September  2022, 27(9): 4875-4890. doi: 10.3934/dcdsb.2021256

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

1. 

Department of Mathematics, Miami University, Hamilton, OH 45011, USA

2. 

Department of Mathematics, North Carolina State University, Raleigh, NC 27695, USA

* Corresponding author: Vahagn Manukian

Received  May 2020 Revised  April 2021 Published  September 2022 Early access  November 2021

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.

Citation: Vahagn Manukian, Stephen Schecter. More traveling waves in the Holling-Tanner model with weak diffusion. Discrete and Continuous Dynamical Systems - B, 2022, 27 (9) : 4875-4890. doi: 10.3934/dcdsb.2021256
References:
[1]

S. AiY. 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. GasullR. 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. LiW. 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.

show all references

References:
[1]

S. AiY. 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. GasullR. 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. LiW. 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.

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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