Stability Switching Dynamics of a Food Chain System Incorporating Gestation Delays
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Stability Switching Dynamics of a Food Chain System Incorporating Gestation Delays Vijay Kumar1,2
· Joydip Dhar3 · Harbax S. Bhatti4
© Foundation for Scientific Research and Technological Innovation 2019
Abstract In the present study, a three species plant-pest-natural enemy model is proposed incorporating gestation delay for pests and natural enemies. The boundedness and positivity of the model are studied and analyzed. 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 coexisting steady state E ∗ is stable for specific threshold parameters τ1 ∈ (0, τ10 ), + ++ τ2 ∈ (0, τ20 ) respectively. If both τ1 , τ2 crosses the threshold parameters i.e., τ1 > τ10 , τ2 > ++ τ20 system perceived oscillating behavior and Hopf bifurcation occurs. Furthermore, the ++ ++ coexisting steady state E ∗ is stable for specific threshold parameters τ1 < τ10 , τ2 > τ20 ++ ++ or τ1 > τ10 , τ2 < τ20 . The sensitivity analysis of the system at interior steady state is presented and identified the sensitive indices of the variables. Finally, simulations are performed to support our analytic results with a distinct set of parametric values. Keywords Plant-pest-natural enemy · Equilibria · Gestation delay · Boundedness · Positivity · Hopf bifurcation · Sensitivity analysis Mathematics Subject Classification 34C23 · 34D20 · 92B05 · 92D30
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Vijay Kumar [email protected] Joydip Dhar [email protected] Harbax S. Bhatti [email protected]
1
Research Scholar, Inder Kumar Gujral-Punjab Technical University, Kapurthala, Punjab 144601, India
2
Beant College of Engineering and Technology, Gurdaspur, Punjab, India
3
ABV-Indian Institute of Information Technology and Management, Gwalior, M.P 474015, India
4
B.B.S.B. Engineering College, Fatehgarh Sahib, Punjab, India
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Differential Equations and Dynamical Systems
Introduction The classification of pests, and detecting natural enemies for the pests, is necessary for the biological control. Thoroughly observe the mites and insects on a particular plant to help discern their activity. For example, the complex of imported parasites, which control alfalfa weevil [1], and another example is the augmentative release of natural enemies, which have been applied in greenhouses in Europe for control of many vegetable pests [2]. The mathematical system could not contain all the features of the real world, but it is important that the model must give the basics, which are expected for the solution of the problem or described up to the desired level. The pests are harmful species for the plants, and their control has become a challenge for authors and researchers, etc. Pests outbreak usually causes severe environmental problems [3,4]. In the real world, several species usually go into two different life steps from birth to death, immature and mature. The immature species needs some time to become mature and this period is known as a delay. Delay is a significant f
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