A comparative study of models for predation and parasitism

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24 This problem will not be discussed any further in this paper. The solution of simultaneous eqs. (4a. la) and (4a. lb) under the assumption of discrete generations was not considered by the original authors. The solution, as I will show in the next section, is in fact possible and is related to the NICHOLSON-BAILEY model. b). The NICHOLSON-BAILEY model This model is known as the 'Competition model'. It is, primarily, constructed for the purpose of demonstrating NICHOLSON'S hypothesis that animal populations are in the state of balance fluctuating around a steady density of each species concerned, and that this steady density (or steady state) is brought about by competition among the members of the parasite species (NICHOLSON 1933). NICHOLSON with the collabo- ration of a mathematician, BAILEY (NICHOLSON and BAILEY 1935), intended to con- struct a model on the assumption of a very simple, idealized situation, concerning a theoretical relationship in densities between host and parasite species. By altering conditions in this simple model, they drew numerous conclusions about the mode of existence of steady states. Whether or not NICHOLSON'S basic philosophy that animal populations are in the state of balance is a useful one, is not of concern here. It is more important to determine whether the basic premises in the NICHOLSON-BAILEY theory can produce a reasonable model for parasitism, so that a comparison between the model and observation can provide any useful direction. original authors. The following is the reasoning by the Let x0 be the number of objects (hosts) originally present in a unit area, and let x be the number left undiscovered after an area s has been traversed (by all parasites concerned). Then the number of previously undiscovered objects discovered in a fraction of area traversed, i.e. ds, is xds. This must be equal to the decrease, -dx, of the number of undiscovered objects per unit area, i.e. -dx--xds (4b. 1), and since x=xo when s=0, integrating eq. (4b. 1) for the range (0, s), and hence (x0, x), we obtain X ~Xoe-* from which we have z/xo: 1- e-' (4b. 2) where z=xo-x. Factor s is called by the authors the 'area traversed', which is the area that is searched effectively by all parasites involved and includes possible overlaps. In passing, the average area traversed by each individual parasite is called the 'area of discovery'. As against the area traversed, the net total area searched by all parasites concerned is called the 'area covered', which excludes areas already searched. Thus, the right-hand side of eq. (4b. 2) shows that the proportion of the 'area covered' increases only asymptotically as the 'area traversed' increases, and that therefore the
number of hosts attacked in terms of a proportion of the initial number present per unit area, i.e. Z/Xo, increases only asymptotically. Hence, the equation shows a simple example of the law of diminishing returns. 25 As it is important to understand the geometric meaning of the above equation in order to see if the assumptions involved are reasonable, an illustration will be given. Before doing so, however, it should be pointed out that NICHOLSON and BAILEY failed to recognize the distinction between the predation and parasitism processes. For the reason already given in w 3, the differential equation as in eq. (4b. 1) is a starting point of deduction in the predation process, whereas NICHOLSON and BAILEY were aiming at constructing a parasitism model. Since I am examining the reasoning of NICHOLSON and BAILEY, their differential equation as a means of deduction has to be taken seriously. Since their reasoning is based on this differential equation, it is unreasonable to use the word 'parasite', and hence, for the remaining part of this section, I shall use the word 'predator' instead. Although the NICHOLSON-BAILEY equation can be regarded as one for parasitism because, as pointed out in w 3, an equation for predation can take the same form as one for parasitism under a particular assumption, the maintenance of consistency between terminology and reasoning is more important here. The case in which the NICHOLSON-BAILEY equation is considered to be a parasitism model will be discussed in w 4g. Suppose a number of prey individuals are scattered at random over a plane where one predator searches with an average speed V, completely independently of the distribution of the prey individuals, from point A to B (see Fig. 4). The path of the predator between A and B is assumed to be rectilinear, and all the prey individuals in the plane remain stationary. (It can be shown that an irregular path may be assumed without influencing the conclusion, or that there is no need to assume a stationary distribution of prey individuals.) As in Fig. 4a, each prey individual has an area around it within, and only within, which the predator can recognize the prey. To simplify [" 9 9 . (a) (b) Fig. 4. A geometric interpretation of the NICHOLSON-BAILEY (1935) model. For explanation see text.
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: course an instantaneous hunting fun
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 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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