# zbMATH — the first resource for mathematics

Periodic solution of a Leslie predator-prey system with ratio-dependent and state impulsive feedback control. (English) Zbl 1377.37117
Summary: In this paper, a Leslie predator-prey system with ratio-dependent and state impulsive feedback control is investigated by applying the geometry theory of differential equation. When the economic threshold level is under the positive equilibrium, the existence, uniqueness and orbital asymptotical stability of order-1 periodic solution for the system can be obtained. When the economic threshold level is above the positive equilibrium, and the positive equilibrium is a focus point, sufficient conditions of the existence, uniqueness and orbital asymptotical stability of order-1 periodic solution for the system are also acquired. Furthermore, when the positive equilibrium is an unstable focus point, the existence of order-1 periodic solution of the impulsive system can be obtained within limit cycle of the continuous system. The mathematical results can be verified by numerical simulations.

##### MSC:
 37N25 Dynamical systems in biology 92D25 Population dynamics (general) 93B52 Feedback control 34A37 Ordinary differential equations with impulses
Full Text:
##### References:
 [1] Stern, VM; Smith, RF; Rosch, VDR; Hagen, KS, The integrated control concept, Hilgardia, 29, 81-101, (1959) [2] Bainov, D.D., Simeonov, P.S.: Impulsive Differential Equations: Periodic Solutions and Applications. Longman, England (1993) · Zbl 0815.34001 [3] Bainov, D.D., Simeonov, P.S.: Impulsive Differential Equations: Asymptotic Properties of the Solutions. World Scientific, Singapore (1995) · Zbl 0828.34002 [4] Simenov, PS; Bainov, DD, Orbital stability of the periodic solutions of autonomous systems with impulse effect, Int. J. Syst. Sci., 19, 2561-2585, (1988) · Zbl 0669.34044 [5] Ciesielski, K, On stability in impulsive dynamical systems, Bull. Polish Acad. Sci. Math., 52, 81-91, (2004) · Zbl 1098.37017 [6] Bonotto, EM; Federson, M, Poisson stability for impulsive semidynamical systems, Nonlinear Anal., 71, 6148-6156, (2009) · Zbl 1194.37030 [7] Bonotto, EM; Federson, M, Limit sets and the PoincarĂ©-Bendixson theorem in impulsive semidynamical systems, J. Differ. Equ., 244, 2334-2349, (2008) · Zbl 1143.37014 [8] Chen, LS, Pest control and geometric theory of semi-continuous dynamical system, J. Beihua Univ. (Nat. Sci.), 12, 1-9, (2011) [9] Chen, LS, Theory and application of semi-continuous dynamical system, J. Yulin Normal Univ. (Nat. Sci.), 34, 1-10, (2013) [10] Liang, ZQ; Pang, GP; Zen, XP; Liang, YH, Qualitative analysis of a predator-prey system with mutual interference and impulsive state feedback control, Nonlinear Dyn., 87, 1495-1509, (2017) · Zbl 1384.92050 [11] Sun, SL; Guo, CH; Qin, C, Dynamic behaviors of a modified predator-prey model with state-dependent impulsive effects, Adv. Differ. Equ., 2016, 50, (2016) · Zbl 1418.92131 [12] Wang, TY; Chen, LS, Nonlinear analysis of a microbial pesticide model with impulsive state feedback control, Nonlinear Dyn., 65, 1-10, (2011) · Zbl 1235.93108 [13] He, ZM, Impulsive state feedback control of a predator-prey system with group defense, Nonlinear Dyn., 79, 2699-2714, (2015) · Zbl 1331.92124 [14] Xiao, QZ; Dai, BX; Xu, BX; Bao, LS, Homoclinic bifurcation for a general state-dependent Kolmogorov type predator-prey model with harvesting, Nonlinear Anal. RWA, 26, 263-273, (2015) · Zbl 1331.34102 [15] Wei, CJ; Chen, LS, Periodic solution and heteroclinic bifurcation in a predator-prey system with allee effect and impulsive harvesting, Nonlinear Dyn., 76, 1109-1117, (2014) · Zbl 1306.92052 [16] Pang, GP; Chen, LS, Periodic solution of the system with impulsive state feedback control, Nonlinear Dyn., 78, 743-753, (2014) · Zbl 1314.93024 [17] Zhao, Z; Pang, LY; Song, XY, Optimal control of phytoplankton-fish model with the impulsive feedback control, Nonlinear Dyn., 88, 2003-2011, (2017) [18] Leslie, PH, Some further notes on the use of matrices in population mathematics, Biometrika, 35, 213-245, (1948) · Zbl 0034.23303 [19] Leslie, PH; Gower, JC, The properties of a stochastic model for the predator-prey type of interaction between two species, Biometrika, 47, 219-231, (1960) · Zbl 0103.12502 [20] Hsu, SB; Huang, TW, Global stability for a class of predator-prey systems, SIAM J. Appl. Math., 55, 763-783, (1995) · Zbl 0832.34035 [21] Chen, LJ; Chen, FD, Global stability of a Leslie-gower predator-prey model with feedback controls, Appl. Math. Lett., 22, 1330-1334, (2009) · Zbl 1173.34333 [22] Singh, MK; Bhadauria, BS; Singh, BK, Qualitative analysis of a Leslie-gower predator-prey system with nonlinear harvesting in predator, Int. J. Eng. Math., 2016, 2741891, (2016) · Zbl 1413.34177 [23] Feng, P; Kang, Y, Dynamics of a modified Leslie-gower model with double allee effects, Nonlinear Dyn., 80, 1051-1062, (2015) · Zbl 1345.92115 [24] Yang, RZ; Zhang, CR, Dynamics in a diffusive modified Leslie-gower predator-prey model with time delay and prey harvesting, Nonlinear Dyn., 87, 863-878, (2017) · Zbl 1372.92098 [25] Cao, JZ; Yuan, R, Bifurcation analysis in a modified lesile-gower model with Holling type II functional response and delay, Nonlinear Dyn., 84, 1341-1352, (2016) · Zbl 1354.92059 [26] Murray, J.D.: Mathematical Biology. Springer, Berlin (1989) · Zbl 0682.92001 [27] May, R.M.: Stability and Complexity in Ecosystems. Princeton University Press, Princeton (2001) · Zbl 1044.92047 [28] Gasull, A; Kooij, RE; Torregrosa, J, Limit cycles in the Holling-tanner model, Publ. Mat., 41, 149-167, (1997) · Zbl 0880.34028 [29] Saez, E; Gonzalez-Olivares, E, Dynamics of predator-prey model, SIAM J. Appl. Math., 59, 1867-1878, (1999) · Zbl 0934.92027 [30] Song, ZG; Zhen, B; Xu, J, Species coexistence and chaotic behavior induced by multiple delays in a food chain system, Ecol. Complex., 19, 9-17, (2014) [31] Guo, L; Song, ZG; Xu, J, Complex dynamics in the Leslie-gower type of the food chain system with multiple delays, Commun. Nonlinear Sci. Numer. Simul., 19, 2850-2865, (2014) [32] Holling, CS, Functional response of predators to prey density and its role in mimicry and population regulation, Mem. Entomol. Soc. Can., 97, 1-60, (1965) [33] Arditi, R; Ginzburg, LR, Coupling in predator-prey dynamics: ratio-dependence, J. Theor. Biol., 139, 311-326, (1989) [34] Abrams, PA; Ginzburg, LR, The nature of predation: prey dependent, ratio-dependent or neither?, Trends Ecol. Evol., 15, 337-341, (2000) [35] Akcakaya, HR; Arditi, R; Ginzburg, LR, Ratio-dependent prediction: an abstraction that works, Ecology, 79, 995-1004, (1995) [36] Liang, ZQ; Pan, HW, Qualitative analysis of a ratio-dependent Holling-tanner model, J. Math. Anal. Appl., 334, 954-964, (2007) · Zbl 1124.34030 [37] Celik, C, Stability and Hopf bifurcation in a delayed ratio dependent Holling-tanner type model, Appl. Math. Comput., 255, 228-237, (2015) · Zbl 1338.34127 [38] Saha, T; Chakrabarti, C, Dynamical analysis of a delayed ratio-dependent Holling-tanner predator-prey model, J. Math. Anal. Appl., 358, 389-402, (2009) · Zbl 1177.34103 [39] Wang, Q; Zhang, YM; Wang, ZJ; Ding, MM; Zhang, HY, Periodicity and attractivity of a ratio-dependent Leslie system with impulses, J. Math. Anal. Appl., 376, 212-220, (2011) · Zbl 1216.34045
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.