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Dynamical analysis arising from the Willamowski-Rössler model. (English) Zbl 1502.34060

Authors consider the Willamowski-Rössler model \begin{align*} & \dot{x}={{\alpha }_{1}}x\left( 1-x-y-z \right)-{{\alpha }_{-1}}{{x}^{2}}-xy-xz, \\ & \dot{y}=xy-{{\alpha }_{3}}y, \\ & \dot{z}=-xz+{{\alpha }_{5}}z\left( 1-x-y-z \right)-{{\alpha }_{-5}}{{z}^{2}} \\ \end{align*} with initial conditions \(x(0)\ge 0\), \(y(0)\ge 0\), and \(z(0)\ge 0\), where the parameters \({{\alpha }_{1}}\), \({{\alpha }_{-1}}\), \({{\alpha }_{3}}\), \({{\alpha }_{5}}\), and \({{\alpha }_{-5}}\) all lie in the interval \((0, 1)\). The authors investigate existence, local stability and the global dynamics of boundary equilibria. Sufficient conditions for the uniform persistence of the model are given. In addition, a transversality condition for the Hopf bifurcation is obtained, which makes it possible to establish the existence of periodic solutions of the system.

MSC:

34C60 Qualitative investigation and simulation of ordinary differential equation models
92C45 Kinetics in biochemical problems (pharmacokinetics, enzyme kinetics, etc.)
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34D20 Stability of solutions to ordinary differential equations
34D05 Asymptotic properties of solutions to ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
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