Modeling Earth Systems and Environment

, Volume 5, Issue 1, pp 59–69 | Cite as

Stage-structured plant-pest-natural enemy interaction dynamics incorporating gestation delay for both pest and natural enemy

  • Vijay KumarEmail author
  • Joydip Dhar
  • Harbax Singh Bhatti
Original Article


The biological control is an alternate of chemical control as well as other pest control methods. In the present study, we propose a stage-structured food chain dynamics of plant-pest-natural enemies with gestation delays for both pests and natural enemies. The positivity and boundedness properties of the system are proved. The existence criteria of the equilibria and corresponding stability conditions are established. The existence of Hopf bifurcation in the system is analyzed. When there are delays in the system, it is established that the steady states \(E_2\) and \(E^*\) are stable for specific threshold values, \(\tau _1\in [0,\tau _{10}^+)\), \(\tau _2\in [0,\tau _{20}^+)\). When \(\tau _1>\tau _{10}^+\), \(\tau _2>\tau _{20}^+\) system is unstable and oscillatory character of the system is observed. Further, the natural enemy free steady state, \(E_2\) and the coexisting steady state, \(E^*\) are stable for specific threshold values \(\tau _1<\tau _{10}^{+}, \ \tau _2>\tau _{20}^{+}\) and \(\tau _1<\tau _{10}^{++}, \ \tau _2>\tau _{20}^{++}\) respectively. If both \(\tau _1, \ \tau _2\) crosses the threshold values, i.e., \(\tau _1>\tau _{10}^{+}, \ \tau _2>\tau _{20}^{+}\) and \(\tau _1>\tau _{10}^{++}, \ \tau _2>\tau _{20}^{++}\), system perceived oscillating behavior and Hopf bifurcation occurs. Calculated the respective sensitive indices of the system parameters at the coexisting steady state by performing the sensitivity analysis. Further numerical experimentations have been carried out to support our analytic findings.


Stage-structured Plant-pest-natural enemy Equilibria Gestation delay Boundedness Positivity Hopf bifurcation Sensitivity analysis 

Mathematics Subject Classification

34C23 34D20 92B05 92D30 



The authors would like to thank the referee and reviewer for their constructive suggestions on improving the presentation of the paper. Also, authors would like to thank, I. K. G. Punjab Technical University, Jalandhar-Kapurthala-144601, Punjab, India for giving the opportunity to pursue the research.


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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.IKG-Punjab Technical UniversityKapurthalaIndia
  2. 2.Department of Applied Sciences Humanities and ManagementBeant College of Engineering and TechnologyGurdaspurIndia
  3. 3.Department of Applied SciencesABV-Indian Institute of Information Technology and ManagementGwaliorIndia
  4. 4.Department of Applied SciencesB.B.S.B. Engineering CollegeFatehgarh SahibIndia

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