A comparative study of models for predation and parasitism

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62 hunting t, and z the total number of hosts parasitized per unit area by the end of that period. If encounters between parasites and hosts are at random (for the defini- tion of random encounter, see w 4e), a further increment of the number of eggs laid, i.e. An, will be distributed with equal probability among the unparasitized and the already parasitized hosts (this statement is not strictly true; see later). Then, letting Az be an increment in the number of hosts parasitized, the ratio Az/Jn must be proportional to the proportion of the hosts unparasitized, i.e. ,~z/,~n = ( X- z) IX and thus for An-~O, the following differential equation is obtained, dz/dn = (X-z)/X (4g. 1) and integrating, we get z = X(1 - e -~/x) (4g. 2). This is THOMPSON'S equation. Note that the meaning of the differential equation in eq. (4g. 1) is entirely different from that in eq. (3. 5) which represents a predation process. Therefore the derivation of a parasite model by means of a differential equation as above may be justified, but this is not inconsistent with my earlier statement in w 3. If eq. (4g. 2) is compared to eq. (3. 20), it will be found that the probability of a host individual receiving no parasite egg, i.e. Pr{0}, is equal to the expression e -'/z which is the 0-term of a PoIssON series. So it becomes clear that THOMPSON'S statement (in italics above) is not quite rigorous but requires an additional condition: the statement is correct if the number of hosts is sufficiently large so that the probability of a given host individual being found by each parasite individual is sufficiently small. In a laboratory experiment, however, the animals are often confined to a small cage, and, unless the density of hosts is sufficiently large, the condition which could ensure the PoIssoN distribution is not satisfied. So, let us look at the problem more closely from the probabilistic point of view. As already mentioned, the precise expression of 'random encounters' is that each host individual in the area concerned has an equal probability of being parasitized. Let this probability be p and the probability that a given host individual does not receive any egg be q, i.e. p+q=l. Then, the frequency distribution of nM eggs over the hosts in area M will be given by the following bionomial series, (p+q)'~ =q'~ +nMq~t-lp/1 !+ nM(nM- 1) q'~-2pz/2 ! + ...... +p~M where q~ is the proportion of hosts unparasitized, so that 1-q "~ is the proportion of the hosts parasitized. If the density of hosts is X, the probability of each host being parasitized (i. e. p) is equally 1/MX, and as q=l-p, we have q~ = (1 - 1/MX) "~. Also, the total number of hosts parasitized per area M (i. e. zM) is the proportion of hosts parasitized (i. e. 1-q "~) multiplied by the total number of hosts in area M (i. e. MX), thus
and so zM=MX {1- (1-1/MX) n~} z =X {1 - (1-1/MX) ~'~} (4g. 3). This is SwoY's model for random and indiscriminate parasitism (appendix to SALT 1932), and holds for all values of p; that is, STOY'S equation is more general than THOMPSON'S. In order to compare THOMPSON'S model with STOY'S, let us calculate the deriva- tive dz/dn in eq. (4g. 3) which is differentiable for all real values of MX larger than 1, and thus dZ (MX ln MX ) X-z dn MX- 1 X (4g. 4). If eqs. (4g. 4) and (4g. 1) are compared, it is at on ceclear that THOMPSON's differential equation is a special case of eq. (4g. 4) in which the expression in the brackets tends to 1. This occurs only when MX tends to infinity, i.e. lim MX ln{MX/(MX-1)} =1 MX~oo and MX~c~ means that p=I/MX->O. This is of course the well-known relationship between the POISSON and bionomial distributions; i.e. the PolssoN distribution is a special case of the binomial distribution in which p is sufficiently small. The above analysis illustrates that although THOMPSON'S reasoning, as it appeared in his differential eq. (4g. 1) under the assumption of random encounters, was appar- ently reasonable, it was in fact not sufficiently precise because it is not obvious that the equation requires the condition p~0. Here again a differential equation was used rather uncritically; it should have been noticed that calculus was not quite an appropriate method of reasoning in a parasitism model. The above argument, however, excludes the possibility that THOMPSON'S eq. (4g. 1) might be more appropriate than STOY'S if, paradoxically, random encounters are not assumed. That is to say, there is a possibility, though not demonstrated here, that some non-random encounters might again satisfy the condition expressed in eq. (4g. 1). One such example is given in my simulation model for parasitism in w 4e, in which the observed frequency was fairly close to that expected in a PoIssoN series, while the underlying mechanism is clearly of a non-PoIssoN type. TORII (1956) has already pointed out that an agreement between the observed and expected frequencies alone would not imply that the same mechanism is involved, as it is possible that different mechanisms could yield an almost identical frequency distribution. suggests that an agreement between the observed and the expected in THOMPSON'S equation could be entirely irrelevant to the test of the hypothesis of random encounters. Conversely, it may be suggested that the assumption of random encounters is not very important. 63 This In passing, we may note that TORII (1956) also showed that the index of disper- sion, i.e. the variance-mean ratio, which is unity in the PolssoN distribution, could statistically be less than unity if the binomial series was involved. This is because,
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A COMPARATIVE STUDY OF MODELS FOR P
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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
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: 61 7; or 1968, figure ii. 2). It is
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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