| Abstract: |
| {\it Wolbachia}, an intracellular bacterium, is well-known for inducing cytoplasmic incompatibility, which has become a promising and environmentally sustainable strategy for controlling pest populations. The strain {\it w}Stri, specifically identified in {\it Nilaparvata lugens} (brown planthopper), has shown potential for such biocontrol applications. In this study, we develop a comprehensive discrete mathematical model to analyze the dynamics of {\it w}Stri spread in a mixed population of {\it w}Stri-infected, {\it w}Lug-infected, and uninfected {\it Nilaparvata lugens} under both constant and periodically varying environmental conditions. Under a constant environment, the model identifies the critical threshold necessary for the successful establishment of {\it w}Stri within the population. Our analysis reveals that the model exhibits a strong Allee effect, where a population must exceed a certain critical density, the Allee threshold, for the {\it w}Stri strain to persist and spread. Below this threshold, the {\it w}Stri strain is likely to be eliminated, failing in pest control efforts. When the environment varies periodically, the model transforms into a non-autonomous periodic discrete model, introducing additional complexity. In this scenario, we derive sufficient conditions that ensure the composition of finitely many Allee maps continues to function as an Allee map. Furthermore, we prove that a unique periodic orbit exists within such a periodic environment. This orbit is characterized as unstable and acts as a threshold, determining whether {\it w}Stri will establish itself in the population or die out over time. The findings from this model provide critical insights into the conditions under which {\it w}Stri can be effectively used to control {\it Nilaparvata lugens}, particularly in environments that are not constant but fluctuate periodically. These insights have significant implications for the practical deployment of {\it Wolbachia}-based biocontrol methods in pest management strategies. |
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