Relaxation oscillations in a spruce-budworm interaction model with Holling's type II functional response

Previous Article
Long-time dynamics of a diffusive epidemic model with free boundaries
DCDS-B Home
This Issue
Next Article
Carrying simplex in the Lotka-Volterra competition model with seasonal succession with applications

April 2021, 26(4): 2173-2199. doi: 10.3934/dcdsb.2021027

Relaxation oscillations in a spruce-budworm interaction model with Holling's type II functional response

Yi-Ming Tai ^1, and Zhengyang Zhang ^2,,

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

^* Corresponding author: Zhengyang Zhang

Dedicated to Prof. Sze-Bi Hsu in acknowledgement of his helpful suggestions

Received July 2020 Revised December 2020 Published April 2021 Early access January 2021

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.

Keywords: Spruce-budworm model, singular perturbation method, relaxation oscillations, Holling's type II functional response.

Mathematics Subject Classification: 34A34, 34C26, 34E15.

Citation: Yi-Ming Tai, Zhengyang Zhang. Relaxation oscillations in a spruce-budworm interaction model with Holling's type II functional response. Discrete and Continuous Dynamical Systems - B, 2021, 26 (4) : 2173-2199. doi: 10.3934/dcdsb.2021027

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.
[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.
[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.

Figure 1. In this figure we show the nullclines when $ \alpha ^2\geqslant 1/27 $. In this case there exists exactly one positive equilibrium. Other parameters are $ \varepsilon = 0.001 $, $ \varrho = 5 $, $ Y_{max} = 20 $, $ A = 1 $

Download full-size image

Download as PowerPoint slide

Figure 2. In this figure we show the nullclines when $ \alpha ^{2} = 0.0013\in (0,1/27) $ and the system of equations (5)-(6) has one positive solution. In this case there exists one positive equilibrium. Other parameters are: $ \varepsilon = 0.001 $, $ \varrho = 5 $, $ Y_{\max } = 20 $, $ A = 1 $

Download full-size image

Download as PowerPoint slide

Figure 3. In this figure we show three possible situations producing only one equilibrium in the strip $ \mathcal{S} $ when $ \alpha ^{2} = 0.0013\in (0,1/27) $. Other parameters are: $ \varepsilon = 0.001 $, $ \varrho = 5 $ and $ Y_{\max } = 10 $, $ A = 1 $ for (a), $ Y_{\max } = 20 $, $ A = 1 $ for (b), $ Y_{\max } = 55 $, $ A = 10 $ for (c)

Download full-size image

Download as PowerPoint slide

Figure 4. In this figure the stability of the quasi-steady state is shown (the tilde over $ X $ and $ Y $ is neglected here for a better look). The lower and higher branch (the blue part) of the quasi-steady state are uniformly asymptotically stable, while the intermediate branch (the pink part between $ X_{1} $ and $ X_{2} $) is unstable. The arrows indicate the domains of attraction of the stable branches. The pink curve segments $ \Gamma_{1}^{*} $, $ \Gamma_{2}^{*} $ and $ \Gamma_{3}^{*} $ indicate the common boundary between the attraction domains of the lower and higher branch (see Remark 3.3 in the following)

Download full-size image

Download as PowerPoint slide

Figure 5. In this figure we show the construction of the closed orbit. The point $ (X_{1}+\delta ,Y_{1}),\ \delta >0 $ is attracted to the higher stable branch of the quasi-steady state, while the point $ (X_{2}-\delta ,Y_{2}),\ \delta >0 $ is attracted to the lower stable branch (indicated in (a)). As $ \delta\rightarrow 0 $, a closed orbit emerges, which represents the relaxation oscillations (indicated in (b))

Download full-size image

Download as PowerPoint slide

Figure 6. In this figure we show the decomposition of the orbit in the calculation of the period of relaxation oscillation

Download full-size image

Download as PowerPoint slide

Figure 7. In this figure we show how we compute the period of relaxation oscillation from numerical simulations of the original system (2). The period is estimated by the difference of time between two maximal (or minimal) points. The parameters are: $ \alpha ^{2} = 0.0013 $, $ \varepsilon = 0.001 $, $ \varrho = 5 $, $ Y_{\max } = 20 $, $ A = 1 $

Download full-size image

Download as PowerPoint slide

Figure 8. In Figure (a) we plot the budworm abundance (the lower curve) and the leaf area abundance (the upper curve) during one oscillation period. In Figure (b) we plot the budworm abundance against leaf area abundance on the phase plane. The closed orbit is displayed in (b). The period is divided into 20 equidistant time intervals indicated by rings. The parameters are: $ \alpha ^{2} = 0.0013 $, $ \varepsilon = 0.001 $, $ \varrho = 5 $, $ Y_{\max } = 20 $, $ A = 1 $

Download full-size image

Download as PowerPoint slide

Figure 9. In this figure we show the numerically estimated period $ T_{num} $ as a function of $ \varepsilon $. $ T_{num} $ converges to $ T_{0} $ as $ \varepsilon\rightarrow 0 $. The parameters are: $ \alpha ^{2} = 0.0013 $, $ \varrho = 5 $, $ Y_{\max } = 20 $, $ A = 1 $

Download full-size image

Download as PowerPoint slide

Figure 10. In this figure we show the variation of the relaxation oscillations with common initial condition $ X(0) = 0.2 $, $ Y(0) = 8 $ and different $ \varepsilon $ values: 0.001, 0.0028, 0.0046, 0.0064, 0.0082 and 0.01. In Figure (a) the evolutions of the budworm and leaf area in time are displayed, while in Figure (b) the corresponding closed orbits in the phase plane are displayed. Notice that the period of oscillation increases with $ \varepsilon $. The other parameters are: $ \alpha ^{2} = 0.0013 $, $ \varrho = 5 $, $ Y_{\max } = 20 $, $ A = 1 $

Download full-size image

Download as PowerPoint slide

Figure 11. In this figure we plot the variation of $ Y_{cri} $ with different values of $ A $. The other parameters are: $ \alpha ^{2} = 0.0013 $. When $ A = 0 $, $ Y_{cri} = 15.2333 $. When $ A\rightarrow\infty $, $ Y_{cri}\rightarrow 18.2349 $

Download full-size image

Download as PowerPoint slide

Figure 12. In this figure we plot the nullclines such that the graph of the function $ X = \Theta (Y;\varrho ,Y_{\max },A) $ passes through the points $ (X_{1},Y_{1}) $ and $ (X_{2},Y_{2}) $. Five values of $ A $ are selected: $ A = $0, 1, 5, 20,500. The other parameters are: $ \alpha ^{2} = 0.0013 $

Download full-size image

Download as PowerPoint slide

Figure 13. In this figure we plot parameter $ \varrho $ versus parameter $ A $ for $ Y_{\max } = 16 $. The red line represents all the parameter pair $ (A,\varrho ) $ such that the nullclines intersect at $ (X_{1},Y_{1}) $. The green line represents all the parameter pair $ (A,\varrho ) $ such that the nullclines intersect at $ (X_{2},Y_{2}) $. Relaxation oscillations are possible when the parameter pair $ (A,\varrho ) $ takes values in the two triangular regions between the two straight lines

Download full-size image

Download as PowerPoint slide

Figure 14. In figure (a) we plot parameter $ \varrho $ versus parameter $ A $ for $ Y_{\max } = 20 $. The upper line represents all the parameter pair $ (A,\varrho ) $ such that the nullclines intersect at $ (X_{1},Y_{1}) $. The lower line represents all the parameter pair $ (A,\varrho ) $ such that the nullclines intersect at $ (X_{2},Y_{2}) $. Relaxation oscillations are possible when the parameter pair $ (A,\varrho ) $ takes values in the quadrilateral region between the two straight lines. In figure (b) we plot the asymptotic period $ T_{0} $ of relaxation oscillations as a function of $ \varrho $ for $ A = 0 $, corresponding to the pink part in (a)

Download full-size image

Download as PowerPoint slide

Figure 15. In this figure we plot parameter $ \varrho $ versus parameter $ A $ for $ Y_{\max } = 20,30,40,100 $. The representations of upper lines and lower lines are the same as in Figure 14-(a)

Download full-size image

Download as PowerPoint slide

Figure 16. In this figure we plot the asymptotic period $ T_{0} $ of relaxation oscillations as a function of $ \varrho $ when $ A = 0 $ and $ Y_{\max } = 20,30,40,100 $, corresponding to the colored part on the $ \varrho $-axis in Figure 15

Download full-size image

Download as PowerPoint slide

Figure 17. In this figure we plot parameter $ \varrho $ versus parameter $ A $ for $ Y_{\max } = 20 $. The representations of upper lines and lower lines are the same as in Figure 14-(a). The colored vertical straight line segments indicate the possible $ \varrho $ values for exhibiting relaxation oscillations for different $ A $ value

Download full-size image

Download as PowerPoint slide

Figure 18. In this figure we plot the asymptotic period $ T_{0} $ of relaxation oscillations as a function of $ \varrho $ when $ A = 0,1,5,10 $ and $ Y_{\max } = 20 $, corresponding to the colored vertical straight line segments in Figure 17

Download full-size image

Download as PowerPoint slide

Figure 19. In this figure we plot the asymptotic period $ T_{0} $ of relaxation oscillations as a function of $ A $ when $ \varrho = 2,4,6,10,20 $ and $ Y_{\max } = 20 $

Download full-size image

Download as PowerPoint slide

Table 1. The biological interpretations of the variables and parameters of model (1)

Variable/parameter	Biological interpretation
$ t $	Time
$ N $	Population density of the larvae
$ S $	Average leaf area of the spruce
$ r $	Intrinsic growth rate of budworm population
$ \rho $	Intrinsic growth rate of spruce leaf area
$ \kappa $	Coefficient measuring to which degree the leaves
	can accommodate the larvae
$ S_{\max } $	Carrying capacity of spruce leaf area
$ \beta $	Maximum consumption rate of budworms
	per budworm-predator per time
$ P $	Budworm population density at maximal predation pressure
$ \eta $	Regulating coefficient of the predation pressure
$ \delta $	Maximal rate of budworm predation pressure
	as spruce leaf area increases
$ K $	Spruce leaf area at half of the maximal predation pressure

Download as excel

Variable/parameter	Biological interpretation
$ t $	Time
$ N $	Population density of the larvae
$ S $	Average leaf area of the spruce
$ r $	Intrinsic growth rate of budworm population
$ \rho $	Intrinsic growth rate of spruce leaf area
$ \kappa $	Coefficient measuring to which degree the leaves
	can accommodate the larvae
$ S_{\max } $	Carrying capacity of spruce leaf area
$ \beta $	Maximum consumption rate of budworms
	per budworm-predator per time
$ P $	Budworm population density at maximal predation pressure
$ \eta $	Regulating coefficient of the predation pressure
$ \delta $	Maximal rate of budworm predation pressure
	as spruce leaf area increases
$ K $	Spruce leaf area at half of the maximal predation pressure

[1]	Qiumei Zhang, Daqing Jiang, Li Zu. The stability of a perturbed eco-epidemiological model with Holling type II functional response by white noise. Discrete and Continuous Dynamical Systems - B, 2015, 20 (1) : 295-321. doi: 10.3934/dcdsb.2015.20.295
[2]	Shuping Li, Weinian Zhang. Bifurcations of a discrete prey-predator model with Holling type II functional response. Discrete and Continuous Dynamical Systems - B, 2010, 14 (1) : 159-176. doi: 10.3934/dcdsb.2010.14.159
[3]	Jun Zhou, Chan-Gyun Kim, Junping Shi. Positive steady state solutions of a diffusive Leslie-Gower predator-prey model with Holling type II functional response and cross-diffusion. Discrete and Continuous Dynamical Systems, 2014, 34 (9) : 3875-3899. doi: 10.3934/dcds.2014.34.3875
[4]	Zengji Du, Xiao Chen, Zhaosheng Feng. Multiple positive periodic solutions to a predator-prey model with Leslie-Gower Holling-type II functional response and harvesting terms. Discrete and Continuous Dynamical Systems - S, 2014, 7 (6) : 1203-1214. doi: 10.3934/dcdss.2014.7.1203
[5]	Jian Zu, Wendi Wang, Bo Zu. Evolutionary dynamics of prey-predator systems with Holling type II functional response. Mathematical Biosciences & Engineering, 2007, 4 (2) : 221-237. doi: 10.3934/mbe.2007.4.221
[6]	Shanshan Chen, Junping Shi, Junjie Wei. The effect of delay on a diffusive predator-prey system with Holling Type-II predator functional response. Communications on Pure and Applied Analysis, 2013, 12 (1) : 481-501. doi: 10.3934/cpaa.2013.12.481
[7]	Kazuhiro Oeda. Positive steady states for a prey-predator cross-diffusion system with a protection zone and Holling type II functional response. Conference Publications, 2013, 2013 (special) : 597-603. doi: 10.3934/proc.2013.2013.597
[8]	Tzung-shin Yeh. S-shaped and broken s-shaped bifurcation curves for a multiparameter diffusive logistic problem with holling type-Ⅲ functional response. Communications on Pure and Applied Analysis, 2017, 16 (2) : 645-670. doi: 10.3934/cpaa.2017032
[9]	Kexin Wang. Influence of feedback controls on the global stability of a stochastic predator-prey model with Holling type Ⅱ response and infinite delays. Discrete and Continuous Dynamical Systems - B, 2020, 25 (5) : 1699-1714. doi: 10.3934/dcdsb.2019247
[10]	Zhifu Xie. Turing instability in a coupled predator-prey model with different Holling type functional responses. Discrete and Continuous Dynamical Systems - S, 2011, 4 (6) : 1621-1628. doi: 10.3934/dcdss.2011.4.1621
[11]	Sze-Bi Hsu, Tzy-Wei Hwang, Yang Kuang. Global dynamics of a Predator-Prey model with Hassell-Varley Type functional response. Discrete and Continuous Dynamical Systems - B, 2008, 10 (4) : 857-871. doi: 10.3934/dcdsb.2008.10.857
[12]	Zhijun Liu, Weidong Wang. Persistence and periodic solutions of a nonautonomous predator-prey diffusion with Holling III functional response and continuous delay. Discrete and Continuous Dynamical Systems - B, 2004, 4 (3) : 653-662. doi: 10.3934/dcdsb.2004.4.653
[13]	Chaoqun Huang, Nung Kwan Yip. Singular perturbation and bifurcation of diffuse transition layers in inhomogeneous media, part II. Networks and Heterogeneous Media, 2015, 10 (4) : 897-948. doi: 10.3934/nhm.2015.10.897
[14]	Sampurna Sengupta, Pritha Das, Debasis Mukherjee. Stochastic non-autonomous Holling type-Ⅲ prey-predator model with predator's intra-specific competition. Discrete and Continuous Dynamical Systems - B, 2018, 23 (8) : 3275-3296. doi: 10.3934/dcdsb.2018244
[15]	Goro Akagi. Doubly nonlinear evolution equations and Bean's critical-state model for type-II superconductivity. Conference Publications, 2005, 2005 (Special) : 30-39. doi: 10.3934/proc.2005.2005.30
[16]	Eduardo González-Olivares, Betsabé González-Yañez, Jaime Mena-Lorca, José D. Flores. Uniqueness of limit cycles and multiple attractors in a Gause-type predator-prey model with nonmonotonic functional response and Allee effect on prey. Mathematical Biosciences & Engineering, 2013, 10 (2) : 345-367. doi: 10.3934/mbe.2013.10.345
[17]	Willian Cintra, Carlos Alberto dos Santos, Jiazheng Zhou. Coexistence states of a Holling type II predator-prey system with self and cross-diffusion terms. Discrete and Continuous Dynamical Systems - B, 2022, 27 (7) : 3913-3931. doi: 10.3934/dcdsb.2021211
[18]	Ilona Gucwa, Peter Szmolyan. Geometric singular perturbation analysis of an autocatalator model. Discrete and Continuous Dynamical Systems - S, 2009, 2 (4) : 783-806. doi: 10.3934/dcdss.2009.2.783
[19]	Weiran Sun, Min Tang. A relaxation method for one dimensional traveling waves of singular and nonlocal equations. Discrete and Continuous Dynamical Systems - B, 2013, 18 (5) : 1459-1491. doi: 10.3934/dcdsb.2013.18.1459
[20]	Freddy Dumortier, Robert Roussarie. Bifurcation of relaxation oscillations in dimension two. Discrete and Continuous Dynamical Systems, 2007, 19 (4) : 631-674. doi: 10.3934/dcds.2007.19.631

2020 Impact Factor: 1.327

Tools

Metrics

Other articles
by authors

on AIMS
Yi-Ming Tai Zhengyang Zhang
on Google Scholar
Yi-Ming Tai Zhengyang Zhang

2020 Impact Factor: 1.327

Tools

Article outline

Figures and Tables

2020 Impact Factor: 1.327

Tools

Figures and Tables

2020 Impact Factor: 1.327

Tools

Metrics

Other articles
by authors

on AIMS
Yi-Ming Tai Zhengyang Zhang
on Google Scholar
Yi-Ming Tai Zhengyang Zhang

[Back to Top]