Slowfast dynamics of Hopfield sprucebudworm model with memory effects
 Na Wang^{1, 2}Email author and
 Maoan Han^{2}
https://doi.org/10.1186/s1366201608048
© Wang and Han 2016
Received: 3 August 2015
Accepted: 7 March 2016
Published: 14 March 2016
Abstract
In this paper we consider a kind of the sprucebudworm system with memory effects. On the basis of geometric singular perturbation theory, the transition of the solution trajectory is illustrated, and the existence of the relaxation oscillation with a rapid movement process alternating with a slow movement process is proved. The characteristic of the relaxation oscillation, it is indicated, is dependent on the structure of the slow manifold. Moreover, the approximate expression of the relaxation oscillation and its period are obtained analytically. Finally we present two simulations to demonstrate the validity of the analytical conclusion.
Keywords
MSC
1 Introduction
To reproduce the dynamics of the sprucebudworm system, the Hopfield sprucebudworm model has been proposed in the work of Rasmusse et al. [1] who showed for a sprucebudworm model that the predator and prey permanently oscillate for any positive initial conditions. Such kinds of sprucebudworm systems exhibit periodic outbreaks at about 40 year intervals in NorthAmerican forests. During outbreaks, populations of the budworm may multiply hundredfold in a few months, causing severe defoliation. Trees are seldom killed by defoliation, and they may live for 100150 years, but it takes them 710 years to fully regrow their leaves.
As pointed out in [4, 5], the existence of such relaxation oscillations implies that the coexistence of predators and prey occurs through a simple periodically alternated twoseason behavior: a poor season, characterized by an almost endemic presence of the prey, alternates with a rich season, during which prey are abundant and predators are regenerated.
If the weight function in model (1.2) is given by (1.3), we have the advantage that it specifies the moment in the past when the quantity of the prey is most important from the point of view of the present growth of the prey. This occurs \(\frac{1}{a}\) units before the present time t (the weight function has a hump at \(z=t\frac{1}{a}\), and going further backwards in time the effect of the past is fading away), the phenomenon is richer. And when the delay kernel function \(W(s)\) takes the form of the ‘strong’ generic kernel function, it is unknown how the delay kernel \(W(s)=a^{2}s e^{a s}\) (\(a>0\)) (i.e., the parameter a) affects the dynamics of system (1.2). Therefore, in this paper, we study mainly the effects of the parameter a in the ‘strong’ delay kernel on the dynamical behaviors of system (1.2).
The present paper is organized as follows. In Section 2, by means of a change of variables, we first transform system (1.2) with the ‘strong’ delay kernel into a fourdimensional system of differential equations. In Section 3, by linearizing the resulting fourdimensional system at the positive equilibrium and analyzing the associated characteristic equation, the Hopf bifurcations are demonstrated. In particular, by applying geometric singular perturbation theory, the approximate expression of the relaxation oscillation and its period are obtained analytically. To verify our theoretical predictions, two numerical simulations are also included in Section 4.
As argued by Ludwig et al. [2], we assume that the predator population in our model has fast dynamics (the predator population grows much faster than those of the predator), i.e., \(\rho\ll r\).
2 The model equation
Remark 1
The model with the delay kernel \(W(s)\) is very hard to analyze. So authors use many methods to eliminate delay. By defining new variables and using the linear chain trick technique, the original model (1.2) can be rewritten as the equivalent ordinary differential equations (2.2) without delay. But the price is that the dimension of the equations would be increased from two to four. Although the model becomes the fourdimensional system (2.2), the variables U and V of the systems do not play a major role. Therefore, we only need to analyze the first two equations of (2.2).
3 Slowfast dynamics
Geometric singular perturbation theory, mainly due to Tikhonov and Fenichel [18–21], is an efficient tool for investigating the slowfast dynamics of the systems with two timescales. According to the geometric singular perturbation theory, the dynamical behavior of the two timescale systems is governed by the structure of the slow manifold including the shape, stability, and bifurcation of the slow manifold, as well as the location and stability of the equilibrium of the two timescale systems.
When \(\varrho<\frac{2a}{X_{0}}\) and \(\varrho X_{0}(22 X_{0}a)+\frac{a Y_{0}}{Y_{\max}}>0\), one has \(b<0\) and \(d<0\), which leads to \(\lambda_{\varepsilon}^{i}<0 \) (\(i=1,2,3,4\)) due to \(\lambda_{\varepsilon}^{1}+\lambda_{\varepsilon}^{2} +\lambda_{\varepsilon}^{3}+\lambda_{\varepsilon}^{4}=b<0\) and \(\lambda_{\varepsilon}^{1}\lambda_{\varepsilon}^{2}(\lambda _{\varepsilon}^{3}+\lambda_{\varepsilon}^{4}) +\lambda_{\varepsilon}^{3}\lambda_{\varepsilon}^{4}(\lambda _{\varepsilon}^{1}+\lambda_{\varepsilon}^{2})=d<0\).
When \(\varrho>\frac{2a}{X_{0}}\) or \(\varrho X_{0}(22 X_{0}a)+\frac{a Y_{0}}{Y_{\max}}<0\), one has \(b>0\) or \(d>0\), which leads to the sign of \(\lambda _{\varepsilon}^{i} \) (\(i=1,2,3,4\)) being either \((,,+,+)\) or \((+,+,+,+)\).
Thus, when the equilibrium point \((Z_{0},Y_{0},V_{0},U_{0})\in M_{2}\) and \(\varrho>\frac{2a}{X_{0}}\) or \(\varrho X_{0}(22 X_{0}a)+\frac{a Y_{0}}{Y_{\max}}<0\), then the equilibrium point located in \(M_{2}\) is unstable. When the equilibrium point \((Z_{0},Y_{0},V_{0},U_{0})\in M_{1}\cup M_{3}\), \(\varrho<\frac{2a}{X_{0}}\), and \(\varrho X_{0}(22 X_{0}a)+\frac{a Y_{0}}{Y_{\max}}>0\), then the equilibrium point located in \(M_{1}\cup M_{3}\) is stable.
Once the structure of the slow manifold M and the position and stability of the equilibriums of equation (2.6) are obtained, the dynamical behaviors of equation (2.6) can be analyzed through the geometric singular perturbation theory.
Conditions \(\varrho<\frac{1}{X_{2}}\) and \(Y_{\max}\varrho>\frac {Y_{2}Y_{1}}{X_{2}X_{1}}\) guarantee that there is an equilibrium point in the slow manifold \(M_{1}\) and \(M_{3}\), respectively, as shown in Figure 3(b). When \(\frac{1}{X_{2}}<\varrho<\frac{1}{X_{1}}\) and \(Y_{\max}\varrho <\frac{Y_{1}}{X_{2}X_{1}}\), there are no equilibrium points in the slow manifold \(M_{1}\cup M_{3}\), as shown in Figure 1. According to the geometric singular perturbation theory, the solution trajectory will be attracted by the stable manifold and repelled by the unstable manifold.
Remark 2
The original model (1.1) in [1] is a 2D system. The Jacobian J of the vector field defining the original system is given by secondorder matrices. The authors used \(\operatorname{tr}(J)\) and \(\operatorname{det}(J)\) to judge on the positive and negative of eigenvalues. However, the corresponding characteristic equation of our model is a quartic equation. Thus, the positive and negative judgment of eigenvalues is more complicated. Through the analysis of the characteristic equation, the positive and negative of characteristic roots were obtained.
Remark 3
Reference [1] used the attraction domain of the upper and lower stable branch of the quasisteady state (16) to judge on the clockwise direction of the relaxation oscillation. In this paper, we use the sign of \(\frac{dY}{d\tau}\) to judge on the clockwise direction of the relaxation oscillation. This method is simpler and clearer in comparison.
In summary, the main results of this paper can be stated as follows.
Proposition
 (H_{1}):

\(\max\{\frac{1}{X_{2}},\frac{2a}{X_{2}},\frac {2a}{X_{1}}\}<\varrho<\frac{1}{X_{1}}\), \(Y_{\max}\varrho<\frac{Y_{1}}{X_{2}X_{1}}\);
 (H_{2}):

\(\frac{1}{X_{2}}<\varrho<\frac{1}{X_{1}}\), \(Y_{\max}\varrho<\frac{Y_{1}}{X_{2}X_{1}}\), and \(\varrho X_{i}(22 X_{i}a)+\frac{a Y_{i}}{Y_{\max}}<0\) (\(i=1,2\)),
Remark 4
Conditions (H_{1}) and (H_{2}) are incompatible.
4 Numerical example
To demonstrate the validity of the analytical results obtained in the previous sections, two specific examples are studied in this section.
Example 1
From equation (3.6), one obtains the approximate period of the relaxation oscillation \(T_{\mathrm{appr}}=1.4124 \), which agrees with the numerical result \(T=1.76242\).
Example 2
Remark 5
Though the two phase planes look very similar for the two cases, they represent two different cases which satisfy the incompatible conditions (H_{1}) and (H_{2}).
5 Summary
In this paper the geometric singular perturbation theory is employed to illuminate the transition of the solution trajectory. The existence of the relaxation oscillation is also proved. Its validity is illustrated by two examples. The whole study is complemented with direct numerical simulations of the dimensionless sprucebudworm model (2.6).
Remark 6
By using the qualitative result (see the proposition), the existence of the relaxation oscillation of this kind of predatorprey system with distributed delay can be determined quickly. The figures of the solution which is obtained by mathematical software can be explained as the periodical change of ecological population, such as the period of pests outbreaks. After understanding the periodic system, we can control the pests outbreaks possibly by adjusting the model’s parameters.
Remark 7
The studies indicate that this kind of predatorprey system with distributed delay can exhibit relaxation oscillation, which shows that the Hopfield predatorprey system has the potential to reproduce the complex dynamics of a real predatorprey system.
Declarations
Acknowledgements
The authors are grateful to the editor and two referees for a number of helpful suggestions, which have greatly improved our original manuscript. This research is supported by National Science Foundation of China (No: 11401385).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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