Relaxation oscillations in spruce-budworm interactions
The authors consider a mathematical model for the spruce-budworm interaction. A normalized form of this prey-predator system reads \[ \begin{aligned}\varepsilon {dZ\over d\tau} &= Y\Biggl[{Z\over Y}\Biggl(1-{Z\over Y}\Biggr)- {1\over Y} {Z^2\over \alpha^2 Y^2+ Z^2}\Biggr],\\ {dY\over d\tau} &= Y\Biggl[1- {Y\over Y_{\max}}- \rho{Z\over Y}\Biggr],\end{aligned}\tag{\(*\)} \] where \(\alpha^2\), \(\rho\) and \(Y_{\max}\) are positive parameters, \(0< \varepsilon\ll 1\). Firstly, number and stability of the equilibria in the positive orthant are investigated. For the main part of their investigations, the authors assume \(0<\alpha^2<{1\over 27}\) and that \((*)\) has a unique positive equilibrium. Using the analytical approach for relaxation oscillation due to \textit{E. F. Mishchenko} and \textit{N. Kh. Rozov} [Differential equations with small parameters and relaxation oscillations. New York, London: Plenum Press (1980; Zbl 0482.34004)], they derive an asymptotic expression for the period of the relaxation oscillation. Numerical simulations are presented.
scientific article; zbMATH DE number 5826296
scientific article