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Relaxation oscillations in a spruce-budworm interaction model with Holling's type II functional response
1. | Department of Mathematics, National Tsing Hua University, No. 101, Sec. 2, Kuang-Fu Road, Hsinchu 300, Taiwan |
2. | School of Mathematical Sciences, Ocean University of China, No. 238 Songling Road, Laoshan District, Qingdao, Shandong Province, China |
In this paper we study the spruce-budworm interaction model with Holling's type II functional response. The existence, number and stability of equilibria are studied. Moreover, we prove the existence of relaxation oscillations by using singular perturbation method and give an asymptotic expression of the period of relaxation oscillations. Finally, the parameter ranges which allow the relaxation oscillations in several scenarios are explored and displayed by conducting numerical simulations.
References:
[1] |
D. Ludwig, D. D. Jones and C. S. Holling, Qualitative analysis of insect outbreak systems: The spruce budworm and forest, J. Anim. Ecol., 47(1) (1978), 315-332. Google Scholar |
[2] |
E. F. Mishchenko and N. Kh. Rozov, Differential Equations with Small Parameters and Relaxation Oscillations, Mathematical Concepts and Methods in Science and Engineering, 13. Plenum Press, New York, 1980.
doi: 10.1007/978-1-4615-9047-7.![]() ![]() |
[3] |
J. Murray, Mathematical Biology, 2$^{nd}$ edition, Springer Verlag, Berlin Heidelberg, 1993.
doi: 10.1007/978-3-662-08542-4. |
[4] |
A. Rasmussen, J. Wyller and J. O. Vik,
Relaxation oscillations in spruce-budworm interactions, Nonlinear Anal. Real World Appl., 12 (2011), 304-319.
doi: 10.1016/j.nonrwa.2010.06.017. |
[5] |
T. Royama,
Population dynamics of the spruce budworm Choristoneura Fumiferana, Ecol. Monogr., 54 (1984), 429-462.
doi: 10.2307/1942595. |
[6] |
T. Royama, Analytical Population Dynamics, Springer, Dordrecht, 1992.
doi: 10.1007/978-94-011-2916-9. |
[7] |
T. Royama, W. E. MacKinnon, E. G. Kettela, N. E. Carter and L. K. Hartling,
Analysis of spruce budworm outbreak cycles in New Brunswick, Canada, since 1952, Ecology, 86 (2005), 1212-1224.
doi: 10.1890/03-4077. |
[8] |
N. Kh. Rozov,
Asymptotic calculation of nearly discontinuous solutions of a second-order system of differential equations, Dokl. Akad. Nauk SSSR, 145 (1962), 38-40.
|
[9] |
N. Wang and M. Han, Slow-fast dynamics of Hopfield spruce-budworm model with memory effects, Adv. Differ. Equ., 2016 (2016), Paper No. 73, 12 pp.
doi: 10.1186/s13662-016-0804-8. |
[10] |
M. I. Zharov, E. F. Mishchenko and N. Kh. Rozov,
Some special functions and constants that arise in the theory of relaxation oscillations (Russian), Dokl. Akad. Nauk SSSR, 261 (1981), 1292-1296.
|
show all references
References:
[1] |
D. Ludwig, D. D. Jones and C. S. Holling, Qualitative analysis of insect outbreak systems: The spruce budworm and forest, J. Anim. Ecol., 47(1) (1978), 315-332. Google Scholar |
[2] |
E. F. Mishchenko and N. Kh. Rozov, Differential Equations with Small Parameters and Relaxation Oscillations, Mathematical Concepts and Methods in Science and Engineering, 13. Plenum Press, New York, 1980.
doi: 10.1007/978-1-4615-9047-7.![]() ![]() |
[3] |
J. Murray, Mathematical Biology, 2$^{nd}$ edition, Springer Verlag, Berlin Heidelberg, 1993.
doi: 10.1007/978-3-662-08542-4. |
[4] |
A. Rasmussen, J. Wyller and J. O. Vik,
Relaxation oscillations in spruce-budworm interactions, Nonlinear Anal. Real World Appl., 12 (2011), 304-319.
doi: 10.1016/j.nonrwa.2010.06.017. |
[5] |
T. Royama,
Population dynamics of the spruce budworm Choristoneura Fumiferana, Ecol. Monogr., 54 (1984), 429-462.
doi: 10.2307/1942595. |
[6] |
T. Royama, Analytical Population Dynamics, Springer, Dordrecht, 1992.
doi: 10.1007/978-94-011-2916-9. |
[7] |
T. Royama, W. E. MacKinnon, E. G. Kettela, N. E. Carter and L. K. Hartling,
Analysis of spruce budworm outbreak cycles in New Brunswick, Canada, since 1952, Ecology, 86 (2005), 1212-1224.
doi: 10.1890/03-4077. |
[8] |
N. Kh. Rozov,
Asymptotic calculation of nearly discontinuous solutions of a second-order system of differential equations, Dokl. Akad. Nauk SSSR, 145 (1962), 38-40.
|
[9] |
N. Wang and M. Han, Slow-fast dynamics of Hopfield spruce-budworm model with memory effects, Adv. Differ. Equ., 2016 (2016), Paper No. 73, 12 pp.
doi: 10.1186/s13662-016-0804-8. |
[10] |
M. I. Zharov, E. F. Mishchenko and N. Kh. Rozov,
Some special functions and constants that arise in the theory of relaxation oscillations (Russian), Dokl. Akad. Nauk SSSR, 261 (1981), 1292-1296.
|



















Variable/parameter | Biological interpretation |
Time | |
Population density of the larvae | |
Average leaf area of the spruce | |
Intrinsic growth rate of budworm population | |
Intrinsic growth rate of spruce leaf area | |
Coefficient measuring to which degree the leaves | |
can accommodate the larvae | |
Carrying capacity of spruce leaf area | |
Maximum consumption rate of budworms | |
per budworm-predator per time | |
Budworm population density at maximal predation pressure | |
Regulating coefficient of the predation pressure | |
Maximal rate of budworm predation pressure | |
as spruce leaf area increases | |
Spruce leaf area at half of the maximal predation pressure |
Variable/parameter | Biological interpretation |
Time | |
Population density of the larvae | |
Average leaf area of the spruce | |
Intrinsic growth rate of budworm population | |
Intrinsic growth rate of spruce leaf area | |
Coefficient measuring to which degree the leaves | |
can accommodate the larvae | |
Carrying capacity of spruce leaf area | |
Maximum consumption rate of budworms | |
per budworm-predator per time | |
Budworm population density at maximal predation pressure | |
Regulating coefficient of the predation pressure | |
Maximal rate of budworm predation pressure | |
as spruce leaf area increases | |
Spruce leaf area at half of the maximal predation pressure |
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