# American Institute of Mathematical Sciences

October  2012, 17(7): 2509-2522. doi: 10.3934/dcdsb.2012.17.2509

## On a reaction-diffusion model for sterile insect release method with release on the boundary

 1 Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, China 2 Department of Applied Mathematics, University of Western Ontario, London, Ontario N6A 5B7

Received  February 2012 Revised  April 2012 Published  July 2012

We consider a partial differential equation model that describes the sterile insect release method (SIRM) in a bounded 1-dimensional domain (interval). Unlike everywhere-releasing in the domain as considered in previous works [17] and [14] , we propose the mechanism of releasing on the boundary only. We show existence of the fertile-free steady state and prove its stability under some conditions. By using the upper-lower solution method, we also show that under some other conditions there may exist a coexistence steady state. Biological implications of our mathematical results are that the SIRM with releasing only on the boundary can successfully eradicate the fertile insects as long as the strength of the sterile releasing is reasonably large, while the method may also fail if the releasing is not sufficient.
Citation: Xin Li, Xingfu Zou. On a reaction-diffusion model for sterile insect release method with release on the boundary. Discrete and Continuous Dynamical Systems - B, 2012, 17 (7) : 2509-2522. doi: 10.3934/dcdsb.2012.17.2509
##### References:
 [1] H. J. Barclay, The sterile release method for population control with interspecific competition, Res. Pop. Ecol., 23 (1981), 145-155. doi: 10.1007/BF02514097. [2] H. J. Barclay, Models for pest control: Complementary effects of periodic release of sterile pests and parasitoids, Theor. Pop. Biol., 32 (1987), 76-89. doi: 10.1016/0040-5809(87)90041-4. [3] H. J. Barclay, Demographic consequences of monogamy and polygamy for a sterile release program, Protection Ecol., 6 (1984), 209-217. [4] H. J. Barclay and M. Mackauer, The sterile insect release method for pest control: A density dependent model, Envir. Entom., 9 (1980), 810-817. [5] H. J. Barclay and M. Mackauer, Effects of sterile insect releases on a population under predation or parasitism, Res. Pop. Ecol., 22 (1980), 136-146. doi: 10.1007/BF02513541. [6] A. H. Baumhover, A. J. Graham, B. A. Bitter, D. E. Hopkins, W. D. New, F. H. Dudley and R. C. Bushland, Screwworm control though release of sterilized flies, J. Econ. Entomol., 48 (1955), 462-466. [7] A. A. Berryman, Mathematical description of the sterile male principle, Can. Entomol., 99 (1967), 858-865. doi: 10.4039/Ent99858-8. [8] K. Dietz, The effect of immigration on genetic control, Theor. Popul. Biol., 9 (1976), 58-67. [9] Y. H. Du, "Order Structure and Topological Methods in Nonlinear PDEs. Vol. 1. Maximum Principle and Applications,'' World Scientific, Singapore, 2005. [10] H. M. Graham, Sterile pink bollworm: Field release for population suppression, J. Econ. Entomol., 71 (1978), 233-235. [11] G. W. Harrison, H. J. Barclay and P. van den Driessche, Analysis of a sterile release model with predation, J. Math. Biol., 16 (1982/83), 33-48. doi: 10.1007/BF00275159. [12] Y. Ito, A model of sterile insect release for eradication of the melon fly, Dacus cucurbitae COQUILLETT. Appl. Ent. Zool., 12 (1977), 303-310. [13] O. Iwahashi, Eradication of the melon fly, Dacus eucurbitae, from Kume Is. Okinawa. with the sterile insect release method, Res. Popul. Ecol., 19 (1977), 87-98. [14] W. Jiang, X. Li and X. Zou, On a reaction-diffusion model for sterile insect release method in a bounded domain, submitted. [15] E. F. Knipling, Possibilities of insect control or eradication through the use of sexually sterile males, J. Econ. Entomol., 48 (1955), 459-462. [16] E. S. Krafsur, H. Townson, G. Davidson and C. F. Curtis, Srewworm eradication is what it seems, Nature, 323 (1986), 495-496. [17] M. A. Lewis and P. van Den Driessche, Waves of extinction from sterile insect release, Math. Biosci., 5 (1992), 221-247. [18] A. W. Lindquist, The use of gamma radiation for control or eradication of the Screw-worm, J. Econ. Entomol., 48 (1955), 467-469. [19] V. S. Manoranjan and P. van den Driessche, On a diffusion model for sterile insect release, Math. Biosci., 79 (1986), 199-208. [20] D. R. Miller and D. E. Weidhaas, Equilibrium populations during a sterile release program, Environ. Entomol., 3 (1974), 211-216. [21] C. V. Pao, "Nonlinear Parabolic and Elliptic Equations,'' Plenum Press, New York, 1992. [22] R. E. Plant and R. T. Cunningham, Analysis of the dispersal of sterile Mediterranean fruit flies (Diptera: Tephritidae) released from a point source, Environ. Entomol., 20 (1991), 1493-1530. [23] R. E. Plant and M. Mangel, Modeling and simulation in agricultural pest management, SIAM Rev., 29 (1987), 235-261. [24] T. Prout, The joint effects of the release of sterile males and immigration of fertilized females on a density regulated population, Theor. Popul. Biol., 13 (1978), 40-71. [25] M. D. Proverbs, J. R. Newton and D. M. Logan, Codling moth control by the sterility method in twenty-one British Columbia orchards, J. Econ. Entomol., 70 (1977), 667-671. [26] H. L. Smith, "Monotone Dynamical System: An introduction to the Theory of Competitive and Cooperative Systems,'' Mathematical Surveys and Monographs, 41, AMS, Providence, RI, 1995.

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##### References:
 [1] H. J. Barclay, The sterile release method for population control with interspecific competition, Res. Pop. Ecol., 23 (1981), 145-155. doi: 10.1007/BF02514097. [2] H. J. Barclay, Models for pest control: Complementary effects of periodic release of sterile pests and parasitoids, Theor. Pop. Biol., 32 (1987), 76-89. doi: 10.1016/0040-5809(87)90041-4. [3] H. J. Barclay, Demographic consequences of monogamy and polygamy for a sterile release program, Protection Ecol., 6 (1984), 209-217. [4] H. J. Barclay and M. Mackauer, The sterile insect release method for pest control: A density dependent model, Envir. Entom., 9 (1980), 810-817. [5] H. J. Barclay and M. Mackauer, Effects of sterile insect releases on a population under predation or parasitism, Res. Pop. Ecol., 22 (1980), 136-146. doi: 10.1007/BF02513541. [6] A. H. Baumhover, A. J. Graham, B. A. Bitter, D. E. Hopkins, W. D. New, F. H. Dudley and R. C. Bushland, Screwworm control though release of sterilized flies, J. Econ. Entomol., 48 (1955), 462-466. [7] A. A. Berryman, Mathematical description of the sterile male principle, Can. Entomol., 99 (1967), 858-865. doi: 10.4039/Ent99858-8. [8] K. Dietz, The effect of immigration on genetic control, Theor. Popul. Biol., 9 (1976), 58-67. [9] Y. H. Du, "Order Structure and Topological Methods in Nonlinear PDEs. Vol. 1. Maximum Principle and Applications,'' World Scientific, Singapore, 2005. [10] H. M. Graham, Sterile pink bollworm: Field release for population suppression, J. Econ. Entomol., 71 (1978), 233-235. [11] G. W. Harrison, H. J. Barclay and P. van den Driessche, Analysis of a sterile release model with predation, J. Math. Biol., 16 (1982/83), 33-48. doi: 10.1007/BF00275159. [12] Y. Ito, A model of sterile insect release for eradication of the melon fly, Dacus cucurbitae COQUILLETT. Appl. Ent. Zool., 12 (1977), 303-310. [13] O. Iwahashi, Eradication of the melon fly, Dacus eucurbitae, from Kume Is. Okinawa. with the sterile insect release method, Res. Popul. Ecol., 19 (1977), 87-98. [14] W. Jiang, X. Li and X. Zou, On a reaction-diffusion model for sterile insect release method in a bounded domain, submitted. [15] E. F. Knipling, Possibilities of insect control or eradication through the use of sexually sterile males, J. Econ. Entomol., 48 (1955), 459-462. [16] E. S. Krafsur, H. Townson, G. Davidson and C. F. Curtis, Srewworm eradication is what it seems, Nature, 323 (1986), 495-496. [17] M. A. Lewis and P. van Den Driessche, Waves of extinction from sterile insect release, Math. Biosci., 5 (1992), 221-247. [18] A. W. Lindquist, The use of gamma radiation for control or eradication of the Screw-worm, J. Econ. Entomol., 48 (1955), 467-469. [19] V. S. Manoranjan and P. van den Driessche, On a diffusion model for sterile insect release, Math. Biosci., 79 (1986), 199-208. [20] D. R. Miller and D. E. Weidhaas, Equilibrium populations during a sterile release program, Environ. Entomol., 3 (1974), 211-216. [21] C. V. Pao, "Nonlinear Parabolic and Elliptic Equations,'' Plenum Press, New York, 1992. [22] R. E. Plant and R. T. Cunningham, Analysis of the dispersal of sterile Mediterranean fruit flies (Diptera: Tephritidae) released from a point source, Environ. Entomol., 20 (1991), 1493-1530. [23] R. E. Plant and M. Mangel, Modeling and simulation in agricultural pest management, SIAM Rev., 29 (1987), 235-261. [24] T. Prout, The joint effects of the release of sterile males and immigration of fertilized females on a density regulated population, Theor. Popul. Biol., 13 (1978), 40-71. [25] M. D. Proverbs, J. R. Newton and D. M. Logan, Codling moth control by the sterility method in twenty-one British Columbia orchards, J. Econ. Entomol., 70 (1977), 667-671. [26] H. L. Smith, "Monotone Dynamical System: An introduction to the Theory of Competitive and Cooperative Systems,'' Mathematical Surveys and Monographs, 41, AMS, Providence, RI, 1995.

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