×

Bifurcation dynamics of a plant-pest-natural enemy system in polluted environment incorporating gestation delays. (English) Zbl 1467.34086

Summary: 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.

MSC:

34K60 Qualitative investigation and simulation of models involving functional-differential equations
92C80 Plant biology
34K21 Stationary solutions of functional-differential equations
34K20 Stability theory of functional-differential equations
34K18 Bifurcation theory of functional-differential equations
34K13 Periodic solutions to functional-differential equations
Full Text: DOI

References:

[1] Thomas, MB; Willis, AJ, Biocontrol-risky but necessarys, Trends Ecol. Evol., 13, 325-329 (1998)
[2] Parrella, MP; Heinz, KM; Nunney, L., Biological control through augmentative releases of natural enemies: a strategy whose time has come, Am. Entomol., 38, 3, 172-179 (1992)
[3] Kishimba, M.; Henry, L.; Mwevura, H.; Mmochi, A.; Mihale, M.; Hellar, H., The status of pesticide pollution in Tanzania, Talanta, 64, 1, 48-53 (2004)
[4] Weaver, RD; Evans, DJ; Luloff, AE, Pesticide use in tomato production: consumer concerns and willingness-to-pay, Agribusiness, 8, 2, 131-142 (1992)
[5] Jain, PC; Bhargava, MC, Entomology: Novel Approaches (2007), New Delhi: New India Publishing Agency, New Delhi
[6] Kar, TK; Ghosh, B., Sustainability and optimal control of an exploited prey predator system through provision of alternative food to predator, BioSystems, 109, 220-232 (2012)
[7] Ghosh, B.; Grogrnard, F.; Mailleret, L., Natural enemies deployment in patchy environments for augmentative biological control, Appl. Math. Comput., 266, 982-999 (2015) · Zbl 1410.93084
[8] Kumar, V.; Dhar, J.; Singh, H.; Bhatti, HS, Plant pest natural enemy dynamics with disease in pest and gestation delay for natural enemy, J. Math. Comput. Sci., 5, 7, 948-965 (2017) · Zbl 1436.34075
[9] Kumar, V., Dhar, J., Bhatti, H.S.: Stability and Hopf-bifurcation dynamics of a food chain system: plant-pest-natural enemy with dual gestation delay as a biological control strategy 4(2), 881-889 (2018) · Zbl 1429.34086
[10] Lian, F.; Xu, Y., Hopf bifurcation analysis of a predator-prey system with Holling type IV functional response and time delay, Appl. Math. Comput., 215, 4, 1484-1495 (2009) · Zbl 1187.34116
[11] Liu, X.; Han, M., Chaos and Hopf bifurcation analysis for a two species predator-prey system with prey refuge and diffusion, Nonlinear Anal. Real World Appl., 12, 2, 1047-1061 (2011) · Zbl 1222.34099
[12] Singh, H.; Dhar, J.; Bhatti, HS, Dynamics of a prey generalized predator system with disease in prey and gestation delay for predator, Model. Earth Syst. Environ., 2, 2, 52 (2016)
[13] Song, Y.; Wei, J., Local Hopf bifurcation and global periodic solutions in a delayed predator-prey system, J. Math. Anal. Appl., 301, 1, 1-21 (2005) · Zbl 1067.34076
[14] Zhao, H.; Lin, Y., Hopf bifurcation in a partial dependent predator-prey system with delay, Chaos Solitons Fractals, 42, 2, 896-900 (2009) · Zbl 1198.34148
[15] Faria, T., Stability and bifurcation for a delayed predator-prey model and the effect of diffusion, J. Math. Anal. Appl., 254, 2, 433-463 (2001) · Zbl 0973.35034
[16] Li, K.; Wei, J., Stability and Hopf bifurcation analysis of a prey-predator system with two delays, Chaos Solitons Fractals, 42, 5, 2606-2613 (2009) · Zbl 1198.34144
[17] Song, Y.; Peng, Y.; Wei, J., Bifurcations for a predator-prey system with two delays, J. Math. Anal. Appl., 337, 1, 466-479 (2008) · Zbl 1132.34053
[18] Xu, C.; Liao, M.; He, X., Stability and Hopf bifurcation analysis for a Lotka-Volterra predator-prey model with two delays, Int. J. Appl. Math. Comput. Sci., 21, 1, 97-107 (2011) · Zbl 1231.34151
[19] Aiello, W.; Freedman, H., A time-delay model of single-species growth with stagestructure, Math. Biosci., 101, 2, 139-153 (1990) · Zbl 0719.92017
[20] Magniusson, k., Destabilizing effect of cannibalism on a structured predator-prey system, Math. Biosci., 155, 1, 61-75 (1999) · Zbl 0943.92030
[21] Wang, W.; Chen, L., A predator-prey system with stage-structure for predator, Comput. Math. Appl., 33, 8, 83-91 (1997)
[22] Xu, R.; Chaplain, M.; Davidson, F., Global stability of a Lotka-Volterra type predator-prey model with stage-structure and time delay, Appl. Math. Comput., 159, 3, 863-880 (2004) · Zbl 1056.92063
[23] Zhang, X.; Chen, L.; Neumann, A., The stage-structured predator-prey model and optimal harvesting policy, Math. Biosci., 168, 2, 201-210 (2000) · Zbl 0961.92037
[24] Alakes, M., Rita, P., Shariful, A.: A ratio-dependent predator-prey model with strong allee effect in the prey and an alternative food source for the predator. Int. J. Res. Eng. Technol. (2016)
[25] Hu, H.; Huang, L., Stability and Hopf bifurcation in a delayed predator-prey system with stage-structure for prey, Nonlinear Anal. Real World Appl., 11, 4, 2757-2769 (2010) · Zbl 1203.34132
[26] Huang, C.; Zhao, M.; Zhao, L., Permanence of periodic predator-prey system with two predators and stage-structure for prey, Nonlinear Anal. Real World Appl., 11, 1, 503-514 (2010) · Zbl 1189.34085
[27] Zhang, H.; Jiao, J.; Chen, J., Pest management through continuous and impulsive control strategies, Biosystems, 90, 2, 350-361 (2007)
[28] Zhang, J.; Jin, Z.; Yan, J.; Sun, G., Stability and Hopf bifurcation in a delayed competition system, Nonlinear Anal. Theory Methods Appl., 70, 2, 658-670 (2009) · Zbl 1166.34049
[29] Shigui, R., Absolute stability, conditional stability and bifurcation in Kolmogorov-type predator-prey systems with discrete delays, Q. Appl. Math., 59, 1, 159-174 (2001) · Zbl 1035.34084
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.