item id number 02977 · reed army medical center. dr. calhoun's training in ecology and mr....

Item ID Number 02977

AllthOT Calhoun, John B.

Corporate Author

RflpOrt/ArtlCiO TltlB Calculation of Home Range and Density of SmallMammals

Journal/Book Title

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Month/Day

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Number of miagos 28

DOSCrlptOll NOtBS Public Health Monograph No. 55, Public Health ServicePublication No. 592

Thursday, November 01, 2001 Page 2977 of 3007

Calculation of Home Range

And Density

Of Small Mammals

John B. Calhoun, Ph.D.

James U. Casby, B.S.

PUBLIC HEALTH MONOGRAPH No. 55

The Authors

Dr. Calhoun is a member of the Laboratory of Psychology of theNational Institute of Mental Health, National Institutes of Health,Public Health Service, Bethesda, Md., and Mr. Casbyis on the staffof the Eastern Pennsylvania Psychiatric Institute, Philadelphia, Pa.The work on this paper was initiated under the auspices of the WalterReed Army Medical Center. Dr. Calhoun's training in ecology andMr. Casby's in mathematics made possible the focus of these twodisciplines on the topics considered in the text.

Public Health Service Publication No. 592

(Issued concurrently with the December 1958 issue of Public Health Reports, vol. 73, No. 12)

(Received for publication March 1958)

Library of Congress Catalog Card No. 58-60088

UNITED STATES GOVERNMENT PRINTING OFFICE : 1958

For sale by the Superintendent of Documents, U. S. Government Printing Office, Washington 25, D. C.- Price 25 cents

Contents

Introduction. 1Home range 1 1

Basic nature 1Source of data 2Expressed as a density function 2Successive capture derivation 6General considerations 7

North American Census of Small Mammals 12Estimation of population density 15Sampling populations of small mammals 20

Adequacy of sampling device 20Variations in size of daily catch- ._ 21Invasion 21Competition between sampling stations 21The "trap-day" index. 21Relative densities 22

Summary 22References 24

Hi

Acknowledgments

The authors are indebted to the following persons who have readand commented upon portions oj this monograph in various stages ofits development: Dr. Don W. Hayne, Michigan State University,East Lansing, Mich.; Dr. John P. Flynn, Naval Medical Re-search Institute, Bethesda, Md.; and Dr. Ardie Lubin, Dr. Ross L.Gauld, and Mr. Lou Joseph, Walter Reed Army Institute of Research,Washington, D. C. In particular we are indebted to Samuel Green-house, chiej, Applied Statistics and Mathematics Section, NationalInstitute of Mental Health, Bethesda, Md. He recommended tests Ito Vfor the assumptions concerning the home range of male harvestmice, and prepared all relevant calculations. Furthermore, Mr.Greenhouse developed for us equations 56 to 63.

We are also indebted to the University of California at Berkeley,under whose auspices the home range data of male harvest mice wereassembled by Daniel Brant, and to Dr. Brant for allowing us to usehis original field records. Much of the success of our effort wasdependent upon Dr. Brant's creativeness in developing a samplingtechnique which amounted to a moving trap colliding with a movingmouse.

Dr. Ruth Louise Nine, of the Game Management Division, Wis-consin Conservation Department, Madison, Wis., also provided uswith an excellent set of data which she had procured while at theUniversity of Wisconsin.

However, the authors assume full responsibility for any defects inthe analysis and interpretation oj the data.

Introduction

Small mammals such as mice and shrews aredifficult to observe directly. Nocturnality,protection by overhanging vegetation, andretirement to nest or burrow upon detection ofan approaching human contribute to this diffi-culty of observation. • Mammalogists have beenforced to develop techniques of trapping (1) inorder to explore the biology of such animals.Where traps that capture the animals alive areused, it is generally found that each animal con-fines its activities to the vicinity of severalneighboring traps. This spatial limitation ofexcursions has become known as home range.Traps are usually relatively sparsely distributedthroughout any home range. Even so, manyanimals are captured. Several methods havebeen proposed for converting records of capturesinto estimates of density. Some of thesemethods ignore the question of the movements

of the animals in relation to the trap. Othersattempt to utilize home range in the calculationof density, that is, number of animals per unitof area, but lack of a mathematical expressionof home range-has hampered such attempts.

The task we set ourselves was a dual one.First, we wished to examine home range withthe view of selecting an equation which wouldapproximately describe it. Second, we wishedto examine the methods of estimating densitywith particular reference to the role of conceptsof home range in making such estimates morelogical and precise.* Although we have at-tempted to take a broad view of these twotopics, our principal effort was directed towardthose aspects of home range and density thatwere of particular relevance to the standardizedsampling procedure used by the North Ameri •can Census of Small Mammals, or NACSM (2).

Home Range

Basic NatureThe home range of an animal is defined as

the area it covers in its day-to-day travels (3).An inherent property of the home range isthat it is fixed, in the sense that the animaldoes not wander through a space at randombut. repeatedly covers the same general area.Attempts to make home range a useful concepthave involved arbitrary delimitations of theextent of the home range. Boundaries havebeen designated by polygons, encompassedby lines drawn between the outermost pointsof capture or by lines drawn halfway betweenthe outermost points of capture and the nextmost peripherally located traps.

Hayne, who has reviewed the concept ofhome range (4), was the first to present a logi-

cal approach to the problem of the relative fre-quency with which different portions of thehome range are visited by a mammal. As areference point, he took the mean coordinatepoint of observation or capture, defined as the"apparent center" of activity within the homerange. The distance of each point of capturefrom its apparent center of activity can thus becalculated for each animal. Hayne then carriedthe concept of home range one step farther.He calculated an index of the relative fre-quency with which an animal is found perunit of area at different radii from the ap-

*This monograph is not intended as a comprehensivereview of home range and the calculation of density-We have cited only those references which are particu-larly cogent to the development of our thesis.

Public Health Monograph No. 55, 1958

parent center of activity. This index equaledthe number of captures at a radius divided bythe number of traps available at that radius.These indices showed that the probability ofan animal being captured in a particular areaof unit size decreased with its distance from theapparent center of activity.

Hayne later presented further but more in-direct evidence concerning the phenomenon ofhome range (5, fig. 2). -Where traps were setin grids, the mean maximum distance from theapparent center of activity increased with thenumber of times the animals were captured.This may be restated as indicating that thelonger the period of observation, the more likelywill the animal be observed at those distantpoints which it visits infrequently. It islogical to assume that the degree to which ananimal interacts with its physical environment,in terms of food consumption for example, isproportional to the time it is present in a givenarea. Therefore, this new concept of homerange developed by Hayne is essentially one ofrelative intensity of usage of the environmentalthough its measurement is in terms of therelative probability of observation of the animalat different places within the home range.

Source of Data

An extensive set of data on the captures ofmale harvest mice (Eeithrodontomys), securedby Brant (6), forms the basis of tins analysis.Three study plots were located in grasslandhabitat near Berkeley, Calif. Details of thehabitat of these plots are given in the 1950Annual Report of the North American Censusof Small Mammals (2, pp. 25-29). The plotswere irregularly hexagonal in shape, and eachplot encompassed approximately 28 acres.

Trapping stations were so distributed as toform a 50-foot grid over each plot. Only one-seventh of the trapping stations were in opera-tion at any one time. Each activated stationcontained two traps. These activated stationswere in groups of four, one at each corner of asquare, each side of which measured 50 feet.The minimum distance between the centers ofactivated groups was 280 feet. At regular in-tervals all eight traps in each activated groupwere inspected. Captured animals were markedand released or, if already marked, their num-

ber designations were noted prior to release.Then the traps were moved to an adjoininggroup of stations. By systematically shiftingthe traps, the entire plot was sampled afterseven shifts (6).

The distance separating the groups of trapswas great enough that most animals had anextremely low probability of exposure to all butone group of traps at any one time. Becauseof their sparse distribution, the traps onlyslightly hindered the movement of the mice.Such a situation is desirable for, as Hayne (o)has demonstrated, the mean distance of capturefrom the apparent center 'of activity decreasesas the traps become more dense.

The data initially selected for study werelimited to males for which (a) there were 3 ormore captures (median=8 captures), and (6)less than 25 percent of the captures were on theperipheral trapping stations. One hundredand fifteen animals met these specifications.The apparent center of the home range wascalculated for each animal, as well as the dis-tance, r, in feet of each capture from theapparent center of activity.

rn=\(xn-W+(yn-yW* [1]

where

(a;, y) is the center of the home range

and

(xn, yn) is the position of the nth capture.

Expressed as a Density Function

In home range studies, "density function" isa mathematical expression representing theprobability of an animal being present in somearbitrarily small area.

From the discussion of concepts of homerange, three assumptions are made:

• The home range is fixed. In other words,the statistics of the home range are stationary,or time independent.

• There is a true center of activity althoughthe apparent center of activity may deviatefrom it.

• The probability of an animal being in aunit of area decreases with increasing distancesfrom the true center of activity. This and thesecond assumption suggest a bivariate normal

Calculation of Home Range and Density of Small Mammals

distribution of the density function (7, fig. 3,ch. 6).,

f(r,y)dxdy=>~ e-^+^l^dxdy [2]

where a is the standard deviation of the dis-tances in the x and y direction and is assumedto be equal for both, and x and y are measuredfrom their respective means. This densityfunction may be used to represent the percent-age of time spent in the area dxdy located at theCartesian coordinates x, y,or, in polar coordinates

f(r,0)rdQdr= [3]

Here the area rdOdr is determined by r.There are an unlimited number of equations

which would fulfill the requirements of thesecond and third conditions. The bivariatenormal distribution given in equations 2 and 3is one such function. We shall examine thehome range data for male harvest mice to seehow well they are approximated by this equa-tion although we are primarily interested in theradial frequency distribution of the averageanimal.

The density function in terms of the Cartesiancoordinates is more meaningful from an ecologi-cal standpoint because it states in comparativeterms the amount of time spent by an animal ina small standard area at any position in the homerange. However, for the initial mathematicalmanipulation it was found more convenient toexpress the density function in terms of polarcoordinates. Then the probability of findingthe animal between the radii r and r-\-dr aboutthe true center of the home range is

[4]

If equation 4 is integrated over the range 0to ff we have

r~ e-«^dr= 1 - <r»= 0.3940 [5]

In the above equations a, the standard devia-tion of the normal distribution function, is thevalue of a radius within which the probabilityof the animal being present is 39.4 percent if itsmovements can be described by a bivariatenormal density function.

Similarly, integrating equation 4 over therange 0 to 2 <r gives

1-6-4/2=0.8645 [6]

Similarly, integrating equation 4 over therange 0 to 3 <r gives

When the recapture data are contaminatedby inclusion of animals which have shiftedtheir home range, that is, those animals withnonstationary home ranges, the observed fre-quency distribution will include too manycaptures at longer radii. This will lead to anoverestimate of the dimensions of the homerange. An examination of the field maps of thehome ranges of those 115 males included in theinitial group of mice selected for this studyrevealed that certain individuals had definitelyshifted the center of their home range. In themore obvious of these shifts the initial group ofcaptures formed a chimp no more than 300 feetin diameter, while the later captures formed asimilar clump several hundred feet away. In

Table 1. Home range parameter a for 25 maleharvest mice

Mouse No.

22984053309277100 .278 _ . .Cl-62 . -380'369242 , ,-. .286403527 ._- -Cl-58 -Ci-18Cl-3T-7256Cl-46149393 . -284T-98Cl-1601-25

Numbercaptures

10101010101010111111121212131414141516161921212224

< r '(feet)

44. 934. 653. 547. 9

2 77.542. 569. 259. 532. 560. 043. 536. 860.032.550.868.561.5

352.571.253.4

< 3 2 . 247.340. 155. 557.7

Number daysbetween first

and lastcapture

8810913976

2378486

145133115172137261177106131218174107201137165205172157

1 Sec equation 9a.4 Minimum.

2 Maximum. 3 Median.

PuWic Health Monograph No. 55, 1958

the light of our hypothesis of a stationary homorange, we thought it wise to exclude animalswhich had definitely exhibited such shifts.The criteria for assembling a more homogeneousgroup were as follows:

• Animal must have been captured 10 ormore times.

• No obvious shifts in home range; that is, astraight line separating.the later captures fromthe earlier ones could not be drawn on the fieldmap.

Twenty-five male mice met these criteria,and they provided 348 capture records, r's(table 1). These records showed no detectableshift in the center of home range.

When it can be assumed that the same bi-variate normal distribution given by equation 2holds for all animals, then the best estimate, s,of a, based on the maximum likelihood estimateof <r2 is:

where: Kt— number of captures of ith animaln— number of animals

nAr= total captures— ]>] Kt

I KLand 2/'= W

ytj and x ij— position of ith animal on jth captureFor the 25 male harvest mice a was estimated

to be 52.7 feet. An estimate, sfl of a was alsomade for each mouse (table 1) :

2(13:,-1) T [9a]

For many purposes in which the ascertainingof home range is useful, the a of the recaptureradii may be obtained by direct use of the radii.In this procedure the map of the area studied islaid out on a large sheet of graph paper that isfinely spaced with vertical and horizontal lines.An overall set of coordinates is established onthis grid and points of capture for each animalare plotted. At the termination of observationthe mean coordinate point of capture for each

animal is recorded on the graph paper andrecapture radii are measured directly from thispoint with a ruler. For practical usages, suchas in the calculation of density (equation 51),the <r calculated from such estimated radii areprobably accurate enough to justify the timesaved, in calculation. Equations 8a and 9abecome:

r « KI

2(N-n)[8b]

[9b]

Equations for a 8a, 9a, 8b, and 9b take intoaccount the fact that distances are measuredfrom apparent rather than true centers, and thesquares of these <r's are unbiased estimates ofthe parameter appearing in equation 4.

Likelihood ratio methods (8, p. 270) wereused to test certain assumptions concerningthe home range of male harvest mice.

Since the complete bivariate normal distribu-tion contains another parameter, pzy, the corre-lation between the x and y coordinates, andsince we wrote equation 2 with />,„=(), \ve testthis assumption jointly in test I and separatelyin test III. The other tests are obvious.

Test I. The hypothesis was that p^=0,<ji

x=<jiv, for the normal bivariate distribution

of captures within an animal

where:

-2 logoff, loge(l-r*)+^ log,

Ki

[I2 fei^l~~ AT,

[10]

[11]

[12]

[13]


Assumption: x—x and y—y are distributednormally with the parameters <r2

z, <r2v, and pxv.

This test was performed for each animal. —2log X is distributed as x3 with 2 degrees of free-dom. The hypothesis was sustained by 22 ofthe 25 animals at the 0.05 level of confidence.

Test II. The hypothesis was that <r2x=cr2

v

for the distribution of captures within ananimal

where:

-2 log, \=K( log,- [14]

Assumptions: Same as in test I. ,—2 log X is distributed as x2 with 1 degree offreedom. The hypothesis was sustained by 24of the 25 animals at the 0.05 level.

Test III. The hypothesis was that the co-efficient of correlation, pzV=Q, for the distribu-tion of captures within an animal

where:

[15]

r is as defined in equation 13.Assumptions: Same as in test I.

—2 log X is distributed as x2 with 1 degree offreedom. The hypothesis was sustained by 23of the 25 animals at the 0.05 level.

Test IV. The hypothesis was that all the (•«.)animals came from a population with the somevariance (Bartlett's test for homogeneity ofvariances)

where:

sented below by equation 22 approximates thefrequency distribution of recapture radii fromthe mean coordinate center of the home range.

Let uw=St

and » = [18]

where ij represents the jfth observation on theitli animal. That is, the variables u and »represent deviations from the respective means,i; e., centers of home ranges, for each animalexpressed in standard deviation units. st* isthe estimate of variance of an individual ob-servation (radius) obtained by pooling thevariance of x and y within each animal (seetest II, which shows that o-x=<ry).

Z5 <*«-_ $ " \

K>

[19]

Of interest to investigators of this subject isthe distribution of radii, that is, of

?•«=(*<*+^)H [20]

However, since the raw observations x andy have different variances (test IV) and dif-ferent means among animals, it would be in-appropriate to compare a radius, rtil of the»th animal with a radius rnj> of the nth animal.Therefore, each radius was expressed in stand-ard form denoted by Zti\

[21]

-2 log, X'= -D]:

71 Jf\

Si[16]

3(n-l)n

"V* 1 1 12(^-1) £ 2(Kt

-1)

[17

Assumption: (x—x) aud (y—y) were1 inde-pendent and normally distributed with thesame variance. —2 log X'/wi, 72.59, is distrib-uted as x2 with n—1 degrees of freedom. Thehypothesis was rejected at the 0.001 level.

Test, V. Hypothesis; Equation 4 as rcprc-

This Ztj was computed for each of the 348observations and put into a frequency distri-bution with class intervals of size 0.3Z, It

then desired to determine whether thewastheoretical frequency function

e-z'* dZ [22]

that is, equation 4 with Z—r/<r, fitted the data.To this end, expected relative frequencies werecomputed by direct integration (table 3). Ex-

Public Health Monograph No. 55, 1958470108—58 2

poctecl frequencies were then obtained (fig. 3),and a x2 goodness of fit test was made of thesein comparison with the observed frequencies for10 class intervals. The hypothesis that equa-tion 22 was consistent with the data was sus-tained at the 0.5 level (x2= 6.287; 9 d. f.).

Equation 22 can also be derived mathe-matically from certain assumptions concerningthe normal and independent distribution ofx and y when it is assumed that the true meansand standard deviations are known. Forexample, let ^, vt and at bo the respectivemeans of x and y and the standard deviationfor the ith animal. Then it is well known that

«Q)!are each distributed likea

X2 with 1 degree of freedom. Hence their sum,

[23]

is distributed like a x2 with 2 degrees of freedom.Further Zw, is independent of Zit and inde-pent of Zni. Therefore, the distribution ofZ?,is

/(Z2)dZ2=Ke-^ dZ* [24]

and it follows immediately that the distri-bution of Z is equation 22.

Successive Capture Derivation

Many persons have published data on themovement of animals as tabulations of the fre-quency of distances between successive captures.Such data provide a basis for obtaining a roughestimate of the group a of the home range.For any two captures the mean coordinatepoint of capture, or calculated center of activity,lies halfway between the two observed points ofcapture. However, the true center of activitymost probably lies to one side of the line con-necting these points. Thus, the radial distance,r, of capture from the true center of activityis somewhat greater than half the distancebetween observed points of capture. Theradial distance of capture from the calculatedcenter of activity must be multiplied by (n/n—l)w, where n is the number of captures for agiven animal, in order to make the recaptureradii, on the average, provide a more accurateestimate of the distance from the true center ofactivity. Where n — 2, as in the case, when

distances, d, between successive captures onlyare considered, /•=0.5rfX1.414=0.707d. Whensuch procedures of calculation are followed, thehome range a-, for practical purposes, should beapproximated by that distance within which0.394 of the r's fall. One note of caution: Whenthe animal is caught at the same station on twosuccessive dates, one must tabulate this simplyas two r's of less than half the distance betweentraps forming the grid.

We wondered how the home range <r of maleReithrodontomys approximated in this waywould compare with that of 52.7 feet estimatedthrough the use of equation 8a. To this end,frequency tabulations were prepared of dis-tances, d, between consecutive captures. Sincethe trapping stations were arranged in a 50-footgrid, only certain discrete d's were possible.This similarly limits the r's possible. These r'sin feet followed by their frequency are: 0 (32);35.4 (97); 50.2 (52); 70.7 (34); 79.1 (52); 100.4(14); 106.0 (10); 111.8 (17); 127.2 (3); 141.2(1); 145.9 (2); 150.1 (2); 158.1 (1); greater than158.1 feet (2). The catch at each of thesedistances should represent the sampling ofanimals traveling within a band whose widthextends halfway to the two contiguous radii.Thus, the widths of bands represented by theabove r's vary from 4 to 25 feet.

If we knew the real home range <r, we couldestimate the expected number of mice in eachof these bands from figure 2. For this purpose,<T=52.7 feet as estimated independently byequation 8a, was used. These expected fre-quencies for the 14 r's were, respectively, 16.9,72.5, 64.8, 46.9, 43.1, 29.3, 8.0, 13.4, 12.8, 3.8,1.9, 1.6, 1.6, and 2.8, Although 3er (158.1 feet)includes the correct number of r's as predictedby equation 7, there are some marked deviationsbetween observed and expected frequencies.In particular, there are too many captures atthe same station and at adjoining stations.Presumably this excess is due to the fact thatDr. Brant prebaited the traps part of the timeand also occasionally left the traps at the fouradjoining stations throughout several days oftrapping. Thus any development of a trap-habit will produce too many short r's. Homerange a estimated as that distance including0.394 of the captures is only 43.25 feet. Like-wise, a-, when estimated as one-half the distance


including 0.865 (from equation 6) of the cap-tures, is 48.2 feet.

Another factor which may contribute to vari-ation in the frequency distribution of r's is thefact that, from any one point of capture, 4 trapslie at some distances and 8 traps lie at others.This may well account for the relative excess ofr's of 79.1 feet, for which there are 8 oppor-tunities for capture, in contrast to the muchfewer captures at 70.7 and 100.4 feet, for eachof which there are only 4 such opportunities.Apparently, any estimation of a from distancesbetween consecutive captures is likely to be amuch cruder one than that derived by the useof equation 8a or 8b.

Some criticism has been leveled at us by oiirmathematical and statistical colleagues for re-stricting our sample to 25 male harvest micewith only 348 captures. For example, 19 othermales with 10 or more captures appeared eitherto have made a major shift in the center of theirhome range or only a small portion of theirhome range la}^ within the trapping plot, as in-dicated by the fact that captures of these micewore confined to the edge of the plot. These 19males provided 216 d's for which derived r'sranged up to 496 feet. Home range <r esti-mated from those distances, including 0.395,0.865, and 0.989 of the captures (equations 5-7)were, respectively, 45.6, 66.5, and 257.5 feet.If we include all 44 mice which had at least 10captures, these three estimates of a based upon535 d's become 42.8, 56.1, and 229 feet.

Some recapture data, particularly those de-pendent upon short-term sampling, supply in-formation primarily on the distance betweentwo consecutive captures. This raises thequestion, "How can the best estimate of thehome range sigma be derived from them?" Astrict rule of thumb would be that cr=(/•i+0.5r2)/2 where rl—0.707d, which includes0.394 of the captures, and r2=0.707^, which in-cludes 0.865 of the captures. However, onewould be hard pressed to make a rigorousmathematical validation of this estimate. Ona logical basis, the estimate minimizes the' effectof the increased frequency of short r's arisingfrom trap habit and from d's of those animalsfor which less than half of the home range isincluded in the sampling plot, and it also ex-cludes the excessively long r's arising from


shifts in the center of the home range. Thisextensive discussion of home range as reflectedby distances between successive captures hasbeen included in this section since it is apparentthat persons involved in some of the more prac-tical problems of wildlife management mustrely on this sort of data for their estimate of thehome range sigma.

General Considerations

From tests I to III, it may be concluded thatfor most of the mice the home range may beconsidered circular. However, the extent ofhome range varied significantly among theanimals (test IV). Yet the pooled data ofstandardized radii (Z) was consistent withequation 22. If this is so, it follows that equa-tion 4 probably describes the home range of anyparticular animal iri terms of its own a. Now,if equation 4 truly represents the probability ofcapturing an animal between radii r and r-\-drabout the true center of home range, it followsthat equations 2 and 3 probably represent agood approximation of the density functionabout the center of home range.

The parameter, <r, fixes all properties of equa-tions 2, 3, and 4. To the extent that theseequations depict home range, the followingaspects of home range may be calculated:

• Idealized frequency of captures betweenany two radii.

• Density function at any radius.• Probability of capturing an animal within

any portion of its home range.Such information may be derived from the

data in tables 2 and 3 and from figures 1 and 2.In figure 1, the curve representing equation 2is the cross section of the density function whena= 1. In order to convert this to any other <r,multiply the abscissa by a- and divide the ordi-nate by a2. The proportionate time an animalspends in any small portion of home range maybe determined by multiplying the area by thedensity function at the radius of the center ofthe area. For example, the proportionate timean animal spends in a square 0.1<r on a side atradius a is 0.00095 and at 2<r is 0.00021. Thetotal amount of time spent in all areas withinthe 7r(3<r)2 range is 0.9888.

Table 2 is a numerical tabulation of the curvein figure 1. To obtain the appropriate values,

Tab'e 2. Normative data for calculating densityfunction1 in terms of area2 for any value of a

Radius(r) in aunits

0.0.3. 6. 9

1.21.51.8

' 1 «-r>2a2v e

2 Equatioi

Densityfunction

d ( r )

0 159152133106077605170315

/2o-2.1*0 .

1.

Radius(/•) in aunits

2. 12.42. 73.03.33.63.9

Densityfunction

d ( r )

0. 0176. 00880. 00415. 00175. 000 G8. 00024. 00008

Table 3. Normative data for calculating expectedcaptures for any value of a

Class inter-val of radius

of a or '£

0. 0 -0, 3. 301- . 6. 601- . 9. 901-1. 2

1. 201-1. 51. 501-]. 81. 801-2. 1

Expectedpropor-tion oftotal

captures

0 044121168180162128087

Class intervalof radius of

a or Z

2. 101-2. 42. 401-2. 72. 701-3. 03. 001-3. 33. 301-3. 63. 601-3. 9

3.901 and over

Expectedpropor-tion oftotal

captures

0 05403001500674002830010400049

the left hand column, the radius, must bemultiplied by <r and the right hand column, thedensity function, must be divided by <r. B}*this procedure the density function is expressedin terms of the probability of capture per thesquare unit of distance in which a is measured(feet, yards, etc.).

Since this curve (fig. 1) of density functionrepresents the relative amount of time ananimal spends per unit of area at various radii,it may represent the impact of the organismupon its environment. This curve may also beuseful in calculating cohabitation. Cohabita-tion may be thought of in two ways. First, itmay concern the total use of an area by all itsinhabitants; this use will be proportional to thesum of the density functions. Second, it m&yconcern the probability of simultaneous pres-ence of individuals in the particular area con-cerned. This will be proportional to theproduct of their density functions at that place.

Figure 2 represents the integral of equation 4and may be reconstructed from the data given

in table 3. This curve is useful in calculatingthe probability of observing an animal up toor beyond a radius or between two radii. Inorder to convert this curve into values for any a,multiply the abscissa by a.

In figure 3 and table 3 the density function interms of radius from center of home rangeassumes the shape of the theoretical or expectedcurve, which represents the probability offinding an animal within increments of radius

Figure 1. Cross section of the density functionof home range in terms of area.

0 .6 1.2 1.8 2.4 3.0 3.6

RADIUS FROM CENTER OF HOME RANGE

IN UNITS OF CT

Figure 2. Accumulated probability of capture ofmice in terms of radius from center of homerange.

0 .6 1.2 1.8 2.4 3,0 3.6

RADIUS FROM CENTER OF HOME RANGEIN UNITS OF cr


figure 3. Observed (histogram) and theoretical dis-tribution of 348 recapture radii (/) of 25 maleharvest mice from the center of their home range.

RADIUS FROM CENTER OF HOME RANGE

from the center of activity. Since the homerange a varied significantly among the 25 mice,each capture of each mouse was reexpressed interms of its own a by the Z transformation dis-cussed in test V. The exact number of observedcaptures in the twelve 0.3 Z length class inter-vals shown are: 20, 38, 68, 65, 47, 49,25,18, 9, 6,2, 1. In the chi square test the captures in thelast three intervals were grouped together.

The method of collecting the present datawas not ideal. In the first place, the traps werefrequently prebaited for 6 days. Traps wereoccasionally left set at the same station for twoor more consecutive days. Under such condi-tions an animal sometimes developed the habitof entering the same trap each consecutive day.We do not know how this biased the movementof the mice, other than to state that it is acommon experience that prebaiting the trapsincreases the number of animals caught. Afrequency plot was made of the recapture radiifollowing prebaiting in contrast to those pre-ceded by no prebaiting. These curves coincidedso closely as to suggest that under the presentconditions prebaiting had little if any effect inaltering the behavior of male harvest mice.Ideally, what is desired is a large number ofobservations of each animal in the sample. Each


such set of observations should cover a shortspan of time, probably 1 to 5 days for harvestmice, and the taking of observations should inno way disturb the animal or interfere with itsmovements. However, the home range ofmany mice was sufficiently stable that evenwhen the observations were dispersed over aperiod of 2.5 to 8.0 months, there was noapparent change in the home range.

An appreciation of the general characteristicsof home range may be secured by examiningfigure 4. Without any alteration in their ori-entation, each of the groups of captures for 25mice were superimposed about a single center.The circles represent radial distances of pro-portions of p where p—<r-j2, the radius en-compassing 0.63 of the captxires nearest thecenter. Traps were prebaited and left unsetand untended for approximately 6 days. Onthe sixth day they were baited and set, andcaptures were recorded on the seventh day.Only captures taken under this regimen wereused in the preparation of figure 4.

It is immediately apparent that the numberof captures per unit of area decreases withdistance from the center. It is also apparentthat it would be possible for considerable over-lap to develop in adjoining home ranges in thearea of l<r to 3o- from their centers without theinhabiting animals very frequently coming into

Figure 4. One hundred and ninety-four captures of25 male harvest mice (Reilhrodonlomys) super-imposed about a single center of home range*

contact or competing in their usage of theenvironment. This raises the question of thenature of the topography of the sximmateddensity functions for all individuals inhabitingan area.

We have not yet solved some of the technicalproblems inherent in integrating sums andproducts of density functions for all membersof the population for every point in space.However, empirical calculations utilizing thedata in table 2 lead to the following generaliza-tions: When home range centers are character-ized by an intervening distance greater than2.0<r, the topography of summated densityfunctions exhibits peaks and valleys. In otherwords, the environment is utilized most in-tensely near the centers of home range andleast midway between centers. As home rangecenters become more uniformly distributed andas the interval between adjoining home rangecenters decreases, utilization of the environmentbecomes both more intense and more uniform.Xearly complete uniformity of usages is attainedwith an inter-home range center interval of 2.0<rdistance. Any further decrease in the inter-home range center interval merely increases theintensity of usage of the environment withoutaltering the uniform usage of the environment.Therefore, one might logically anticipate theevolutionary process to have culminated in be-havior patterns assuring development of uni-form distribution of home range centers havingan inter-home range center interval of approx-imately 2.Otr. Where population density in-creases beyond this level and is accompanied byfurther contraction of home range size over-utilization of the environment must result. Asa consequence the static characteristics of homerange must break down and result in migratoiybehavior that is regularly observed among suchrodents as lemmings during periods of highpopulation density (9, 10).

Since, according to the concept presentedhere, there is actually no boundary or finitelimits to home range, we propose to assign anarbitrary limit determined by the radius 3o-from the center of activity. In such an area,0.9888 of the observations will be made. Fur-thermore, as the present data on Keithrodonto-mys show, it is practical to obtain observationsof home range up to this limit at least. This

method of describing the home range shouldprovide figures similar to those obtained withthe "minimum area" method commonly usedby mammalogists, in which an area is deter-mined by the polygon encompassing the outer-most points of observation. At least, this simi-larity in representation should result if no shiftin centers of activity occurred in the periodduring which the "minimum area" was deter-mined. If a shift in the center of activity takesplace, an overestimate of the size of the homerange would result.

Such a comparison may be made by utilizingthe data presented by Hayne for Microtus, forwhich a 60-foot spacing interval of traps wasused (4, 5). The average estimate for indi-viduals with three or more captures was 0.194acre for females and 0.429 acre for males. The avalues were calculated from table 1 of Hayne'sdata (4) for males, after excluding all animalscaught at only one trapping station. Thisgave a <r of 9.5 feet for females, with a 3<r circulararea of 0.059 acre, and a a of 12.7 feet for males,with a 3<r circular'area of 0.104 acre.

In these calculations approximately 12 per-cent of the observations for each sex fell beyond3o-. This suggests that about 10 percent of theobservations represented cases in which shiftsin the center of activity had occurred. It wasundoubtedly the inclusion of these 10 percentof the observations in Hayne's calculationswhich produced the markedly greater estimateof the area of the home range by the use of his"minimum area" method. For this reason, webelieve that the method of determining thearea of the home range presented in this paperis a more realistic one for estimating the actualarea used.

Furthermore, the intensity of usage of differ-ent portions of the home range may be demon-strated by calculating the density function interms of area. Such a comparison of densityfunction was made for both male and femaleMicrotus, as well as for male Reithrodontomys,by utilizing the above values of or, calculatedfrom Hayne's data. These values are shown infigure 5, which indicates that small differencesin the home range parameter, <7, produce muchgreater differences in the probability of observ-ing the animals per unit of area.

10 Calculation of Home Range and Density of Small Mammals

• Although the curves in figure 5 represent therelative amount of time the animal spends perunit of area from the center of the home range, acertain amount of caution needs to be exercisedin utilizing this density function as an indexof the relative impact of an organism upon itsenvironment. First, different behaviors, suchas those involved in securing nesting materialor food or in the investigation of the environ-ment, may each have their own density func-tion. Second, when comparing two differentspecies, density function cannot be utilized forequating their impact upon the environmentunless both species are similar in all their prop-erties. For example, a mouse and a deer mightexhibit the same density function curve but,because of differences in size and behavior, theimpact of each species upon the environmentwould be different, both qualitatively and quan-titatively.

Figure 5. Comparative probabilities of being in anarea of 1 square foot for three groups of mice.

§ be> *< o

* z5 2- to *"z z

UJ o0 «f& =•fc §3 u.ffi U< Kffi 40 =It 0Q. in

'

.001,75

.001,50

.001,25

.001,00

.000,75

.000,50

.000,25

n

1 1 1 ,—— | 1 1 1 1 1 1 | |

\ .

( g RE/THRODONTOMYS. \ jf MICROTUS

\

\ ? MICROTUS\

- \\

I \CT-9.SFT.

\ t- \\ -

\\CT-ll.7FT.

\

\'\\. <T'Si.7FT.

0 10 20 3C 40 50 60 70 80 90 100 110 120 130 140DISTANCE FROJ1 CENTER OF HOME RANGE (N FEET

i;

There is still mtlch to be learned regarding thedensity function of different behaviors occurringwithin the home range. Several other questionsare also much in need of clarification:

• To what extent does the calculated centerof the home range actually represent one ormore major goals, such as a place of harborage orsource of water, about which the individualanimal orients its other activities?

• What is the character of the paths of loco-motion from and toward the center of activity?Are the outward and return trips identical induration and in the types of activities exhibitedduring them?

• How does the structure of the environmentaffect the density function? That it must affect

it is shown by the data presented by Stickel (11)for deer mice (Peromyscus) inhabiting bottom-land and upland forests. The mice in the latterhabitat regularly exhibited larger home rangesthan did those inhabiting the bottomland.Both the quantity of a goal per unit of areaand the distance separating goals should be im-portant variables. Blair presents similar datafor Microtus (12).

• How does social behavior alter the densityfunction of the home range? Where there is noterritorial behavior and the environment isuniform, the number of centers of the homerange of individual animals or colonies per unitof area should be distributed at random, thatis, a Poisson distribution. As territorialbehavior approaches a maximum, one wouldexpect the nearest approximation to a uniformdistribution of centers of activity, that is, therewould be a minimum variance in the distancebetween centers of adjoining home ranges.Clark and Evans have presented a method ofanalyzing this question (13).

There is much in the literature which permitsone to voice opinions about some of theseproblems. However, not until they are investi-gated systematically and quantitatively willit be possible to develop an understanding ofthe ecology of home range. Such an under-standing has considerable relevance to theproblem of population density.

More effective utilization of space is a basicproblem in human sociology, animal husbandry,and wildlife management. In each of theseareas, the circumstances of the present ageexert increasing pressure for greater popula-tion densities. The approach toward greaterdensities involves three processes:

• Development of a smaller home range tr.• Simultaneous coexistence of more indi-

viduals within the same home range.• A more uniform distribution of centers

of home range.However, although the development of a

science of home range will assist in producingdesired densities, it is well to bear in mind thatdetermination of numbers of animals is onlyone of several goals. Biomass of the com-munity, growth of the individual, the socialstability of the group, and the psychologicalwell-being of the individual are other goals

Public Health Monograph No. 55, 1958 11

(14), and the degree of attainment of any oneof these goals necessarily modifies the degreeof attainment of the others. Homo range maythus be seen to form an important concept inequating these goals in relation to whatevervalue system we wish to impose upon them.

Dice and Clark (15) have presented a treat-ment of home range which closely parallelsours in its approach to the problem. Theirdata included 32 deer mice, Peromyscus, whichwere captured on an average of 5.5 times each.Males, females, adults, and juveniles wereincluded in the sample. Each recapture radiusfrom the calculated center of activity was

Table 4. Home range of Peromyscus

Accumu-lated

probabilityof capture

0.25.50.75.95.99

Hadial distance (feet)

Observed

4677

116185255

Calculated by method of —

Dice andClark >

5079

118193260

Calhoun andCasby 2

5277

109161214

1 Reference 15.2 Public Health Monograph No. 55, figure 2

subjected to a square root transformation andthe resultant data were tested for conformityto a Pearson's type III probability function.With 15 class intervals of recapture radii thereresulted a chi square of 20.08, which indi-cates significant heterogeneity from thathypothesized.

We subjected the data of Dice and Clark toa test for conformity with the bivariate normaldistribution of home range which we haveproposed. The standard deviation was approx-imated as that radius, 65 feet, within which0.394 of the captures fell (equation 5). Thenumbers of captures in the 10 class intervals(table 3) from 0 to 3<r were, respectively: 6, 28,28, 34, 18, 20, 25, 6, 7, and 2, while beyond 3<rthere were 9 captures. Chi square was 40.4,with 24.5 of this contributed by capturesbeyond 3<r. Considering the fact that the smallnumber of captures per individual precludeddetermination of possible shifts in center ofhome range, and that animals with different-sized home ranges must have been included,it is concluded that for this small set of datano equations can • be found which fit betterthan do Dice's and Clark's (15) or ours andthat either at least roughly describes homerange as exhibited by these data. A comparisonof the results provided by these two methodsis presented in table 4.

North American Census of Small Mammals

A number of mammalogists have cooperatedin the utilization of a standardized procedurewhich provides data on relative densities ofsuch genera as the mice Peromyscus, Cleth-rionomys, Microtus, Reithrodontomys, and Sig-tnodon, as well as the shrews Blarina and Sorex(2). This procedure consists of placing threetraps within a radius of 2.5 to 5.0 feet from astation marker. Traps are left set for 3 con-secutive days. The animals are killed and re-moved each day. A straight line of 20 stationsforms the basic sampling unit. An interval of50 feet between stations is customarily used.A summary of results from 744 traplines runfrom 1948 through 1951 is given in table 5.

The concept behind placing three traps at

each station was that, even after one or moreanimals had been caught, there would still be aset trap available for capturing another indi-

Table 5. Total captures on 744 North AmericanCensus of Small Mammals traplines'

Number of stations capturing—

Day

12 . . --3

0 animals

12, 24412,83913, 243

1 animal

2,0301, 6921, 368

2 animals

483284223

3 animals

1236546

1 Each trapline consisted of 20 stations with 3 trapsper station. On 77 percent of the lines, the intervalbetween stations was 50 feet; on the remainder, 25 feet.


Figure 6. Funclional efficiency of trapping stations.

lOjOOOl

4677 (.716 )

1,000

o:LUCD

1247 (.192 )

418 (.064)

190 (.029)

100 10 2 3

TRAPS UNSET PER STATION

vidual. As soon as three mice had been caughtat a station during a day, any others which ap-proached the traps would be turned away.When all traps functioned perfectly, only asmall number of stations (0.01352 of total) hadthe opportunity of turning mice away (table 5).Actually, a somewhat larger number of stationsturned mice away because some traps weresprung accidentally. Data were available froma portion of the traplines showing the number of

traps unset at each station each day (fig. 6).Based upon this sample of the number of trapsfound unset within a 24-hour period, it is ap-parent that the efficiency of the trapping sta-tions was not unduly lowered by traps becomingunset bjMvind, rain, larger animals, and so on.Assuming that the data in table 5 are repre-sentative, the sampling device has a high oper-ational efficiency.

An approximation of the number of animalswhich may be turned away by unset traps onthe first day of trapping may be made (table 6)by taking into consideration the data in figure 6along with that in table 6 which concerns thenumber of stations catching 0, 1, 2, or 3 animals.These are probably slight underestimates.

Figure 7. Accumulated captures of mice along allNorth American Census of Small Mammals trap-lines from end of trapline to center, 1948-51.

0 1+20 2+19 3+18 4+17 5+16 6+15 7+WEND STATION NUMBERS

8+16 «+IZ 10+11CENTER

Table 6, Trapline data from North American Census of Small Mammals, 1948-51

Total 3-daycatch

(range)

Total...

0-910-1920-29 .30-3940-59 ....50 and over

Numberof

trap-lines

845

57516655181516

Mean daily catch

Day 1

4.25

1.76. 1

10.613.918. 131. 1

Day 2

3.22

1.34.27.2

11.314.622. 6

Day 3

2.45

0.93.25.98.9

11.322. 1

Three-daytotal

(mean)

9.92

3.913.523.734. 144.075.8

Number stations with catch onday 1 of —

0

14, 083

10,6092,460

64818112956

1

2, 190

8277263351159592

2

502

601201045751

110

3

125

414137

2562

Approxi-mate pro-portion ofapproach-ing ani-

malsturnedaway

on day 1

0.033.040.047.050.073. 150


Figure 8. Observed and expected catcli of smallmammals taken during 8 consecutive days of I rap-ping along 12 North American Census of SmallMammals traplines.

100

CATCH

T8

PER

DAY

> (118)• ' OBSERVEDA = E X P E C T E D

• 79

65 .66

48. 55

> 60

.4743.8 *

32.8*

26.2*21.8 A

18.7 A 16,3 A

DAY

However, it is apparent that when the total3-day catch on a trapline exceeds 35 animals,not more than 95 percent of the animals ap-proaching the trap on day 1 are caught. Whenthe anticipated catch for 3 days exceeds 35 ani-mals per trapline, 4 or more traps per stationare needed.

Only rarely is it necessary to take the precau-tion of providing additional traps since the3-day catch for a trapline will exceed 35 animalsonly 5 percent of the time (tahle 6). Eachsuccessive day the catch will decline. There isno systematic difference in the rate of declinein the daily catch within the range of 4 to 44animals per trapline for 3 days,

In this study, the largest number of animalswas captured on the two terminal stations,Nos. 1 and 20 (fig. 7). A smaller number weretaken on stations Nos. 2 and 19, which are

adjacent to the terminal ones. The other 16internal stations had approximately equivalentcatches but less than the 4 stations at or nextto the end of the trapline, indicating that a 50-foot interval between stations is too short toprevent competition between adjacent stationsfor the available animals. The 16 centrallylocated trapline stations primarily drew victimsonly from either side of the trapline, whereasthe two terminal stations also drew victims wholived beyond the ends of the lines.

Twelve traplines were run for periods longerthan the usual 3 days (16). One would antici-pate that continued trapping and removal ofthe animal population would lead to a continueddecline in the number of animals caught onsuccessive days, but such is not the case whenthe trapline is surrounded by extensive similarhabitat (fig. 8). Apparently animals whichinitially have an extremely low probability ofexposure to the traps are attracted into thevicinity of the trapline as a result of the removalof the previous residents. During the last 6days of trapping, the total observed catch wasover twice that of the expected total for thisperiod. This increase in the observed catclipresumably was caused by invasion by animalsfrom surrounding areas, and by expansion ofhome range by certain subordinate members ofthe community. Full documentation of thelatter concept will be presented in anotherpublication at a later date. The expectedcatch per day was calculated by using equation54.

The preceding paragraphs describe a widelyused procedure for sampling populations ofsmall mammals. We shall now proceed toshow how the captures on days 1 and 2 inconjunction with data on home range can beused to estimate density.


Estimation of Population Density

The data from the North American Census ofSmall Mammals were derived from a sequenceof daily samples from the animal population.Several persons have estimated density fromsuch a sequence of samples (17-21). Through-out these papers there is a common rationale,which we have paraphrased as follows:

Initial population—NProportion of remaining animals captured per

night=P

First night's capture, C^= [25]

Residual population after first night, N—NP=N(l-P) [26]

Second night's capture, C2=PN(l-P) [27]

Residual population after second night,N(l~P)-PN(l-P)=-N(l-P)s [28]

By extension:Capture on nth night, Cn=PN(l-P)n-1 [29]

Residual population after (n—1)*' nightP)n~i [30]

Therefore, total capture up to but not includingnth night, Tn

•P)"-1 • [31]

•P)"-1] [32]

Both Cn and Tn contain the common param-eter, (1— P)*~l. We eliminate the parameterby substitution.

[33]

[34]Cn=P(N-Tn)

This is the equation of a straight line (fig. 9),as proposed by Hayne (19). Our equation 29is essentially that presented by Moran (20),and our equation 33 is essentially that presentedby Leslie and Davis (17), DeLury (18), andHayne (19). Hayne's method is applicable onlyto those conditions in which all animals havethe same probability of capture by the samplingdevices. Our equations 34 and 35 are applicableto this curve.


From equations 25 and 27:

KT Ol

1 62/61 [35]

The questions we posed initially were: Areequations 33 and 34 applicable to the datasecured by the North American Census of SmallMammals? Equation 33 applies only to thepopulation in an enclosed area in which at thebeginning of each sampling period each animalhas the same probability of capture.

Moran (20) was cognizant of this limitationto the use of equation 33, for he says: "Thelast and perhaps the most important reserva-tion about the above theory is that it assumesthat the chance of being trapped is the same foreach animal." Zippin (21) accepts the logicpropounded by Moran, Leslie and Davis,DeLury, and Hayne, with its recognition thatsuch procedures for estimating density are validonly if "the probability of capture during atrapping is the same for all animals."

The following combinations of sampling pro-cedures and characteristics of movement bymembers of the animal population represent themajor conditions giving rise to a constant proba-bility of capture:

A. If there are sufficient sampling devices in

Figure 9. Graphic method of calculating density ofmice (N) according toJHaytie.

ONP

Q

CCLJQ_

IO

O

Cn=P(N-Tn)

ACCUMULATEDCATCH J

N

PRIOR

15

the area to insure that all areas are equallydepleted, equation 33 applies.

B. If each animal moves at random throughthe area, equation 33 applies regardless of thepattern in which the devices are set, as long asno animals are rejected for want of opportunityof entering devices already full.

C. If the sampling devices arc placed eachday without regard to the position they held theprevious day, equation 33 applies, regardless ofwhether the individuals move at random or havesome stationary statistics such as in equation 2.

The sampling procedure utilized by the NorthAmerican Census of Small Mammals and thecharacteristics of movement of the animalsstudied are such as rarely to satisfy any of theseconditions. The "A" conditions are violatedbecause traplines are customarily placed inenvironments so extensive that many animalshave a very low probability of encounteringtraps because they live at such long distancesfrom the traplines. In other words, for prac-tical purposes we are not dealing with enclosedareas. The "B" conditions are violated by thefact that small mammals do not move throughthe habitat at random but rather have a homerange similar to that represented by equation 2.The "C" conditions are violated by the factthat the traps were left in the same position onsuccessive days. Thus, it is apparent thatequation 33 will not serve satisfactorily in esti-mating density from the day-to-day removalcatches obtained by the North AmericanCensus of Small Mammals.

The procedure of fixed trapping stations hasbeen used because it facilitates conformity ofsampling by different persons. Furthermore,many traplines have been run two or more timesa year for several years. Where this has beendone, it is possible to make analyses of theeffect of local habitat characteristics in deter-mining the presence of animals.

It seems reasonable to us that an animalliving far away from a trapping station has alower probability of capture than does oneliving nearby. In fact, we assume that theprobability of capture of an animal at a trappingstation is proportional to its density functionat the trapping station (see equation 3). In theecological sense, this density function repre-sents the amount of time an animal spends per

unit of area within its home range. Where thesampling device remains at the same positionthroughout the taking of the several consecutivesamples, and where, for practical purposes, thehabitat about the sampling device is infinitewith reference to the animal's daily movements,the probability of capture requires furtherconsideration.

If, as' we believe, the hazard area about atrapping station is relatively small, the numberof times an animal encounters a station will beproportional to the density function of theanimal at the station. In the vicinity of anyone trapping station located so as not to be incompetition with any other station, there willbe distributed animals whose centers of homerange lie at varying distances from the trappingstation. Therefore, each of these animals willhave a particular expectation of capture depend-ent upon the distance of the center of its homerange from the trapping station. The expec-tation of capture (E(W)')* is used in the senseof the percentage of nights an animal would becaptured in a trap provided it were releasedeach morning and no learning by the animalwere involved.

Suppose a trap is situated in an animal'shome range at the coordinate (x'', y'}, which isat distance (a'2-!-?/'2)1'2— W from the center.The density function expressing the probabilityof the animal being at this point (xr, y'} is either

or2T(T2

1

27r<r2

[36]

[37]

The expectation of capture of this particularanimal will be proportional to the density ofthe species at that coordinate and will be

K[38]

Since a priori there is an equal likelihood ofa center of home range occurring at any pointin the environment, it follows that the number

*Our use of the word "expectation" and the symbolE(]V) do not exactly correspond with the customaryuse of these terms in statistics and probability. In thelatter sense E(W)=fW f(W)dW. Here, E(]V) moreclosely corresponds to a binomial expectation in that itrepresents the relative frequency of captures.


of animals, AT(ir)> whose centers of home rangefall within a ring about the trapping station ofradius TT~ and width dW is

[39]

where c is number of animals per unit area.Then the expected capture of animals from

the ring will be the product of equations 38and 39:

K2™*' [40]

Finally, the expected capture on day I , GI,from all territory surrounding the trappingstation will be the integral of equation 40.

/'Jo 27TO-2[41]

The expected residual population in the ringat W after removing the captures on the first •night will be

=N(W)[l-EW)]dW [42]

and the expected capture on the second nightwill be

Of= f "E(W) N(W) [l-E(W)]<W7 [43]Jo

and, in general, the expected capture on the nthnight will be

n= rE(WJo

It follows that

nT451[45]

The above equations may be simplified so asto solve for c, the animals per unit of area:Expected captures on day 1 :

Expected captures on day 2:

Expected captures on day 3:

The functions xc2o-2 and K occur as param-

eters of these equations. It will be helpfulin interpreting these equations if we can gainmore insight into the meaning of these twofunctions.

Since c represents the animals per unit ofarea and <r2 represents the range of each animalin units of area, c<r2 is an expression of the in-tensity of occupancy in terms of the staticcharacteristics of extent of home range. Inother words, c<r2 denotes the extent of overlapof home ranges and the scarcity of space un-occupied by animals. K represents a kind ofvelocity with which an animal covers its home

TT

range and — -^~i the frequency with which it

visits a particular place. The latter we desig-nate as the visitation frequency. This param-eter is essentially that treated in militaryobservation problems, such as detection ofsubmarines from airplanes during World WarII (**).

Thus, each day's catch is compounded of theintensity of occupancy function, cer2 and the

jrvisitation frequency, -TfT However, the ratio,

Cg/Ci depends entirely upon the visitationfrequency:

K

[44] ThereforeK

[49]

[50]

and so, if a is known, c (animals per unit ofarea) can be calculated using equation 46.

[51]

where d and C-2 represent the captures pertrapping station on nights 1 and 2.

In handling actual sets of data, there arisesthe problem of predicting what the actual catchwould be provided sampling were continuedwithout any change in the activities of theanimals remaining within an area.


Figure 10. Comparison of observed and expecteddaily catches of mice along 744 North AmericanCensus of Small Mammals trapliiics during 1918-51, based on two hypotheses.

LDAY i (3265) • = OBSERVED CATCH

* i EXPECTED

• DAY 2 ( 2 4 5 5 )2. Cn = 14609

DAY 3 (1952)

, (1530)

*(I267)(1070)

«(93C

AND

C20=326C40 = 163

C80 = 81

105):ftS'2)

10• £cn

" 0 2 " 4 6 8 10

ACCUMULATED PRIOR CATCH IN THOUSANDS13167= N-jf^

Let U~- K

%,

[52]

=2(1- ft/a)* [53]

and, then from equations 45 and 46

.-(1-Z7)"] [54]

By calculating U from the catches on thefirst 2 days of trapping and inserting thisquantity in equation 54, the predicted catclion any following day may bo estimated. Be-cause of variability in C\ and C2 resulting fromchance and vagaries of the weather, severalsets of such data are needed for use of equation54.

The observed catch for 3 days and the ex-pected numbers of mice caught through a longersequence of time are shown in figure 10. Theexpected numbers indicated by the straight lineare derived from equation 35, which assumes aconstant probability of capturing all animals.This hypothesis produces a lower calculatedcatch than the expected numbers derived fromequation 54, shown as triangles in figure 10.Tiiis latter hypothesis assumes that animalsliving farther from the trapping station havelower probabilities of capture.

From equation 38, U represents the expecta-tion of capture of a particular animal whosecenter of home range is at 0 distance from thetrapping station; therefore, U must be between*See equation 50.

0 and 1. If this is so, then from equation 53,i must be between 0.5 and 1. Similarly.

Ca must lie between 2/3 and 1 ; C4/C3 between3/4 and 1, and so on to Cn+i/Cn betweenn/n+1 and 1.

As shown in test IV in the section, "Deriva-tion of Home Range," a varies significantlyamong animals, and in equation 38 we wereforced to use a grouped o- (s as in equation 8a).Determination of the distribution of a- requiresmore data than is now at hand. However, ifthis distribution were known, equation 38 couldbe replaced by

[55]

Then equations 39 to 51 would be correspond-ingly altered toward greater accuracy in esti-mating density. Likewise, the reliability ofestimates of density requires a knowledge of thedistribution of CyCi where (73 is not independentof Ci. For the purposes of our formulationswe have considered small mammals as a biologi-cal entity. Further use of these formulationsdemands that restrictions of species, sex, habi-tat, and perhaps season be placed on Ca/Ci aswell as the home range <r.

In the preceding formulations we were con-cerned primarily with the development of meansof estimating density from removal trappingsuch as is represented by the procedure utilizedin the North American Census of Small Mam-mals. This effort culminated in equation 51,which utilized only data for captures on the first2 days of trapping since continued removalthereafter is usually accompanied by invasionby animals from other areas or by expansion ofhome range of survivors (fig. 5).

However, it is theoretically possible to obtaindata on. density of animals without altering pat-terns of movement. Were all animals markedfor visual identification, the number of indi-viduals appearing at a particular point for thefirst time on each successive day could be re-corded, when capture-mark-release-and-recap-ture procedures are used, the number of un-marked animals entering traps each day can berecorded. However, two precautions must beexercised. First, there must be enough livetraps at each station to capture every animalwhich approaches a trap. Second, stations


must be far enough apart so as not to competefor the opportunity of capturing animals livingbetween stations. For rodents such as Cleth-rionomys, Peromyscus, Reithrodvntomys, andSigmodon, which generally appear to form, atmost, family groups or other small aggregates,there should be at least 5 traps at each station,and stations should be located at least 4 to 6home range <r apart. For example, stationsshould be 60 to 80 feet apart for Microtus and240 to 360 feet apart for Peromyscus. We areaware of no live-trapping data which fulfill theserequirements.

"Where the procedures of sampling do notdisrupt the stationary statistics of home range,i. e., invasion and alteration of <r, a further esti-mate of density, c, follows from equation 45.

In such situations, when it is reasonable toassume that c and K remain constant for cap-tures on all n days, it should be pointed out thatthese parameters can be estimated directly byutilizing all captures on all n days, rather thanfrom the first and second night captures only.Equation 45, which contains the parametersnonlinear!}-, can be differenced so that the loga-rithm of the differences in captures on successivenights is linear in c and K.

The steps are as follows:From equation 45

Therefore

[l -(l -

Thus

[56]

[57]

Therefore( l~

-l) log (l~^ 5)

[59]

Now plot ting the logarithm of [nCn~ (n—l) Ca-i\against n— 1 should result in a straight linewhose intercept and slope can be estimated byleast squares or graphically. In practice, inorder to avoid negative differences of [nCn—

(n—l)0n-i], it may be necessary to fit asmoothed curve through the capture data.Then each Cn and its Cn~\ would be read fromthe smoothed curve. The intercept of courseis equal to log cK and the slope is equal to log

1—^—5 )• Taking antilogs, we find cK andMirff /J£

n—5 and hence for known <rz, c and K may be'TTff

calculated. For example, if the intercept andslope are denoted by a and b respectively, wehave

(

andlog cK—a

log (l-;

From equation 61

and substituting in equation 60

antilog ac=

—antilogi)

[60]

[61]

[62]

[63]

Determination of density by equation 63has the advantage in that it is applicable to asingle consecutive series of samplings, whereasequation 51 demands several sets of data onCi and (72 in order to cancel variability causedby weather conditions which modify theactivity of the animals.

In the text directly following equation 48brief mention was made of what we understandK to represent. Now that it is possible to esti-mate K, further comment concerning what issubsumed under it is justified. K is a kind ofvelocity in that it probably includes:

• Actual rate of travel.• The perception swath the animal "cuts"

through its environment. If an animal candetect a trap or similar stimulus 10 feet away,it will cut a 20-foot perception swath alongits route of travel. Thus, where all other fac-tors are equal, an animal cutting a 20-foot per-ception swath will be more likely to be cap-tured than one cutting only a 10-foot swath.

• The number of trips per unit of time awayfrom home and back again.

• The pattern of movement. An animalwhich retraces homeward the same route it


Table 7. Captures of male Reithrodontomys fromthe North American Census of Small Mammalscensus tract California 1—I and II'

Day

Total....

12.3

Number of stations withcatches of —

0

348

99123126

1

117

533331

2

13

742

3

2

101

Totalcatch

149

704138

1 Run by Dr. Brant 4 times during 1950 and 1951.This was the tract from which the home rangeparameter, <r, of 52.7 feet was determined. (See equa-tion 8a.)

traveled outward will be less likely to encounterany particular point than one which movesoutward radially, traverses an arc, and thenreturns homeward along a different radius.

K might will be more appropriately desig-nated as a scanning constant. It is presentedas a necessary postulate to understanding homerange dynamics. All four variables whichcontribute to it may be measured on a practicalbasis.

Table 7 presents the kind of data requisite tocalculating population density. Since in the

8 traplines there were 100 trapping stations,each run 3 nights, the mean catch per trappingstation on day 1 was 0.437, 70/160; and

C2/C1=0.585; and <r=52.7 feet [64]

Therefore, from equation 51

c=0.437/4;r(52.7)2(l -0.585) [65]

=0.0000302 male Reithrodontomys persquare foot, or 1.316 per acre.

In the censuses run by Dr. Brant (6) therewere also female Reithrodontomys, as well asmale and female Peromyscus and Microtus.The total catches for the 8 traplines in thissample were 43, 29, 85, 63, 78, 70, 49, 19.Thus, in addition to the 149 male Reithrodon-tomys, 287 other animals entered the traps.Some animals must have been turned awaybecause all three traps at a station were full.

This turning away of animals may haveproduced a slight error in the calculation ofthe population density of male Reithrodontomys.It again emphasizes the necessity of having asufficient number of traps per trapping stationso that no animal will be turned away (tables6 and 7).

Sampling Populations of Small Mammals

Adequacy of Sampling DeviceWhenever animals are turned away from a

sampling station because of the inadequacy ofthe sampling device, too small a Ci will result.The animals turned away will be likely toaugment <72, ^3> and so on. C7a/<71 will be toolarge, and an overestimate of density will result.The preceding example of the calculation ofdensity of Reithrodontomys certainly involvedsuch an error.

Hine's data* illustrate the aberrations which.may result from an inadequate sampling device.She ran 103 traplines for 3 consecutive days.Several localities were represented, and eachwas sampled during several months of the year.*See Acknowledgments.

Thus, any favoring of an increased catch onany one of the 3 days because of weather wasprobably cancelled. Each trapline consisted of50 traps placed in a line, with 1 trap per stationand with 100 feet between stations. On eachof the first, second, and third days of trapping,5,150 stations were represented, for which thetotal daily catches of 1 animal per station were519, 546, and 495, respectively, for each of the3 days.

Considering the large size of these samples,and that all animals were killed and removedfrom the traps each day, the fact that the catchfor each consecutive day did not decreasepresents an apparent anomaly. However, sincethe animal population sampled consisted of


small mammals, such as Peromyscus, Blarina,and Microtus, as was also true of the popula-tions for which data are presented in the NorthAmerican Census of Small Mammals, therewas every reason to believe that 2 and some-times 3 animals did arrive at many of thestations during a single day. When thishap'pened, the extra 1 or 2 individuals musthave been turned away from these single-traptrapping stations.

If the animals which are turned away fromthe trapping stations keep returning to the new-found source of food until they find a trapunoccupied, one may anticipate a series ofcatches over the 3 days much like the catchesfound by Dr. Hine. In fact, by taking the datain table 5 for known numbers of second andthird captures and assuming that the popula-tion was, instead, sampled by only one trap perstation, and furthermore, that the animalsturned away kept coming back to the samestation until they found a trap empty, one willderive catches Ci, C2, and C3, which are verysimilar percentagewise to those of Dr. Hine.

Variations in Size of Daily Catch

Both random sampling error and temporalchanges in the weather produce variations inCi and C2. We have not as yet attempted toestimate how many determinations of Ci andC2 are required for adequate averaging of suchvariations. Until the effect of fluctuations inweather on this variation is known, precautionmust be taken to include several pairs of differ-ent days when using equation 51. The follow-ing is an excellent example of how weather mayoccasionally modify the relationship betweenCj and C2.

Only infrequently is C2 greater than C\. YetWilliam L. Webb reported that each ofeight North American Census of Small Mam-mals traplincs set out on October 10, 1951, hadgreater catches on the second day of trappingthan on the first day (2}. Totals were :0i, 19;andC2, 52. Eight other lines, 4 set out September 4,and 4 set out September 10, in a neighboringhabitat all gave the usual higher Ci. Totalswere: Ci, 116; and C2,67. Although the weatherchanges on these dates are not definitely known,it is unlikely that the greater C2's on the lines setout October 10 were due to chance error.

Invasion

The problem of invasion has already beenmentioned in the description of the NorthAmerican Census of Small Mammals. Thenumbers of animals taken in and removed fromtraps on days 3 through 8 (fig. 8) were con-siderably larger than the numbers anticipatedby the use of equation 54. Verification of thevalidity of equation 54 probably cannot beaccomplished by procedures which kill the ani-mals captured by the sampling device, becauseinvasion results. Testing the accuracy ofequation 54 probably can be accomplished onlyby marking and releasing the animals. Thesequence of captures of unmarked individuals isthe one to compare with this equation. It isprobable that even Ci and C2 are slightly in-creased by invasion when kill-trapping isemployed.

Competition Between Sampling Stations

In areas where stations are too close togetherthe mean number of captures per station willbe reduced (fig. 7) since some animals whichwoxild be taken by a particular station are in-stead removed by neighboring ones. Whenusing equation 51, this leads to an underesti-mate of density. Where home range has beendetermined, and stations are placed 6<r apart,GI and C2 will have.only a very small errorresulting from competition between adjoiningstations for the available population. The50-foot interval between stations used by theNorth American Census of Small Mammalsprobably leads to a 10 to 20 percent under-estimate of Ci and C2, and thus to an under-estimate of density when Ci is taken as theaverage number of animals per station.

The Trap-Day Index

A great many papers have been publishedwhich present information' concerning the den-sity of small mammals. It is rare to find anindication in these papers that the authors areaware that the procedures used may modify theresults. Historically, knowledge of the densityof small mammals was a byproduct of trappingto obtain specimens for museums. The collectorcustomarily followed an irregular path throughthe habitat. Traplines were formed by placing


traps at irregular intervals at points where itwas thought specimens were most likely to becaught. These traplines were left set until thediminishing catch dictated removal of the trapsto a new site. Relative density was expressed as

Xnumber .of daystraps were set

number of animals caughtnumber of traps

or, in other words, the number of animals pertrap-day.

From data alread3r presented, it is apparentthat density of traps and the number of daystraps are left set will markedly modify thisindex of density. Yet this index of densitycontinues to be used by some investigators, forexample, Beer and his co-workers (28), both fortraplines and for grid trapping.

Relative DensitiesRelative rather than absolute density has

been assumed to suffice for solving many biolog-ical problems, but the area factor in density hasbeen ignored. If all procedures are maintainedconstant, including the time over which samplesare taken, it is assumed that the relative den-

sities obtained will suffice for comparing differ- ,ent habitats or the same habitat at different ;times. However, these relative densities can be \misleading. For example, on the traplines runby the North American Census of Small Mam-mals from 1948 to 1951, 1,901 male and 1,521 ;female deer mice (Peromyseus) were trapped.One might conclude from this that males inthese areas were 25 percent more abundant thanfemales, but this is probably not so. It is morelikely that more males than females were caughtbecause males have a larger home range than dofemales (11) and, therefore, more males thanfemales are exposed to the traplines. Similarinaccuracies of assuming that relative densitiesare proportionate to true densities also apply tocomparisons of the density of different generaor of the same genus in different habitats.Until much more is known concerning the in-fluence exerted upon home range statistics byspecies, sex, arid habitat, it is well to use cautionin drawing conclusions from relative densitiesother than for those densities which concern asingle species and sex from different times iti thesame habitat.

Summary

The principal objective of this monographhas been the development of a method ofestimating density of small mammals in habitatssufficiently extensive so that a considerable partof a habitat lies peripheral to the habitat inwhich the sampling stations are located. Ourmajor premise was that the expectation ofcapturing an animal at a particular stationdepends upon the distance of its center ofactivity from the trapping station. This prem-ise required that an equation be found whichapproximated the density function of the ani-mal about its center of activity. In general,the statistics of the homo range were found tobe stationary and to be approximated by thebivariato normal distribution of the densityfunction:

=~ e dxdy

where /' is the radial distance of the point ofcapture at the coordinate, xy, from the centerof activity.

The probability of finding an animal betweenthe radii r and r-\-dr about the true center ofits home range is

=- e rdr

According to this equation, 0.394 of thecaptures fall within a distance of la- from thecenter of the home range, 0.8645 within la,and 0.9888 within 3cr. Tables and figures arepresented which enable us to calculate the pro-portion of expected captures within any bandabout the center of the home range.

Several methods are presented for estimatingthe standard deviation of this bivariate normaldistribution from recapture data. An unbiased


estimate of the standard deviation, <r, of thehome range for a single animal in its simplestform is

K

§"

where there are a K number of j captures.For male Reithrodontomys the best estimate

of the grouped a=52.7 feet. Sigma variedsignificantly among animals, that is, all maleReithrodontomys do not have the same-sizedhome range. Therefore all r's had to be divided"by their own CT'S before radii from different micecould be pooled for comparing the observed andtheoretical frequencies. These closely approx-imated each other.

It was further found that the number, c, ofanimals per unit area could be estimated by

where Ci and C2 are the average captures pertrapping station for day 1 and day 2, respectively.Precautions to be considered when utilizingthis equation include:

• The less one interferes with the normal pat-tern of movement in obtaining the observationsupon which the home range parameter, <r, isbased, the more accurate is this measurement.

• Inaccuracies in estimating a will be min-imized if the observations upon which a isbased are made over a span of time which isshort in relation to the life span. This willincrease the likelihood of excluding from theestimate animals which have shifted their centerof home range.

• Sigma varies with sex, species, and habitat.Therefore, density measurements must includeonly one category of each of these conditions.

• Each sampling station must be adequateto provide the opportunity of sampling eachanimal which approaches it during each periodof sampling.

• Sampling stations should be almost 6<rapart if underestimation of density arising fromcompetition between sampling stations is to beminimized.

• Where C\ and <7S represent animals whichhave been removed from the habitat, the time


between samplings should not exceed 24 hours,because C2 in particular is likely to be aug-mented by animals which shift the centers oftheir home ranges toward the area about thesampling station that is becoming depleted ofresidents.

• Several samples of <7j and C2 must be takenin order to average out random error and varia-tions due to changes in the weather.

The above approach to estimating density ismost applicable to techniques of samplinginvolving removal trapping at fixed stations inextensive environments.

Where marking and releasing procedures areemployed, the number, c, of animals per unitarea may be estimated by

antilog alira* (1—antilog b)

where <r is the standard deviation of the homerange, log a is the intercept, and log b is theslope of a line formed by plotting the loga-rithms of the differences in captures betweensuccessive days, n, against n—l. For thismethod of estimating density to be valid twoconditions must be met:

• Every animal approaching an observationor trapping station must have the opportunityof being sampled.

• Stations must be at least 60- apart.Details are presented of the development of

the equations for estimating <r, the standarddeviation of the home range, and for estimatingdensity.

• Considerable discussion is devoted to howour formulation of home range may be employedfor elaborating a more detailed understandingof this phenomenon and to how the habits ofanimals and the procedures of sampling affectthe applicability of our equations for estimat-ing density.

Several authors have estimated density byequations equivalent to

where JV=initial population, and Ci and Cz arethe numbers of animals taken on the first andsecond days of sampling. Such equations areapplicable only in those special situations in

23

which the probability of capture remains con-stant on both the first and second days as wellas on following days. In general, these equa-

tions are not applicable when animals are re-moved from fixed stations about which someanimals live closer than others.

References

(1) Manville, R. H.: Techniques for capture andmarking of mammals. J. Mammalogy 30: 27-33(1949).

(#) Calhoun, J. B,, Ed.: Annual reports of the NorthAmerican Census of Small Mammals, 1948-56.Distributed by the editor, National Institutesof Health, Bethesda 14, Md.

(3) Dice, L. R.: Natural communities. Ann Arbor,University of Michigan Press, 1952, 547 pp.

(4) Hayne, D. W.: Calculation of size of home range.J. Mammalogy 30: 1-18(1949).

(5) Hayne, D. W.: Apparent home range of Microtusin relation to distance between traps. J.Mammalogy 31: 26-39 (1950).

(6) Brant, D.: Small rodent populations near Berkeley,California: Reithrodontomys, Peromyscus, Micro-tus. Ph.D. thesis, Museum of Vertebrate Zo-ology, University of California. Berkeley, Calif.

(7) Hoel, P. G.: Introduction to mathematical statis-tics. New York, N. Y., John Wiley & Sons,Inc., 1951, 258 pp.

(8) Mood, A. McF.: Introduction to the theory ofstatistics. New York, N. Y., McGraw-HillBook Co., 1956, 433 pp.

(!f) Elton, C.: Voles, mice and lemmings. Prob-lems in population dynamics. Oxford, England,Clarendon Press, 1942, 496 pp.

(10) Kalela, O.: Uber Fjeldlemming-invasionen undandere irregulare Tierwanderungen. Ann. Zool.Soc. Zool.-Bot.-Penn. Vanamo. 13: 1-82;14:1-90 (1949).

(11) Stickel, L. F.: The trap line as a measure of smallmammal populations. J. Wildlife Management12: 153-161 (1948).

(12) Blair, W. F.: Home ranges and populations ofthe meadow vole in southern Michigan. J.Wildlife Management 4: 149-161 (1940).

(13) Clark, P. J., and Evans, F. C.: Distance to near-est neighbor as a measure of spatial relationshipin populations. Ecology 35: 445-453 (1954).

(14) Calhoun, J. B.: The social aspects of populationdynamics. J. Mammalogy 33: 139-159(1952).

(15) Dice, L. R., and Clark, P. J.: The statistical con-cept of home range as applied to the recaptureradius of the deermouse (Peromyscus). Contri-butions Lab. Vertebrate Biol. 62: 1-15 (1953).

(16) Calhoun, J. B., and Webb, W. L.: Induced emi-grations of small mammals. Science 117:358-360 (1953).

(17) Leslie, P. H., and Davis, D. H. S.: An attempt todetermine the absolute number of rats on agiven area. J. Animal Ecol. 8: 94-113 (1939).

(18) DeLury, D. B.: On the estimation of biologicalpopulations. Biometrics 3: 145-167 (1947).

(19) Hayne, D. W.: Two methods for estimatingpopulation from trapping records. J. Mam-malogy 30: 399-411 (1949).

(20) Moran, P. A. P.: A mathematical theory of animaltrapping. Biometrika 38: 307-311 (1951).

(21) Zippin, C.: An evaluation of the removal methodof estimating animal populations. Biometrics12: 163-189 (1956).

(22) Morse, P. M., and Kimball, G. E.: Methods ofoperations research. New York, N. Y., JohnWiley & Sons, Inc., 1951, 158 pp.

(23) Beer, J. R., Lukens, P., and Olson, D.: Smallmammal populations on the islands of BasswoodLake, Minnesota. Ecology 35: 437-445 (1954).

O


LISPvARY£-T.\.:-: UMWERSITY