A deterministic approach to modelling the dynamics of vulnerabilities, cyberattacks, and security hardening using Lotka–Volterra equations

Abstract In the rapidly evolving cybersecurity landscape, comprehending the dynamic interactions among system vulnerabilities, cyberattacks, and security hardening measures is essential for building resilient systems. Cyber-ecosystems exhibit inherent complexity, characterized by nonlinear feedback loops, adaptive adversaries, and evolving defences. This study proposes a deterministic mathematical framework, grounded in the Lotka–Volterra equations, to model these dynamics, conceptualizing vulnerabilities as "prey", cyberattacks as "predators", and security hardening as a regulatory "third species" that suppresses both vulnerabilities and attacks. By adapting this ecological model, the study captures the competitive and symbiotic relationships driving cybersecurity environments, incorporating parameters such as vulnerability discovery rates, attack sophistication, and the efficacy of security measures. The Systemic Behavioural Analysis section investigates emergent behaviours, including phase transitions between stable and chaotic states, hysteresis effects in response to parameter changes, and bifurcations that arise from delayed hardening efforts. These qualitative insights are juxtaposed with the quantitative findings from the Deterministic Analysis, which employs Jacobian matrices, eigenvalue stability criteria, and Lyapunov exponents to identify equilibrium points and classify system trajectories. Key regimes—such as the Optimum Ecosystem (balanced coexistence), Extinction Status (collapse of vulnerabilities/attacks), and Enhancement Status (runaway growth of threats)—are rigorously mapped through phase-space curves and validated via Runge–Kutta discretization. A critical contribution of this work lies in reconciling systemic behavioural predictions with deterministic results. For instance, the Systemic Analysis hypothesizes that delayed security interventions could destabilize the system, a claim quantitatively confirmed by Lyapunov stability metrics showing sensitivity to hardening response times. Similarly, phase-space plots corroborate the Systemic Analysis’s prediction of oscillatory dynamics under high attack rates. This synergy between qualitative and quantitative methods strengthens the model’s predictive power.