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January  2015, 20(1): 161-171. doi: 10.3934/dcdsb.2015.20.161

Traveling waves for the Lotka-Volterra predator-prey system without diffusion of the predator

Yuzo Hosono 1,

1. 

Department of Mathematics, Kyoto Sangyo University, Kyoto 603, Japan

Received  November 2013 Revised  July 2014 Published  November 2014

This paper investigates the existence of traveling waves and their propagation speeds for the Lotka-Volterra predator-prey reaction-diffusion models with no predator diffusion. We prove the existence of traveling waves with any positive speed. Our mathematical tool is the shooting argument in the phase space based on the Wazewski theorem.
Keywords: the Wazewski theorem., propagation speed, Traveling waves, the Lotka-Volterra predator-prey system.
Mathematics Subject Classification: Primary: 35C07, 35K57; Secondary: 92D2.
Citation: Yuzo Hosono. Traveling waves for the Lotka-Volterra predator-prey system without diffusion of the predator. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 161-171. doi: 10.3934/dcdsb.2015.20.161
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.  Google Scholar

[2]

S. R. Dunbar, Travelling wave solutions of diffusive Lotka-Volterra equations, J. Math. Biology, 17 (1983), 11-32. doi: 10.1007/BF00276112.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

P. Hartman, Ordinary Differential Equations, Wiley and Sons, Inc., New York-London-Sydney, 1964.  Google Scholar

[9]

R. Hengeveld, Dynamics of Biological Invasions, Chapman and Hall, London, 1989. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

J. D. Murray, Mathematical Biology, Springer Verlag, Heiderberg, 1989. doi: 10.1007/978-3-662-08539-4.  Google Scholar

[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.  Google Scholar

[16]

N. Shigesada and K. Kawasaki, Biological Invasions: Theory and Practice, Oxford University Press, Oxford, 1997. Google Scholar

[17]

J. Smaller, Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[18]

A. I. Volpert, Vitaly A. Volpert and Vladimir A. Volpert, Travelling Wave Solution of Parabolic Systems, American Mathematical Society, Rhode Island, 1994.  Google Scholar

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.  Google Scholar

[2]

S. R. Dunbar, Travelling wave solutions of diffusive Lotka-Volterra equations, J. Math. Biology, 17 (1983), 11-32. doi: 10.1007/BF00276112.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

P. Hartman, Ordinary Differential Equations, Wiley and Sons, Inc., New York-London-Sydney, 1964.  Google Scholar

[9]

R. Hengeveld, Dynamics of Biological Invasions, Chapman and Hall, London, 1989. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

J. D. Murray, Mathematical Biology, Springer Verlag, Heiderberg, 1989. doi: 10.1007/978-3-662-08539-4.  Google Scholar

[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.  Google Scholar

[16]

N. Shigesada and K. Kawasaki, Biological Invasions: Theory and Practice, Oxford University Press, Oxford, 1997. Google Scholar

[17]

J. Smaller, Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[18]

A. I. Volpert, Vitaly A. Volpert and Vladimir A. Volpert, Travelling Wave Solution of Parabolic Systems, American Mathematical Society, Rhode Island, 1994.  Google Scholar

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