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Modelling bark beetle flight: a review

Modelling bark beetle flight: a review Helland, I. S..Anderhram.O, and Hoff. J . M . 1989, Modelling bark beetle flight: a review. - Holarct, Ecol. 12: 427-431. In studying the attraction of spruce bark beetles Ips typographus to a pheromone trap, we have proposed a simple diffusion model with drift as a description of the flight of the bark beetles. The model predictions were compared with Ihc results of a release-recapture experiment, and here we recapitulate the main results from the fitting of models to the data. Several modifications of the basic model were necessary in order to describe the data adequately. /. 5. Helland, Dept of Mathematical Sciences, Agricultural University of Norway, N-1432 As-NLH. Norway. O. Anderhrant, Dept of Ecology. University of Lund, Ecology Building, S-223 62 Lund, Sweden. J. M. Hoff, IBM. Dronning Maud.'i gt. 10, N-0250 Oslo 2. Norway. Introduction The purpose of the present paper is to sum up some of the experiences we gathered when we tried to find mathematical models to describe tbe flight of bark beetles, and when we tested these models by an experiment during tbe summer 1981. The intensions behind tbis endeavour went in several directions. First, we wanted to get some basic information about the behaviour of tbe bark beetles. Next, we wanted to test out a class of mathematical models in a concrete biological situation. Finally, one can bope tbat a model of tbis type, if it is simple enough and at the same time approximates the behaviour of the beetles sufficiently well, can be used as one of tbe building blocks in a larger simulation model describing the whole population dynamics of the bark beetles. Tbe flight of insects is a very complex pbenomenon, and any detailed mathematical model of sucb a fligbt must necessarily be an idealization. In the present investigation we have tried to model tbe fligbt of bark beetles that are attracted to a single pberomone source. In tbe experiment designed to test tbe models we used a pberomone trap as tbe attraction centre, but it seems reasonable to assume tbat the same models can be used to describe the flight of beetles attracted to an infested tree. The basic mathematical model used is a diffusion © HOLARCirC ECOLOGY model (random walk with small steps) where the beetles have a drift (bias in the random walk) pointing towards the pberomone source. Tbe magnitude of tbis drift term is inversely proportional to the distance from the source. In Fig. 1 we see some fligbt patterns simultated from sucb a simplified model, A diffusion process is in general defined as a continuous Markov process. In a model for tbe fligbt of bark beetles, this means that the bark beetles start afresh at eacb instant of time, tbus forgetting tbe past. Of course this is an idealization, but fortunately it needs only be approximately true. Some other examples of use of diffusions in biology are given in Okubo (1980). Tbe main advantage of the special model used bere is tbat practically every quantity of interest can be calculated explicitly from the model equations. In particular, one can predict tbe expected catch in a passive trap at any position relative to the release point and tbe attractive pberomone trap. The experiment To test the models and estimate the parameters of tbe models, a release-recapture experiment was performed. Spruce bark beetles were caught in window and drainpipe traps in Lardal. soutbern Norway, during the end of May and beginning of June 1981, and kept in a refrigerator at 4°C until about 1 h before release. The HOLARCTtC ECOLOGY 12;4 <1989) mone trap. The setup is shown in Fig. 2, where also the cateh in the passive traps in one experiment is indieated. Release was performed when weather conditions permitted flight, i.e. negligible wind and air temperature above 20°C. These conditions were fulfilled on 23 June and 8 July, 1981, and one release experiment was done each day. Around 6(KK) beetles were released from each of four platforms (NE, NW, SW and SE). at 12 m distance on 23 June and at 20 m distance on 8 July. The beetles were marked with fluorescent powder with a different colour for each release platform, and released simultaneously. The catch in the central trap was registered every 2. 5 or 10 min, and that ofthe passive trap 3.5 h after release, when the activity had ceased. Fig. 1. Two simulated traces (diffusion with drift) of beetles released at A and attracted towards a pheromone trap at B. Experimental results and model fitting traps were loaded with strips of Ipslure® containing 15(K) mgmethylbutenol, 70 mg cw-verbenol and 15 mg ipsdienol per metre. The release experiment was done in an open plain near Stensoffa Ecological Research Station outside Lund, southern Sweden. The experimental setup consisted of one central plastic drainpipe trap baited with a I m Ipslure® dispenser. Passive trap stations were placed in eight concentric rings around the central pheromone trap, the distance between the rings being 3 m. The number of trap stations per ring was eight, placed at ± 22.5°, ± 67.5°. ± 112.5° and ± 157.5''with respect to the line between the release point and the phero- Sca/e.- I, D - 5 o , Fig. 2. Design of experiment with 4 release platforms, one pheromone trap (centre) and 64 passive traps. The total catch in the passive traps is also indicated. The release points are marked by open crosses. From Helland et al. (1984). The detailed results from the experiment can be found in Helland et al. (1984). Here we only recall some ofthe main points. In addition to the central pheromone trap we also placed a pheromone trap of the same type 500 m from the experimental area in each of four directions to check for long-distance dispersal. These distant traps caught 89 beetles on 23 June and 127 beetles on 8 July, indicating that long distance dispersal did occur. This is not inconsistent with the prediction from the diffusion model, as indicated by the rightmost simulated trace in Fig. I. The main information from the experiment was given by the cateh in the passive traps. The original data from the first experiment are indicated in Fig. 2. It is seen that we did not only catch beetles on the downwind side of the pheromone trap, and this was also true if we concentrate on beetles from the downwind release platform. Thus the flying beetles are not only concentrated in the pheromone plume downwind from the central trap. This is consistent with the findings of Elkinton ct al. (1984). They concluded from experiments with male gypsy moth Lymaniriu dispar that moths were responding to the peak instantaneous concentrations rather than average concentration of pheromone predicted by time-average Gaussian plume models. It is possible to include the effect of wind in the diffusion models that we discuss here. However, then we will have difficulties in arriving at analytical results, and model predictions can only be made by simulation studies requiring vast computer time. Therefore we decide to try to eliminate the effect of wind on the data from this experiment hy making the catches symmetrical before analysing the data. This is done by identifying the capture point ofthe beetle by the distance from the central trap and the angle between the release point and the capture point, as seen from the central trap. Catches of coloured beetles from all four release platforms with the same distance and angle were added together. The result is shown by the hatched histograms in Fig. 3. HOLARCTIC ECOLOGY 12:4 i lyH")) Tab. I. The maximum likelihood estimates for three parameters A, C and b; using k = 2.26. Date 23 June 8 July 60.4 ± 5.9 67.0 ± 7.2 91.8 ±7.6 26.6 ± 3.3 6= (m') 0,121 ± ,008 0.160 ± .012 D, = (3.46 ± 1.59) m^ min'. The fit turned out to be better if we allowed for "flight exercise" in the model, /oo i.e., a state where the beetles moved according to a Z/r diffusion equation without drift before they started to respond to the pheromone. Taking such a state into account gives model equations that are more complicated than (1); for details, see Helland et al. (1984). Fig. 3. Comparison of predicted and observed catch (made In the simulations in Fig. 1, we have only simulated symmetric with respect to wind) in the passive traps. Vertical the first phase, and we have neglected the "flight exeraxis: Number of beetles. Horizontal axis: Angle between re- cise". The crucial constant here is k. In the simulations lease point and passive trap. From Holland ct al. (1984). we have used k = 2, which is within three standard deviations from the estimated value. When comparing these data with model predictions, If we use k = 2.26 in equation (1) and find maximum we must take into account that the pheromone trap is likelihood estimates for A, C and b, we arrive at the not 1(K)% efficient. Only 10.01% and 7.31% of the results of Tab. 1. The fit is shown by the black hisbeetles released were caught in the two experiments. tograms in Fig. 3. Although the deviations from the From experiments with male pea moths, Cydia nigri- observed values are significant by a formal x"-test. the cana. Wall and Perry (1982) report that even when visual fit seems good. One has to keep in mind that any moths came within f m of the trap, only 40% were model is an idealization, and that a x'-test with a large caught. If we assume that beetles reaching the central number of observations is extremely sensitive to deviatrap without being caught continue their flight with a tions from the model. new diffusion coefficient D, (phase two), we arrive at One observes that the estimated constants C and A the model prediction for the catch in the passive trap are of the same order of magnitude. Since N, must with polar coordinates (r,6) as necessarily be less than N,, this must mean that D, is less than Dj, i.e., the motility of the beetles decreased after they ceased responding to pheromone. Another complicating factor is that one may have loss AK,, (b,r) (1) of beetles during the phase when they were attracted Here a is the release distance (12 m or 20 m), k is a towards the pheromone. Such a loss turned out to be dimensionless constant characterizing the strength of negligible. Hence the total loss of beetles can be attrithe attraction towards the pheromone trap and K,, is a buted to two factors: (1) Some beetles do not respond at so-called modified Bessel function of the second kind. all or move away at onee; (2) Some beetles reach the The constants A. C and b, are given by C= N,E A= N,E (2) Tab. 2. The sex ratio of beetles caught in the pheromone trap and the passive traps at various distances on 23 June. Males Pheromone trap (sample) Passive traps with radius (m) where N, is the number of beetles released minus the number that fly out of the region at once, and N. is the number of beetles that enter phase two near the pheromone trap. Furthermore, D, is the diffusion coefficient during the first phase (diffusion with drift towards the pheromone trap), E is the efficiency parameter for the passive trap, D; is the diffusion coefficient in phase two, and Cl is the loss of beetles per unit time during this phase. By using the model equations, one ean also arrive at a prediction of the catch in the pheromone trap. Comparing these predictions with data gave R — 2.26 ± 0.12 and ]ioLARrnc ECOLOGY 12:4 mm Females % males S i m p l e Drift = const./dist. m o d e l s Threshold (example) Drift = const / ^ release i \....,___^ • E x p e r i m e n t s Fig. 4. Total catch per ring predicted for some diffusion models (above) and the experimental results (below). In the rightmost "threshold model" the beetles move randomly without drift until they meet a certain pheromone concentration and then fly directly to the odour source. Catc Distance from pheromone source S | holds when the number of traps per ring is relatively large. In Fig. 4 some of these model predictions are shown together with the experimental results from the present experiment. The leftmost graph gives the prediction of the main model of the present paper when phase two is neglected. The central graph is similar, except that we assume that the drift of the beetles is inversely proportional to the square of the distance from the pheromone source. As discussed in Helland (1983), such a model may result if we assume that the drift is proportional to the gradient of pheromone density, and a simple, but not unreasonable, form of this density is assumed. The third graph gives an example of the expected catch that may result from a model of a totally different kind: All the bark beetles start in a state without any drift, i.e., unaffected by the pheromone. Thus at the beginning they move according to an ordiOther attraction models nary Brownian motion. After a certain random time, The basic diffusion model used in this paper is about the however, each beetle enters a second state, where it simplest one that can be imagined if we want to take flies in straight line towards the pheromone trap. The drift towards a pheromone source into account. It probability per unit time of entering into this seeond turned out that several modifications of this basic model state is inversely proportional to the square of the diswere necessary in order to get a reasonable fit to the tance from the pheromone trap, i.e., greater the closer data (i.e.. symmetrization of the catches to eliminate the beetle is to the trap. This may be thought of as a wind effect, two phases, different diffusion coefficients simple "threshold" model for the flight of the beetles. Another two-state mode! for this first phase was disin the two phases). cussed in Helland et al. (1984). Further refinements of in Helland (1983) several other diffusion models were discussed, for instance models with a drift term in- the basic model are discussed in Helland (1982). versely proportional to the square of the distance to the The difference between theory and experiment in pheromone trap. In all these models it turned out that these models can be attributed to (at least) two factors: one can find an approximate expression for the ex- (1) The models have not taken phase two into account, pected total catch per ring when passive traps are dis- where some beetles fly out again after having been tributed along a ring with its centre at the pheromone attracted towards the pheromone trap. (2) In the model trap, as in the present experiment. The approximation predictions a great part of the contribution comes from HOLARCTIC ECOLOGY 12 pheromone trap but are not caught (phase two). The relative importance of these two factors is difficult to determine, hut both seem to contribute. One class of results that was not discussed in Helland et al. (1984). was the sex determination of the beetles caught. The sex ratios for the experiment on 23 June are given in Tab. 2. The sex ratio is similarly biased towards females in all the traps. At first glance this is surprising, since Schlyter et al. (1987) found a decreasing proportion of males landing on and in a pheromone baited pipe trap. However, the beetles used in our experiments had already responded to pheromone. and a selection might have taken place when they were caught the first time. passive traps near the release point, in particular with angle 6 near 0. In the present experiment the smallest angle was 22.5°. If this or a similar experiment should be repeated, one should also try to include passive traps along the line between the pheromone trap and the release platform. If this could be done, there would also be some possibilities of discriminating between the different models from catch data. An attraction model of a quite different type has been proposed by Perry and Wall (1984). This was a simulation model where the shape of the so-called threshold contour for pheromone was taken into account. The model had several parameters tbat partly depended upon the micrometeorological conditions. Perry and Wall s model considerations were based on the observation by David et al. (1982) that moths fly upwind if they deteet pheromone and crosswind otherwise. For beetles flying over open ground with constant wind direction, a model of this type will predict a movement along straight lines in the upwind direction. We must have in mind that Perry and Wall's mode! also represents an idealization. Since it is a simulation model, it is difficult to estimate the parameters from experiments of the type that we bave performed. Some of their model predictions, for instance on the interaetion between phcromone traps placed near eaeh other (Wall and Perry 1978) will also be predicted by our diffusion models with drift. Which idealization that will be most fruitful, remains to be determined. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Ecography Wiley

Modelling bark beetle flight: a review

Ecography , Volume 12 (4) – Dec 1, 1989

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Wiley
Copyright
Copyright © 1989 Wiley Subscription Services, Inc., A Wiley Company
ISSN
0906-7590
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1600-0587
DOI
10.1111/j.1600-0587.1989.tb00918.x
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Abstract

Helland, I. S..Anderhram.O, and Hoff. J . M . 1989, Modelling bark beetle flight: a review. - Holarct, Ecol. 12: 427-431. In studying the attraction of spruce bark beetles Ips typographus to a pheromone trap, we have proposed a simple diffusion model with drift as a description of the flight of the bark beetles. The model predictions were compared with Ihc results of a release-recapture experiment, and here we recapitulate the main results from the fitting of models to the data. Several modifications of the basic model were necessary in order to describe the data adequately. /. 5. Helland, Dept of Mathematical Sciences, Agricultural University of Norway, N-1432 As-NLH. Norway. O. Anderhrant, Dept of Ecology. University of Lund, Ecology Building, S-223 62 Lund, Sweden. J. M. Hoff, IBM. Dronning Maud.'i gt. 10, N-0250 Oslo 2. Norway. Introduction The purpose of the present paper is to sum up some of the experiences we gathered when we tried to find mathematical models to describe tbe flight of bark beetles, and when we tested these models by an experiment during tbe summer 1981. The intensions behind tbis endeavour went in several directions. First, we wanted to get some basic information about the behaviour of tbe bark beetles. Next, we wanted to test out a class of mathematical models in a concrete biological situation. Finally, one can bope tbat a model of tbis type, if it is simple enough and at the same time approximates the behaviour of the beetles sufficiently well, can be used as one of tbe building blocks in a larger simulation model describing the whole population dynamics of the bark beetles. Tbe flight of insects is a very complex pbenomenon, and any detailed mathematical model of sucb a fligbt must necessarily be an idealization. In the present investigation we have tried to model tbe fligbt of bark beetles that are attracted to a single pberomone source. In tbe experiment designed to test tbe models we used a pberomone trap as tbe attraction centre, but it seems reasonable to assume tbat the same models can be used to describe the flight of beetles attracted to an infested tree. The basic mathematical model used is a diffusion © HOLARCirC ECOLOGY model (random walk with small steps) where the beetles have a drift (bias in the random walk) pointing towards the pberomone source. Tbe magnitude of tbis drift term is inversely proportional to the distance from the source. In Fig. 1 we see some fligbt patterns simultated from sucb a simplified model, A diffusion process is in general defined as a continuous Markov process. In a model for tbe fligbt of bark beetles, this means that the bark beetles start afresh at eacb instant of time, tbus forgetting tbe past. Of course this is an idealization, but fortunately it needs only be approximately true. Some other examples of use of diffusions in biology are given in Okubo (1980). Tbe main advantage of the special model used bere is tbat practically every quantity of interest can be calculated explicitly from the model equations. In particular, one can predict tbe expected catch in a passive trap at any position relative to the release point and tbe attractive pberomone trap. The experiment To test the models and estimate the parameters of tbe models, a release-recapture experiment was performed. Spruce bark beetles were caught in window and drainpipe traps in Lardal. soutbern Norway, during the end of May and beginning of June 1981, and kept in a refrigerator at 4°C until about 1 h before release. The HOLARCTtC ECOLOGY 12;4 <1989) mone trap. The setup is shown in Fig. 2, where also the cateh in the passive traps in one experiment is indieated. Release was performed when weather conditions permitted flight, i.e. negligible wind and air temperature above 20°C. These conditions were fulfilled on 23 June and 8 July, 1981, and one release experiment was done each day. Around 6(KK) beetles were released from each of four platforms (NE, NW, SW and SE). at 12 m distance on 23 June and at 20 m distance on 8 July. The beetles were marked with fluorescent powder with a different colour for each release platform, and released simultaneously. The catch in the central trap was registered every 2. 5 or 10 min, and that ofthe passive trap 3.5 h after release, when the activity had ceased. Fig. 1. Two simulated traces (diffusion with drift) of beetles released at A and attracted towards a pheromone trap at B. Experimental results and model fitting traps were loaded with strips of Ipslure® containing 15(K) mgmethylbutenol, 70 mg cw-verbenol and 15 mg ipsdienol per metre. The release experiment was done in an open plain near Stensoffa Ecological Research Station outside Lund, southern Sweden. The experimental setup consisted of one central plastic drainpipe trap baited with a I m Ipslure® dispenser. Passive trap stations were placed in eight concentric rings around the central pheromone trap, the distance between the rings being 3 m. The number of trap stations per ring was eight, placed at ± 22.5°, ± 67.5°. ± 112.5° and ± 157.5''with respect to the line between the release point and the phero- Sca/e.- I, D - 5 o , Fig. 2. Design of experiment with 4 release platforms, one pheromone trap (centre) and 64 passive traps. The total catch in the passive traps is also indicated. The release points are marked by open crosses. From Helland et al. (1984). The detailed results from the experiment can be found in Helland et al. (1984). Here we only recall some ofthe main points. In addition to the central pheromone trap we also placed a pheromone trap of the same type 500 m from the experimental area in each of four directions to check for long-distance dispersal. These distant traps caught 89 beetles on 23 June and 127 beetles on 8 July, indicating that long distance dispersal did occur. This is not inconsistent with the prediction from the diffusion model, as indicated by the rightmost simulated trace in Fig. I. The main information from the experiment was given by the cateh in the passive traps. The original data from the first experiment are indicated in Fig. 2. It is seen that we did not only catch beetles on the downwind side of the pheromone trap, and this was also true if we concentrate on beetles from the downwind release platform. Thus the flying beetles are not only concentrated in the pheromone plume downwind from the central trap. This is consistent with the findings of Elkinton ct al. (1984). They concluded from experiments with male gypsy moth Lymaniriu dispar that moths were responding to the peak instantaneous concentrations rather than average concentration of pheromone predicted by time-average Gaussian plume models. It is possible to include the effect of wind in the diffusion models that we discuss here. However, then we will have difficulties in arriving at analytical results, and model predictions can only be made by simulation studies requiring vast computer time. Therefore we decide to try to eliminate the effect of wind on the data from this experiment hy making the catches symmetrical before analysing the data. This is done by identifying the capture point ofthe beetle by the distance from the central trap and the angle between the release point and the capture point, as seen from the central trap. Catches of coloured beetles from all four release platforms with the same distance and angle were added together. The result is shown by the hatched histograms in Fig. 3. HOLARCTIC ECOLOGY 12:4 i lyH")) Tab. I. The maximum likelihood estimates for three parameters A, C and b; using k = 2.26. Date 23 June 8 July 60.4 ± 5.9 67.0 ± 7.2 91.8 ±7.6 26.6 ± 3.3 6= (m') 0,121 ± ,008 0.160 ± .012 D, = (3.46 ± 1.59) m^ min'. The fit turned out to be better if we allowed for "flight exercise" in the model, /oo i.e., a state where the beetles moved according to a Z/r diffusion equation without drift before they started to respond to the pheromone. Taking such a state into account gives model equations that are more complicated than (1); for details, see Helland et al. (1984). Fig. 3. Comparison of predicted and observed catch (made In the simulations in Fig. 1, we have only simulated symmetric with respect to wind) in the passive traps. Vertical the first phase, and we have neglected the "flight exeraxis: Number of beetles. Horizontal axis: Angle between re- cise". The crucial constant here is k. In the simulations lease point and passive trap. From Holland ct al. (1984). we have used k = 2, which is within three standard deviations from the estimated value. When comparing these data with model predictions, If we use k = 2.26 in equation (1) and find maximum we must take into account that the pheromone trap is likelihood estimates for A, C and b, we arrive at the not 1(K)% efficient. Only 10.01% and 7.31% of the results of Tab. 1. The fit is shown by the black hisbeetles released were caught in the two experiments. tograms in Fig. 3. Although the deviations from the From experiments with male pea moths, Cydia nigri- observed values are significant by a formal x"-test. the cana. Wall and Perry (1982) report that even when visual fit seems good. One has to keep in mind that any moths came within f m of the trap, only 40% were model is an idealization, and that a x'-test with a large caught. If we assume that beetles reaching the central number of observations is extremely sensitive to deviatrap without being caught continue their flight with a tions from the model. new diffusion coefficient D, (phase two), we arrive at One observes that the estimated constants C and A the model prediction for the catch in the passive trap are of the same order of magnitude. Since N, must with polar coordinates (r,6) as necessarily be less than N,, this must mean that D, is less than Dj, i.e., the motility of the beetles decreased after they ceased responding to pheromone. Another complicating factor is that one may have loss AK,, (b,r) (1) of beetles during the phase when they were attracted Here a is the release distance (12 m or 20 m), k is a towards the pheromone. Such a loss turned out to be dimensionless constant characterizing the strength of negligible. Hence the total loss of beetles can be attrithe attraction towards the pheromone trap and K,, is a buted to two factors: (1) Some beetles do not respond at so-called modified Bessel function of the second kind. all or move away at onee; (2) Some beetles reach the The constants A. C and b, are given by C= N,E A= N,E (2) Tab. 2. The sex ratio of beetles caught in the pheromone trap and the passive traps at various distances on 23 June. Males Pheromone trap (sample) Passive traps with radius (m) where N, is the number of beetles released minus the number that fly out of the region at once, and N. is the number of beetles that enter phase two near the pheromone trap. Furthermore, D, is the diffusion coefficient during the first phase (diffusion with drift towards the pheromone trap), E is the efficiency parameter for the passive trap, D; is the diffusion coefficient in phase two, and Cl is the loss of beetles per unit time during this phase. By using the model equations, one ean also arrive at a prediction of the catch in the pheromone trap. Comparing these predictions with data gave R — 2.26 ± 0.12 and ]ioLARrnc ECOLOGY 12:4 mm Females % males S i m p l e Drift = const./dist. m o d e l s Threshold (example) Drift = const / ^ release i \....,___^ • E x p e r i m e n t s Fig. 4. Total catch per ring predicted for some diffusion models (above) and the experimental results (below). In the rightmost "threshold model" the beetles move randomly without drift until they meet a certain pheromone concentration and then fly directly to the odour source. Catc Distance from pheromone source S | holds when the number of traps per ring is relatively large. In Fig. 4 some of these model predictions are shown together with the experimental results from the present experiment. The leftmost graph gives the prediction of the main model of the present paper when phase two is neglected. The central graph is similar, except that we assume that the drift of the beetles is inversely proportional to the square of the distance from the pheromone source. As discussed in Helland (1983), such a model may result if we assume that the drift is proportional to the gradient of pheromone density, and a simple, but not unreasonable, form of this density is assumed. The third graph gives an example of the expected catch that may result from a model of a totally different kind: All the bark beetles start in a state without any drift, i.e., unaffected by the pheromone. Thus at the beginning they move according to an ordiOther attraction models nary Brownian motion. After a certain random time, The basic diffusion model used in this paper is about the however, each beetle enters a second state, where it simplest one that can be imagined if we want to take flies in straight line towards the pheromone trap. The drift towards a pheromone source into account. It probability per unit time of entering into this seeond turned out that several modifications of this basic model state is inversely proportional to the square of the diswere necessary in order to get a reasonable fit to the tance from the pheromone trap, i.e., greater the closer data (i.e.. symmetrization of the catches to eliminate the beetle is to the trap. This may be thought of as a wind effect, two phases, different diffusion coefficients simple "threshold" model for the flight of the beetles. Another two-state mode! for this first phase was disin the two phases). cussed in Helland et al. (1984). Further refinements of in Helland (1983) several other diffusion models were discussed, for instance models with a drift term in- the basic model are discussed in Helland (1982). versely proportional to the square of the distance to the The difference between theory and experiment in pheromone trap. In all these models it turned out that these models can be attributed to (at least) two factors: one can find an approximate expression for the ex- (1) The models have not taken phase two into account, pected total catch per ring when passive traps are dis- where some beetles fly out again after having been tributed along a ring with its centre at the pheromone attracted towards the pheromone trap. (2) In the model trap, as in the present experiment. The approximation predictions a great part of the contribution comes from HOLARCTIC ECOLOGY 12 pheromone trap but are not caught (phase two). The relative importance of these two factors is difficult to determine, hut both seem to contribute. One class of results that was not discussed in Helland et al. (1984). was the sex determination of the beetles caught. The sex ratios for the experiment on 23 June are given in Tab. 2. The sex ratio is similarly biased towards females in all the traps. At first glance this is surprising, since Schlyter et al. (1987) found a decreasing proportion of males landing on and in a pheromone baited pipe trap. However, the beetles used in our experiments had already responded to pheromone. and a selection might have taken place when they were caught the first time. passive traps near the release point, in particular with angle 6 near 0. In the present experiment the smallest angle was 22.5°. If this or a similar experiment should be repeated, one should also try to include passive traps along the line between the pheromone trap and the release platform. If this could be done, there would also be some possibilities of discriminating between the different models from catch data. An attraction model of a quite different type has been proposed by Perry and Wall (1984). This was a simulation model where the shape of the so-called threshold contour for pheromone was taken into account. The model had several parameters tbat partly depended upon the micrometeorological conditions. Perry and Wall s model considerations were based on the observation by David et al. (1982) that moths fly upwind if they deteet pheromone and crosswind otherwise. For beetles flying over open ground with constant wind direction, a model of this type will predict a movement along straight lines in the upwind direction. We must have in mind that Perry and Wall's mode! also represents an idealization. Since it is a simulation model, it is difficult to estimate the parameters from experiments of the type that we bave performed. Some of their model predictions, for instance on the interaetion between phcromone traps placed near eaeh other (Wall and Perry 1978) will also be predicted by our diffusion models with drift. Which idealization that will be most fruitful, remains to be determined.

Journal

EcographyWiley

Published: Dec 1, 1989

There are no references for this article.