## Multiplicity of positive almost periodic solutions in a delayed Hassell-Varley-type predator-prey model with harvesting on prey.(English)Zbl 1292.34077

Summary: We consider a delayed Hassell-Varley-type predator-prey model with harvesting on prey. By means of Mawhin’s continuation theorem of coincidence degree theory, some new sufficient conditions are obtained for the existence of at least two positive almost periodic solutions for the aforementioned model. An example is employed to illustrate the result of this paper. Copyright $$\copyright 2013$$ John Wiley & Sons, Ltd.

### MSC:

 34K60 Qualitative investigation and simulation of models involving functional-differential equations 34K14 Almost and pseudo-almost periodic solutions to functional-differential equations 92D25 Population dynamics (general) 47N20 Applications of operator theory to differential and integral equations
Full Text:

### References:

 [1] Lotka, Elements of Physical Biology (1925) [2] Volterra, Fluctuations in the abundance of species considered mathematically, Nature 118 pp 558– (1926) · JFM 52.0453.03 [3] Ma, Mathematical Modelling and Study of Species Ecology (1996) [4] Chen, Mathematical Models and Methods in Ecology (2003) [5] Huo, Stable periodic solution of the discrete periodic Leslie-Gewer predator-prey model, Mathematical and Computer Modelling 40 pp 261– (2004) · Zbl 1067.39008 [6] Hsu, Global analysis of the Michaelis-Menten type ratio-dependent predator-prey system, Journal of Mathematical Biology 42 pp 489– (2001) · Zbl 0984.92035 [7] Liu, A stage-structured predator-prey model of Beddington-DeAngelis type, SIAM Journal on Applied Mathematics 66 pp 1101– (2006) · Zbl 1110.34059 [8] Fan, Dynamics of a nonautonomous predator-prey system with the Beddington-DeAngelis functional, Journal of Mathematical Analysis and Applications 295 pp 15– (2004) · Zbl 1051.34033 [9] Wang, Asymptotic behavior of solutions in nonautonomous predator-prey patchy system with beddington-type functional response, Applied Mathematics and Computation 172 (1) pp 122– (2006) · Zbl 1102.34030 [10] Wang, Dispersal permanence of periodic predator-prey model with Ivlev-type functional response and impulsive effects, Applied Mathematical Modelling 34 pp 3713– (2010) · Zbl 1201.34077 [11] Ding, Periodic solutions for a semi-ratio-dependent predator-prey system with nonmonotonic functional response and time delay, Nonlinear Analysis: Real World Applications 9 pp 762– (2008) · Zbl 1152.34046 [12] Wei, Existence of multiple positive periodic solutions to a periodic predator-prey system with harvesting terms and Holling III type functional response, Communications in Nonlinear Science and Numerical Simulation 16 pp 2130– (2011) · Zbl 1221.34114 [13] Liu, Positive periodic solutions for neutral delay ratio-dependent predator-prey model with Holling type III functional response, Applied Mathematics and Computation 218 pp 4341– (2011) · Zbl 1260.92023 [14] Hassell, New inductive population model for insect parasites and its bearing on biological control, Nature 223 pp 1133– (1969) [15] Cosner, Effects of spatial grouping on the functional response of predators, Theoretical Population Biology 56 pp 65– (1999) · Zbl 0928.92031 [16] Hsu, Global dynamics of a predator-prey model with Hassell-Varley type functional response, Journal of Mathematical Biology 10 pp 1– (2008) · Zbl 1160.34046 [17] Wang, Periodic solutions to a delayed predator-prey model with Hassell-Varley type functional response, Nonlinear Analysis: Real World Applications 12 pp 137– (2011) · Zbl 1208.34130 [18] Dai, Coexistence region and global dynamics of a harvested predator-prey system, SIAM Journal on Applied Mathematics 58 pp 193– (1998) · Zbl 0916.34034 [19] Xiao, Bifurcations of a ratio-dependent predator-prey system with constant rate harvesting, SIAM Journal on Applied Mathematics 65 pp 737– (2005) · Zbl 1094.34024 [20] Kar, Non-selective harvesting in prey-predator models with delay, Communications in Nonlinear Science and Numerical Simulation 11 pp 499– (2006) · Zbl 1112.34057 [21] Kar, Dynamic behaviour of a delayed predator-prey model with harvesting, Applied Mathematics and Computation 217 pp 9085– (2011) · Zbl 1215.92065 [22] Xia, Multiple periodic solutions of a delayed stage-structured predator-prey model with non-monotone functional responses, Applied Mathematical Modelling 31 pp 1947– (2007) · Zbl 1167.34342 [23] Chen, Multiple periodic solutions of delayed predator-prey systems with type IV functional responses, Nonlinear Analysis: Real World Applications 5 pp 45– (2004) · Zbl 1066.92050 [24] Zhang, Multiple positive periodic solutions of a delayed discrete predator-prey system with type IV functional responses, Applied Mathematics Letters 20 pp 1031– (2007) · Zbl 1142.39015 [25] Zhang, Multiple positive periodic solutions for a generalized predator-prey system with exploited terms, Nonlinear Analysis: Real World Applications 9 pp 26– (2008) · Zbl 1145.34051 [26] Fang, Existence of multiple periodic solutions for delay Lotka-Volterra competition patch systems with harvesting, Applied Mathematical Modelling 33 pp 1086– (2009) · Zbl 1168.34349 [27] Zhang, Multiple periodic solutions of a delayed predator-prey system with stage structure for the predator, Nonlinear Analysis: Real World Applications 11 pp 4109– (2010) · Zbl 1205.34111 [28] Wei, Existence of multiple positive periodic solutions to a periodic predator-prey system with harvesting terms and Holling III type functional response, Communications in Nonlinear Science and Numerical Simulation 16 pp 2130– (2011) · Zbl 1221.34114 [29] Fink, Almost Periodic Differential Equation (1974) · Zbl 0325.34039 [30] He, Almost Periodic Differential Equations (1992) [31] Gaines, Coincidence Degree and Nonlinear Differential Equations (1977)
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.