A comparative study of models for predation and parasitism

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Hence, the fitting of an empirical equation to observed relationships in certain subjects, and I imply that animal predation is one such subject, has a limited value theoretically. In the following, another method, i.e. analogies by attributes, will be explored. Normally, reasoning starts from a set of tentative propositions. This set of propositions is one kind of hypothesis. Because it is only tentatively assumed, it does not necessarilly and immediately postulate mechanisms underlying the subject. Often, an early, tentative hypothesis is a mere collection of all the factors that can be conceived, whereas what one can observe is the integrated complex of factors interacting with each other. It is, however, difficult in many cases to extract each component of the subject to compare with an assumed one purely by the observational method. It is possible, instead, to integrate the assumed components on a theoretical basis so that the assumption-system can be compared with the observed whole. The difficulty is that a mere list of components will not necessarily provide the method of integration. By some means, we have to assume the structure as well. It is at this stage that analogies can play a role, and it is the structure thus derived from analogies (or some known examples) that I call a 'model' here. When the model to adopt is determined, a method of calculating the model's attributes will follow. An analytic (i. e. mathematical) method can be used for the calculation, or the method commonly called 'Monte Carlo simulation' may be useful. Here, mathematics is used not as a convenient means of description, but as a means of inference. Now, we recognize two early stages of inferences; the collection of components, and the arrangement and integration of them by a tentative model. The tentative model may be called a hypothesis, but it should be borne in mind that it is only tentative and not more than a convenient assumption. Such tentative hypotheses do not enable us to postulate the mechanism of the subject. The tentative hypothesis, however, is now compared with observation and will in general need refinement, as it often does not agree with the facts with a desirable degree of precision. A refinement will be made through alteration of the arrangement, adding some more components which have previously been missed, etc. As the stage of refinement advances, the hypothesis would enable one to postulate more confidently. Finally, as the degree of agreement with the facts increases, the postulational hypothesis would eventually emerge as a theory or even a principle. There are three important points in the gradual process of inferences mentioned above; they will be discussed more in detail below. First is the formulation of a tentative hypothesis; second, the evaluation of agreement ~nd disagreement between the theoretical and the observed; third, the fact-observation relationship. The third one is a question of whether an observation can be accepted as fact. For the following discussion, some symbols will be used as defined below: 0 : the result of observation, Ko :the set of all major components involved (not particularly known) in the
observed system, So : the structure of the observed system, KA : the set of all assumed components in the model system, S~ : the structure of the model system, E : theoretical expectation deduced from KA and S.x. When E and 0 are compared, we will get either an agreement or a disagreement, i.e. E=O or Er respectively, to which various conditions (causes) contribute as below: Conditions C1. K~ and S~ are involved in Ka and So respectively (so that both K~4 and S~ are, at least, not false). cu. if KA and S~ are both sufficient, then E=O. c~2. if either KA or S:~ is inadequate, then E~ O. c~a. if O is false or inadequate under c~I, then E:~O. Cz. Ko does not involve the whole of K~, and/or So does not involve the whole of S.~ (so that KA and/or S.n are/is, at least partly, false), c21. if false parts of Ka and Sn, or false parts of O and K~, (or S:,z), are adjusted so that they cancel out each other, then E-O. c~2. if not c21, then Er Now, one can claim that his hypothesis is right only when c, under C~ holds. However, the fact that an agreement (E~O) exists is not sufficient to establish the hypothesis, since E=O also occurs when c~1 under C2 is involved. Therefore, if a comparison between E and O is the only available method, we have to be contented with an assessment of the relative credibility of these causes. The assessment can be done much the same way as for the calculation of the LAPLACIAN probability (see BURNSIDE 1928 ; POL~CA 1955). Let Pr {E=O} be the probability of event (E=O) taking place. As it takes place either when cu or when C~l is involved (the probability of which will be written as Pr {(E=O) ]c~} and Pr {(E=O) Icy} respectively), we get Pr{E=O} =Pr{(E=O) i c~} +Pr{(E=O) !c2~. Also, as C~l is dependent on C~, Pr{(E=O) [c,} =Pr~c,}Pr{C~} and similarly, Pr { (E= O) I ce~} =Pr {c~x} Pr {C~}. From these formulae, the following conclusions are drawn. If Ka is comprised of only those components which are either axiomatic, a priori (known to be true without appeal to the particular facts of evidence), or can be deduced from concepts already known to be true, Ka must be involved in Ko. In other words, Pr {C~} is high but Pr {C2~ is low. Therefore. if an agreement (E=O) was observed under these circumstances, Pr{(E-O) [cH[ is high as compared with Pr {(E=O) lc2~} ; i. e. the credibility of reasoning that the agreement is due to a right hypothesis is com-
Page 1 and 2: A COMPARATIVE STUDY OF MODELS FOR P
Page 3 and 4: the model. The mathematical model r
Page 5: etween the assumption (CHAPMAN's (1
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 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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A comparative study of models for predation and parasitism

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