Extinction and permanence of the predator-prey system with general functional response and impulsive control. (English) Zbl 1481.92106
Summary: Traditional approach for modelling the evolution of populations in the predator-prey ecosystem has commonly been undertaken using specific impulsive response function, and this kind of modelling is applicable only for a specific ecosystem under certain environmental situations only. This paper attempts to fill the gap by modelling the predator-prey ecosystem using a ‘generalized’ impulsive response function for the first time. Different from previous research, the present work develops the modelling for an integrated pest management (IPM) especially when the stocking of predator (natural enemy) and the harvesting of prey (pest) occur impulsively and at different instances of time. The paper firstly establishes the sufficient conditions for the local and the global stabilities of prey eradication periodic solution by applying the Floquet theorem of the Impulsive different equation and small amplitude perturbation under a ‘generalized’ impulsive response function. Subsequently the sufficient condition for the permanence of the system is given through the comparison techniques. The corollaries of the theorems that are established by using the ‘general impulsive response function’ under the locally asymptotically stable condition are found to be in excellent agreement with those reported previously. Theoretical results that are obtained in this work is then validated by using a typical impulsive response function (Holling type-II) as an example, and the outcome is shown to be consistent with the previously reported results. Finally, the implication of the developed theories for practical pest management is illustrated through numerical simulation. It is shown that the elimination of either the preys or the pest can be effectively deployed by making use of the theoretical model established in this work. The developed model is capable to predict the population evolutions of the predator-prey ecosystem to accommodate requirements such as: the combinations of the biological control, chemical control, any functional response function, the moderate impulsive period, the harvest rate for the prey and predator parameter and the incremental stocking of the predator parameter.
MSC:
| 92D25 | Population dynamics (general) |
| 34A37 | Ordinary differential equations with impulses |
| 34C60 | Qualitative investigation and simulation of ordinary differential equation models |
References:
| [1] | Zhang, M.; Gao, L., Input-to-state stability for impulsive switched nonlinear systems with unstable subsystems, Transactions of the Institute of Measurement and Control, 40, 7, 2167-2177 (2018) |
| [2] | Zhu, H.; Li, P.; Li, X.; Akca, H., Input-to-state stability for impulsive switched systems with incommensurate impulsive switching signals, Communications in Nonlinear Science and Numerical Simulation, 80, 104969 (2020) · Zbl 1451.93335 |
| [3] | Yuan, J.; Zhang, X.; Zhu, X.; Feng, E.; Yin, H.; Xiu, Z., Pathway identification using parallel optimization for a nonlinear hybrid system in batch culture, Nonlinear Analysis: Hybrid Systems, 15, 112-131 (2015) · Zbl 1370.92063 |
| [4] | Gao, L.; Zhang, M.; Yao, X., Stochastic input-to-state stability for impulsive switched stochastic nonlinear systems with multiple jumps, International Journal of Systems Science, 50, 1860-1871 (2019) · Zbl 1483.93681 |
| [5] | Yuan, J.; Wu, C.; Ye, J.; Xie, J., Robust identification of nonlinear state-dependent impulsive switched system with switching duration constraints, Nonlinear Analysis: Hybrid Systems, 36, 100879 (2020) · Zbl 1441.93068 |
| [6] | Yuan, J.; Xie, J.; Liu, C.; Teo, K.; Huang, M.; Fan, H.; Feng, E.; Xiu, Z., Robust optimization for a nonlinear switched time-delay system with noisy output measurements using hybrid optimization algorithm, Journal of the Franklin Institute, 356, 9730-9762 (2019) · Zbl 1423.93111 |
| [7] | Liu, C.; Gong, Z.; Wing, H.; Lee, J.; Teo, K., Robust bi-objective optimal control of 1, 3-propanediol microbial batch production process, Journal of Process Control, 78, 170-182 (2019) |
| [8] | Yao, Z.; Xie, S.; Yu, N., Dynamics of cooperative predator-prey system with impulsive effects and beddington-deangelis functional response, Journal of the Egyptian Mathematical Society, 21, 213-223 (2013) · Zbl 1284.34122 |
| [9] | Yuan, J.; Zhang, Y.; Ye, J.; Xie, J.; Teo, K.; Zhu, X.; Feng, E.; H, Y.; Xiu, Z., Robust parameter identification using parallel global optimization for a batch nonlinear enzyme-catalytic time-delayed process presenting metabolic discontinuities, Applied Mathematical Modelling, 00, 1-24 (2017) |
| [10] | J. Yuan, J. Xie, M. Huang, H. Fan, E. Feng, Z. Xiu, Robust optimal control problem with multiple characteristic time points in the objective for a batch nonlinear time-varying process using parallel global optimization, in: Optimization and Engineering, Https://link.springer.com/article/10.1007 |
| [11] | Wang, Z.; Xie, Y.; Lu, J.; Li, Y., Stability and bifurcation of a delayed generalized fractional-order prey-predator model with interspecific competition, Applied Mathematics and Computation, 347, 360-369 (2019) · Zbl 1428.92094 |
| [12] | Wang, X.; Wang, Z.; Shen, H., Dynamical analysis of a discrete-time SIS epidemic model on complex networks, Applied Mathematics Letters, 94, 292-299 (2019) · Zbl 1411.92295 |
| [13] | Pervez, A.; Singh, P.; Bozdoǧan, H., Ecological perspective of the diversity of functional responses, European Journal of Environmental Sciences, 8, 97-101 (2018) |
| [14] | 68-65 · Zbl 1540.34048 |
| [15] | Li, W.; Ji, J.; Huang, L., Global dynamic behavior of a predator-prey model under ratio-dependent state impulsive control, Applied Mathematical Modelling, 77, 1842-1859 (2020) · Zbl 1481.92107 |
| [16] | Hu, J.; Liu, J.; Zhao, Q., Dynamic analysis of pest-epidemic model with impulsive control, Mathematics in practice and theory, 49, 2, 304-310 (2019) |
| [17] | Li, J.; Zhu, X.; Lin, X.; Li, J., Impact of cannibalism on dynamics of a structured predator-prey system, Applied Mathematical Modelling, 78, 1-19 (2020) · Zbl 1481.92105 |
| [18] | Hu, J.; Liu, J., A novel pulsing rodent control method equipped with ratio-dependent reaction system, Journal of Shanxi Agricultural University (Nature Science Edition), 38, 9, 37-42 (2018) |
| [19] | Kumar, C. V.; Reddy, K. S.; Srinivas, M., Dynamics of prey predator with holling interactions and stochastic influences, Alexandria Engineering Journal, 57, 1079-1086 (2018) |
| [20] | Izquierdo, A.; Len, M.; Mayado, M., A generalized holling type II model for the interaction between dextral-sinistral snails and pareas snakes, Applied Mathematical Modelling, 73, 459-472 (2019) · Zbl 1481.92096 |
| [21] | Li, S.; Liu, W., A delayed holling type III functional response predator-prey system with impulsive perturbation on the prey, Advance in different equations, 42, 42-62 (2016) · Zbl 1418.92108 |
| [22] | Wang, X.; Wei, J., Dynamics in a diffusive predator-prey system with strong allee effect and ivlev-type functional response, Journal of Mathematical Analysis and Applications, 422, 1447-1462 (2015) · Zbl 1310.35032 |
| [23] | Liu, J.; Hu, J.; Zhao, Q., Seasonally perturbed ratio-dependent predator-prey system with impulsive control, Journal of Biomathematics, 33, 2, 160-170 (2018) · Zbl 1438.34153 |
| [24] | Liu, B.; Teng, Z.; Chen, L., Analysis of a predator-prey model with holling II functional response concerning impulsive control strategy, Journal of Computational and Applied Mathematics, 193, 347-362 (2006) · Zbl 1089.92060 |
| [25] | Liu, Z.; Zhang, L.; Bi, P.; Pang, J.; Li, B.; Fang, C., On the dynamics of one-prey-n-predator impulsive reaction-diffusion predator-prey system with ratio-dependent functional response, Journal of Biological Dynamics, 12, 551-576 (2018) · Zbl 1447.92352 |
| [26] | Lin, X.; Jiang, Y.; Wang, X., Existence of periodic solutions in predator-prey with watt-type functional response and impulsive effects, Nonlinear Analysis: Theory, Methods and Applications, 73, 6, 1684-1697 (2010) · Zbl 1203.34073 |
| [27] | Kim, H.; Baek, H., The dynamical complexity of a predator-prey system with hassell-varley functional response and impulsive effect, Mathematics and Computers in Simulation, 94, 1-14 (2013) · Zbl 1499.92068 |
| [28] | Wang, X.; Huang, C., Permanence of a stage-structured predator-prey system with impulsive stocking prey and harvesting predator, Applied Mathematics and Computation, 235, 32-42 (2014) · Zbl 1334.92375 |
| [29] | Liu, Z.; Tan, R., Impulsive harvesting and stocking in a monod-haldane functional response predator-prey system, Chaos Solitons Fractals, 34, 454-464 (2007) · Zbl 1127.92045 |
| [30] | Baek, H.; Kim, S.; Kim, P., Permanence and stability of an ivlev-type predator-prey system with impulsive control strategies, Applied Mathematical Modelling, 50, 1385-1393 (2009) · Zbl 1185.34067 |
| [31] | Ling, C.; Tang, S., Seasonally perturbed the monod-haldance type predator-prey model with impulsive, Journal of Biomathematics, 28, 499-504 (2013) · Zbl 1313.92060 |
| [32] | Shi, Z.; Cheng, H.; Li, Y.; Li, Y., A cydia pomonella integrated management predator-prey model with smith growth and linear feedback control, IEEE Access, 7, 126066-126076 (2019) |
| [33] | Pervez, A.; Singh, P.; Bozdoǧan, H., Ecological perspective of the diversity of functional responses, European Journal of Environmental Sciences, 8, 97-101 (2018) |
| [34] | Liu, J.; Zhao, Q.; Chen, C., Stability of two prey and one predator systems with impulsive effects, Jiangsu agricultural science, 46, 18, 94-97 (2018) |
| [35] | Liu, J.; Hu, J.; Zhao, Q., Dynamic analysis of the predator-prey system in a polluted environment with impulsive, Mathematics in practice and theory, 49, 22, 294-298 (2019) · Zbl 1449.34148 |
| [36] | Xie, Y.; Liu, J.; Wang, Z., Stability analysis of a fractional-order diffused prey-predator model with prey refuges, Physica A: Statistical Mechanics and its Applications, 526, 120773 (2019) · Zbl 07566377 |
| [37] | Wang, G.; Liu, Y.; Lu, J.; Wang, Z., Stability analysis of totally positive switched linear systems with average dwell time switching, Nonlinear Analysis: Hybrid Systems, 36, 100877 (2020) · Zbl 1441.93262 |
| [38] | Yuan, J.; Zhang, X.; Liu, C.; Chang, L.; Xie, J.; Feng, E.; Yin, H.; Xiu, Z., Robust optimization for nonlinear time-delay dynamical system of dha regulon with cost sensitivity constraint in batch culture, Communication in Nonlinear Science and Numerical Simulation, 38, 140-171 (2016) · Zbl 1471.35284 |
| [39] | Lakshmikantham, V.; Bainov, D.; Simeonov, S., Theory of Impulsive Differential Equations (1989), World Scientific Publisher: Singapore · Zbl 0719.34002 |
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.
