Article Contents
Article Contents

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

• * Corresponding author: Vahagn Manukian
• 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.

 Citation:

• 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.

•  [1] S. Ai, Y. Du and R. Peng, Traveling waves for a generalized Holling-Tanner predator-prey model, J. Differential Equations, 263 (2017), 7782-7814.  doi: 10.1016/j.jde.2017.08.021. [2] H. Cai, A. Ghazaryan and V. Manukian, Fisher-KPP dynamics in diffusive Rosenzweig-MacArthur and Holling-Tanner models, Math. Model. Nat. Phenom., 14 (2019), Art. 404, 21 pp. doi: 10.1051/mmnp/2019017. [3] C. Chicone, Ordinary Differential Equations with Applications, 2$^{nd}$ edition, Springer Texts in Applied Mathematics, 34, Springer, New York, 2006. [4] A. Ducrot, Z. Liu and P. Magal, Large speed traveling waves for the Rosenzweig-MacArthur predator-prey model with spatial diffusion, Phys. D, 415 (2021), 132730, 14 pp. doi: 10.1016/j.physd.2020.132730. [5] N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Differential Eqs., 31 (1979), 53-98.  doi: 10.1016/0022-0396(79)90152-9. [6] A. Gasull, R. E Kooij and J. Torregrosa, Limit cycles in the Holling-Tanner model, Publ. Mat., 41 (1997), 149-167.  doi: 10.5565/PUBLMAT_41197_09. [7] A. Ghazaryan, V. Manukian and S. Schecter, Traveling waves in the Holling-Tanner model with weak diffusion, Proc. A., 471 (2015), 20150045, 16 pp. doi: 10.1098/rspa.2015.0045. [8] C. S. Holling, The characteristics of simple types of predation and parasitism, Canadian Entomologist, 91 (1959), 293-320. [9] S.-B. Hsu and T.-W. Huang, Global stability for a class of predator-prey systems, SIAM J. Appl. Math, 55 (1995), 763-783.  doi: 10.1137/S0036139993253201. [10] S.-B. Hsu and T.-W. Hwang, Uniqueness of limit cycles for a predator-prey system of Holling and Leslie type, Canad. Appl. Math. Quart., 6 (1998), 91-117. [11] S.-B. Hsu and T.-W. Hwang, Hopf bifurcation analysis for a predator-prey system of Holling and Leslie type, Taiwanese J. Math., 3 (1999), 35-53.  doi: 10.11650/twjm/1500407053. [12] C. K. R. T. Jones, Geometric singular perturbation theory, Dynamical Systems (Montecatini Terme, 1994), Lecture Notes in Math., 1609, Springer, (1995), 44–118. doi: 10.1007/BFb0095239. [13] C. Kuehn, Multiple Time Scale Dynamics, Springer, New York, 2015. doi: 10.1007/978-3-319-12316-5. [14] X. Li, W. Jiang and J. Shi, Hopf bifurcation and Turing instability in the reaction-diffusion Holling-Tanner predator-prey model, IMA J. Appl. Math., 78 (2013), 287-306.  doi: 10.1093/imamat/hxr050. [15] R. M. May, On relationships among various types of population models, American Naturalist, 107 (1973), 46-57.  doi: 10.1086/282816. [16] R. M. May,  Stability and Complexity in Model Ecosystems, Princeton University Press, Princeton, 1974. [17] J. D. Murray, Mathematical Biology, Springer, Berlin, 1989. doi: 10.1007/978-3-662-08539-4. [18] L. Perko, Differential Equations and Dynamical Systems, Third edition. Texts in Applied Mathematics, 7, Springer, New York, 2001. doi: 10.1007/978-1-4613-0003-8. [19] E. Renshaw,  Modelling Biological Populations in Space and Time, Cambridge University Press, Cambridge, 1991.  doi: 10.1017/CBO9780511624094. [20] E. Sáez and E. González-Olivares, Dynamics of a predator-prey model, SIAM J. Appl. Math., 59 (1999), 1867-1878.  doi: 10.1137/S0036139997318457. [21] J. A. Sherratt and M. J. Smith, Periodic travelling waves in cyclic populations: Field studies and reaction-diffusion models, J. R. Soc. Interface, 5 (2008), 483-505.  doi: 10.1098/rsif.2007.1327. [22] J. T. Tanner, The stability and the intrinsic growth rates of prey and predator populations, Ecology, 56 (1975), 855-867.  doi: 10.2307/1936296.

Figures(7)