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Boundedness and asymptotic behavior in a predator-prey model with indirect pursuit-evasion interaction
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 |
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.
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. |







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