In this paper, we consider a diffusive predator-prey system where the prey exhibits the herd behavior in terms of the square root of the prey population. The model is supposed to impose on homogeneous Neumann boundary conditions in the bounded spatial domain. By using the abstract Hopf bifurcation theory in infinite dimensional dynamical system, we are capable of proving the existence of both spatial homogeneous and nonhomogeneous periodic solutions driven by Hopf bifurcations bifurcating from the positive constant steady state solutions. Our results allow for the clearer understanding of the mechanism of the spatiotemporal pattern formations of the predator-prey interactions in ecology.

In this paper, we are concerned with a homogeneous reaction-diffusion Atkinson oscillator system subject to homogeneous Neumann boundary conditions on a bounded spatial domain. Using the comparison principle and the techniques of invariant rectangle, we prove the existence of the attraction region of the solutions. We thus prove that under certain conditions, the solutions of the PDE system converge to the unique positive equilibrium solutions. We also derive precise conditions such that the system does not have nonconstant positive steady-state solutions. Finally, we use the bifurcation technique to show the existence of Turing patterns. The results provide a clearer understanding of the mechanism of formations of patterns.

This study analyzes the process of corporate reputation (CR) propagation. We consider that different positions play different roles via social word of mouth (sWOM). In essence, our interest lies in the following question: how does sWOM affect the process of CR propagation? As a benchmark, we develop a dynamic model to reflect the factors and laws influencing corporate reputation propagation. We find that the stability of the system is not what corporations want to see in the propagation of positive CR after analyzing two basic reproduction numbers and conducting a numerical simulation. Another important finding is that stability is what corporations expect in the propagation of negative CR. This study makes the logical analysis of the relationship between CR and consumer loyalty more complete and helps us to clearly understand the overall process of CR propagation. The study provides a new way to analyze the law of CR propagation and can provide a reference for decision-making for corporations.