Xiangming Zhang

Insect populations, which are diverse and widespread, provide a principal area of utilization of the stage-structured modeling approach. In this paper, housefly populations incorporating a stage-structured model are investigated theoretically and graphically. First, stability charts and rightmost characteristic roots of the positive equilibrium are elucidated analytically and numerically. Furthermore, the Hopf bifurcation at the positive equilibrium is derived employing geometric stability switch criterion. Second, the properties of Hopf bifurcation are determined using the center manifold theorem and by reducing the equation to the Poincaré normal form. Finally, the correctness of the theoretical derivation is confirmed using a numerical simulation based on specific parameter values. Our results show that with an increase in delay [ ], the unique positive equilibrium may undergo two stability switches: from stable to unstable, and from unstable to stable. Interestingly, the characteristic equation has pure imaginary roots at the second pair and subsequent critical values. However, Hopf bifurcation theorem is not satisfied because all other characteristic roots of the characteristic equation at these critical values do not have strictly negative real parts, except the pure imaginary roots. We also simulate the unstable periodic solutions at the second pair of critical values through a bifurcation diagram. Therefore, a pair of supercritical Hopf bifurcations appear around the positive equilibrium of the housefly population stage-structured model.

In this paper, we make a mathematical analysis of an age-structured model with diffusion including a generalized Beverton-Holt fertility function. The existence of periodic wave train solutions of the age structure model with diffusion are investigated by using the theory of integrated semigroup and a Hopf bifurcation theorem for second order semi-linear equations. We also carry out numerical simulations to illustrate these results.

A three-dimensional elastic-plastic finite element model was established in finite element software ANSYS to investigate the interfacial stress of carbon fiber patch bonded to transverse cracked steel plate reinforcement. The patch and adhesive were modeled as linear elastic. The steel plate was modeled as elastic-plastic. The load-deflection curve of the steel was obtained by bending test. The influence of crack depth, adhesive thickness, patch thickness and patch length on reinforcing efficiency was studied. The results show that the peel stress of the adhesive was distributed symmetrically along the center line, mainly concentrated on the crack center, near the crack and the patch ends. The peel stress at the crack center was always greater than the near crack. The shear stress is anti-symmetric along the center line, and the shear stress at the crack center is zero, mainly concentrated near the crack and plate end. The debonding may be initiated from the vicinity of the crack. Increasing the crack depth can significantly increase the peel stress and shear stress at the crack, but has little effect on the stresses at the end. Increasing the thickness of the adhesive can reduce the peel stress and shear stress at the crack center and near the crack, and the shear stress at the ends, but has little effect on the peel stress at the ends. Increasing the patch thickness can reduce the stress at the crack but increase the stress at the end. Increasing the patch length cannot reduce the stress at the crack, but significantly reduce the stress at the end.