In this study, a three species plant-pest-natural enemy compartmental model incorporating gestation delays for both pests and natural enemies in a polluted environment is proposed. The boundedness and positivity properties of the model are established. Equilibria and their stability analysis are carried out for all possible steady states. The existence of Hopf bifurcation in the system is analyzed. It is established that the natural enemy free steady state \(E_2\) is stable for specific threshold parameter values \(\tau _1\in (0,\tau _{10}^+)\), i.e., gestation delay for pest species belongs to zero and it’s own critical value, \(\tau _{10}^+\) and the coexisting steady state \(E^*\) is stable for specific threshold parameter values \(\tau _1\in (0,\tau _{10}^+)\) and \(\tau _2\in (0,\tau _{20}^+)\), i.e., gestation delay for pest species belongs to zero and it’s own critical value, \(\tau _{10}^+\) and gestation delay for natural enemies belongs to zero and it’s own critical value, \(\tau _{20}^+\). If both gestation delays for pest and natural enemies, i.e., \(\tau _1, \tau _2\) respectively cross their threshold parameter values, i.e., \(\tau _1>\tau _{10}^{+},\tau _2>\tau _{20}^{+}\), then the system perceived oscillating behavior and Hopf bifurcation occurs in the system. The sensitivity analysis of the system at interior steady state is presented and the sensitive indices of the variables are identified. Finally, simulations are performed to support our analytic results with a distinct set of parametric values.

在这项研究中，提出了一种三类植物-虫-天敌隔间模型，该模型结合了受污染环境中有害生物和天敌的妊娠延迟。建立了模型的有界性和正性。对所有可能的稳态进行平衡及其稳定性分析。分析了系统中Hopf分叉的存在。建立了天敌自由稳态\（E_2 \）对于特定阈值参数值\（\ tau _1 \ in（0，\ tau _ {10} ^ +）\）是稳定的，即害虫的妊娠延迟物种属于零，并且它是自己的临界值\（\ tau _ {10} ^ + \），并且对于特定阈值参数值，并存的稳态\（E ^ * \）是稳定的\（\ tau _1 \ in（0，\ tau _ {10} ^ +）\）和\（\ tau _2 \ in（0，\ tau _ {20} ^ +）\），即害虫的妊娠延迟物种属于零，并且它是自己的临界值\（\ tau _ {10} ^ + \），而天敌的妊娠延迟也就是零，并且它是自己的临界值\（\ tau _ {20} ^ + \）。如果害虫和天敌的妊娠延迟（即\（\ tau _1，\ tau _2 \））分别超过其阈值参数值，即\（\ tau _1> \ tau _ {10} ^ {+}，\ tau _2> \ tau _ {20} ^ {+} \），则系统会感知到振荡行为，并且Hopf分叉会在系统中发生。给出了系统在内部稳态下的灵敏度分析，并确定了变量的敏感指标。最后，进行仿真以使用一组独特的参数值来支持我们的分析结果。