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

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

Vahagn Manukian 1,, and  Stephen Schecter 2,

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.

Keywords: Geometric singular perturbation theory, traveling wave, compactification, prey-predator model, diffusive Holling-Tanner model, blow-up.
Mathematics Subject Classification: Primary: 35B25, 35B32, 35K57; Secondary: 35B36, 92D25.
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. 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.

show all references

References:
[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.

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.
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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.
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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.
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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
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Figure A.1.  (a) $ \gamma_0 $ and $ \mathcal{K} $. The disk $ \mathcal{D} $ is shaded. (b) $ \gamma_\epsilon $ and $ \mathcal{K} $.
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Figure B.1.  The flow in the quadrant $ X\ge0, \; Y\ge0 $ of the Poincaré sphere when $ \beta\delta>2 $
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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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