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