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Insect pest control by a spatial barrier
Er.oI,OF IIL mODELLInG ELSEVIER
Ecological Modelling 75/76 (1994) 203-211
Insect pest control by a spatial barrier R a l f M a r s u l a *, Christian Wissel Centerfor Encironmental Research UFZ, Department of Ecosystem Studies (OSA), Postfach 2, 04301 Leipzig, Germany
Abstract
A model is introduced which determines the dynamics of a pest population with respect to control by the sterile insect technique (SIT). Random spreading of the population is described by diffusion. The model determines the conditions for stopping an invasion of the pest population by a barrier. The optimization with respect to the costs of the barrier is discussed. There is a conflict between costs and safety of a barrier. We give a proposal which mitigates this conflict. Key words: Pest control; Population dynamics; Spatial barrier; Sterile insect technique
I. Introduction
Point of departure of our discussion is an insect pest control project against the screw worm fly (Cochliomyia hominivorax, Diptera: Calliphoridae). Its larvae parasite in wounds of mammals, often with lethal consequences. Humans are infested too (Wall and Stevens, 1990). The screw worm has a enormous economic importance (about $100 million loss to the livestock industry in 1955 in the USA). Therefore the sterile insect technique (SIT) was used to eliminate this insect (Franz and Krieg, 1982; Mackley and Brown, 1985). The concept of the SIT is based on the factory rearing of millions of flies, which are sterilized by T-rays and released. Wild females mating with sterile males lay infertile eggs. Thus the size of the native population is reduced until it is eventually eradicated (Knipling, 1982). After a successful elimination of the screw worm using the SIT in the USA a barrier at the border U S A - M e x i c o was established to control the seasonal invasion
* Corresponding author. 0304-3800/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI 0304-3800(93)E0129-Q
R. Marsula, C. Wissel/ Ecological Modelling 75 / 76 (1994) 203-211
204
of screw worms from Mexico. (A barrier is a strip of land were steriles are released in order to stop the spread of the insect population from one side to the other.) However, the barrier was not placed in an optimal position. Steriles had to be released in a region which has - from the American east coast to the west coast an extension of more than 3000 km. Therefore it is intended to move the barrier to the south. If the political situation is stable, the Darien-Gap in Panama will be reached in a few years. Then the whole of Central America will be free of the screw worm (Wendell Snow et al., 1985). This project represents a new quality in biological pest control because of its spatial dimension (over a whole continent), its time horizon (struggle against the screw worm since 37 years) and its optimistic aim (control of a permanent invasion). But there are still questions of principal interest which have not been discussed in the literature so far. This p a p e r we will pose two questions: 1. In what conditions are spatial barriers able to control the invasion of an insect pest? 2. Is it possible to find an optimal strategy for an SIT barrier?
2. The model
To discuss these questions we develop a model which uses essential features of the screw worm population dynamics. We reveal some general trends which could be important for other pest control projects too. The model is developed in three steps.
2.1. Model without sterile insects We start with a larval stage and an imaginal stage, denoted by n t and N t, respectively, at the discrete time, t. The following two equations describe a density-dependent regulation at the larval stage which is a plausible assumption for a lot of insect species.
n, = F N , ,
(1)
Nt+ 1 = o ( n t ) n '.
(2)
F denotes the fertility, i.e. the mean number of eggs multiplied by the probability that an egg develops into a larva, o ( n t) is the probability that a larva develops into an imago. This probability depends on the actual number of larvae, n r The function o ( n t) requires a submodel for the larval stage. For this submodel we choose the approach: 1 dnt(~- ) nt(r ) d~"
c1(1 +Cznt(~')).
(3)
The idea of this approach (Prout, 1978): Between two discrete time steps, t and t + 1, a continuous time, ~- runs from 0 to T. ~- is the point in time in the larval
R. Marsula, C. Wissel/ Ecological Modelling 75/76 (1994)203-211
205
stage. Then the mortality rate (left side of the equation) increases lineary with the number of larvae (right side). The equation can be read, on the one hand, as the most simple mathematical approach and as a fit on data which are available for some insect species on the other (Hassel, 1975). The solution of Eq. 3 is given by (Bronshtein and Semendyayev, 1985):
nt(O ) exp(-c1~" ) nt('r) = 1 + (1 - e x p ( - c , 7 ) ) C z n t ( O ) "
(4)
Using Eq. 4, the probability o(n,) results as the ratio of the number of larvae which arrive at the pupal stage to the larvae number which enter at the larval stage.
o(,,,) = n,( T) /nt(O).
(5)
With the help of Eqs. 1, 2, 4 as well as 5 and with the two definitions R = exp(-cl~" )
and
1 K = (1 - e x p ( - c l T ) ) c 2 F '
(6)
exp(-clT)F-
(7)
the following equation results: RN t N,+, = l+(R-
Art . l)~-
(8)
This equation is the well-known time-discrete analogue of the logistic equation. K represents the equilibrium level or the capacity, R the rate of reproduction.
2.2. Release of steriles To include the SIT into the model we use two assumptions which are in good agreement with the biology of the screw worm (Barclay, 1982, 1984; Peterson et al., 1983; Krieg and Franz, 1989). 1. Wild females mate only once. 2. Wild and sterile males show equal competitive ability. In this case the fertility F has to be substituted by F ( N , / ( N , + S)), where S describes the annual number of steriles which are released, and N J ( N , + S) equals the probability that a female of the wild population mates with a fertile male. This leads to the following equation:
N, RNt N~ + S Ut+ 1 =
I+(R_a)
Aft K Nt+S
gt
(9)
R. Marsula, C. Wissel/ Ecological Modelling 75/76 (1994) 203-211
206
0 Fig. 1. Sketch of a barrier. The left hand curve symbolizes a pest population which spreads from left to right, x represents the spatial coordinate. In the barrier (x > 0) the density S of sterile insects is released.
2.3. Spreading of insects To discuss pest control by a barrier, we have to include the spreading of insects. Fig. 1 illustrates how a barrier works. We have to concider two zones. The pest population develops entirely in the left hand zone (x < 0). x represents the spatial coordinate. Steriles are released on the right hand side (for x > 0). The barrier starts at x = 0. The pest population drifts from the left hand side to the right hand side into the barrier (Fig. 1). According to empirical investigations (Welch, 1989) we suggest the spread of the screw worm population to be a random process. This is taken into account by adding a diffusion term D(d2/dx2)Nt to the right hand side of Eq. 9 (D -= diffusion constant) (Manorajan and Van den Driessche, 1986).
N, RNt N t + S
N,+t
I+(R-1)
Nt
d2 Art
+Ddx~ .---=N.
(10)
K Nt+S Eq. 10 models the pest control by a barrier. Figs. 2-5 show some typical numerical solutions. 3. N u m e r i c a l solutions
In Fig. 2 a pest population starts with an initial density distribution (the left hand curve) and drifts from left to right into the barrier where the pest is stopped. NtIKJ .751.0 . ~ ~ ~ ~ 1 .5.25-30 -20 -10
0
1'0 20 x
Fig. 2. Spatial distribution (density) of an insect pest population (in units of K) versus x (in units of v ~ ) after 0, 10, 20.... generations (S = 0.5K, R = 2). The left hand curve represents the initial distribution. The spread of the pest population is stopped by the barrier.
R. Marsula, C. Wissel/ Ecological Modelling 75 / 76 (1994) 203-211
207
Nt/K, 1.025.5.25. 0
-25
0
25
50
7~
100 x
Fig. 3. Spatial distribution after 0, 2, 4,... generations (S = 0.5K, R = 2) starting from the right hand curve. The pest population has collapsed after nearly 8 generations.
Nt/K, 1.0.'/5 .5" .250
0
50
700
750 x
Fig. 4. Distribution (from left hand to right hand) after 0, 2, 4 .... generations at S = 0.05K, R = 2. After nearly 8 generations the population density has taken on the shape of a step function.
x represents the spatial coordinate. The barrier starts at x = 0. Between two curves ten generations of the population pass. If wild insects are present in the barrier before steriles are released, the initial population (right hand curve) will collapse after eight generations (Fig. 3). In Figs. 2 and 3 the density of steriles has a value of S = 0.5K. The features of the solution change totally if S is reduced to S = 0.05K as shown in the next two figures. Then the density of the wild population rises and forms a step function (Fig. 4). Thereafter the population propagates in slow motion from left to right through the barrier (Fig. 5). In this case the control does not work. The barrier fails.
NtlK 1.0 -.75.5.250
6
s'0
160
150",
Fig. 5. Distribution as in Fig. 4 after 10, 20,... generations (S = 0.05K, R = 2). T h e population is spreading in slow-motion from left to right through the barrier. In this case the control does not work. The release of steriles results in the slight decrease of the wild population at x = 0 and the slow-down of the spread.
208
R. Marsula, C. Wissel/Ecological Modelling 75/76 (1994)203-211
We worked out by numerical investigations that the control of the pest is possible only if the following condition is valid: S - -
R-1
> 0.2105K.
(11)
4. Stochastical effects
Eq. 11 is a first step to answer the question: In what conditions are barriers able to control the invasion of an insect pest ? However, Eq. 11 arises from a deterministic model and it is well known that stochastical effects can often change the characteristics of a deterministic model totally. We found that a barrier which is reliable in a deterministic sense fails if stochastic effects are taken into account. A master equation with stochastical birth and death processes as well as random propagation (in the sense of a discrete jump process) has been investigated (Gardiner, 1983; Wissel, 1989; Marsula, 1992). The following points explain and illustrate this result. A barrier being reliable in a deterministic sense has the following stochastical features: 1. The mean population density asymptotically tends towards the exponential function: N ( x ) ~ e x p ( - V / ~ / D x ) . At no point in the barrier is N ( x ) zero, at which the population would be extinct (/~ = individual mortility rate, D = diffusion constant). 2. Consequently a barrier of any width is only reliable with a probability less than 1. To calculate this probability we determine the width of the barrier by demanding that N = N at the right hand border of the barrier for an arbitrary but fixed value of A~ (i.e., the insect release is stopped at that place were = A~). Then the barrier will fail with the probability: W= 1 - exp(- ~AI).
(12)
W increases if the mean live span 1//z or the diffusion constant D increases. W decreases if A~ decreases. This result gives our deterministic model (Eq. 10) a new interpretation. There is always a probability that the barrier may fail, but it is possible to control this probability by the parameter /V (or equivalently by the place where the insect release is stopped).
5. O p t i m i z a t i o n
Now we are going to discuss our second question: Is it possible to find an optimal strategy for an SIT barrier minimizing the costs? (Taylor, 1976). We assume that the costs for each factory-reared insect are constant. Therefore the total costs of controlling per year are: S . width of the barrier.
R. Marsula, C. Wissel/ Ecological Modelling 75/76 (1994) 203-211
209
18
16
14
12
I
0.3
014
015
s/K Fig. 6. Costs as a function of the density of steriles (Sopt = 0.226K, optimal costs = 12.28Kv~, R = 2, = 10-4K).
Table 1 Sopt and the optimal costs as a function of A3 N/K
Sont / K
costs
10 -4 5.10 4 10-3 5.10 -3 10 -z
0.226 0.246 0.254 0.266 0.274
12.28 4.05 2.97 2.02 1.81
Fig. 6 shows the result of a first optimization. T h e costs versus the density S of steriles are shown. T h e r e is a n o p t i m u m at Soot = 0.226K. If S is r e d u c e d below S = 0.2105K, the b a r r i e r will fail. F o r this calculation the rate of r e p r o d u c t i o n was fixed at R = 2 a n d N had the value of 10-4K. If A~ is varied as shown in T a b l e 1, the safety varies a n d the costs vary too. T h e table d e m o n s t r a t e s the conflict b e t w e e n costs a n d safety of a barrier. This seems to be trivial a n d intuitively clear, b u t we are able to p r o p o s e a b e t t e r SIT-strategy which mitigates this conflict. S i m u l a t i o n s lead us to the suggestion that a b a r r i e r should consist of two parallel substrips (Fig. 7).
strip 1
strip 2
i
IIx
Fig. 7. S k e t c h o f a b a r r i e r c o n s i s t i n g o f t w o strips.
210
R. Marsula, C. W~tssel/ Ecological Modelling 75/76 (1994) 203-211
• The spatial extention of the first strip (i.e., the strip with the high population density) should be chosen in a way that the pest population at the right hand border of this strip is reduced to 1 - 5 % of the capacity. The second strip controls the rest population. • The density S of steriles in the first strip should be of the order of 1 / 4 K ( R - 1). • In the second strip the density of steriles has to be as small as possible. If S is too small, the barrier will fail, but it is profitable to choose a density near this critical density. This optimized two-strip strategy reduces the costs (in comparison with the control by only one strip) to 14.7% (by a factor 6.8). Moreover, the conflict between costs and safety is mitigated. If A~ is changed from N = 10-2K to /V = 10-4K, the safety is increased significantly (Eq. 12) while the costs only grow slightly (by 19.2%). Naturally it is possible to improve the proposed SIT-strategy by a barrier of three or more substrips. But a detailed numerical investigation shows that a further reduction of the costs by more than 50% is not possible.
6. Summary A model has been introduced which describes an insect pest control strategy by a barrier. We consider the insect population to consist of a larval and an imaginal stage. Intraspecific competition takes place at the larval stage. We were able to determine effective conditions for the control of the pest by the barrier (Eq. 11). A barrier which is reliable in a deterministic sense will fail if stochastical birth and death processes as well as stochastical propagation of the population are discussed (Eq. 12). It is possible to calculate the probability that a barrier fails, but if the safety of the barrier is increased, the costs increase too. We propose a better SIT strategy which mitigates this conflict between costs and safety. We suggest the division of the barrier into two strips with different densities of released steriles. This two-strip strategy can be of practical interest: It is simple to implement and has a dramatic effect on the reduction of costs.
References Barclay, H.J., 1982. The sterile release method with unequal male competitive ability. Ecol. Modelling, 15: 251-263. Barclay, H.J., 1984. Demographic consequences of monogamy and polygamy for a sterile release program. Prot. Ecol., 6: 209-217. Bronshtein, I.N. and Semendyayev, K.A., 1985. Handbook of Mathematics. Verlag Harry Deutsch, Frankfurt. Franz, J.M. and Krieg, A., 1982. Biologische Sch~idlingsbek~impfung. Paul Parey, Berlin. Gardiner, C.W., 1983. Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences. Springer, Berlin. Hassel, M.P., 1975. Density-dependence in single-species populations. J. Anim. Ecol., 44: 283-295.
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Knipling, E.F., 1982. Present status and future trends of the SIT approach to the control of arthropod pests. In: Sterile Insect Technique and Radiation in Insect Control. IAEA-SM-255/1, IAEA, Vienna, pp. 3-23. Krieg, A. and Franz, J.M., 1989. Lehrbuch der Biologischen Schiidlingsbek~impfung. Paul Parey, Berlin. Mackley, J.W. and Brown, H.E., 1985. Screwworm research 1976-84: attractants, behavior, ecology, and development of survey and control technologies. Miscellaneous Publications, Entomol. Soc. Am., 62: 49-55. Manoranjan, V.S. and Van den Driessche, P., 1986. On a diffusion model for sterile insect release. Math. Biosci., 79: 199-208. Marsula, R., 1992. Modelle zur Kontrolle von Sch~idlingen durch eine riiumliche Barriere. Ph.D. Thesis, University of Marburg, Germany. Peterson, R., Ocambo-Candido, A. and Del Var Petersen, H., 1983. Longevity and mating capacity of male screwworms. J. Econ. Entomol., 76: 1262-1264. Prout, T., 1978. The joints effect of the release of sterile males and immigration of fertilized females on a density regulated population. Theor. Popul. Biol., 13: 40-71. Taylor, C.R., 1976. Determining optimal sterile male release strategies. Environ. Entomol., 5: 87-95. Wall, R. and Stevens, J., 1990. The turn of the screw worm. New Sci., 1720: 54-57. Welch, J., 1989. Observations on dispersal of sterile screwworms in relation to a clear-cut utility right of way in Southern Mexico. J. Econ. Entomol., 82: 1580-1584. Wendell Snow, J., Whitten, C.J., Salinas, A., Fetter, J. and Sudlow, W.H., 1985. The screwworm, Cochliomyia hominil,orax (Diptera: Calliphoridae), in Central America and proposed plans for its eradication south to the Darien Gap in Panama. J. Med. Entomol., 22: 353-360. Wissel, C., 1989. Theoretische Okologie - Eine Einfiihrung. Springer, Berlin.
Ecological Modelling 75/76 (1994) 203-211
Insect pest control by a spatial barrier R a l f M a r s u l a *, Christian Wissel Centerfor Encironmental Research UFZ, Department of Ecosystem Studies (OSA), Postfach 2, 04301 Leipzig, Germany
Abstract
A model is introduced which determines the dynamics of a pest population with respect to control by the sterile insect technique (SIT). Random spreading of the population is described by diffusion. The model determines the conditions for stopping an invasion of the pest population by a barrier. The optimization with respect to the costs of the barrier is discussed. There is a conflict between costs and safety of a barrier. We give a proposal which mitigates this conflict. Key words: Pest control; Population dynamics; Spatial barrier; Sterile insect technique
I. Introduction
Point of departure of our discussion is an insect pest control project against the screw worm fly (Cochliomyia hominivorax, Diptera: Calliphoridae). Its larvae parasite in wounds of mammals, often with lethal consequences. Humans are infested too (Wall and Stevens, 1990). The screw worm has a enormous economic importance (about $100 million loss to the livestock industry in 1955 in the USA). Therefore the sterile insect technique (SIT) was used to eliminate this insect (Franz and Krieg, 1982; Mackley and Brown, 1985). The concept of the SIT is based on the factory rearing of millions of flies, which are sterilized by T-rays and released. Wild females mating with sterile males lay infertile eggs. Thus the size of the native population is reduced until it is eventually eradicated (Knipling, 1982). After a successful elimination of the screw worm using the SIT in the USA a barrier at the border U S A - M e x i c o was established to control the seasonal invasion
* Corresponding author. 0304-3800/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI 0304-3800(93)E0129-Q
R. Marsula, C. Wissel/ Ecological Modelling 75 / 76 (1994) 203-211
204
of screw worms from Mexico. (A barrier is a strip of land were steriles are released in order to stop the spread of the insect population from one side to the other.) However, the barrier was not placed in an optimal position. Steriles had to be released in a region which has - from the American east coast to the west coast an extension of more than 3000 km. Therefore it is intended to move the barrier to the south. If the political situation is stable, the Darien-Gap in Panama will be reached in a few years. Then the whole of Central America will be free of the screw worm (Wendell Snow et al., 1985). This project represents a new quality in biological pest control because of its spatial dimension (over a whole continent), its time horizon (struggle against the screw worm since 37 years) and its optimistic aim (control of a permanent invasion). But there are still questions of principal interest which have not been discussed in the literature so far. This p a p e r we will pose two questions: 1. In what conditions are spatial barriers able to control the invasion of an insect pest? 2. Is it possible to find an optimal strategy for an SIT barrier?
2. The model
To discuss these questions we develop a model which uses essential features of the screw worm population dynamics. We reveal some general trends which could be important for other pest control projects too. The model is developed in three steps.
2.1. Model without sterile insects We start with a larval stage and an imaginal stage, denoted by n t and N t, respectively, at the discrete time, t. The following two equations describe a density-dependent regulation at the larval stage which is a plausible assumption for a lot of insect species.
n, = F N , ,
(1)
Nt+ 1 = o ( n t ) n '.
(2)
F denotes the fertility, i.e. the mean number of eggs multiplied by the probability that an egg develops into a larva, o ( n t) is the probability that a larva develops into an imago. This probability depends on the actual number of larvae, n r The function o ( n t) requires a submodel for the larval stage. For this submodel we choose the approach: 1 dnt(~- ) nt(r ) d~"
c1(1 +Cznt(~')).
(3)
The idea of this approach (Prout, 1978): Between two discrete time steps, t and t + 1, a continuous time, ~- runs from 0 to T. ~- is the point in time in the larval
R. Marsula, C. Wissel/ Ecological Modelling 75/76 (1994)203-211
205
stage. Then the mortality rate (left side of the equation) increases lineary with the number of larvae (right side). The equation can be read, on the one hand, as the most simple mathematical approach and as a fit on data which are available for some insect species on the other (Hassel, 1975). The solution of Eq. 3 is given by (Bronshtein and Semendyayev, 1985):
nt(O ) exp(-c1~" ) nt('r) = 1 + (1 - e x p ( - c , 7 ) ) C z n t ( O ) "
(4)
Using Eq. 4, the probability o(n,) results as the ratio of the number of larvae which arrive at the pupal stage to the larvae number which enter at the larval stage.
o(,,,) = n,( T) /nt(O).
(5)
With the help of Eqs. 1, 2, 4 as well as 5 and with the two definitions R = exp(-cl~" )
and
1 K = (1 - e x p ( - c l T ) ) c 2 F '
(6)
exp(-clT)F-
(7)
the following equation results: RN t N,+, = l+(R-
Art . l)~-
(8)
This equation is the well-known time-discrete analogue of the logistic equation. K represents the equilibrium level or the capacity, R the rate of reproduction.
2.2. Release of steriles To include the SIT into the model we use two assumptions which are in good agreement with the biology of the screw worm (Barclay, 1982, 1984; Peterson et al., 1983; Krieg and Franz, 1989). 1. Wild females mate only once. 2. Wild and sterile males show equal competitive ability. In this case the fertility F has to be substituted by F ( N , / ( N , + S)), where S describes the annual number of steriles which are released, and N J ( N , + S) equals the probability that a female of the wild population mates with a fertile male. This leads to the following equation:
N, RNt N~ + S Ut+ 1 =
I+(R_a)
Aft K Nt+S
gt
(9)
R. Marsula, C. Wissel/ Ecological Modelling 75/76 (1994) 203-211
206
0 Fig. 1. Sketch of a barrier. The left hand curve symbolizes a pest population which spreads from left to right, x represents the spatial coordinate. In the barrier (x > 0) the density S of sterile insects is released.
2.3. Spreading of insects To discuss pest control by a barrier, we have to include the spreading of insects. Fig. 1 illustrates how a barrier works. We have to concider two zones. The pest population develops entirely in the left hand zone (x < 0). x represents the spatial coordinate. Steriles are released on the right hand side (for x > 0). The barrier starts at x = 0. The pest population drifts from the left hand side to the right hand side into the barrier (Fig. 1). According to empirical investigations (Welch, 1989) we suggest the spread of the screw worm population to be a random process. This is taken into account by adding a diffusion term D(d2/dx2)Nt to the right hand side of Eq. 9 (D -= diffusion constant) (Manorajan and Van den Driessche, 1986).
N, RNt N t + S
N,+t
I+(R-1)
Nt
d2 Art
+Ddx~ .---=N.
(10)
K Nt+S Eq. 10 models the pest control by a barrier. Figs. 2-5 show some typical numerical solutions. 3. N u m e r i c a l solutions
In Fig. 2 a pest population starts with an initial density distribution (the left hand curve) and drifts from left to right into the barrier where the pest is stopped. NtIKJ .751.0 . ~ ~ ~ ~ 1 .5.25-30 -20 -10
0
1'0 20 x
Fig. 2. Spatial distribution (density) of an insect pest population (in units of K) versus x (in units of v ~ ) after 0, 10, 20.... generations (S = 0.5K, R = 2). The left hand curve represents the initial distribution. The spread of the pest population is stopped by the barrier.
R. Marsula, C. Wissel/ Ecological Modelling 75 / 76 (1994) 203-211
207
Nt/K, 1.025.5.25. 0
-25
0
25
50
7~
100 x
Fig. 3. Spatial distribution after 0, 2, 4,... generations (S = 0.5K, R = 2) starting from the right hand curve. The pest population has collapsed after nearly 8 generations.
Nt/K, 1.0.'/5 .5" .250
0
50
700
750 x
Fig. 4. Distribution (from left hand to right hand) after 0, 2, 4 .... generations at S = 0.05K, R = 2. After nearly 8 generations the population density has taken on the shape of a step function.
x represents the spatial coordinate. The barrier starts at x = 0. Between two curves ten generations of the population pass. If wild insects are present in the barrier before steriles are released, the initial population (right hand curve) will collapse after eight generations (Fig. 3). In Figs. 2 and 3 the density of steriles has a value of S = 0.5K. The features of the solution change totally if S is reduced to S = 0.05K as shown in the next two figures. Then the density of the wild population rises and forms a step function (Fig. 4). Thereafter the population propagates in slow motion from left to right through the barrier (Fig. 5). In this case the control does not work. The barrier fails.
NtlK 1.0 -.75.5.250
6
s'0
160
150",
Fig. 5. Distribution as in Fig. 4 after 10, 20,... generations (S = 0.05K, R = 2). T h e population is spreading in slow-motion from left to right through the barrier. In this case the control does not work. The release of steriles results in the slight decrease of the wild population at x = 0 and the slow-down of the spread.
208
R. Marsula, C. Wissel/Ecological Modelling 75/76 (1994)203-211
We worked out by numerical investigations that the control of the pest is possible only if the following condition is valid: S - -
R-1
> 0.2105K.
(11)
4. Stochastical effects
Eq. 11 is a first step to answer the question: In what conditions are barriers able to control the invasion of an insect pest ? However, Eq. 11 arises from a deterministic model and it is well known that stochastical effects can often change the characteristics of a deterministic model totally. We found that a barrier which is reliable in a deterministic sense fails if stochastic effects are taken into account. A master equation with stochastical birth and death processes as well as random propagation (in the sense of a discrete jump process) has been investigated (Gardiner, 1983; Wissel, 1989; Marsula, 1992). The following points explain and illustrate this result. A barrier being reliable in a deterministic sense has the following stochastical features: 1. The mean population density asymptotically tends towards the exponential function: N ( x ) ~ e x p ( - V / ~ / D x ) . At no point in the barrier is N ( x ) zero, at which the population would be extinct (/~ = individual mortility rate, D = diffusion constant). 2. Consequently a barrier of any width is only reliable with a probability less than 1. To calculate this probability we determine the width of the barrier by demanding that N = N at the right hand border of the barrier for an arbitrary but fixed value of A~ (i.e., the insect release is stopped at that place were = A~). Then the barrier will fail with the probability: W= 1 - exp(- ~AI).
(12)
W increases if the mean live span 1//z or the diffusion constant D increases. W decreases if A~ decreases. This result gives our deterministic model (Eq. 10) a new interpretation. There is always a probability that the barrier may fail, but it is possible to control this probability by the parameter /V (or equivalently by the place where the insect release is stopped).
5. O p t i m i z a t i o n
Now we are going to discuss our second question: Is it possible to find an optimal strategy for an SIT barrier minimizing the costs? (Taylor, 1976). We assume that the costs for each factory-reared insect are constant. Therefore the total costs of controlling per year are: S . width of the barrier.
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209
18
16
14
12
I
0.3
014
015
s/K Fig. 6. Costs as a function of the density of steriles (Sopt = 0.226K, optimal costs = 12.28Kv~, R = 2, = 10-4K).
Table 1 Sopt and the optimal costs as a function of A3 N/K
Sont / K
costs
10 -4 5.10 4 10-3 5.10 -3 10 -z
0.226 0.246 0.254 0.266 0.274
12.28 4.05 2.97 2.02 1.81
Fig. 6 shows the result of a first optimization. T h e costs versus the density S of steriles are shown. T h e r e is a n o p t i m u m at Soot = 0.226K. If S is r e d u c e d below S = 0.2105K, the b a r r i e r will fail. F o r this calculation the rate of r e p r o d u c t i o n was fixed at R = 2 a n d N had the value of 10-4K. If A~ is varied as shown in T a b l e 1, the safety varies a n d the costs vary too. T h e table d e m o n s t r a t e s the conflict b e t w e e n costs a n d safety of a barrier. This seems to be trivial a n d intuitively clear, b u t we are able to p r o p o s e a b e t t e r SIT-strategy which mitigates this conflict. S i m u l a t i o n s lead us to the suggestion that a b a r r i e r should consist of two parallel substrips (Fig. 7).
strip 1
strip 2
i
IIx
Fig. 7. S k e t c h o f a b a r r i e r c o n s i s t i n g o f t w o strips.
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• The spatial extention of the first strip (i.e., the strip with the high population density) should be chosen in a way that the pest population at the right hand border of this strip is reduced to 1 - 5 % of the capacity. The second strip controls the rest population. • The density S of steriles in the first strip should be of the order of 1 / 4 K ( R - 1). • In the second strip the density of steriles has to be as small as possible. If S is too small, the barrier will fail, but it is profitable to choose a density near this critical density. This optimized two-strip strategy reduces the costs (in comparison with the control by only one strip) to 14.7% (by a factor 6.8). Moreover, the conflict between costs and safety is mitigated. If A~ is changed from N = 10-2K to /V = 10-4K, the safety is increased significantly (Eq. 12) while the costs only grow slightly (by 19.2%). Naturally it is possible to improve the proposed SIT-strategy by a barrier of three or more substrips. But a detailed numerical investigation shows that a further reduction of the costs by more than 50% is not possible.
6. Summary A model has been introduced which describes an insect pest control strategy by a barrier. We consider the insect population to consist of a larval and an imaginal stage. Intraspecific competition takes place at the larval stage. We were able to determine effective conditions for the control of the pest by the barrier (Eq. 11). A barrier which is reliable in a deterministic sense will fail if stochastical birth and death processes as well as stochastical propagation of the population are discussed (Eq. 12). It is possible to calculate the probability that a barrier fails, but if the safety of the barrier is increased, the costs increase too. We propose a better SIT strategy which mitigates this conflict between costs and safety. We suggest the division of the barrier into two strips with different densities of released steriles. This two-strip strategy can be of practical interest: It is simple to implement and has a dramatic effect on the reduction of costs.
References Barclay, H.J., 1982. The sterile release method with unequal male competitive ability. Ecol. Modelling, 15: 251-263. Barclay, H.J., 1984. Demographic consequences of monogamy and polygamy for a sterile release program. Prot. Ecol., 6: 209-217. Bronshtein, I.N. and Semendyayev, K.A., 1985. Handbook of Mathematics. Verlag Harry Deutsch, Frankfurt. Franz, J.M. and Krieg, A., 1982. Biologische Sch~idlingsbek~impfung. Paul Parey, Berlin. Gardiner, C.W., 1983. Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences. Springer, Berlin. Hassel, M.P., 1975. Density-dependence in single-species populations. J. Anim. Ecol., 44: 283-295.
R. Marsula, C. Wissel/ Ecological Modelling 75/76 (1994) 203-211
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Knipling, E.F., 1982. Present status and future trends of the SIT approach to the control of arthropod pests. In: Sterile Insect Technique and Radiation in Insect Control. IAEA-SM-255/1, IAEA, Vienna, pp. 3-23. Krieg, A. and Franz, J.M., 1989. Lehrbuch der Biologischen Schiidlingsbek~impfung. Paul Parey, Berlin. Mackley, J.W. and Brown, H.E., 1985. Screwworm research 1976-84: attractants, behavior, ecology, and development of survey and control technologies. Miscellaneous Publications, Entomol. Soc. Am., 62: 49-55. Manoranjan, V.S. and Van den Driessche, P., 1986. On a diffusion model for sterile insect release. Math. Biosci., 79: 199-208. Marsula, R., 1992. Modelle zur Kontrolle von Sch~idlingen durch eine riiumliche Barriere. Ph.D. Thesis, University of Marburg, Germany. Peterson, R., Ocambo-Candido, A. and Del Var Petersen, H., 1983. Longevity and mating capacity of male screwworms. J. Econ. Entomol., 76: 1262-1264. Prout, T., 1978. The joints effect of the release of sterile males and immigration of fertilized females on a density regulated population. Theor. Popul. Biol., 13: 40-71. Taylor, C.R., 1976. Determining optimal sterile male release strategies. Environ. Entomol., 5: 87-95. Wall, R. and Stevens, J., 1990. The turn of the screw worm. New Sci., 1720: 54-57. Welch, J., 1989. Observations on dispersal of sterile screwworms in relation to a clear-cut utility right of way in Southern Mexico. J. Econ. Entomol., 82: 1580-1584. Wendell Snow, J., Whitten, C.J., Salinas, A., Fetter, J. and Sudlow, W.H., 1985. The screwworm, Cochliomyia hominil,orax (Diptera: Calliphoridae), in Central America and proposed plans for its eradication south to the Darien Gap in Panama. J. Med. Entomol., 22: 353-360. Wissel, C., 1989. Theoretische Okologie - Eine Einfiihrung. Springer, Berlin.