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More traveling waves in the Holling-Tanner model with weak diffusion

  • * Corresponding author: Vahagn Manukian

    * Corresponding author: Vahagn Manukian 
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  • We identify two new traveling waves of the Holling-Tanner model with weak diffusion. One connects two constant states; at one of them, the model is undefined. The other connects a constant state to a periodic wave train. We exploit the multi-scale structure of the Holling-Tanner model in the weak diffusion limit. Our analysis uses geometric singular perturbation theory, compactification and the blow-up method.

    Mathematics Subject Classification: Primary: 35B25, 35B32, 35K57; Secondary: 35B36, 92D25.


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  • Figure 2.1.  Equilibria of (2.8): (a) $ 0<\delta<1 $. If $ (\alpha,\beta) $ is in $ \mathcal{R}_2 $ and $ 0<\delta<\delta_h<1-\alpha $, then $ \tilde A $ is an attractor. Otherwise $ \tilde A $ is a repeller. (b) $ \delta>1 $. $ \tilde A $ is a repeller.

    Figure 3.1.  Equilibria and positive closed orbits of (2.8) in two cases. (a) A repelling relaxation oscillation for small $ \delta>0 $. (b) Two closed orbits with $ \delta_h<\delta<\delta_t $ in the case of a supercritical Hopf bifurcation.

    Figure 3.2.  The flow in the quadrant $ X\ge0, \; Y\ge0 $ of the Poincaré sphere when positive closed orbits are present, in which case we must have $ \beta\delta<2 $. The flow inside the outermost closed orbit is not shown since it can vary.

    Figure 3.3.  The flow near the degenerate equilibrium $ (0,0) $ of (3.4) in polar coordinates when $ \beta\delta<2 $. So that the reader can more easily compare this figure with Figure 3.2, the circle $ r = 0 $ is shown upside down, with the point $ (\bar x, \bar y) = (0,1) $ at the bottom of the circle. With some abuse of notation, the equilibria are labeled $ E_1 $ and $ E_2 $ to correspond to the equilibria in the two affine coordinate systems

    Figure A.1.  (a) $ \gamma_0 $ and $ \mathcal{K} $. The disk $ \mathcal{D} $ is shaded. (b) $ \gamma_\epsilon $ and $ \mathcal{K} $.

    Figure B.1.  The flow in the quadrant $ X\ge0, \; Y\ge0 $ of the Poincaré sphere when $ \beta\delta>2 $

    Figure B.2.  The flow near the degenerate equilibrium $ (0,0) $ of (3.4) in polar coordinates when $ \beta\delta>2 $. Compare Figure 3.3.

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