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June  2015, 20(4): 1135-1154. doi: 10.3934/dcdsb.2015.20.1135

Global dynamics and traveling wave solutions of two predators-one prey models

1. 

Department of Mathematics, National Central University, Chung-Li 32001, Taiwan

2. 

College of Mathematics and Information Science, Wenzhou University, Wenzhou, 325035

3. 

Department of Mathematics and Information Science, Wenzhou University, Zhejiang Province, 325035

4. 

Department of Mathematics, Tamkang University, Tamsui, Taipei County 25137

Received  January 2014 Revised  June 2014 Published  February 2015

In this work, we consider an ecological system of three species with two predators-one prey type without or with diffusion. For the system without diffusion, i.e. a system of three ODEs, we clarify global dynamics of all equilibria and find an exact condition to guarantee the existence and global asymptotic stability of the positive equilibrium. However, when the corresponding condition does not hold, the prey becomes extinct due to the over exploitation. On the other hand, for the system with diffusion, using the cross iteration method we find the minimum speed $c_*$. The existence of traveling wave front connecting the trivial solution and the coexistence state with some sufficient conditions is verified if the wave speed is large than $c_*$ and we also prove the nonexistence of such solutions if the wave speed is less than $c_*$. Finally, numerical simulations of system without or with diffusion are implemented and biological meanings are discussed.
Citation: Jian-Jhong Lin, Weiming Wang, Caidi Zhao, Ting-Hui Yang. Global dynamics and traveling wave solutions of two predators-one prey models. Discrete & Continuous Dynamical Systems - B, 2015, 20 (4) : 1135-1154. doi: 10.3934/dcdsb.2015.20.1135
References:
[1]

A. Boumenir and V. M. Nguyen, Perron theorem in the monotone iteration method for traveling waves in delayed reaction-diffusion equations,, Journal of Differential Equations, 244 (2008), 1551. doi: 10.1016/j.jde.2008.01.004. Google Scholar

[2]

C.-C. Chen, L.-C. Hung, M. Mimura and D. Ueyama, Exact travelling wave solutions of three-species competition-diffusion systems,, Discrete and Continuous Dynamical Systems - B, 17 (2012), 2653. doi: 10.3934/dcdsb.2012.17.2653. Google Scholar

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Y. Du and R. Xu, Traveling wave solutions in a three-species food-chain model with diffusion and delays,, International Journal of Biomathematics, 5 (2012). doi: 10.1142/S1793524511001350. Google Scholar

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N. J. Ford, P. M. Lumb and E. Ekaka-a, Mathematical modelling of plant species interactions in a harsh climate,, Journal of Computational and Applied Mathematics, 234 (2010), 2732. doi: 10.1016/j.cam.2010.01.025. Google Scholar

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C.-H. Hsu, J.-J. Lin and T.-H. Yang, Travelling wave solutions for Kolmogorov-type delayed lattice reaction-diffusion systems,, IMA Journal of Applied Mathematics, 28 (2014). doi: 10.1093/imamat/hxu054. Google Scholar

[7]

J. Huang, G. Lu and S. G. Ruan, Traveling wave solutions in delayed lattice differential equations with partial monotonicity,, Nonlinear Analysis. Theory, 60 (2005), 1331. doi: 10.1016/j.na.2004.10.020. Google Scholar

[8]

J. Huang and X. Zou, Traveling wavefronts in diffusive and cooperative Lotka-Volterra system with delays,, Journal of Mathematical Analysis and Applications, 271 (2002), 455. doi: 10.1016/S0022-247X(02)00135-X. Google Scholar

[9]

J. Huang and X. Zou, Existence of traveling wavefronts of delayed reaction diffusion systems without monotonicity,, Discrete and Continuous Dynamical Systems, 9 (2003), 925. doi: 10.3934/dcds.2003.9.925. Google Scholar

[10]

Y.-L. Huang and G. Lin, Traveling wave solutions in a diffusive system with two preys and one predator,, Journal of Mathematical Analysis and Applications, 418 (2014), 163. doi: 10.1016/j.jmaa.2014.03.085. Google Scholar

[11]

L.-C. Hung, Traveling wave solutions of competitive-cooperative Lotka-Volterra systems of three species,, Nonlinear Analysis: Real World Applications, 12 (2011), 3691. doi: 10.1016/j.nonrwa.2011.07.002. Google Scholar

[12]

N. Krikorian, The Volterra model for three species predator-prey systems: Boundedness and stability,, Journal of Mathematical Biology, 7 (1979), 117. doi: 10.1007/BF00276925. Google Scholar

[13]

K. Q. Lan and J. H. Wu, Travelling wavefronts of scalar reaction-diffusion equations with and without delays,, Nonlinear Analysis. Real World Applications, 4 (2003), 173. doi: 10.1016/S1468-1218(02)00020-2. Google Scholar

[14]

W.-T. Li, G. Lin and S. Ruan, Existence of travelling wave solutions in delayed reaction-diffusion systems with applications to diffusion-competition systems,, Nonlinearity, 19 (2006), 1253. doi: 10.1088/0951-7715/19/6/003. Google Scholar

[15]

G. Lin, W.-T. Li and M. Ma, Traveling wave solutions in delayed reaction diffusion systems with applications to multi-species models,, Discrete and Continuous Dynamical Systems - B, 13 (2010), 393. doi: 10.3934/dcdsb.2010.13.393. Google Scholar

[16]

G. Lv and M. Wang, Traveling wave front in diffusive and competitive Lotka-Volterra system with delays,, Nonlinear Analysis : Real World Applications, 11 (2010), 1323. doi: 10.1016/j.nonrwa.2009.02.020. Google Scholar

[17]

S. Ma and J. Wu, Existence, uniqueness and asymptotic stability of traveling wavefronts in a non-local delayed diffusion equation,, Journal of Dynamics and Differential Equations, 19 (2007), 391. doi: 10.1007/s10884-006-9065-7. Google Scholar

[18]

R. McGehee and R. A. Armstrong, Some mathematical problems concerning the ecological principle of competitive exclusion,, Journal of Differential Equations, 23 (1977), 30. doi: 10.1016/0022-0396(77)90135-8. Google Scholar

[19]

T. Namba and K. Tanabe, Omnivory and stability of food webs,, Ecological Complexity, 5 (2008), 73. doi: 10.1016/j.ecocom.2008.02.001. Google Scholar

[20]

E. R. Pianka, On $r$- and $K$-selection,, American Naturalist, 104 (1970), 592. doi: 10.1086/282697. Google Scholar

[21]

K. W. Schaaf, Asymptotic behavior and traveling wave solutions for parabolic functional-differential equations,, Transactions of the American Mathematical Society, 302 (1987), 587. doi: 10.2307/2000859. Google Scholar

[22]

J. W.-H. So and X. Zou, Traveling waves for the diffusive Nicholson's blowflies equation,, Applied Mathematics and Computation, 122 (2001), 385. doi: 10.1016/S0096-3003(00)00055-2. Google Scholar

[23]

D. Tilman, Resource Competition and Community Structure,, Princeton University Press, (1982). Google Scholar

[24]

E. Trofimchuk, M. Pinto and S. Trofimchuk, Traveling waves for a model of the Belousov-Zhabotinsky reaction,, Journal of Differential Equations, 254 (2013), 3690. doi: 10.1016/j.jde.2013.02.005. Google Scholar

[25]

J. Wu and X. Zou, Asymptotic and periodic boundary value problems of mixed FDEs and wave solutions of lattice differential equations,, Journal of Differential Equations, 135 (1997), 315. doi: 10.1006/jdeq.1996.3232. Google Scholar

[26]

J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with delay,, Journal of Dynamics and Differential Equations, 13 (2001), 651. doi: 10.1023/A:1016690424892. Google Scholar

[27]

Z.-X. Yu and R. Yuan, Traveling waves of delayed reaction-diffusion systems with applications,, Nonlinear Analysis: Real World Applications, 12 (2011), 2475. doi: 10.1016/j.nonrwa.2011.02.005. Google Scholar

[28]

X. Zou and J. Wu, Existence of traveling wave fronts in delayed reaction-diffusion systems via the monotone iteration method,, Proceedings of the American Mathematical Society, 125 (1997), 2589. doi: 10.1090/S0002-9939-97-04080-X. Google Scholar

show all references

References:
[1]

A. Boumenir and V. M. Nguyen, Perron theorem in the monotone iteration method for traveling waves in delayed reaction-diffusion equations,, Journal of Differential Equations, 244 (2008), 1551. doi: 10.1016/j.jde.2008.01.004. Google Scholar

[2]

C.-C. Chen, L.-C. Hung, M. Mimura and D. Ueyama, Exact travelling wave solutions of three-species competition-diffusion systems,, Discrete and Continuous Dynamical Systems - B, 17 (2012), 2653. doi: 10.3934/dcdsb.2012.17.2653. Google Scholar

[3]

Y. Du and R. Xu, Traveling wave solutions in a three-species food-chain model with diffusion and delays,, International Journal of Biomathematics, 5 (2012). doi: 10.1142/S1793524511001350. Google Scholar

[4]

N. J. Ford, P. M. Lumb and E. Ekaka-a, Mathematical modelling of plant species interactions in a harsh climate,, Journal of Computational and Applied Mathematics, 234 (2010), 2732. doi: 10.1016/j.cam.2010.01.025. Google Scholar

[5]

J. E. Guyer, D. Wheeler and J. A. Warren, FiPy: Partial Differential Equations with Python,, Computing in Science & Engineering, 11 (2009), 6. doi: 10.1109/MCSE.2009.52. Google Scholar

[6]

C.-H. Hsu, J.-J. Lin and T.-H. Yang, Travelling wave solutions for Kolmogorov-type delayed lattice reaction-diffusion systems,, IMA Journal of Applied Mathematics, 28 (2014). doi: 10.1093/imamat/hxu054. Google Scholar

[7]

J. Huang, G. Lu and S. G. Ruan, Traveling wave solutions in delayed lattice differential equations with partial monotonicity,, Nonlinear Analysis. Theory, 60 (2005), 1331. doi: 10.1016/j.na.2004.10.020. Google Scholar

[8]

J. Huang and X. Zou, Traveling wavefronts in diffusive and cooperative Lotka-Volterra system with delays,, Journal of Mathematical Analysis and Applications, 271 (2002), 455. doi: 10.1016/S0022-247X(02)00135-X. Google Scholar

[9]

J. Huang and X. Zou, Existence of traveling wavefronts of delayed reaction diffusion systems without monotonicity,, Discrete and Continuous Dynamical Systems, 9 (2003), 925. doi: 10.3934/dcds.2003.9.925. Google Scholar

[10]

Y.-L. Huang and G. Lin, Traveling wave solutions in a diffusive system with two preys and one predator,, Journal of Mathematical Analysis and Applications, 418 (2014), 163. doi: 10.1016/j.jmaa.2014.03.085. Google Scholar

[11]

L.-C. Hung, Traveling wave solutions of competitive-cooperative Lotka-Volterra systems of three species,, Nonlinear Analysis: Real World Applications, 12 (2011), 3691. doi: 10.1016/j.nonrwa.2011.07.002. Google Scholar

[12]

N. Krikorian, The Volterra model for three species predator-prey systems: Boundedness and stability,, Journal of Mathematical Biology, 7 (1979), 117. doi: 10.1007/BF00276925. Google Scholar

[13]

K. Q. Lan and J. H. Wu, Travelling wavefronts of scalar reaction-diffusion equations with and without delays,, Nonlinear Analysis. Real World Applications, 4 (2003), 173. doi: 10.1016/S1468-1218(02)00020-2. Google Scholar

[14]

W.-T. Li, G. Lin and S. Ruan, Existence of travelling wave solutions in delayed reaction-diffusion systems with applications to diffusion-competition systems,, Nonlinearity, 19 (2006), 1253. doi: 10.1088/0951-7715/19/6/003. Google Scholar

[15]

G. Lin, W.-T. Li and M. Ma, Traveling wave solutions in delayed reaction diffusion systems with applications to multi-species models,, Discrete and Continuous Dynamical Systems - B, 13 (2010), 393. doi: 10.3934/dcdsb.2010.13.393. Google Scholar

[16]

G. Lv and M. Wang, Traveling wave front in diffusive and competitive Lotka-Volterra system with delays,, Nonlinear Analysis : Real World Applications, 11 (2010), 1323. doi: 10.1016/j.nonrwa.2009.02.020. Google Scholar

[17]

S. Ma and J. Wu, Existence, uniqueness and asymptotic stability of traveling wavefronts in a non-local delayed diffusion equation,, Journal of Dynamics and Differential Equations, 19 (2007), 391. doi: 10.1007/s10884-006-9065-7. Google Scholar

[18]

R. McGehee and R. A. Armstrong, Some mathematical problems concerning the ecological principle of competitive exclusion,, Journal of Differential Equations, 23 (1977), 30. doi: 10.1016/0022-0396(77)90135-8. Google Scholar

[19]

T. Namba and K. Tanabe, Omnivory and stability of food webs,, Ecological Complexity, 5 (2008), 73. doi: 10.1016/j.ecocom.2008.02.001. Google Scholar

[20]

E. R. Pianka, On $r$- and $K$-selection,, American Naturalist, 104 (1970), 592. doi: 10.1086/282697. Google Scholar

[21]

K. W. Schaaf, Asymptotic behavior and traveling wave solutions for parabolic functional-differential equations,, Transactions of the American Mathematical Society, 302 (1987), 587. doi: 10.2307/2000859. Google Scholar

[22]

J. W.-H. So and X. Zou, Traveling waves for the diffusive Nicholson's blowflies equation,, Applied Mathematics and Computation, 122 (2001), 385. doi: 10.1016/S0096-3003(00)00055-2. Google Scholar

[23]

D. Tilman, Resource Competition and Community Structure,, Princeton University Press, (1982). Google Scholar

[24]

E. Trofimchuk, M. Pinto and S. Trofimchuk, Traveling waves for a model of the Belousov-Zhabotinsky reaction,, Journal of Differential Equations, 254 (2013), 3690. doi: 10.1016/j.jde.2013.02.005. Google Scholar

[25]

J. Wu and X. Zou, Asymptotic and periodic boundary value problems of mixed FDEs and wave solutions of lattice differential equations,, Journal of Differential Equations, 135 (1997), 315. doi: 10.1006/jdeq.1996.3232. Google Scholar

[26]

J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with delay,, Journal of Dynamics and Differential Equations, 13 (2001), 651. doi: 10.1023/A:1016690424892. Google Scholar

[27]

Z.-X. Yu and R. Yuan, Traveling waves of delayed reaction-diffusion systems with applications,, Nonlinear Analysis: Real World Applications, 12 (2011), 2475. doi: 10.1016/j.nonrwa.2011.02.005. Google Scholar

[28]

X. Zou and J. Wu, Existence of traveling wave fronts in delayed reaction-diffusion systems via the monotone iteration method,, Proceedings of the American Mathematical Society, 125 (1997), 2589. doi: 10.1090/S0002-9939-97-04080-X. Google Scholar

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