Competing probabilistic models for catch-effort relationships in wildlife censuses

Ecological Modelling, 19 (1983) 299-307

299

Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands

C O M P E T I N G PROBABILISTIC M O D E L S FOR C A T C H - E F F O R T R E L A T I O N S H I P S IN WILDLIFE C E N S U S E S

JOHN R. SKALSKI, D O U G L A S S. ROBSON * and CARRIE L. MATSUZAKI

Pacific Northwest Laboratory, Richland, Washington 99352 (U.S.A.) • Biometrics Unit, Cornell University, Ithaca, New York 14853 (U.S.A.) (Accepted for publication 21 February 1983)

ABSTRACT Skalski, J.R., Robson, D.S. and Matsuzaki, C.L., 1983. Competing probabilistic models for catch-effort relationships in wildlife censuses. Ecol. Modelling, 19: 299-307. Two probabilistic models are presented for describing the chance that an animal is captured during a wildlife census, as a function of trapping effort. The models in turn are used to propose relationships between sampling intensity and catch-per-unit-effort (C.P.U.E.) that were field tested on small mammal populations. Capture data suggests a model of diminishing C.P.U.E. with increasing levels of trapping intensity. The catch-effort model is used to illustrate optimization procedures in the design of mark-recapture experiments for censusing wild populations.

INTRODUCTION

Considerable attention has been directed at the use of catch-effort models in the study of exploited populations in commercial fisheries. The focus has been on the relationship between sequential sampling effort and the subsequent patterns in the sizes of the catch (Leslie and Davis, 1939; DeLury, 1947; Ricker, 1958; Chapman, 1961). The models have generally assumed that the size of the sample catch is proportional to sampling effort and population size. Alternatively, sampling has been regarded as a Poisson process with regard to effort. Few structured experiments have actually been conducted, however, to determine the empirical relationship between catchper-unit-effort (C.P.U.E.) and sampling intensity, and to assess its fit to proposed models. In the study of terrestrial populations, catch-effort models have been largely ignored, with the one exception of constant effort removal sampling techniques (Hayne, 1949; Zippin, 1956, 1958). Rarely have the variable

300 effort sampling techniques used in fisheries research been employed on terrestrial populations (Eberhardt et al., 1963; Lewis and Farrar, 1968). Barbehenn (1974) used different trap spacings to determine their effect on the trap response of rodents and shrews and the precision of abundance estimates. Hansson (1974) developed a model which assumed that the size of the sample catch was directly proportional to the number of animals exposed to the effort, but failed to demonstrate such a relationship. Optimal design of wildlife censuses is dependent on knowing the relationship between trapping effort and sampling precision. Sampling precision in turn is a function of population abundance, usually proportional to plot size and capture probabilities. Therefore, analytic expressions for the relationship between trapping effort and capture probabilities were investigated and field tested. The resulting catch-effort model can be used to estimate the optimal size of research study areas and trap densities for fixed research costs. ALTERNATIVE CATCH-EFFORT MODELS

Model A Assume that for each individual in a population of size N, there is an effective area within the home range where, if a trap is located within such a zone, the animal will be captured with positive probability. Let a be the size of this catch zone for each animal in the population. The probability that any randomly distributed trap will be located in the catch zone of any particular animal is then a/A where A is the size of the study plot. Therefore, the probability the trap will lie outside the catch zone of that animal is equal to ( 1 - a/A). Similarly, the probability that f randomly distributed traps will all fall outside the catch zone of that animal is (1 -

a/A) /

and so its complement,

p=[l-(l-a/A)/] is the probability that one or more traps will fall inside the catch zone and is equivalent to the probability that the animal will be captured. Hence, the probability of capture ( p ) is a function of the trapping effort, f , exerted upon a population. This probability of capture can be written in an equivalent form p = (1 -

e :~"~l-a/A>)

or more simply as p = (1 - e - c : )

(1)

301

where c is the Poisson "catchability" coefficient (Seber, 1973, p. 246). An assumption in the use of the capture probability, p, is that the units of effort (the traps) are independent, i.e., the traps do not compete with each other. If the N individuals in the population have equal and independent probabilities of capture, a binomial distribution describes the probability that n of the N animals are captured, where the likelihood function can be written as

The expected number of animals captured with f units of effort is then E ( n l / , N ) = U(1 - e-
E(n,lf,.)

= #x (1 -e -<7' )

(2)

where ~ ( i = 1.... ,s) is the number of traps allocated to each of the s replicates. Using Taylor series expansion of the form e - x - 1 - X+ (X2/2) the expected catch (2) can be approximated by the quadratic regression equation

~C2f 2

E(,,,IZ) = . ¢ f , -

(3)

or dividing through by fi

z ( n , / f , lf,) =

-

2

"'

(4)

the C.P.U.E. is approximately a linear function in f~. As ~ ( i = 1. . . . . s), the n u m b e r of traps set at a plot, increases, the C.P.U.E. decreases, resulting in a diminishing return with each additional unit of effort.

Model B Assume each trap has an effective radius with a resulting area of size a'. The probability that a specific animal ventures into this effective area, and is captured, is then a'/A. Consequently, the probability that an animal does not land in the effective trap area and so escapes capture is (1 - a'/A). The probability that none of the N animals of a population is captured by the

302 trap is then a t

N

and so its complement

p;lis the probability that one or more animals will land within the effective trap zone and a capture will occur. As in eq. 2, this probability can be written in an equivalent form p=(1-

e-C'i).

(5)

If f traps are allocated to the plot and each has an equal and independent probability of capturing an animal, a binomial distribution once again describes the probability n of N animals will be captured. The likelihood function can be written as

L(nlf, N)=(fn)(1-e-c'N)'(e-c'N)f-n The expected number of animals captured with f units of effort (traps) under this alternative model is then

E(nlf, U ) = f ( 1 - e -c'u) or, in general, for i(i = 1,..., s)

replicate plots, the expected value is

E(n,lf,) = f,.(1 - e-C'~).

(6)

U n d e r this model the expected C.P.U.E. is E ( n J f , lf, ) = 1 - e -C'~

(7)

which is a constant for all values of f,. FIELD VERIFICATION STUDIES To determine which, if either, of the competing catch-effort models (2 or 6) may approximate actual census conditions for wildlife populations, field verification studies were conducted. On replicate 1 ha study plots, small m a m m a l populations were censused using baited Museum Special snap traps. Study areas were located in sagebrush-bunchgrass (Artemisia tridentata-Agropyron spicatum) communities at the Arid Lands Ecology (ALE) Reserve in southeastern Washington state (U.S.A.). Different levels of trap density (Table I) were randomly assigned to the plots prior to each study. Trapping was conducted simultaneously at all plots, and numbers and

3O3 TABLE I Alternative trap densities and grid arrangements used in the small mammal censuses on 1 ha plots N u m b e r of traps ( L )

Grid size

Trap spacing (m)

36 64 100 196

6x 8× 10 x 14×

18~0 12.8 10.0 6.9

6 8 10 14

species composition of animals captures were recorded. Traps were arranged in systematic rectangular grid systems. Systematic arrangements were used because of their common use in small mammal studies.

Experiment 1 Three plots serially located 0.4 km apart were assigned trap densities of 36, 196 and 64 traps/ha, respectively. A fourth plot (100 traps), 1.1 km from the next nearest plot, was initially included in the experiment but was subsequently excluded after species composition was found to be different. Removal trapping was conducted for three consecutive nights (23-25 May 1981).

Experiment 2 Small mammal populations were retrapped (30 May-1 June 1982) on the three plots used in experiment 1 plus a newly established 1 ha plot. The new plot was located 0.4 km from the next nearest site. The four trap densities (Table I) were re-randomized before assignment to the study plots in 1982. The purpose of the new randomization was to assure that an observed relationship between trapping effort (f,) and C.P.U.E. (ni/f,) was not caused by an unrelated phenomena. RESULTS

A total of 142 mice were captured on the three 1 ha replicate plots in May 1981, and 52 mice on the four replicate plots in May 1982 (Table II). The predominant species captured was the Great Basin pocket mouse (Perognathusparvus) which accounted for 63% and 87% of the catch in 1981 and 1982, respectively. The deer mouse (Peromyscus manieulatus) and the north-

304 T A B L E II T r a p p i n g effort ( f i ) expressed as n u m b e r of traps, catch of small m a m m a l s C.P.U.E. (nJfi) o n 1 h a plots in Southeastern W a s h i n g t o n

(ni)

a n d observed

Replicate

M a y 1981

May 1982

plot

N u m b e r of traps ( f i )

Catch (r/i)

C.P.U.E. (t/i/L)

N u m b e r of traps ( f i )

Catch (ni)

C.P.U.E.

1 2 3 4

36 196 64 -

26 77 39 -

0.722 0.393 0.609 -

196 36 100 64

20 9 14 9

0.102 0.250 0.140 0.141

(n,/f,)

ern grasshopper mouse (Onychomys leucogaster) constituted the remaining species captured. Capture numbers (n,) reported in Table II are the sum of the three nights of removal trapping. Inspection of Table II indicates a substantial decline in C.P.U.E. (ni/f,) with increased levels of effort (f~). The competing catch-effort models were tested using the trap data (Table II) and assuming the observed ni were Poisson distributed with mean X = g . f, under the null hypothesis (H0). The data were tested against the alternative hypothesis (Ha) that the n i were Poisson distributed with mean X = ~cf,-/xcZfi2/2. The resulting likelihood ratio tests were approximately chi-square distributed with one degree of freedom under H 0. The constant C.P.U.E. model was rejected in favor of (3, 4) with both the 1981 data ( P ( X 2 >19.03)= 0.003) and 1982 data ( P ( X ( >/3.35) = 0.067). Overall significance level for the two tests (P(X22 >/ 12.39) = 0.002) suggests a rejection of model (5). The catch data (Table II) indicate that on the average, greater numbers of mice were caught on plots with higher trap densities (effort), but there was a diminishing return in C.P.U.E. with additional units of effort. The decrease in observed C.P.U.E. with increasing levels of trap effort is consistent with model (2) and its linear approximation (3). USE OF THE CATCH-EFFORT

M O D E L IN D E S I G N O P T I M I Z A T I O N

The single m a r k - r e c a p t u r e or Petersen method (Petersen, 1896) is among the simplest and most useful methods (Seber, 1973, p. 450) for making a census of wild populations. By using the catch-effort model we found to be plausible, (2), estimates of the optimal plot size and trapping effort to employ in a wildlife census can be calculated for fixed total study costs. The Petersen method uses two sampling periods, whereby population abundance ( N ) is estimated from an observed m a r k - r e c a p t u r e ratio. Animals

305 in the population are assumed to have equal and independent probabilities of capture p~(= 1 - q ~ ) during the first sampling period, when animals are captured, marked and returned to the population. Animals are then recaptured with probability P2(= 1 - q2) during a second period in order to estimate the ratio of marked to unmarked animals in the population. When the area (A) to be censused is known and a n i m a l abundance ( N ) is estimated by the Petersen method (Seber, 1973, p. 59), the variance of the estimator of animal density ( b = N / A ) can be written as Var( N / A )

Uqlq2 A2plp2

or simplified to D(lp_ V a r ( N ) = ~-

)2 1

(8)

when capture probabilities are assumed constant (p~ = P2)Optimal plot size and trapping effort are estimated by minimizing the variance estimator (8), taking into account the relationship between effort and capture probabilities (1) and a relationship between effort and cost. Assume that the cost of a census is a function of the area to be censused, A, and the trapping intensity, f, defined as the number of traps per hectare, such that cost = d + A ( b , + bzf )

(9)

or

f=

cost - d - Ab~ Ab 2

If an estimate of animal density is available, optimal plot size can be estimated by further assuming that population abundance is directly proportional to plot size (i.e. N ~xA). The variance (8) is minimized with respect to (1) and (9) by finding the value of f which satisfies the equation 2c 1 -

e -c!

b2

bl + bzf"

(10)

Plot size A is then found by substitution of the value f into eq. 9. The estimates A and f are useful in situations where a study of population dynamics is to be undertaken, but the specific population to be censused has not yet been determined. Optimization of the study design will help assure the greatest precision for the economic resources available. When a specific population (N) has been specified, Robson and Regier (1964) give methods

306 to determine optimal sample sizes (ni) in a mark-recapture program for fixed costs. DISCUSSION The principal distinction between Models A and B is in their assumptions about random mixing. Model A assumes that the animals of a population have well established and fixed home range areas. The units of effort, the trapping devices, are then assumed to be randomly distributed on the study plots. The greater the number of traps distributed on the study plot, the more home ranges will be located by the traps, with subsequently more animals captured. As the study plot becomes "saturated" by traps, the C.P.U.E. diminishes as fewer animals remain to be caught. The field experiments conducted were designed purposely with systematic trap grids to determine whether Model A would be applicable to traditional designs. Random trap placement takes more preparation than the easily established grid arrangements. The consequences of random trap arrangements need further consideration in the theory and practice of censusing wild populations. It is reassuring, however, that Model A seems applicable with the simpler grid arrangement of traps. Model B assumes all individuals within a population have equal access to all traps. This assumption is equivalent to assuming random mixing of the animals within the population and across the plot. However, when the home range of the animal is smaller than the study plot and the distribution of traps, Model B will not be appropriate, The restricted movements of mice may explain the lack-of-fit of Model B to the small mammal data (Table II). In this paper we have used the catch-effort model (2) in the optimization of wildlife censuses. In a subsequent paper we will illustrate how (2) can be used to estimate mean abundance among replicate populations in a geographic region. Advances in the design of efficient wildlife census programs will require additional research on catch-effort relationships and the development of realistic cost functions. ACKNOWLEDGEMENT This research was supported by the U.S. Department of Energy under contract D E - A C O 6 - 7 6 R L O 1830. REFERENCES Barbehenn, K.R., 1974. Estimating density and home range size with removal grids: the rodents and shrews of Guam. Acta Theriologica, 19: 191-234.

307 Chapman, D.G., 1961. Statistical problems in the dynamics of exploited fish populations. Proc. 4th Berkeley Symp., 1960, 4: 153-168. DeLury, D.B., 1947. On the estimation of biological populations. Biometrics, 3: 145-167. Eberhardt, L.L., Peterle, T.J. and Schofield, R., 1963. Problems in a rabbit population study. Wild1. Monogr., 10: 1-51. Hansson, L., 1974. Influence area of trap stations as a function of number of small mammals exposed per trap. Acta Theriologica, 19: 19-25. Hayne, D.W., 1949. Two methods for estimating populations from trapping records. J. Mammal., 30:399-411. Leslie, P.H. and Davis, D.H.S., 1939. An attempt to determine the absolute number of rats on a given area. J. Animal Ecol., 8: 94-113. Lewis, J.C. and Farrar, J.W., 1968. At attempt to use the Leslie census method on deer. J. Wildl. Manage., 32." 760-764. Petersen, C.G.J., 1896. The yearly immigration of young plaice into the Limfjord from the German Sea. Rep. Danish Biol. Sta., 6: 1-48. Ricker, W.E., 1958. Handbook of computations for biological statistics of fish populations. Bull. Fish. Res. Board Can., ll9: 1-300. Robson, D.S. and Regier, H.A., 1964. Sample size in Petersen mark-recapture experiments. Trans. Am. Fish. Soc., 93: 215-226. Seber, G.A.F., 1973. The estimation of animal abundance and related parameters. Charles Griffin, London, 506 pp. Zippin, C., 1956. An evaluation of the removal method of estimating animal populations. Biometrics, 12: 163-189. Zippin, C., 1958. The removal method of population estimation. J. Wildl. Manage., 22: 32-98.