Security systems are essential for maintaining public safety and protecting critical assets, yet most existing evaluation approaches remain static, capture only partial interactions among assets, security systems, and offenders, and lack qualitative analyses of long-term stability. To address these limitations, this study introduces an integrated Petri net–ordinary differential equation framework that combines the discrete-event modeling strength of Petri nets with the continuous-time analysis capability of ordinary differential equations, thereby capturing the dynamic interplay among assets, security systems, and offenders and enabling equilibrium and stability analysis. Numerical simulations are conducted under different parameter settings to validate both the stability of the framework and the role of the security effectiveness parameter. The results demonstrate that enhancing this parameter improves protection capability, suppresses offender activity, reduces asset losses and promotes long-term system stability. By capturing interdependent dynamics within a unified model, the proposed approach advances beyond traditional methods and provides both a theoretical foundation and practical guidance for optimizing resource allocation and designing protection strategies.

Assessing the security risk of projects in high-risk areas is particularly important. This paper develops a security risk assessment model for projects in high-risk areas based on the target loss probability model and Bayesian game model. This model is modeled from the perspective of attack-defense confrontation and addresses the issue that traditional risk assessment focuses on the analysis of the attacker yet neglects to analyze the defender—the defender’s optimum defensive information is not quantitatively determined. The risk level, optimum defensive resource value, and optimum defensive strategy of the project are determined through the analysis of a project in the high-risk area. This enables the project’s risk manager to adjust the defensive resources reasonably and optimally, confirming the objectivity and feasibility of the model and offering a new benchmark for security risk assessment, which has significant practical implications.

This paper is concerned with the following biharmonic problem \begin{equation}\label{ineq} \begin{cases} \Delta^2 u=|u|^{\frac{8}{N-4}}u &\text{ in } \ \Omega\backslash \overline{{B(\xi_0,\varepsilon)}}, u=\Delta u=0 &\text{ on } \ \partial (\Omega \backslash \overline{{B(\xi_0,\varepsilon)}}), \end{cases} \end{equation} where $\Omega$ is an open bounded domain in $\mathbb{R}^N$, $N\geq 5$, and $B(\xi_0,\varepsilon)$ is a ball centered at $\xi_0$ with radius $\varepsilon$, $\varepsilon$ is a small positive parameter. We obtain the existence of solutions for problem (\ref{ineq}), which is an arbitrary large number of sign-changing solutions whose profile is a superposition of bubbles with alternate sign which concentrate at the center of the hole.