Finite-time stability analysis of gene regulatory network in bacteria-host interactions with spatial diffusion term

This study explores the finite-time stable behavior of time-delay genetic regulatory networks incorporating spatial diffusion in the interactional between zebrafish and Escherichia coli under Dirichlet boundary conditions. The interaction between these two systems is examined in the context of E. coli invading the zebrafish body. The system representing the zebrafish is stable, while the E. coli system is initially unstable. After their interaction, both systems gradually reach a stable state. We propose an interacting coupled model, construct a novel Lyapunov-Krasovskii functional, and utilize the secondary delay partitioning method to derive stability criteria for the interacting genetic regulatory networks. The stability criteria we establish are less conservative than existing criteria, allowing the upper bound of the time delay derivative to be not less than 1. According to the defined evaluation criteria, the stronger the interaction between bacteria and the host, the longer the time required to suppress the bacteria or for treatment. Finally, numerical simulations are conducted to illustrate the mRNA and protein concentration trajectories, verifying the accuracy of the proposed criteria.