# American Institute of Mathematical Sciences

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

 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.

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.

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##### 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.
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 $\varepsilon = 0.001$, $\varrho = 5$, $Y_{max} = 20$, $A = 1$
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: $\varepsilon = 0.001$, $\varrho = 5$, $Y_{\max } = 20$, $A = 1$
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)
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)
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))
In this figure we show the decomposition of the orbit in the calculation of the period of relaxation oscillation
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: $\alpha ^{2} = 0.0013$, $\varepsilon = 0.001$, $\varrho = 5$, $Y_{\max } = 20$, $A = 1$
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: $\alpha ^{2} = 0.0013$, $\varepsilon = 0.001$, $\varrho = 5$, $Y_{\max } = 20$, $A = 1$
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: $\alpha ^{2} = 0.0013$, $\varrho = 5$, $Y_{\max } = 20$, $A = 1$
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: $\alpha ^{2} = 0.0013$, $\varrho = 5$, $Y_{\max } = 20$, $A = 1$
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 $A\rightarrow\infty$, $Y_{cri}\rightarrow 18.2349$
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: $\alpha ^{2} = 0.0013$
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
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)
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)
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
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
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
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 $Y_{\max } = 20$
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
 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
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