Relaxation oscillations in predator-prey model with distributed delay. (English) Zbl 1400.34131
From the summary: A predator-prey model with distributed delay is stated in present paper. On the basis of geometric singular perturbation theory, the existence of the relaxation oscillation is proved. An approximate expression of the relaxation oscillation and its period are obtained analytically.
MSC:
| 34K60 | Qualitative investigation and simulation of models involving functional-differential equations |
| 92D25 | Population dynamics (general) |
| 34C26 | Relaxation oscillations for ordinary differential equations |
| 34E15 | Singular perturbations for ordinary differential equations |
References:
| [1] | Chen, Y; Yu, J; Sun, C, Stability and Hopf bifurcation analysis in a three-level food chain system with delay, Chaos Solitions Fract, 31, 683-694, (2007) · Zbl 1146.34051 · doi:10.1016/j.chaos.2005.10.020 |
| [2] | Fenichel, N, Asymptotic stability with rate conditions, Indiana Univ Math J, 23, 1109-1137, (1974) · Zbl 0284.58008 · doi:10.1512/iumj.1974.23.23090 |
| [3] | Fenichel, N, Geometric singular perturbation theory for ordinary differential equations, J Differ Equ, 31, 53-98, (1979) · Zbl 0476.34034 · doi:10.1016/0022-0396(79)90152-9 |
| [4] | Freedman, HI; Wolkowicz, GSK, Predator-prey systems with group defence: the paradox of enrichment revisited, Bull Math Biol, 48, 493-508, (1986) · Zbl 0612.92017 · doi:10.1007/BF02462320 |
| [5] | Holling, CS, Some characteristics of simple types of predation and parasiti sm, Can Entomol, 91, 385-398, (1959) · doi:10.4039/Ent91385-7 |
| [6] | Jones C (1994) Geometric singular perturbation theory in dynamical systems. Springer, Berlin |
| [7] | Liu, WS; Xiao, DM; Yi, YF, Relaxation oscillations in a class of predator-prey systems, J Differ Equ, 188, 306-331, (2003) · Zbl 1094.34025 · doi:10.1016/S0022-0396(02)00076-1 |
| [8] | Ludwig, D; Jones, D; Holling, CS, Qualitative analysis of insect outbreak systems: the spruce budwormand forest, J Anim Ecol, 47, 315-332, (1978) · doi:10.2307/3939 |
| [9] | Ma ZP, Liu HFHCY (2009) Stability and Hopf bifurcation analysis on a predator-prey model with discrete and distributed delays. Nonlinear Anal Real Word Appl 2009(10):1160-1172 · Zbl 1167.34382 |
| [10] | Mishchenko EF, Rozov KhN (1980) Differential equations with small parameters and relaxation oscillations. Plenum, New York · Zbl 1234.34047 |
| [11] | Muratori, S; Rinaldi, S, Remarks on competitive coexistence, SIAM J Appl Math, 49, 1462-1472, (1989) · Zbl 0681.92020 · doi:10.1137/0149088 |
| [12] | Rasmussen, A; Wyller, J; Vik, JO, Relaxation oscillations in sprucecbudworm interactions, Nonlinear Anal Real World Appl, 12, 304-319, (2011) · Zbl 1225.34057 · doi:10.1016/j.nonrwa.2010.06.017 |
| [13] | Rizaner, FB; Rogovchenko, SP, Dynamics of a single species under periodic habitat fluctuations and allee effect, Nonlinear Anal Real Word Appl, 13, 141-157, (2012) · Zbl 1239.37013 · doi:10.1016/j.nonrwa.2011.07.021 |
| [14] | Rozov, NKh, Asymptotic computation of solutions of systems of second-order differential equations close to discontinuous solutions, Soviet Math Doklady, 3, 932-934, (1962) · Zbl 0135.31002 |
| [15] | Sun, C; Lin, Y; Han, M, Stability and Hopf bifurcation for an epidemic disease model with delay, Chaos Solitions Fract, 30, 204-216, (2006) · Zbl 1165.34048 · doi:10.1016/j.chaos.2005.08.167 |
| [16] | Tikhonov, A, Systems of differential equations containing a small parameter multiplying the derivative, Mat Sb, 31, 575-586, (1952) · Zbl 0048.07101 |
| [17] | Yu, W; Cao, J, Stability and Hopf bifurcation analysis on four-neuron BAM neural network with time delays, Phys Lett, 351, 64-78, (2006) · Zbl 1234.34047 · doi:10.1016/j.physleta.2005.10.056 |
| [18] | Zharov, MI; Mishchenko, EF; Rozov, NKh, On some special functions and constants arising in the theory of relaxation oscillations, Soviet Math Doklady, 24, 672-675, (1981) · Zbl 0504.34031 |
| [19] | Zheng, YG; Bao, LJ, Slowcfast dynamics of tri-neuron Hopfield neural network with two timescales, Commun Nonlinear Sci Numer Simulat, 19, 1591-1599, (2014) · Zbl 1457.92016 · doi:10.1016/j.cnsns.2013.09.001 |
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.
