Bifurcation and Stability of a Haptotaxis Mathematical Model for Complex MAP

This paper studies the bifurcation and stability of a parabolic–ordinary-parabolic system to a model with three reactive species: molecular Anchors [Formula: see text], Nanoparticulates [Formula: see text] and Matrix constituents [Formula: see text] ([Formula: see text]). Global asymptotic stability is studied for both the ODE system and the reaction–diffusion system. The results show that the positive equilibrium is globally asymptotically stable in both the ODE and the corresponding reaction–diffusion systems. Furthermore, by selecting the haptotaxis coefficient as a bifurcation factor, we analyze the existence, direction, and stability of the Hopf bifurcation. The results also reveal that haptotaxis is crucial in determining the stability and bifurcation behavior of the model, suggesting that it has a destabilizing effect and can undergo Hopf bifurcation. Moreover, numerical simulations are presented to verify and illustrate the theoretical results.