A predator-prey model is used to investigate the interactions between phages and bacteria by considering the lytic and lysogenic life cycles of phages and the prophage induction. We provide answers to the following conflictual research questions: (1) what are conditions under which the presence of phages can purify a bacterial infected environment? (2) Can the presence of phages triggers virulent bacterial outbreaks? We derive the basic offspring number |$\mathcal N_0$| that serves as a threshold and the bifurcation parameter to study the dynamics and bifurcation of the system. The model exhibits three equilibria: an unstable environment-free equilibrium, a globally asymptotically stable (GAS) phage-free equilibrium (PFE) whenever |$\mathcal N_0<1$|⁠, and a locally asymptotically stable environment-persistent equilibrium (EPE) when |$\mathcal N_0>1$|⁠. The Lyapunov–LaSalle techniques are used to prove the GAS of the PFE and estimate the EPE basin of attraction. Through the center manifold approximation, topological types of the PFE are precised. Existence of transcritical and Hopf bifurcations are established. Precisely, when |$\mathcal N_0>1$|⁠, the EPE loses its stability and periodic solutions arise. Furthermore, increasing |$\mathcal N_0$| can purify an environment where bacteriophages are introduced. Purposely, we prove that for large values of |$\mathcal N_0$|⁠, the overall bacterial population asymptotically approaches zero, while the phage population sustains. Ecologically, our results show that for small values of |$\mathcal N_0$|⁠, the existence of periodic solutions could explain the occurrence of repetitive bacteria-borne disease outbreaks, while large value of |$\mathcal N_0$| clears bacteria from the environment. Numerical simulations support our theoretical results.

中文翻译：

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噬菌体-细菌相互作用模型与原噬菌体诱导的分叉分析

通过考虑噬菌体的裂解和溶原性生命周期以及原噬菌体诱导，使用捕食者-猎物模型来研究噬菌体和细菌之间的相互作用。我们为以下相互矛盾的研究问题提供答案：（1）噬菌体的存在可以净化细菌感染环境的条件是什么？(2) 噬菌体的存在是否会引发剧毒细菌爆发？我们推导出基本后代数|$\mathcal N_0$| 它作为阈值和分岔参数来研究系统的动力学和分岔。该模型表现出三个平衡：不稳定的无环境平衡，全局渐近稳定（GAS）无噬菌体平衡（PFE）每当|$\mathcal N_0<1$|⁠，以及当|$\mathcal N_0>1$|⁠时局部渐近稳定的环境持久平衡（EPE）。Lyapunov-LaSalle 技术用于证明 PFE 的 GAS 并估计 EPE 的吸引力盆地。通过中心流形近似，精确确定了 PFE 的拓扑类型。建立了跨临界分岔和 Hopf 分岔的存在。准确地说，当|$\mathcal N_0>1$|⁠ 时，EPE 失去稳定性并出现周期性解。此外，增加|$\mathcal N_0$| 可以净化引入噬菌体的环境。我们特意证明，对于较大的|$\mathcal N_0$|⁠，总细菌种群渐近地接近于零，而噬菌体种群则维持不变。在生态学上，我们的结果表明，对于小值|$\mathcal N_0$|⁠，周期解的存在可以解释重复性细菌传播疾病爆发的发生，而大值|$\mathcal N_0$| 清除环境中的细菌。数值模拟支持我们的理论结果。