DYNAMICS OF FIELD POPULATION OF PINE LOOPER
223
of nymphs per funnel. This was obtained by fitting the observed frequency distribution against the Poisson series and negative binomial.
Figure 6 shows that both distributions agree fairly well with the
observed distribution.
In &B(the Poisson series) P(r = x) = e-np npx,’x!, where E(3) = np = 0.73
and 4.2 for 1967 and 1962, respectively.
In 6-C(the negative binomial) P( g = x) = (k+$-l)px(l + p)-k-x, where E(x)
= kp = 0.73 and 4.2 for 1967 and 1962, respectively, and var . (g) = kp(1 + p)
= 1.08 and 7.2 for 1967 and 1962, respectively.
To study the dispersion of nymphs we used the coefficient of dispersion defined by Fisher (see Milne, 1964). This is the quotient of variance
and mean, c = $/it, which amounts to unity when the distribution is
random, and is more than unity when the objects are aggregated or
underdispersed.
If we sample from a Poisson distribution, the stochastic c has a normal distribution, with mean = 1 and variance s% = 2n/(n - l)a. Thus,
if the estimated value of c is outside the ran,ge 1 f 2sC, then the probability of the observed distribution being random is less than 5%. If
c > 1 + 21/{2n/(n -
As shown in Table V (columns 1-6) in all years studied the coefficient
of dispersion is more than unity and, moreolrer, in 8 out of 12 years its
value is outside the range. Consequently, th13 nymphs are more or less
aggregated, and it is to be expected that the negative binomial gives a
better fit.
The negative binomial distribution has been fitted to the observed
distributions each year, by taking E(g) = i i = (zxi)/n, and var . (x)
= s$ = { zx; - ( 1xr)2/n}/(n - 1). Two examples of this distribution
are given in Fig. 6-C. The “goodness of fit” of these results were tested
by the chi-square method (Table VI). It shows that in 1 out of 11 cases
there is a significant deviation (level of significance a = 0.05) where we
can expect this to occur once in 20 cases. Consequently, the agreement
is satisfactory.
The mean of a sample taken from a negative binomial, provided the
sample is sufficiently large, has approximately a normal distribution;
therefore, where the sample size is more than 50, the 95% confidence
limits can be given by two times the standard deviation of the
mean: it f 2sz (see Table V, columns 7 and 8).
then the objects ,are underdispersed.
-
c. DENSITY O F PUPAE
During the winters 1950-51 to 1953-54 inclusive, pupal density was
assessed in December, i.e. shortly after pupation, and in April, i.e. just
prior to emergence. In later years only the April census was made, because winter mortality of pupae proved to be very low, and moreover,
223
of nymphs per funnel. This was obtained by fitting the observed frequency distribution against the Poisson series and negative binomial.
Figure 6 shows that both distributions agree fairly well with the
observed distribution.
In &B(the Poisson series) P(r = x) = e-np npx,’x!, where E(3) = np = 0.73
and 4.2 for 1967 and 1962, respectively.
In 6-C(the negative binomial) P( g = x) = (k+$-l)px(l + p)-k-x, where E(x)
= kp = 0.73 and 4.2 for 1967 and 1962, respectively, and var . (g) = kp(1 + p)
= 1.08 and 7.2 for 1967 and 1962, respectively.
To study the dispersion of nymphs we used the coefficient of dispersion defined by Fisher (see Milne, 1964). This is the quotient of variance
and mean, c = $/it, which amounts to unity when the distribution is
random, and is more than unity when the objects are aggregated or
underdispersed.
If we sample from a Poisson distribution, the stochastic c has a normal distribution, with mean = 1 and variance s% = 2n/(n - l)a. Thus,
if the estimated value of c is outside the ran,ge 1 f 2sC, then the probability of the observed distribution being random is less than 5%. If
c > 1 + 21/{2n/(n -
As shown in Table V (columns 1-6) in all years studied the coefficient
of dispersion is more than unity and, moreolrer, in 8 out of 12 years its
value is outside the range. Consequently, th13 nymphs are more or less
aggregated, and it is to be expected that the negative binomial gives a
better fit.
The negative binomial distribution has been fitted to the observed
distributions each year, by taking E(g) = i i = (zxi)/n, and var . (x)
= s$ = { zx; - ( 1xr)2/n}/(n - 1). Two examples of this distribution
are given in Fig. 6-C. The “goodness of fit” of these results were tested
by the chi-square method (Table VI). It shows that in 1 out of 11 cases
there is a significant deviation (level of significance a = 0.05) where we
can expect this to occur once in 20 cases. Consequently, the agreement
is satisfactory.
The mean of a sample taken from a negative binomial, provided the
sample is sufficiently large, has approximately a normal distribution;
therefore, where the sample size is more than 50, the 95% confidence
limits can be given by two times the standard deviation of the
mean: it f 2sz (see Table V, columns 7 and 8).
then the objects ,are underdispersed.
-
c. DENSITY O F PUPAE
During the winters 1950-51 to 1953-54 inclusive, pupal density was
assessed in December, i.e. shortly after pupation, and in April, i.e. just
prior to emergence. In later years only the April census was made, because winter mortality of pupae proved to be very low, and moreover,