (1-Pr {nd), and so eq. (4e. 1) becomes
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 BAILEY but redefined in w 43 as
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 ~

A comparative study of models for predation and parasitism

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