Res. Popul. Ecol. (1974) 15, 121--137.

AN APPROACH TO THE EDGE EFFECT IN PROOF

OF THE VALIDITY OF DICE'S ASSESSMENT

LINES IN SMALL-MAMMAL CENSUSING 1

Ryo TANAKA

Higashi-Jinzenji 17, Kochi 780, Japan

INTRODUCTION

Much attention is now focussed on the subject of edge effect among the population

ecologists of small mammals. Needless to say, they at tempt making progress of

methodology by which to determine justifiable assessment lines that del imitate the

addit ional boundary strip, on which trapping exerts actual effect, outside a sampl ing

quadrat or on both sides of a trap-line. It is desirable that the procedure to reach

the objectives should be simple as much as possible, for we will need, as a rule, to

determine dist inct assessment line in accordance with different population densities.

In the preceding paper (TANAKA 1972), I introduced a theoretical ground for

the supposit ion that no such edge effect as discovered from disproport ional ly large

catches per trap-stat ion at or near the outermost (edge) trap-rows of a quadrat, as

compared to those in inner square, would be l ikely to occur only if no animals shift

into the quadrat. Therefore, by way of helping confusion, here I shall discuss

exclusively the edge effect aside from the influence of invasion, i.e. shift of range

into a plot by an animal, which effect will be discussed distinctively.

Fur ther in 1972 I advocated DICE's assessment lines merely on the grounds that

his means is simple only if average range size is given to a population and that

there has been not yet any conclusive disproof of its validity, provided one overlooks

some information of the animal movement being of overdispersed pattern within

range.

In the second study for edge effect, nevertheless, I am led to the conclusion in

posit ive support of DICE's on the basis of trap-revealed range data of a vole

population, gathered in 1971, through the medium of notions of MARTEN (1972) and

WIERZBOWSKA (1972).

STUDY PLAN AND METHODS

The f ieldwork of mark-and-release in 1971 was executed at about the same time

of the year on the same plots (A, B) in the grassland within the enclosure of

Sugadaira Biological Laboratory, Tokyo Kyoiku University, Nagano Pref. in succession

to that in 1970 (TANAKA 1972). But the plan in 1971 was made rather s impler in

1 Contribution from JIBP-PT No. 167, carried out by the grant from the expenditure of Education Department to the specific study on "Dynamics of Biosphere".

122

that only the inner square (60 of the preceding work was used, the t rap

spacing ( interva l between trap-stat ions) being kept at 5 m through the study per iod

of ten days.

The predominant species (Microtus montebelli) fo rmed a large major i ty of al l the captured an ima ls ; all the handled voles, tota l ing 116, were regarded as adult

(20 g and above) in v iew of body-we ight cr i ter ion.

In this s tudy alike, the check of traps was made twice dai ly in the morn ing

and in the evening. The populat ion densi ty turned out to be reduced to two- th i rds

of that in the prev ious year ;hence the sample avai lable for range analys is was

much smal ler but it was never insufficient to fo rm conclusions.

As for the range analysis, stil l near ly the same means was adopted at the start

so as to bui ld up the provis ional aspect of cruis ing range for the present populat ion ;

namely the range contour, d rawn by del ineat ing a scatter of capture loci of a s ingle

vole, was g iven to every specimen, captured three t imes or more.

Es t imat ing of the populat ion parameters was done in quite the same way as before

except that all the t rapp ing days could be t reated as a single census period.

POPULATION PARAMETERS

Capture records f rom the two plots were pooled to es t imate several parameters

of the vole populat ion inhabit ing the grass land on purpose to raise the accuracy of

Table 1. Capture records and estimates of population parameters from the pooled data of both plots (A, B), see text as to explanation.

] A Date Time ~"i Pi ~i (m#FO (Aug. 13-22, 1971) (i) mi

I (13) { MC**EC

II (14) ( MC EC

III (15) ~ MC ( EC

IV (16) [ MC / EC

V (17) [ MC EC

VI (18) { MC EC

VII (19) f M C t EC

VIII(20) { MeEC

IX (21) [ MC / EC

X (22) I MC EC t

1 2

5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

Ui Ri

11 63 0 0 5 6 0 0 5 2 0 0 3 1 0 0 3 1 0 0 4 0 0 0 2 1 1 0 5 0 0 0 2 0 0 0 1 0 0 0

0 41 71 52 78 47 81 47

89 5O

88(1)* 64

91 (1)* 57 9O 5O

97(1)* 6O 94 69

0 73. O0 73. O0 84. OO

82. 10 _+ O. 32 89. O0

89.10 _+ O. 32 92.98 _+ 1.40

91. O2 _+ O. 15 94. 90 _+ 1.32

83. O0 96. OO

96. 05 _+ O. 24 97.70_+ 1.10 97.23 O. 50

99. O0 99. O9_+0. 3O 96. 61 1. O0 95. 40 3= O. 75

O. 986 1. 000 1. OOO

O. 977 + O. 004 0.999_+0. OO4 1.001 + O. 004 O. 999 :t: O. 015 O. 979 + O. 015 O. 999 _+ O. 014 O. 980 _+ O. 014

1.000 1. 001 _+ O. 003 O. 996 011 O. 985 _+ O. 012 O. 968 _+ O. OO5 1.001-+ O. 003 O. 965 _+ O. 010 O. 987 _+ O. 013

0. 56 0. 97 0.62 0. 95 0. 53 0.91 0.51

0. 98 0. 53 O. 95 0. 67 0.95 0. 58 0. 93 0. 51 0. 98 0. 62 0. 99

Total

* one animal dead ** MC for morning

42 74

in captivity in the group. check, EC for evening check.

123

estimation. The values in Table 1 were calculated by means of JOLLY'S equations (1965), because these are concerned with the recaptures alone, but the total popula-

tion at the initial time of sampling was estimated by the maximum likelihood

method conforming to the subsequent removal census equation :

C,= (N-S,_,)p. (1)

The notation for variables and parameters in the table is basically the same as

that in the previous paper (1972), but the interpretation is rather simplified according

to the present study plan as following;

U~ : number of new captures, at time i, that had never been caught before the

start of this study.

R~ : number of recaptures, at time i, which had been caught for the first time

before the start.

ms: number of the marked animals, which were handled for the period from

time 1 to i -1 irrespective of whether they belong to the U or the R group, in the

ith sample. /x

~: estimated total number of the marked animals in the population at time

i (i=2, 3 . . . . T - l ) .

/3 : estimated probability that one of the marked animals alive and released at

time i will survive till the time of capture in the i+ l th sample. A

rr~ : estimated probability that one of the marked animals alive at time i will be

caught in the ith sample.

As explained above and also in 1972, there exist two groups of captures which

are regarded as new in the meaning that they are not yet handled so far as the

present work goes, but those of U group are literally new and unmarked as contrasted

to those of R group that had already been marked one month or longer before the

initial day (Aug. 13). Since both groups respond to traps in quite different way

from each other, they are treated separately as objectives of census to reach the

total population at the start of sampling.

Let the initial size of U and R groups be No and Nn respectively, and Nu is

estimated by applying actual values of Us in MC only on each day to C~ of eq. (1),

where -Nv and P~o had better be determined by means of SUGIYAMA's method rather

than MORAN's (1951) or ZH'PIN's (1956) (see appendix); as for the present data of

R group including no captures in EC on each day, Nn and ~r~ (probability of

recapture) can be worked out by ZIPPIN's (Fig. 1).

As a consequence, the parameter estimates based on pooled data of both plots

were obtained as follows:

p~v=0. 17+0. 03 trio=0. 76

Thus the total initial population for both plots proves to be 123. How to change it

into density is the main subject of this article, which will be discussed later on.

124

15

50

25

N~ N~ 25 50 75

I I ] I I [ i I0 20 ~0 40 50

Fig. 1. Regression lines determined by the maximum likelihood estimates (Aro=49, A A p~a=O. 17) for U group and by those (NR=74, =_~0=0.76) for R group by following eq. (1) resting on combined MC data of both plots ; C~ from NR or No is on the ordinate and Si-i on the abscissa.

The values of ~, in Table 1 turn out to remain invariable both in series of MC

and in EC, averaging 0. 96 and 0. 57 respectively, and it is justifiable in the light of

the trapping design that these are approximate to those in Ser. 3 of the 1970 work.

It is also noteworthy that n~o of R group is on the order of a mean of n,-values /x A

for MC and EC, while P~a of U group is definitely smaller than any of =, as recognized in the previous work.

Let the rate of loss due to death and emigration in marked voles be ~, at time

i. The survival probability (rate), P~, is denoted as 1 -~, ; all of /6, except those

at times i=4, 15 and 17, are of no significant difference from unity. Since the

number of dead voles, confirmed halfway through the capture-recapture work, was

only three, the loss of marked animals will chiefly be ascribable to emigration.

Aside from the significant losses, the total number (Q) of lost voles works out at

20 according to the formula X{(R~+U~+~'~)-~ =Q.

On the other hand, the estimated survival number of marked animals holds at

100 or thereabout towards the end (times i=16 and 17) of study period. Thereby

the remainder of the total initial population (]~r=123) minus the cumulative number

of lost marked voles (0=20) must be coincident with the number of survivals at

the end (~16,17~100). Such was the case. The agreement demonstrates that the

total population estimates by means of eq. (1) are not incompatible with the estimates

of marked survivors reached by JOLLY's method.

HOME RANGES AS OUTLINES OF CAPTURE LOCI

As mentioned above, as the first step toward attaining natural range, a polygonal

range contour, that is formed by circumscribing an assemblage of capture loci, of

125

every vole captured three t imes or more was constructed on section paper. As the

second step, out of these voles, those specimens that were captured ten t imes or

more and survived within plots for seven days or longer, i.e., precisely speaking,

the interval between the first and the last Capture was seven days or longer, were

selected. Further , as the last step, those which satisfy such conditions that 80 per

cent or more of single-vole's ~ capture loci are distr ibuted on the trap-rows exclusive

of the edge rows (dashed lines in Figs. 2, 3), were selected out of the specimens

picked out at the second s tep .

The range outlines given to the last 24 voles (18 females, 6 males), selected

with the above rigid conditions, were considered to reveal almost entirely their

natural home ranges in terms of size and shape (Figs. 2, 3). These ranges must

be comparable to those of Type c in pat terns of range-concerned behaviors (TANAKA

1972).

Af ter the manner in the previous study, observed range length and width (ORL

and ORW) of the outl ined ranges of the select voles were measured on the conviction

T

i

J r

f l I

Fig. 2. Map of home range contours of the select specimens with such conditions that they show natural, entire ranges in Plot A: dashed lines stand for the edge trap-rows of the grid; thick contour lines for male ranges and thin lines for female ones; the distribution of capture loci (crosses for males, dots for females) is given to some voles' ranges.

126

I [ . . . . f Q ~0 20~

I Fig. 3. Map of home range contours of the select specimens in Plot B: other

illustrations are the same as in Fig. 2,

that these dimensions are most reliable on purpose to approach size and shape of

their natural ranges, with the result that these averages each denote no significant

difference from those for Type c in 1970 (Table 3 of TANAKA 1972). The respective

averages of ORL (2a), ORW (2b) and range area (nab) are as fol lows:

I Range area Eccentricity ~* ORL (m) ORW (m) (acres) (c/a)

Female 18 20,33_+1.20 I 12.57+0.76 0.05 0.79 7

Male 6 32.04+1.24 / 17.42_+1.33 0.11 0.84

* n : number of specimens

Doubtless, the above still offers evidence in support of the oblong shape of

natural ranges. The tendency for the range size of either sex to surpass slightly

that in I970 will reflect the reduced population density in 1971.

THEORETICAL APPROACH TO RANGE SIZE ON THE RANDOM-WALK ASSUMPTION

The mathematical methods of estimating range size, whether the range is presumed to be a circle or an ellipse, relying on bivariate normal distribution of points around a

single center of activity have proved to be inappropriate for mammal populations, parti-

cularly of small rodents (TANAKA 1963, 1972; SINIFF and JESSEN 1969, WIERZBOWSKA

1972).

127

Alternat ively, however, several means of assessing range size on the supposit ion

of uniform probabi l i ty of occurrence, to put it in another way, random visit to every

point, by an animal over its home range are furnished by Polish and Japanese

ecologists. Among these methods, WIERZBOWSKA's (1972) which was for the first

t ime in 1966 (ADAMCZYK el al.), seems to be of high uti l i ty in that it is very simple, whereas MORISITA's two methods which were worked out upon excellent idea are disadvantageous to us in that the process is very laborious. As these are not yet

open to the public as a paper (MoRISITA and MURAKAMI 1968), the first of these shall be briefly introduced subsequently.

It is founded on his supposed val id i ty of the empir ical equation as below :

s~=S~ (1 -e -b~)

where S, represnts intra-quadrat part of an animal 's home range in terms of

acreage, t its number of captures, st mean area counted from every combination of

t capture loci among a given assemblage of capture loci within quadrat, and b a

constant ; here the range model is considered a circle in shape and the range area

is determined by the inclusive boundary str ip method (STIcKEL 1953). By extra- polation of the observed values for st, St can be estimated, the est imate S~, thus

obtained, being comparable to, but more rel iable than, the range size which is

reached after a certain number of successive captures in the routine method (STIcKEL

1953).

Further, MORISITA proceeds to change S, into the range size, a imed at, by use of the following formula :

E(S~) "rch~L/(L+2h)32 when L>_2h

in which L is side length of the quadrat and h stands for radius of the circular

range, hence the range size being rchL The last equation implies that mean area of

the intra-quadrat part of animal ranges whose centers are located within the whole

area including the original quadrat and the addit ional boundary str ips h in width

(otherwise speaking, the enlarged quadrat is del ineated with DIcE's assessment l ines).

The second means of his is s imilar to the first in principle, but it is rather

s impler because its range model is assumed to be a square.

Two difficulties are considered in realization of his assumptions involved in the

first means: (a) the range size calculated by the inclusive boundary str ip method is

dist inct ly dependent upon the trap spacing in the grid (STICKEL 1953, TANAKA 1961) ;

(b) the home range that is located over the boundary of a quadrat is apt to shift

toward its interior in the process of sampling, as confirmed in the preceding study

(TANAKA 1972), SO that considerable overest imation of range size will be yielded by

appl ication of his last equation.

In WIERZBOWSKA'S method, it is, in effect, recommended to est imate range size of an animal from solving the ensuing equation so as to reach the value of r, that

is range size measured in terms of number of all the trap-stat ions situated within

128

its home range; this is referred to as moment method :

where variable X,,r represents number of different stations visited by the animal in

k successive capture s . The estimated range size in terms of acreage may be

calculated as S: rd 2, d being trap spacing in the grid. They state that averages of the realized values (x,) for X,,r, gathered from individuals whose range is located

in the interior of a quadrat, are available for practical range estimating. Further

they point out its advantageous feature that range size is assessed on the basis of

as few as several captures, without being affected by abnormal movements like sally

or by the supposition of range shape.

In spite of their statement, according to their z2-test, the fittness of the theore-

tical distribution of xk to the observed in their field data of a vole population is

verified only in a few cases of smaller k's (4~7 for females). The estimated range

size expressed as r-values may lead to indefinite results depending on the way of

arrangement of unit squares (d 2) of the grid.

Anyhow, the fundamental condition of the method is characterized by the

random distribution of animal's occurrence at every point within range.

On the basis of telemetry data of hares, foxes or racoons, SINIFF and JESSEN

(1969) come to believe that in general the overall distributional pattern of fixes on

the home ranges is not random but contagious. Besides they add "Because the

edge of the range lacks definition to an observer, no observed value was possible

for number of squares containing no fixes. Thus, if one is to fit the negative

binominal to such data, it must be truncated to allow for the zero frequency missing".

How about the distributional pattern of fixes on home ranges of small rodents ?

Sufficient telemetry data are not yet provided but lots of data of radio-active finds

within home ranges of a vole are furnished by AMBROSE (1969). Seeing his eight

configurations enclosing scatters of finds, one is given the impression that a truncated

negative binominal or Poisson distribution would be applicable to those depending on

how the borderlines are determined.

So far as my knowledge goes, there have been not a few papers (TANAKA and

KANAMORI 1969, WALLIN 1971) on the distributional pattern of trappabil ity within

a quadrat or plot under study but scarcely any students have discussed on the basis

of trapping data that of occurrence probability by an animal within its home range

for small rodent populations. The fact would be correlated with the debatable

subject whether home range is delimited or indefinite in extent.

As explained in the preceding section, I determined, by way of conventional

rule, the borderline of home range by delineating a scatter of capture loci of an

animal so as to construct a convex polygon, and investigated the distributional pattern

of trappabil ity over all the trap-stations within its range contour of the select 24 voles

through the In index of MORISITA (1959), with the result that for every vole except

129

only two having plainly overdispersed pattern (L=1.99, 3.43), the non-truncated

random type could not be rejected at 5% significance level.

The result looks to be incompatible with what was remarked by SINIFF and

JESSEN. Moreover, in the light of cumulative knowledge about behavioral patterns

in small mammals, i.e. different response to live traps between marked and unmark-

ed animals, innate heterogeneity among a population, the phenomena called neophobia

and neopholia, territoriality, social hierarchy and so on, we are disposed to consider

the probability of visiting each point by an animal within its home range not to be

the same; thus it may occur that an animal, adhering to a certain trap site is

repeatedly captured there, whereas another tends to avoid visiting a nearby trap

site so as never to be caught there.

Putting the subsequent interpretation on the above result in the present work,

the inconsistence seems to be solved to some degree.

Lately IWAO (1968) developed a new, comprehensive method using the regression

of mean crowding against mean density to analyze aggregation patterns of animal

populations, allowing for the disadvantage that L in an overdispersed distribution

may largely be affected by rise of mean density without any density-related change

in the dispersion pattern. There he, adducing the case of eggs of the azuki bean

weevil as an example for analysis, demonstrated by the method that the distribution

of eggs per bean is of uniform pattern at the initial phase of low egg densities, but

that, as the density increases, it approaches to randomness and finally it shows

overdispersed pattern at very high densities.

Apart from the initial phase of the weevil egg, the analogy of its second and

last phase to the two phases of the vole may take place, viz., the intra-range

capture distribution of the vole may change from the random to the overdispersed

pattern with increase of number of captures per trap station. In other words,

the random distribution is revealed at only such capture densities as treated here

(see Table 2), and an overdispersed one would be realized at such higher densities

that are shown in finds or fixes gathered with isotope or by telemetry. It will

be difficult to substantiate by means of trapping data alone because a long continui-

tion of trapping to get lots of captures of an animal within its settled range is

likely to induce shift of the range.

Thus it has turned out that the distribution of captures per station within its

home range is considered to be random altogether in each of the 24 specimens.

Therefore it will be logically justifiable to apply the random pattern to home ranges

of all members of the population.

Then I attempted estimating the range size of the select specimens using

WIERZaOWSKA's tables (1972) after his formula (2), the result being shown in Table

2. From the r-values in the table, we can perceive these to be stable in the range

from k=5 to 20 for females and k=8 to 20 for males with smaller sample ; hence,

130

Table 2. Measurements of ~ and estimates of r in eq. (2); Nk is number of specimens; see text as to other notation.

Female Male k A / - .

Nk Y'k r N~ ~k r

4 5 6

7 8

9

10 11 12 13 14 15 16 17 18 19 20

18 3.278 8 18 4.222 12 18 4.833 11 18 5.333 11 18 5.777 10 18 6.389 11 18 6.833 11 18 7.111 11 17 7.412 I i 16 7.875 11 16 8.188 11 14 8.643 12 12 9.083 12 10 9.400 12 10 9.700 12

8 9.750 12 7 10.429 13

6 3.167 7 6 4.167 11 6 5.167 16 6 5.833 16 6 6.667 19 6 7.333 19 6 8.000 20

6 8.333 17 6 8.833 17 6 9.500 18 6 10.167 19.5 6 10.833 21 6 11.000 19 5 11.600 20 5 12.000 20 5 12.800 21.5 4 13.500 23

averaging the stable values, the most t rustworthy estimate (let it be r *) for the

range size measured in terms of station number is given as follows :

r *= l l .4 for females; r*=19. 5 for males

Next, I proceed to comparing of these with the oblong range sizes counted from

ORL and ORW. Seeing that the range size expressed as r -values may be largely

affected by how the unit squares are arranged, in order to test if both estimates

induced by the different two means lead to agreement, we should attempt arranging

as many unit squares as r*-values so as to form an ellipse (Fig. 4).

If, according to the above values of r*, the eleven unit squares for females and

the twenty for males are arranged respectively in the manner as i l lustrated in Fig. 4,

a figure, which is surely symmetr ic in two directions and shaped something like an

ellipse, can be construcked in either sex. Besides, there occurs a surprising coin-

cidence, also in both sexes, such that the elliptic range contour, which is drawn

from the scale of nab based on respective averages of ORL (2a) and ORW (2b) of

the select voles, covers with considerable accordance the assemblage of centers of

unit squares.

The considerable coincidence of the routine observational method, accumulat ing

successive observed capture loci, with the theoretical method in results for both sexes

can convince us that (a) the assumption, underlying the second method, of random

distribution of captures on home range holds true, and that (b) the supposition that

a natural range is oblong rather than circular in shape never involves any discordance,

131

emale ~ ~ Y

Male

\

/ 9

Fig. 4. Showing elliptic range outlines, constructed by the formula ~ab from respective averages of ORL (2a) and ORW (2b) of the select specimens, circumscribing the assemblage of centers (dots) of unit squares that are arranged so as to shape something like an ellipse;note that the unit square is d*=25 sq. m in area.

for, if a circular range were assumed, such a reasonable coincidence would be by

no means furnished.

VALIDITY OF DICE'S ASSESSMENT LINES

DICE's assessment lines for density determinat ion will have been introduced on

the ground that the uniform distr ibut ion of an animal 's occurrence on any diameter

of its home range is supposed, hence the mean of its locality, expressed in terms

of abscissa, leading to r (radius of a circular range) that is regarded as width (h)

of addit ional boundary strip. The theory is valid for the case where the random

distr ibut ion in number of captures per station within range is approved.

At present, however, the ell iptic range prevai l ingly obtains; thereby the correct

1 (a+b) in TANAKA (1972), because E(r 2) = width (h) should be 1 /~, instead of

ab and then E(r) " .V '~ when r is defined as radius vector of an ell iptic range. The density per acre (D) can be calculated after DICE'S rule from the est imated

total population (N=123) by dealing with female and male groups separately as

fol lowing :

First , /V is divided into 74.2 (females) and 48.8 (males) using the sex ratio in

the actual captures, and then we have

132

for females : N=74. 2, a=10. 17 m, b=6.29 m and h=l /ab =8. 0m, hence D=26. 2

for males : Ar=48.8, a=16.02m, b=8.71m and h=l /ab=l l . 8m, hence D=14.4.

Thus the total density per acre proves to be 40. 6.

Now, on purpose to prove the val idity of DICE's procedure, one needs to at tempt

approaching the density in a way quite different from the assessment line method

and the like.

MARTEN (1972) may be the first to have appl ied tracks on smoked paper to

censusing of small rodents. He, adopt ing the technique to overcome a difficulty

involved in usual t rapping methods, concluded that populat4on est imates rest ing on

t rack ing can be highly precise and are far superior to those based on t rapping

alone.

Despite his aff irmative conclusion, we still consider that the essential difficulty

of t racking technique consists in how to surely identify a scatter of t racks around

a station with those left by a single or mult iple animals, because we usually a dual

system with fore and hind feet to mark animals dist inct ively from each other. To

my regret, there seems to be no sat isfactory explanation coming to the point of the

methodology in his paper.

But his new, suggestive idea is discovered in another respect that the population

density can be direct ly counted from the samples gathered on a quadrat, without

sett l ing any assessment lines, on the ground of mouse-equivalents in terms of number

of tracks. The idea must be avai lable for actual censusing on the str ict condition

that his mouse-equivalent is represented by a definite extent corresponding to home

range size of an animal. His paper appears to be deficient in any convincing

evidence in favour of the condition.

The r -va lue of WIERZBOWSKA's is suitably a measure of range size, in terms of

number of stations, which is completely equivalent to range area. Accordingly we

can try enforcing the idea of MARTEN by means of observed r -va lues from nearly

all the members of the population, which are considered to have been marked

judging from the census result in the foregoing section (Table 3).

Table 3. Sums of observed vlues of r in WIERZBOWSKA'S for three subsamples, the combined sample being formed of nearly all the marked captures in both plots.

Subsamples*

I

II III

combined

Female ! Male i

Subsample Xr } size

18 197.5 29 224.0 16 71.5 63 493.0

11.0

Subsample Xr r size

6 115.0 19.2 21 227.0 13 69.5 40 411.5

* see text as to these explanation.

133

These subsamples in the table are interpreted as follows :

Subsample I. It is the group of the select specimens which were considered to have

revealed almost entirely their natural home ranges, so that the mean r may

indicate the vole-equivalent of range size with reason; note that r is nearly

equal to r*.

Subsample II. The specimens were all captured ten t imes or more and supposed

to have stayed for a week or longer l ikewise with above, but the part greater

than 20% of each vole's capture loci are distr ibuted on the edge t rap- rows; of

these, two were omitted, for they moved about so extensively that r was counted

at as large as 40 or 65.

Subsample III. The specimens were all captured 3 to 9 t imes at stat ions more or

less involved in the edge rows ; in three of these, one capture locus each was

omitted because of these loci indicative of sally.

Of all the captures, which were util ized for censusing, eleven specimens besides

the two, referred to in the remark of subsample II, were excluded from the sample

for the r -va lue analysis on the ground that they are unqualified to supply information

for i t ; for instance, they include specimens captured less than three times.

Provided that ~; of subsample I is used as "vole-equivalent", the r -va lue observed

in each vole of subsamples II and II I must be a fraction of the equivalent. Conse-

quently, after the notion of MARTEN, the population density per acre (D) is s imply

calculated as the number of vole-equivalents on the quadrat areas (both plots) from

~Vr for the combined sample divided by r as below :

for females _N=493. 0/11.0 = 44. 8, for males -hr=411.5/19. 2 = 21.4, total ing to

66. 2, hence D =37. 2.

Thus we may well say that this is considerably coincident with that (40.6)

reached by DICE's method in the l ight of some reduction (13 voles) in sample size

used for MARTEN'S method.

What will be meant by the coincidence ? Before going further, it is worth

mention the effect of range shift on density estimation. The animals of subsamples

II and I I I are supposed to hold their ranges over the borderl ines of plots, so that

their ranges tend to remove inward, as remarked in my crit ique on MORISITA's

means. The effect may induce some degree of overest imation of r -values, hence of

densities, but the like overest imation must be seldom avoidable in usual censuses

using traps with bait helping enough to al lure animals into plots. Nevertheless, I

am inclined to suppose that such overest imation is balanced with underest imat ion due to something of unexposed population.

The coincidence indicates that MARTEN's method as well as DmE's has proved

to be t rustworthy through the mediation of WIERZBOWSKA's. It is, however, desirable

to accumulate more of empir ical evidence for these methods by f ieldwork of obtaining

data from plots, laid within an extensive mouse-proof enclosure, including a semi- natural population with a known density.

134

By way of reference, I would like to mention the elaborate method (relying on

linear regressions of cumulative catch against distances from a given origin) for

density determination offered by SMITH et al. (1971) and KAUFMAN et al. (1971).

Although I should highly evaluate their theoretical ground, it seems to be lacking

in empirical justifiability because there are infeasible assumptions, radically required,

that the individual location is of uniform or random distribution on the census area

and it is kept fixed as long as a month. Be that as it may, the method is very

laborious in its procedure.

CONCLUDING REMARKS

Up to the present, I have maintained study plans on the same lines to utilize

DICE's method for density calculation from estimated populations, but everyone,

inclusive of me, has failed to offer positive proofs or reasons in favour of it. Now

I have largely come to the conclusion that his assessment lines stand on justifiable

basis from theoretical and empirical angles. The method is distinctive in simplicity

only if one is able to determine average range size, which will be in practice easily

made when he follows such ways of study as in this work.

SUMMARY

A second fieldwork for the sake of solving the edge-effect subject was carried

out in almost the same ways and plans in summer of 1971 in succession to that of

1970. It was found out that the population density was reduced to two-thirds, but

the data were Sufficient to form the subsequent conclusions :

(1) The initial population estimated by the removal census method turned out

to be well compatible with the numbers of survivals and emigrants of marked voles

calculated by JOLLY's method.

(2) The natural, entire home ranges of both sexes reached by determining

observed range length and width with 24 voles, selected on rigid conditions, agreed

with those of the range-conservative type in 1970 in both acreage and elliptic shape.

(3) In the majority of the select specimens, considered to reveal the natural

ranges, the random distribution in number of captures per station within range was

proved by means of I~ so far as such capture density as treated here is concerned.

On the ground of the empirical evidence, the above range size approached by the

routine observational method could be made surprisingly accordant with the range

size in terms of r-values calculated by WIERZBOWSKA's method.

(4) From these proofs, the validity of DICE's assessment lines makes evident,

and besides, it could be further substantiated by use of MARTEN'S notion of mouse-

equivalents through the mediation of WIERZBOWSKA's method. (5) Consequently, I have largely come to the conclusion that DICE's assessment

lines stand on justifiable basis from theoretical and empirical angles.

135

ACKNOWLEDGEMENT: It is a pleasure to thank Mr. M. KANAMORI of Sugadaira Biological Labora-

tory, Tokyo Kyoiku University, for his energetic cooperation with me in performing the fieldwork.

REFERENCES

ADAMCZYK, K., M. JANION, L. RYSZKOWSKI and T. WIERZBOWSKA (1966) Number of traps visited in

recaptures of rodents. Bull. Poland Acad. Sci. C1. II 14 : 697-701.

AMBROSE III, H.W. (1969) A comparison of Microtus pennsylvanicus home ranges as determined

by isotope and live trap methods. Amer. Midl. Nat. 81: 535-555.

IWAO, S. (1968) A new regression method for analyzing the aggregation pattern of animal

populations. Res. Popul. Ecol. 10: 1-20.

JOLLY, G.M. (1965) Explicit estimates from capture-recapture data with both death and immigra-

tion-stochastic model. Biometrika 52: 225-247.

KAUFMAN, D.W., G.C. SMITrq R. M. JONES, J.B. GENTRY and M. H. SMITH (1971) Use of assessment

lines to estimate density of small mammals. Acta theriol. 16: 127-147.

MARTEN, G.G. (1972) Censusing mouse populations by means of tracking. Ecology 53 : 859-867.

MORAN, P. A. P. (1951) A mathematical theory of animal trapping. Biometrika 0~ : 307-311.

MORISITA, M. (1959) Measuring of the dispersion of individuals and analysis of the distributional

patterns. Mere. Fac. Sci., Kyusyu Univ., Ser. E (Biol.) 2 : 215-235.

MORISITA, M. and O. MURAKAMI (1968) Methods of estimating home range size in small mammals.

Lecture at 15th Ann. Meet. Ecol. Soc. Japan.

SImFF, D.B. and C.R. JSSSEN (1969) A simulation model of animal movement patterns. Adv.

Ecol. Res. 6: 185-219.

SMITH, M.H., R. BLESSING, J.G. CKELTON, J.B. GENTRY, F.n. GOLLEy and J.T. McG1NNIS (1971)

Determining density for small mammal populations using a grid and assessment lines. Acta

theriol. 16 : 105-125.

STICKEL, L.F. (1954) A comparison of certain methods of measuring ranges of small mammals.

J. Mature. 33 : 1-15.

TANAKA, R. (1961) A field study of effect of trap spacing upon estimates of ranges and populations

in small mammals by means of a Latin square arrangement of quadrats. Bull. Kochi Women's

Univ., Set. Nat. Sci. 11: 8-16.

TANAKA, R. (1963) Truthfulness of the delimited area concept of home range in small mammals.

Bull. Kochi Women's Univ., Ser. Nat. Sci. 11: 6-11.

TANAKA, R. and M. KANAMOR! (1969) Inquiry into effects of prebaiting on removal census in a vole population. Res. Popul. Ecol. 11: 1-13.

TANAKA, R. (1972) Investigation into the edge effect by use of capture-recapture data in a vole

population. Res. Popul. Ecol. 13: 127-151.

WALLm, L. (1971) Spatial pattern of trappability of two populations of small mammals. Oikos

22 : 221-224.

WI~RZBOWSKA, T. (1972) Statistical estimation of home range size of small rodents. Ekol. Polska

20 : 782-831.

ZIPPIN, C. (1956) An evaluation of the removal method of estimating animal populations.

Biometrics 12 : 163-189.

136

APPENDIX

The advantage peculiar to SUGIYAMA'S method to determine maximum likelihood estimates of

parameters on the theory of the removal census equation [eq. (1) in the text] has not yet been

fully made clear since it was published by mimeograph in 1953. Here it will be accounted for

using the following example of trapping record (Table 4 in TANAKA 1972), to which it is definitely

appropriate ; Ci is number of new captures at t ime i and S,-~=CI+C2+ .. . . . +Ci-~.

Date (1970)

Aug. 8

MC ~6 Aug. 9 Aug. 10 Aug. 11

MC Ed MC EC MC Ed Total (I) Time (i) 1 2 3 4 5 6 7 1 8

C~ 49 (14) 23 (4) 19 (7) 16 i (2) 107

Si_~ 0 (49) 63 (86) I 90 (101) 116 t (132) 269

Under the conditions that trappability in EC is much lower than in MC, it is desirable to

estimate the parameters on the basis of such incomplete data that are not inclusive of every

catch in EC (within parenthesis). Thus the respective totals of C~ and S~-1 work out by omitting

the figures in EC on each day.

In theory we can estimate parameters by applying eq. (1) to even the incomplete data. In

practice of estimating, we may guess at very gross approximates by fitting a regression line by

eye to the plotted data. To approach nearer the true value, we should calculate maximum

likelihood estimates by which the best regression line can be determined.

In the stochastic process, the likelihood (L) of the present data is as below:

LI=(4Ng) p4, qN-..

L~=(N2363 ) p23 qN-63-ZS

L=LI. L8 Ls. LT=-~( N--cSi-l l Pc, qN-St-I-C, (2')

In order to maximize L, putting OL/Op=O and OL/ON=O, we get easily from the first differential equation

St - i& -ss , _? (3')

where i=4, Si=107 and vS,-a=269, but the subsequent formula (4')

~= s~ 1 _~i ' (4')

where i=4 and S~=107, is obtained from the second by the intense approximation (Stirling's

and neglection of its first term) that is permissible only on the condition N-->oo (MoR~,N 1951).

On the other hand, by way of determinism, if eq. (1) (and hence Ci=Npq i-l) holds with a complete catch sries, the equation S i=N(1-q~) , which is essentially the same with eq. (4'), is

readily introduced; in effect, strictly speaking, eq. (4') is invalid for an incomplete catch series

as in this example that is far from the condition N->co. Both MOAN'S successive approximation

method (1951) and ZmmN's convenient one (1956) rest on eq. (4') as well as on eq. (3'). Thus

their methods had better not be used unless a complete series is available or N is very large.

In SUGIYAMA'S, however, we calculate through the direct count of L only relying on eqs. (2')

and (3').

137

The further process of SUOIYAMA'S is as follows ; taking log of both sides of eq. (2~), we get

=Zlog(Nc -,)+(log whose first term turns into 21og (N-Si_~)! -Zlog(N-Si_~-Ci)! -Z log C~!. Next, we read an approximate (let it be N~) to /V, aimed at, from an eye-fitted regression line conforming to eq.

(1). For instance, if we are given N~=150, p~ is computed at 0. 3233 by substituting N~ into (3'),

and then, using these paired values, log L is counted as (log p) ZCi=-52.47173, (log q) 2 ' (N -

2'1og (Nct " i -~) - =85.46032, resulting in affording log L=--5.00181. In S~_I--C~) = --37. 99040 and

the same way, adopting other arbitrary N-values near N~ so as to search after /~r (to maximize

log L), several values for p and log L against each of them will be given as following :

N 150 160 165 166 167

p 0. 3233 0. 2884 0. 2737 0. 2709 0. 2682

log L --5. 00181 --4. 67191 --4. 63318 --4. 63084 -4 . 63197

Of the five N-values, 166 proves to maximize log L ; thus the maximum likelihood estimates

/~r=166 and p=0. 2709 are definitely decided.

In conclusion, SC61YAMA'S method is no less laborious than MORAS'S, but it is specially useful

for such incomplete catch data as in this example.

~ I~I~l~"E1971~ 8 ~-~]~, ,~]~l~l~-~iUSz~: l~ j~! i | 2 ~-U~ ,5o ~

~" ~ --~-i~:~'" (7)~gs ~"C, DICE ~j~O) )T-~]~;O~ ~_~.~e~ ~ :~ ~':o U_ ~ (, ~ ~ :~ ~" "~, DtCE ~:j~$]~Ij~I~IE