For decades, software development has been dominated by application-centric paradigms in which systems are designed as bounded artifacts with clearly defined lifecycles. Even as architectures evolved from monolithic applications to distributed and service-oriented systems, the fundamental assumption remained unchanged: software is deployed, executed, and eventually replaced, rather than continuously existing as an adaptive entity. The recent emergence of persistent AI agents challenges this assumption by introducing software components that maintain identity, memory, and behavior across extended periods of operation. This paper proposes a shift from traditional application models toward living software systems, defined as persistent ecosystems of AI agents that continuously interact, evolve, and coordinate within a shared software environment. Unlike stateless services or short-lived processes, persistent AI agents retain contextual knowledge over time, enabling systems to exhibit long-term behavior, adaptation, and emergent dynamics. This transformation has profound implications for software development models, which must move beyond code-centric abstractions toward behavior-centric design and lifecycle-aware architectures. The study examines how persistent AI-agent ecosystems differ fundamentally from monolithic and conventional distributed systems, both conceptually and architecturally. It introduces a software development perspective that treats persistence, coordination, and behavioral evolution as first class concerns. Rather than focusing on individual algorithms or agent implementations, the paper analyzes how development models, architectural structures, and lifecycle practices must change to support living systems at scale. The contributions of this work are threefold. First, it provides a clear conceptual distinction between traditional software systems and living systems composed of persistent AI agents. Second, it outlines software development models that enable the design, implementation, and evolution of such systems. Third, it discusses the broader implications of persistent AI-agent ecosystems for software architecture, orchestration, and long-term system governance. By reframing software as a continuously evolving ecosystem rather than a static product, this paper offers a new foundation for AI-native software development.