×

zbMATH — the first resource for mathematics

Permanence of periodic Holling type predator–prey system with stage structure for prey. (English) Zbl 1111.34039
Summary: We study the permanence of the following periodic Holling-type predator-prey system with stage structure for prey \[ \begin{aligned} \dot x_1(t) & =a(t)x_2(t)-b(t)x_1(t) -d(t)x^2_1(t)-\frac{e(t)x^\gamma_1(t)} {p(t)+ x^\gamma_1(t)}y(t),\\ \dot x_2(t)& =c(t)x_1(t)-f(t)x^2_2(t),\\ \dot y(t) &=y(t)\left(-g(t)+\frac{h(t)x_1^\gamma(t)} {p(t)+x^\gamma_1(t)}-q(t)y(t)\right).\end{aligned} \] A sufficient and necessary condition which guarantees the predator and the prey species to be permanent is obtained. Some new results are obtained.

MSC:
34D05 Asymptotic properties of solutions to ordinary differential equations
34C60 Qualitative investigation and simulation of ordinary differential equation models
92D25 Population dynamics (general)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Cui, J.A.; Song, X.Y., Permanence of a predator – prey system with stage structure, Discret. contin. dyn. syst., ser. B, 4, 3, 547-554, (2004) · Zbl 1100.92062
[2] Chen, F.D., Periodic solutions of a delayed predator – prey model with stage structure for predator, J. appl. math., 2005, 2, 153-169, (2005)
[3] Wang, W.; Mulone, G.; Salemi, F.; Salone, V., Permanence and stability of a stage-structured predator – prey model, J. math. anal. appl., 262, 2, 499-528, (2001) · Zbl 0997.34069
[4] Y. Xiao, Study on the eco-epidemiology dynamical system, Ph.D thesis, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, 2001.
[5] Xu, R.; Chaplain, M.A.J.; Davidson, F.A., A lotka – volterra type food chain model with stage structure and time delays, J. math. anal. appl., 315, 1, 90-105, (2006) · Zbl 1096.34055
[6] Holling, C.S., The functional response of predators to prey density and its role in mimicry and population regulations, Mem. entomol. soc. can., 45, 3-60, (1965)
[7] Butler, G.J.; Freedman, H.I., Periodic solution of a predator – prey system with periodic coefficients, Math. biosci., 55, 27-38, (1981) · Zbl 0471.92020
[8] Cushing, J.M., Periodic time-dependent predator – prey systems, SIAM J. appl. math., 32, 82-95, (1977) · Zbl 0348.34031
[9] Ding, T.; Huang, H.; Zanolin, F., A periori bounds and periodic solution for a class of planar systems with applications to lotka – volterra equations, Discret. contin. dyn. syst., 1, 103-117, (1995) · Zbl 0877.34035
[10] Teng, Z., Uniform persistence of the periodic predator – prey lotka – volterra systems, Appl. anal., 72, 339-352, (1998) · Zbl 1031.34045
[11] Chen, F.D., Persistence and periodic orbits for two-species non-autonomous diffusion lotka – voltrra models, Appl. math. J. chin. univ. ser. B., 19, 4, 359-366, (2004) · Zbl 1074.34053
[12] Chen, F.D., On a nonlinear non-autonomous predator – prey model with diffusion and distributed delay, J. comput. appl. math., 180, 1, 33-49, (2005) · Zbl 1061.92058
[13] Chen, F.D.; Shi, J.L., Periodicity in a nonlinear predator – prey system with state dependent delays, Acta math. appl. sin., 26, 1, 49-60, (2005) · Zbl 1096.34050
[14] Sugie, J.; Kohno, R.; Miyazaki, R., On a predator – prey system of Holling type, Proc. am. math. soc., 125, 2041-2050, (1997) · Zbl 0868.34023
[15] Sugie, J.; Katayama, M., Global asymptotic stability of a predator – prey system of Holling type, Nonlinear anal., 38, 105-121, (1999) · Zbl 0984.34043
[16] Chen, F.D.; Shi, J.L.; Chen, X.X., Persistence and global stability for two-species nonautnomous predator – prey system with diffusion and time delay, Ann. differ. equ., 20, 2, 111-117, (2004) · Zbl 1059.34054
[17] Sun, D.X.; Chen, F.D., Positive periodic solution of an integro-differential predator – prey system with infinite delays, Ann. differ. equ., 20, 1, 77-85, (2004) · Zbl 1059.45006
[18] Chen, F.D., Existence, uniqueness and stability of periodic solution for a nonlinear prey-competition model with delays, J. comput. appl. math., 194, 2, 368-387, (2006) · Zbl 1104.34050
[19] Wang, L.L.; Li, W.T., Periodic solutions and permanence for a delayed nonautonomous ratio-dependent predator – prey model with Holling type functional response, J. comput. appl. math., 162, 341-357, (2004) · Zbl 1076.34085
[20] Chen, F.D.; Xie, X.D., Permanence and extinction in nonlinear single and multiple species system with diffusion, Appl. math. comput., 17, 1, 410-426, (2006) · Zbl 1090.92046
[21] Fan, M.; Wang, K., Periodicity in a delayed ratio-dependent predator – prey system, J. math. anal. appl., 262, 179-190, (2001) · Zbl 0994.34058
[22] Xiao, D.; Li, W., Stability and bifurcation in a delayed ratio-dependent predator – prey system, Proc. Edinburgh math. soc., 46, 205-220, (2003) · Zbl 1041.92028
[23] Zhao, X.Q., The qualitative analysis of N-species lotka – volterra periodic competition systems, Math. comp. modeling, 15, 3-8, (1991) · Zbl 0756.34048
[24] Zhang, Z.Q.; Zeng, X.W.; Wang, Z.C., Periodic solution of a non-autonomous diffusive food chain system of three species with time delays, Acta math. appl. sin., English ser., 19, 4, 691-702, (2003) · Zbl 1059.34048
[25] Teng, Z.; Chen, L., The positive periodic solutions in periodic Kolmogorov type systems with delays (in Chinese), Acta math. appl. sin., 22, 446-456, (1999) · Zbl 0976.34063
[26] Wang, S., Research on the suitable living environment of the rana temporaria chensinensis larva, Chin. J. zool., 32, 1, 38-41, (1997)
[27] Zhang, X.; Chen, L.; Neumann, A.U., The stage-structured predator – prey model and optimal harvesting policy, Math. biosci., 101, 139-153, (2000)
[28] Cui, J.; Chen, L.; Wang, W., The effect of dispersal on population growth with stage-structure, Comput. math. appl., 39, 91-102, (2000) · Zbl 0968.92018
[29] Bernard, O.; Souissi, S., Qualitative behavior of stage-structure populations: application to structural validation, J. math. biol., 37, 291-308, (1998) · Zbl 0919.92035
[30] Aiello, W.G.; Freedman, H.I.; Wu, J., Analysis of a model representing stage- structure population growth with state-dependent time delay, SIAM J. appl. math., 52, 855-869, (1992) · Zbl 0760.92018
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.