Journal of Mathematical Biology

, Volume 70, Issue 5, pp 1177–1206

# On an integro-differential model for pest control in a heterogeneous environment

Article

## Abstract

Insect pests pose a major threat to a balanced ecology as it can threaten local species as well as spread human diseases; thus, making the study of pest control extremely important. In practice, the sterile insect release method (SIRM), where a sterile population is introduced into the wild population with the aim of significantly reducing the growth of the population, has been a popular technique used to control pest invasions. In this work we introduce an integro-differential equation to model the propagation of pests in a heterogeneous environment, where this environment is divided into three regions. In one region SIRM is not used making this environment conducive to propagation of the insects. A second region is the eradication zone where there is an intense release of sterile insects, leading to decay of the population in this region. In the final region we explore two scenarios. In the first case, there is a small release of sterile insects and we prove that if the eradication zone is sufficiently large the pests will not invade. In the second case, when SIRM is not used at all in this region we show that invasions always occur regardless of the size of the eradication zone. Finally, we consider the limiting equation of the integro-differential equation and prove that in this case there is a critical length of the eradication zone which separates propagation from obstruction. Moreover, we provide some upper and lower bound for the critical length.

35K57 35K55

### References

1. Anaman KA, Atzeni MG, Mayer DG, Stuart MA (1994) Benefit–cost analysis of the use of sterile insect technique to eradicate screwworm fly in the event of an invasion of Australia. Preven Veter Med 20:79–98
2. Atzeni MG, Mayer DG, Butler DG (1992) Sterile insect release method-optimal strategies for eradication of screwworm fly. Math Comput Simul 33:445–450
3. Barbosa P, Castellanos I (2005) Top-down forces in managed and unmanaged habitats. Ecology of predator–prey interactions. Oxford University Press, OxfordGoogle Scholar
4. Barclay HJ, Mackauer M (1980) The sterile insect release method for pest control: a density dependent model. Environ Entomol 9:810–817
5. Barclay HJ (1981) The sterile release method for population control with interspecific competition. Res Popul Ecol 23:145–155
6. Barclay HJ (1986) Models for pest control: complementary effects of periodic releases of sterile pests and parasitoids. Theor Popul Biol 32:76–89
7. Barclay HJ, Matlock R, Gilchrist S, Suckling DM, Reyes J, Enkerlin W, Vreysen MJB (2011) A conceptual model for assessing the minimum size area for an area-wide integrated pest management program. Intern J Agron 2011:1–12
8. Bates PW, Fife PC, Ren X, Wang X (1997) Traveling waves in a convolution model for phase transitions. Arch Ratl Mech Anal 138:105–136
9. Benedict MQ, Robin AS (2003) The first releases of transgenic mosquitoes: an argument for the sterile insect technique. Tren Parasitol 19(8):349–355
10. Berestycki H, Diekmann O, Nagelkerke CJ, Zegeling PA (2009) Can a species keep pace with a shifting climate? Bull Math Biol 71(2):399–429
11. Berestycki H, Nadal J-P (2010) Self-organised critical hot spots of criminal activity. Eur J Appl Math 21(4–5):371–399
12. Berestycki H, Rodríguez N, Ryzhik L (2013) Traveling wave solutions in a reaction–diffusion model for criminal activity. Multisc Model Simul 11(4):1097–1126
13. Berestycki H, Rodríguez N (2013) Non-local reaction–diffusion equations with a barrier. PreprintGoogle Scholar
14. Berryman AA (1967) Mathematical description of the sterile male principle. Can Entomol 99:858–865
15. Bushland RC (1985) Eradication program in the southeastern US. Misc Publ Entomol Soc Am 62:12–15Google Scholar
16. Carr J, Chmaj A (2004) Uniqueness of travelling waves for nonlocal monostable equations. Proc Amer Math Soc 132(8):2433–2439 (electronic)
17. Chen X (1997) Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations. Adv Diff Equ 2(1):125–160
18. Costello WG, Taylor HM (1975) Mathematical models of the sterile male technique of insect control. In: Charnes A, Lynn WR (eds) Mathematical analysis of decision problems in ecology, 5th edn. Springer Verlag, Berlin, pp 8–59Google Scholar
19. Coville J, Dupaigne L (1994) On a nonlocal reaction–diffusion equation arising in population dynamics. Proc Roy Soc Edinb Sect A 137(4):727–755
20. Coville J, Dupaigne L (2007) On a non-local equation arising in population dynamics. Proc Roy Soc Edinb Sect A 137(4):727–755
21. Ermentrout GB, McLeod JB (1993) Existence and uniqueness of travelling waves for a neural network. Proc Roy Soc Edinb 123A:461–478
22. Esteva L, Mo H (2005) Mathematical model to assess the control of Aedes aegypti mosquitoes by the sterile insect technique. Math Biosci 198(2):132–147
23. Fife PC, Peletier LA (1980) Clines induced by variable migration. In: Biological growth and spread (Proceedings Conference, Heidelberg, 1979), vol 38 of Lecture Notes in Biomathematics. Springer, Berlin, pp 276–278Google Scholar
24. Fife PC, Wang X (1998) A convolution model for interfacial motion: the generation and propagation of internal layers in higher space dimensions. Adv Diff Equ 3(1):85–110
25. Gatehouse AG (1997) Behavior and ecological genetics of wind-borne migration by insects. Ann Rev Entomol 42:475–502
26. Hendrichs J, Ortiz G, Liedo P, Schwarz A (1982) 6 years of successful medfly program in Mexico and Guatemala. In: Fruit Flies of Economic Importance. CEC/IOBC International Symposium, pp 16–19Google Scholar
27. Hutson V, Martinez S (2003) The evolution of dispersal. J Math Bio 47:483–517
28. Hutson V, Grinfeld M (2006) Non-local dispersal and bistability. Eur J Appl Math 17:221–232
29. JohnM Kean, SukLing Wee, AndréaEA Stephens (2008) Modelling the effects of inherited sterility for the application of the sterile insect technique. Agricu For Entomol 10(2):101–110
30. Johnson CG (1969) Migration and dispersal of insects by flight. Methuen & Co. Ltd, LondonGoogle Scholar
31. Klassen W, Curtis CF (2005) History of the sterile insect technique. In: Dyck et al (eds) Sterile insect technique. Principles and practice in areawide integrated pest management. Springer, Netherlands, pp 3–36Google Scholar
32. Klassen W, Curtis CF (2005) History of the sterile insect technique. Ster Ins Tech, pp 3–36Google Scholar
33. Knipling EF (1955) Possibilities of insect control or eradication through the use of sexually sterile males. J Econ Entomol 48:459–462
34. Kogan M (ed) (1986) Ecological theory and integrated pest management practice. John Wiley & Sons, New YorkGoogle Scholar
35. Kojima KI (1971) Stochastic models for efficient control of insect populations by sterile insect release methods. In sterility principle for insect control or eradication. International Atomic Energy Agency, ViennaGoogle Scholar
36. Lawson FR (1967) Theory of control of insect populations by sexually sterile males. Ann Entomol Soc Amer 60(4):713–722
37. Lee SS, Baker RE, Gaffney EA, White SA (2013) Optimal barrier zones for stopping the invasion of Aedes aegypti mosquitoes via transgenic or sterile insect techniques. Theor Ecol 6(4):427–442
38. Lewis MA, van den Driessche P (1993) Waves of extinction from sterile insect release. Math Biosci 116:221–245
39. Lewis TJ, Keener JP (2000) Wave-block in excitable media due to regions of depressed excitability. SIAM J Appl Math 61(1):293–316
40. Manoranjan VS, Van Den Driessche P (1986) On a diffusion model for sterile insect release. Math Biosci 79(2):199–208
41. Medows ME (1985) Eradication program in the southeastern US. Misc Publ Entomol Soc Am 62:8–11Google Scholar
42. Miller DR, Weidhaas DE (1974) Equilibrium populations during a sterile-male release program. Enviorn Entomol 3(2):211–216
43. Pedigo LP (1998) Entomology and pest management. Prentice-Hall International, Hemel HempsteadGoogle Scholar
44. Prout T (1978) The joint effects of the release of sterile males and immigration of fertilized females on a density regulated population. Theor Popul Biol 13(1):40–71
45. Serebrovskii AS (1940) On the possibility of a new method for the control of insect pests (in Russian). Zoologicheskii Zhurnal 19:618–630Google Scholar
46. Vanderplank FL (1944) Hybridization between Glossina species and suggested new method for control of certain species of Tsetse. Nature 154(3915):607–608
47. White Steven M, Pejman Rohani (2010) Modelling pulsed releases for sterile insect techniques: fitness costs of sterile and transgenic males and the effects on mosquito dynamics. J Appl Ecol 47(6):1329–1339