Proceedings of the First Symposium on Marsupials in New Zealand
Application of the Poisson Model to the Bait Interference Method of Possum Trichosurus Vulpecula Assessment
Application of the Poisson Model to the Bait Interference Method of Possum Trichosurus Vulpecula Assessment
The bait interference method of possum assessment put forward by Bamford (1970) has been tested in a range of areas and the simple model found inadequate. However if time is added as a variable such that:
Relative density = -1/i loge (1-f) × 100
where i is a time estimator
and f is the frequency of interference with bait stations, a stable estimate of relative density is obtained.
Sample design is strongly constrained by the method of baiting and other practical considerations. A number of problems may also arise in the design of the bait stations and choice of bait used.
For many years it has been common practise for field officers involved in control of common brushtail possums Trichosurus vulpecula to estimate killing rates from the difference between takes on bait acceptance lines before and after a poison operation. Bamford (1970) developed this practise into a bait interference method which involved a device which would exclude other animals such as rats from the baits. He defined the sample strategy by drawing an analogy between the encounters with bait stations and trapping models where the Poisson model is applied to estimate density (Seber 1973):
Relative density d = - loge (1-f)
where f is the frequency of baits taken.
A major problem is that if bait stations are placed too close together an animal is able to follow along the lines. However Bamford (1970) showed that if bait stations are placed at distances of 40 yards there was little evidence of such contagion.
The bait interference method has been used extensively by N.Z. Forest Products Ltd. and more recently in the Wellington Conservancy of the N.Z. Forest Service to estimate poisoning success. Baiting was usually carried out page 186 nightly, over extended periods of weeks or even months. Both operators found that the frequency of baits taken rose with each assessment, with a break occurring at the date of poisoning. This led to two ways of estimating the kill rate: (1) Use of the mean number of baits taken over several days before and after poisoning, from which estimated densities were calculated; (2) Use of only those values immediately before and after poisoning if the rise in baits taken over successive nights prior to the kill was steep. Neither approach was very satisfactory; with the first method the frequency of baits taken was often still rising immediately before poisoning and almost always rose from the low point after poisoning - it usually gave a value well below visual estimates of kill; with the second method many data were ignored and it had characteristics of a 'shot in the dark' estimate. Neither method gave a value as high as that obtained from trapping or spotlighting where comparison could be made (Table 1). Also the larger labour content involved in baiting lines over an extended period (arising from the uncertainty of the poison date) posed a serious limitation for large operations, such as at Kaingaroa where up to 48000 ha are poisoned each year.
Table 1. Relative density1 and percent kill estimates of possums in trials at Kaingaroa and Western Bays during 1976.
Kaingaroa and West Taupo Trials
In an attempt to reduce the labour content and apparent contagion between assessments two changes were made when the bait interference method was introduced to Kaingaroa and Western Bays, Taupo in 1976: (1) baits were set out and left over two nights before assessment; (2) there were only two baitings before and two after poisoning, separated by at least two weeks. All baits were removed between assessments. This kept the number of visits low and aimed to reduce or eliminate the rise in baits taken between assessments. Some preliminary trials were carried out at Kaingaroa in April/May but high variability in the data made the results inconclusive, so at Kaingaroa (but not Western Bays) bait interference assessment was run in parallel with mark/recapture trapping and spotlight counts. 1200 ha were assessed at Kaingaroa and 6800 ha at Western Bays in 1976. In each area 30 lines of 20 bait stations were used, and the results are shown in Table 1.
The method was unsatisfactory in failing to give a consistent estimate of relative density over the first two (pre-kill) assessments - in both areas there was a marked increase. Moreover at Kaingaroa the percent kill derived from bait interference was lower than estimates based on trapping and spotlight counts.
A New Model
Criticism of our change from daily assessment (Bamford's (1970) recommendation) to assessment over two nights at a time led to our including time in the basic model. The density should remain constant from night to night (or at least vary randomly) so it was possible to demonstrate (Appendix) a Poisson process where:
Relative density dt = - 1/t (loge (1-ft))
Where t is the time interval from the first assessment and
ft is the frequency of baits taken at time t.
When this model was tested with several sets of available data dt remained relatively stable.
Table 2 shows a typical data set from N.Z. Forest Products Ltd. for their 1974 poison area, and an un-poisoned control area. In column (4) dt can be seen to vary considerably and yet shows no systematic variation. The break at the poison date is sharp and results in a new level being established.page 188
|POISONED AREA||NON-POISONED AREA|
|DAYS||BAITS TAKEN %||Relative Density||BAITS TAKEN %||Relative Density||INTERVAL|
|Kill %||Av. Max.||33
Three estimators of relative density were calculated from the % of baits taken:
d = -100 loge (1-f/100)
dt = -100/t loge (1-f/100); t = time interval (column 1)
di = -100/i loge (1-f/100); i = no. of baitings (column 10)
Av. is the kill % from the average of baits taken before and after poisoning
Max. is the kill % from density on days 49 and 56 only (immediately before and after poisoning).page 189
Table 3 from 1972 data shows a similar pattern in the poisoned area data but in the control area dt falls sharply after day 2 and even more sharply after each break in baitings. Examination of the data on interference frequency (column 4) shows that almost all baits were taken from the fourth baiting onwards however and so a limit had soon been reached. Data from this point onwards should be excluded from analysis as it is no longer possible to estimate dt.
Further analysis of the 1976 Kaingaroa and Western Bays data (Table 4) showed difficulties with this model since dt (middle columns) fell rapidly, with a large decline after the poison date - yet at Western Bays dt rose again over the February baitings. If only the number of nights (i) over which the lines were baited are accumulated, then the new estimate di is a good fit and the estimate of kill in better agreement with that obtained from trapping.page 190 page 191
When di is calculated for N.Z. Forest Products data (Table 2) the fit was not good; there tended to be a rise in baits taken at the beginning of each baiting session followed by a decline. However at Kaingaroa and Western Bays baits were removed from bait stations between assessments, but at N.Z. Forest Products Ltd. baits were left and hence the total number of baited nights approximated the total time lapse. Finally it was found after analysis that at the last assessment at Western Bays baits were not removed from the bait stations between assessments and so i (the time interval) was not 10 but 8 + 14 = 22 days (as shown in Table 4 where the revised estimates of di are given in parentheses).
Since di is a Poisson variable the lower the value of di the lower the variance and the smaller the sample required (Johnston & Kotz 1969). Now di is determined from a number of observations of baits taken (ni) and the larger the number taken at time (i) the better the estimate of di. These two variables can be controlled separately. The di is determined by animal numbers and may be varied by changing bait station design to make the station less conspicuous and possibly by varying bait type or the use of lures. The ni can be varied by altering the number of nights (i) the baits are set at each assessment and removing or leaving baits between assessments. Since much of the labour is in establishing bait lines it is usually more efficient to make several assessments on the same lines rather than to increase the number of bait stations. Samples of less than 100 stations however appear too small.
This requirement of 100 stations is difficult to satisfy in one line as at a spacing of 40 m between stations it represents 4 km. Such a long line of uniform habitat is usually difficult to achieve or traverse even in exotic forest. Often 1 km is a more practical distance and so sets of 20 or 25 stations are more realistic. In most areas lines will need to be grouped on the basis of uniform habitat or density. The value of this stratification is evident in the Western Bays data (Table 4) where in different habitats rates of kill and of subsequent re-invasion vary. The highest kill was obtained in the high density areas (as expected) but recolonization was also most rapid in such areas, presumably because of the high residual population in adjacent areas.
At Kaingaroa, birds, including robins Petroica australis and possibly House Sparrows Passer domesticus, left bill marks in incompletely taken baits, especially in open areas. On one occasion a feral cat was observed following the person setting out the baits, undoing his work. On other occasions high levels of baits taken (often 100%) were preceded and followed by relatively low levels (e.g. Table 3, day 24, poisoned area). On pastoral land stray stock could be responsible for taking baits in other cases rabbits Oryctolagus cuniculus or deer Cervus sp. would seem more likely culprits. This can be minimised by designing a bait station inaccessible to other animals (e.g. by placing the bait platform in a tube or under low cover) and by selecting a bait which is unpalatable to say cats and rabbits (a pollard or lured jam bait may be appropriate).
The bait interference method was introduced to provide a simple and easily executed alternative to trapping for estimating possum densities. Faecal pellet counts and spotlight counts (Batcheler 1970) have also been tried in exotic forests. Low faecal pellet counts (less than 5% frequency) and highly variable decay rates have made pellet counts unsatisfactory. Spotlight counts are unsuitable due to physical constraints (need for open areas) resulting in an unrepresentative or biased sample (as indicated by the high kills estimated in Table 1). Thus the bait interference method seems to be a feasible alternative to the well-tried trapping method. The most time-consuming operation is the establishment of the trap or bait interference lines since this involves marking the lines and setting up traps or bait stations. However if there is an adequate sample intensity (as determined from binomial tables) a greater number of assessment visits (say 4 or 5) on a bait interference line will give a better estimate of kill at a lower cost. In indigenous forest tracts this advantage is very evident. Similarly in sampling small areas of say 200 ha, or in low density areas such as farmland, insufficient animals may be trapped to obtain a reliable estimate of density but it would still be possible to obtain a minimum of 100 bait station sets.
The bait interference method therefore offers an alternative or even first choice for assessment of possum density. However the sample design must be chosen with care to obtain consistent estimates of relative density and one must be sure that the possum is the sole animal using the baits.
Data used for Tables 2 and 3 were kindly contributed by N.Z. Forest Products Ltd. and special thanks are due to Dr I. Andrew and Dr J.D. Coleman for constructive discussion leading to the development of the ideas presented in this paper.
Bailey, N.T.J. 1964. The elements of stochastic processes. Wiley, New York.
Bamford, J. 1970. Evaluating opossum poisoning operations by interference with non-toxic baits. Proceedings of the N.Z. Ecological Society 17: 118–125.
Batcheler, C.L. 1973. Estimation of population density changes. In Assessment and management of introduced animals in New Zealand forests. F.R.I. Symposium 14.
Cox, D.R. & Miller, H.D. 1965. The theory of stochastic processes. Chapman & Hall, London.
Jane, G.T. 1979. Opossum density assessment using the bait interference method. N.Z. Journal of Forestry 24: 61–66.
Johnston, N.L. & Kotz, S. 1969. Discrete distributions. Wiley, New York.
Seber, G.A.F. 1973. The estimation of animal abundance and related parameters. Griffin, London.
Statistical basis for the model
From the first premise that the density and hence probability of encounter remains constant from night to night, and animals are simply remembering bait station locations, we have:
P1(0) = P2(0) ----------------------= Pt(0)
where Pt(0) is the probability of non-encounter at time t.
Now n1/N = n2/n1-----------------------------------= nt/nt-1
where nt is the no. of baits remaining at night t, and N is the total bait stations set.page 194
This is a Poisson process, obtained from the superimposition of several independent Poisson processes (Bailey 1964, Cox and Miller 1965). The distribution function can be regarded as a composite one of a Poisson density function and an exponential time function (Bailey 1964) representing the time between encounters.
Thus an estimate of density dt can be obtained from (1):
MORGAN. Could you say how confident you are with a 40 m spacing between baits to overcome contagion?
JANE. Reasonably confident. You usually get contagion where a single animal has gone along a line, probably a larger animal.
MORGAN. Have you done any night observations at bait lines?
JANE. No I have not had the time. I gather from your own night observation work that random encounters may be involved rather more than the possum specifically following along the lines. J. Jolly mentioned earlier the scent signals left by animals, and this could draw them back to the same bait station later on.
WOOLLONS. Unless you can show the location of baits is purely random, all those equations could unfortunately be very, very biased, either way.
JANE. I think the night observation work probably helps interpretation regarding randomness. Are you worried about the successive nights being different from the initial night?
WOOLLONS. I'm worried about the basic model which you really have not changed. You have certainly made the operation far easier and that is a very good thing, but does the basic model hold? I don't think you have shown this.
JANE. Well I admit I have some rather limited data. I have incorporated bait interference lines on which are only 20 bait stations. This I feel is far too few and one probably requires nearer 100. To my mind if one carries out an operation a little more precisely, then we can probably get back to getting a reasonable linear relationship between bait interference and trapping estimates.
ANONYMOUS. In other words you are relying on trapping as your calibration. That raises the question of how reliable is trapping?
JANE. True.page 196
1 Only a relative density is given, not an absolute measure of numbers per unit area; the author has subsequently compared such an index with actual animal numbers using trap density estimates - see Jane (1979).