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.
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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 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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Equilibria of (2.8): (a)
Equilibria and positive closed orbits of (2.8) in two cases. (a) A repelling relaxation oscillation for small
The flow in the quadrant
The flow near the degenerate equilibrium
(a)
The flow in the quadrant
The flow near the degenerate equilibrium