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In the ecosystem, species need time to transform resources into energy for growth. In addition, the degradation period after species' mortality will lead to the reuse of resources. The two phenomena above contribute to the delay effects during the species' evolution. We here develop a turbidostat model incorporating delay effects to investigate the impact of resource consumption and reusing period on population evolution. We first analyze the existence and the stability of the positive equilibrium. Our results reveal the perturbation effect of delay parameters on its dynamics. When the time lag periods cross the critical values, the positive equilibrium becomes unstable, and periodic solutions are produced. The populations of the species will oscillate along with the resource consumption and recycling period. In addition, the multiple delay effect will decrease the negative effect of delay phenomenon on positive solutions. By controlling the delay period of decomposition, we can moderate the oscillations caused by the delay effect in resource consumption. To further understand the dynamics of periodic solutions, the direction of the Hopf bifurcation and the periodic solutions' stability are also discussed here. Several numerical examples are tested to validate the theoretical results in this work. [ABSTRACT FROM AUTHOR]

COVID-19 vaccine-induced protection declines over time. This waning of immunity has been described in modelling as a lower level of protection. This study incorporated fine-scale vaccine waning into modelling to predict the next surge of the Omicron variant of the SARS-CoV-2 virus. In Hong Kong, the Omicron subvariant BA.2 caused a significant epidemic wave between February and April 2022, which triggered high vaccination rates. About half a year later, a second outbreak, dominated by a combination of BA.2, BA.4 and BA.5 subvariants, began to spread. We developed mathematical equations to formulate continuous changes in vaccine boosting and waning based on empirical serological data. These equations were incorporated into a multi-strain discrete-time Susceptible-Exposed-Infectious-Removed model. The daily number of reported cases during the first Omicron outbreak, with daily vaccination rates, the population mobility index and daily average temperature, were used to train the model. The model successfully predicted the size and timing of the second surge and the variant replacement by BA.4/5. It estimated 655,893 cumulative reported cases from June 1, 2022 to 31 October 2022, which was only 2.69% fewer than the observed cumulative number of 674,008. The model projected that increased vaccine protection (by larger vaccine coverage or no vaccine waning) would reduce the size of the second surge of BA.2 infections substantially but would allow more subsequent BA.4/5 infections. Increased vaccine coverage or greater vaccine protection can reduce the infection rate during certain periods when the immune-escape variants co-circulate; however, new immune-escape variants spread more by out-competing the previous strain., Competing Interests: The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper., (© 2024 The Authors.)