In this chapter, the authors have proposed a SIT model to eradicate the pest population. It has been assumed that the females after mating with wild males grow logistically. Pest population is being controlled with the release of sterile insects in their habitat. The model is formulated with the system of differential equations, and the authors have discussed the local stability analysis of deterministic logistic growth rate model. Further, they have also obtained a potential function by incorporating one-dimensional insect release with an invasion on patch size L, which has a toxic exterior as its surrounding. It has been obtained that, in the presence of spatial spread over a finite patch size, the sterile release of the insects produces a sudden declination of the pest population. Finally, the authors have obtained the optimal production of sterile male population using Pontryagin's maximum principle. The applicability of the proposed model is finally illustrated through numerical solution.
TopIntroduction
The sterile insect technique or more commonly known as SIT, was originally invented by (Knipling,1955) It is the process of introducing a large number of sterile male insects into a wild population so that it ceases reproduction when these sterile species mate with the female species. Hence, as an outcome, the growth rate in pest population is diminished. SIT seems to be more effective when only sterile males are released for the following reasons:
- 1.
Males are more efficient carriers of the sterility trait, as they often copulate more than once, whereas females may be restricted in the number of their mating.
- 2.
When only males are released, assortative mating amongst the laboratory-reared insects can be avoided, increasing the exploitation of the sterile sperm.
- 3.
When no females are released, males disperse more rapidly in the native (wild) population.
- 4.
Female insects might damage crops even if they are sterile, as they could cause oviposition wounds.
- 5.
Rearing only males reduces the cost of mass-rearing, provided females can be eliminated early during the rearing process.
The objective of SIT is the interaction of females with the sterile males such that it leads to no further reproduction, and thereby, decreasing the population of the next generation. Hence, the main challenge is faced between the wild and the sterile male to mate with the female. It is further noticed that over low population density, continuous deliverance of sterile males controls the pest population, but for economic purposes dense populations are an ideal situation for pest control before the release. The introduction of large number of sterile males leads to the emergence of Allee effect (Courchamp, Berec & Gascoigne, 2008; Liebhold & Tobin, 2008; Tobin, Berec & Liebhold, 2011) (a phenomenon in biology which is represented by the interdependence of the per capita population growth rate and population density or size of the species). SIT is preferred over insecticide application as it is harmless for farmer’s health (Klassen & Curtis, 2005), does not cause pollution to the environment and is not aimed at other insect population. Along with applying SIT for pest management, it is also important to obtain the optimal rate of production of sterile male and the optimal pest population.
Optimal control theory deals with the problem of dynamic optimization and provides a powerful tool for understanding the dynamic systems. Several papers on application of optimal control theory in epidemiological models exist in the literature (Agusto, 2013; Okosun, Rachid & N.Marcus, 2013; Agusto & Adekunle, 2014; Apreutesei & Strugariu, 2014; Zhou, Liang & Wu, 2014). In fact, recently various papers have also discussed optimality of SIT technique (Sergio Ramirez & Luis F.Gordillo, 2016, Luis F. Gordillo, 2015, Luis F. Gordillo, 2014). Due to the dynamic nature of the present problem and loses due to pest damage, we formulate a optimal control problem to find an optimal production rate of sterile male population over time so that the cost of producing sterile male and feeding cost can be minimized.