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Traveling waves for the Lotka-Volterra predator-prey system without diffusion of the predator
1. | Department of Mathematics, Kyoto Sangyo University, Kyoto 603, Japan |
References:
[1] |
P. L. Chow and W. C. Tam, Periodic and traveling waves solutions to Lotka-Volterra equations with diffusion, Bull. Math. Biol., 38 (1976), 643-658.
doi: 10.1007/BF02458639. |
[2] |
S. R. Dunbar, Travelling wave solutions of diffusive Lotka-Volterra equations, J. Math. Biology, 17 (1983), 11-32.
doi: 10.1007/BF00276112. |
[3] |
S. R. Dunbar, Traveling wave solutions of diffusive Lotka-Volterra equations: A heteroclinic connection in $ R^4$, Trans. American Math. Society, 286 (1984), 557-594.
doi: 10.2307/1999810. |
[4] |
S. R. Dunbar, Traveling waves in diffusive predator-prey equations: Periodic orbits and point-to-periodic heteroclinic orbits, SIAM. J. Appl. Math., 46 (1986), 1057-1078.
doi: 10.1137/0146063. |
[5] |
C. S. Elton, The Ecology of Invasions by Animals and Plants, Methuen and Company, London, 1958.
doi: 10.1007/978-94-009-5851-7. |
[6] |
W. F. Fagan and J. Bishop, Trophic interactions during primary succession: Herbivores slow a plant reinvasion at Mount St. Helens, Amer. Nat., 155 (2000), 238-251.
doi: 10.1086/303320. |
[7] |
R. A. Fisher, The wave of advance of advantageous genes, Ann. Eugenice, 7 (1937), 335-369.
doi: 10.1111/j.1469-1809.1937.tb02153.x. |
[8] |
P. Hartman, Ordinary Differential Equations, Wiley and Sons, Inc., New York-London-Sydney, 1964. |
[9] |
R. Hengeveld, Dynamics of Biological Invasions, Chapman and Hall, London, 1989. |
[10] |
C.-H. Hsu, C.-R. Yang, T.-H. Yang and T.-S. Yang, Existence of traveling wave solutions for diffusive predator-prey type systems, J. Differential Equations, 252 (2012), 3040-3075.
doi: 10.1016/j.jde.2011.11.008. |
[11] |
J. Huang, G. Lu and S. Ruan, Existence of traveling wave solutions in a diffusive predator-prey model, J. Math. Biol., 46 (2003), 132-152.
doi: 10.1007/s00285-002-0171-9. |
[12] |
W.-T. Li and S.-L. Wu, Traveling waves in a diffusive predator-prey model with Holling type-III functional response, Chaos, Solitons and Fractals, 37 (2008), 476-486.
doi: 10.1016/j.chaos.2006.09.039. |
[13] |
S. Ma, Traveling wavefronts for delayed reaction-diffusion systems via a fixed point theorem, J. Differential Equations, 171 (2001), 294-314.
doi: 10.1006/jdeq.2000.3846. |
[14] |
J. D. Murray, Mathematical Biology, Springer Verlag, Heiderberg, 1989.
doi: 10.1007/978-3-662-08539-4. |
[15] |
M. R. Owen and M. A. Lewis, How predation can slow, stop or reverse a prey invasion, Bulletin of Mathematical Biology, 63 (2001), 655-684.
doi: 10.1006/bulm.2001.0239. |
[16] |
N. Shigesada and K. Kawasaki, Biological Invasions: Theory and Practice, Oxford University Press, Oxford, 1997. |
[17] |
J. Smaller, Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, New York, 1994.
doi: 10.1007/978-1-4612-0873-0. |
[18] |
A. I. Volpert, Vitaly A. Volpert and Vladimir A. Volpert, Travelling Wave Solution of Parabolic Systems, American Mathematical Society, Rhode Island, 1994. |
show all references
References:
[1] |
P. L. Chow and W. C. Tam, Periodic and traveling waves solutions to Lotka-Volterra equations with diffusion, Bull. Math. Biol., 38 (1976), 643-658.
doi: 10.1007/BF02458639. |
[2] |
S. R. Dunbar, Travelling wave solutions of diffusive Lotka-Volterra equations, J. Math. Biology, 17 (1983), 11-32.
doi: 10.1007/BF00276112. |
[3] |
S. R. Dunbar, Traveling wave solutions of diffusive Lotka-Volterra equations: A heteroclinic connection in $ R^4$, Trans. American Math. Society, 286 (1984), 557-594.
doi: 10.2307/1999810. |
[4] |
S. R. Dunbar, Traveling waves in diffusive predator-prey equations: Periodic orbits and point-to-periodic heteroclinic orbits, SIAM. J. Appl. Math., 46 (1986), 1057-1078.
doi: 10.1137/0146063. |
[5] |
C. S. Elton, The Ecology of Invasions by Animals and Plants, Methuen and Company, London, 1958.
doi: 10.1007/978-94-009-5851-7. |
[6] |
W. F. Fagan and J. Bishop, Trophic interactions during primary succession: Herbivores slow a plant reinvasion at Mount St. Helens, Amer. Nat., 155 (2000), 238-251.
doi: 10.1086/303320. |
[7] |
R. A. Fisher, The wave of advance of advantageous genes, Ann. Eugenice, 7 (1937), 335-369.
doi: 10.1111/j.1469-1809.1937.tb02153.x. |
[8] |
P. Hartman, Ordinary Differential Equations, Wiley and Sons, Inc., New York-London-Sydney, 1964. |
[9] |
R. Hengeveld, Dynamics of Biological Invasions, Chapman and Hall, London, 1989. |
[10] |
C.-H. Hsu, C.-R. Yang, T.-H. Yang and T.-S. Yang, Existence of traveling wave solutions for diffusive predator-prey type systems, J. Differential Equations, 252 (2012), 3040-3075.
doi: 10.1016/j.jde.2011.11.008. |
[11] |
J. Huang, G. Lu and S. Ruan, Existence of traveling wave solutions in a diffusive predator-prey model, J. Math. Biol., 46 (2003), 132-152.
doi: 10.1007/s00285-002-0171-9. |
[12] |
W.-T. Li and S.-L. Wu, Traveling waves in a diffusive predator-prey model with Holling type-III functional response, Chaos, Solitons and Fractals, 37 (2008), 476-486.
doi: 10.1016/j.chaos.2006.09.039. |
[13] |
S. Ma, Traveling wavefronts for delayed reaction-diffusion systems via a fixed point theorem, J. Differential Equations, 171 (2001), 294-314.
doi: 10.1006/jdeq.2000.3846. |
[14] |
J. D. Murray, Mathematical Biology, Springer Verlag, Heiderberg, 1989.
doi: 10.1007/978-3-662-08539-4. |
[15] |
M. R. Owen and M. A. Lewis, How predation can slow, stop or reverse a prey invasion, Bulletin of Mathematical Biology, 63 (2001), 655-684.
doi: 10.1006/bulm.2001.0239. |
[16] |
N. Shigesada and K. Kawasaki, Biological Invasions: Theory and Practice, Oxford University Press, Oxford, 1997. |
[17] |
J. Smaller, Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, New York, 1994.
doi: 10.1007/978-1-4612-0873-0. |
[18] |
A. I. Volpert, Vitaly A. Volpert and Vladimir A. Volpert, Travelling Wave Solution of Parabolic Systems, American Mathematical Society, Rhode Island, 1994. |
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