Hopf bifurcation and fixed-time stability of a reaction-diffusion echinococcosis model with mixed delays

In this paper, a model with spatial diffusion and mixed delays is presented to describe the spread of echinococcosis between dogs and livestock. Firstly, the local stability is investigated using the Routh-Hurwitz criterion. Furthermore, when considering time delays as bifurcation parameters, the conditions for the occurrence of Hopf bifurcation are discussed based on the linear approximation of the nonlinear system. Lastly, the fixed -time stability is studied under the implementation of appropriate control measures. Numerical simulations are provided to give a better understanding of the theoretical results.