A comparative study of models for predation and parasitism

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64 in the binomial series, the variance is npq as against the mean which is np, and so the variance-mean ratio is npq/np=q (where n is the number of eggs laid, p the probability of a given host receiving one egg, and q=l-p). small, the variance-mean ratio, which is q=l-p, If p is not sufficiently must be smaller than unity. This refutes the common belief that if the ratio is less than unity the parasites concerned are of the discriminate type. The interpretation of frequency distribution of parasite progeny on hosts should therefore be made with the utmost caution. Another relevant point here is the use of the negative binomial series developed by BLISS and FISHER (1953). The expectation of the 0-term (the proportion of hosts receiving no parasite eggs) in the negative binomial series is given by Pr{0} = (1 +m/k) --k (4g. 5) where m is the mean number of eggs per host, which in my notation is n/X. Sub- stituting n/X for m in eq. (4g. 5) we have Pr {0} = (1+ n/kX) -k (4g. 6). Here k is a positive constant, and if k-~oo the negative binomial distribution coincides with the PoISsON distribution, and if k-~0, it becomes the logarithmic series (BLISS and FISHER 1953). When we choose an appropriate value of k, the negative binomial series describes various types of non-random distribution (excluding an even distribution). Substituting the right-hand side of eq. (4g. 6) for Pr{O} in eq. (3.20) we have z =X {1 - (1 +n/kX) - k} (4g. 7). GRIFFITHS and HOLLING (1969) proposed the use of eq. (4g. 7) for parasitism in which the parasite egg distribution is not random (or, more precisely, the variance- mean ratio is larger than unity). The negative binomial series, however, does not distinguish the type of underlying mechanisms involved, and so it is purely a descrip- tive formula. Although the negative binomial distribution is identical with what is known as the POLYA-EGGENBERGER distribution (IT(5 1963) which has a specific model structure, I found it difficult to relate this model structure to the process of parasitism. In passing, although GRIFFITHS and HOLLING (lOt, cit.) suggested the use of eq. (4g. 7) also for predation, it is not legitimate to do so. As already discussed in w 3, predation and parasitism models will not differ from each other in the form of the instantaneous hunting function, i.e. eq. (3. 1) holds for both cases as long as the prey or host density can be considered to be constant. In parasitism, the overall equation in general form is eq. (3.22), in which f(X) can be anything as long as it is not influenced by the pattern of encounters between parasites and hosts. Therefore, THOMPSON did not have to assume a particular form for f(X) but only needed to say that n was the number of eggs laid. It does not matter whether n is an observed value or an assumed function of X, as long as one can justify the assumption of the POISSON or similar distribution. Consequently, if f(X) Yt in eq. (3. 1) is used as the theoretical expectation of n, the following simul- taneous equations will hold, ,~n =f(X) Y, Jt
Az/An = (X-z)/X from which we get dz/dt = (X- z)f(X) Y/X and integrating we have z =X(1- e -f(I) r~/I) (4g. 8). Clearly, HOLLING'S introduction of the factor h, or IVLEV'S equation justified in a toss-a-ring model, does not influence the assumption of Po~ssoN-type encounters, and so eqs. (4c. 10) or (3.24) as specific forms of eq. (4g. 8) are obtained for these two cases respectively (n in STOY'S equation can for the same reason be replaced by f (x) Yt). In the NICHOLSON-BAILEY predation model, however, it is crucial to assume a particular type of f(X), because the evaluation of the overall hunting equation is influenced by f(X), even if encounters are made at random. As already shown, if f(X) is a linear function of X, the overall equation is coincidentally of the same form as THOMPSON'S, but if f(X) is of HOLLING'S type or IVLEV'S, the overall equation is eq. (4c. 9) or (3. 12), which are quite different from eqs. (4c. 10) and (3. 24) respectively. THOMPSON (1939) argued against NICHOLSON-BAILEY (1935) and stated that, while the NICHOLSON-BAILEY assumption of random searching was not justifiable, the fact that THOMPSON himself arrived at the same equation "merely illustrates the well- known fact that identical quantitative relationship may be developed from biologically different postulates, since these postulates are not, in their ontological significance, incorporated in the formula". Now it is clear that THOMPSON was mistaken in that he was comparing incomparables, i.e. predation and parasitism, and that the resem- blance does not signify anything. The ontological significance for the two postulates becomes obvious under general circumstances in which f(X) is not a linear function of X. WATT (1959), in his review of various predation and parasitism models, made similarly erroneous comments that the NICHOLSON-BAILEY and THOMPSON equations are identical, and furthermore, that THOMPSON'S equation should have a constant factor in front of the exponent, to express the efficiency of different parasites. The suggestion is nonsensical because the exponent n/X (in my notation) is just a straight- forward "mean number of parasite eggs per host" laid by all the parasite individ- uals for the entire observation period, and the mean number is a mean number no matter how efficient are the parasites concerned. plied by a constant factor signify ? 65 What does a mean number multi- A correct interpretation is as follows. (1) If n is an observed value, it should be observed under standard conditions in which the time of observation and the densities of both host and parasite populations, i.e. t, X, and Y respectively, are fixed (a standard may be determined conveniently); then differences between values of n for different parasite species reflect differences in efficiency between the species. (2) If n is an expected value, i.e. a theoretical expec- :tation when t, X, and Y are known, it should be replaced by f(X) Yt as in eq.
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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 and 62: 61 7; or 1968, figure ii. 2). It is
Page 63: and so zM=MX {1- (1-1/MX) n~} z =X
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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