October  2017, 37(10): 5433-5454. doi: 10.3934/dcds.2017236

Existence and stability of traveling waves for Leslie-Gower predator-prey system with nonlocal diffusion

1. 

School of Mathematics and Statistics, Shandong Normal University, Jinan 250014, China

2. 

School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China

* Corresponding author: Hongmei Cheng

Received  June 2016 Revised  June 2017 Published  June 2017

This paper will mainly study the information about the existence and stability of the invasion traveling waves for the nonlocal Leslie-Gower predator-prey model. By using an invariant cone in a bounded domain with initial function being defined on and applying the Schauder's fixed point theorem, we can obtain the existence of traveling waves. Here, the compactness of the support set of dispersal kernel is needed when passing to an unbounded domain in the proof. Then we use the weighted energy to prove that the invasion traveling waves are exponentially stable as perturbation in some exponentially as $x\to-\infty $. Finally, by defining the bilateral Laplace transform, we can obtain the nonexistence of the traveling waves.

Citation: Hongmei Cheng, Rong Yuan. Existence and stability of traveling waves for Leslie-Gower predator-prey system with nonlocal diffusion. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5433-5454. doi: 10.3934/dcds.2017236
References:
[1]

M. Aziz-Alaoui and M. D. Okiye, Boundedness and global stability for a predator-prey model with modified Leslie-Gower and Holling-type Ⅱ schemes, Appl. Math. Lett., 16 (2003), 1069-1075. doi: 10.1016/S0893-9659(03)90096-6. Google Scholar

[2]

P. W. BatesP. C. FifeX. Ren and X. Wang, Traveling waves in a convolution model for phase transitions, Arch. Rational Mech. Anal., 138 (1997), 105-136. doi: 10.1007/s002050050037. Google Scholar

[3]

J. Carr and A. Chmaj, Uniqueness of travelling waves for nonlocal monostable equations, Proc. Amer. Math. Soc., 132 (2004), 2433-2439. doi: 10.1090/S0002-9939-04-07432-5. Google Scholar

[4]

X. Chen, Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations, Adv. Differential Equations, 2 (1997), 125-160. Google Scholar

[5]

H. Cheng and R. Yuan, The spreading property for a prey-predator reaction-diffusion system with fractional diffusion, Frac. Calc. Appl. Anal., 18 (2015), 565-579. doi: 10.1515/fca-2015-0035. Google Scholar

[6]

C. Conley and R. Gardner, An application of the generalized Morse index to travelling wave solutions of a competitive reaction-diffusion model, Indiana Univ. Math. J., 33 (1984), 319-343. doi: 10.1512/iumj.1984.33.33018. Google Scholar

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J. Coville, On uniqueness and monotonicity of solutions of non-local reaction diffusion equation, Ann. Mat. Pura Appl., 185 (2006), 461-485. doi: 10.1007/s10231-005-0163-7. Google Scholar

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J. CovilleJ. Dávila and S. Martínez, Nonlocal anisotropic dispersal with monostable nonlinearity, J. Differential Equations, 244 (2008), 3080-3118. doi: 10.1016/j.jde.2007.11.002. Google Scholar

[9]

J. Coville and L. Dupaigne, Propagation speed of travelling fronts in non local reaction-diffusion equations, Nonlinear Anal., 60 (2005), 797-819. doi: 10.1016/j.na.2003.10.030. Google Scholar

[10]

J. Coville and L. Dupaigne, On a non-local equation arising in population dynamics, Proc. Roy. Soc. Edinburgh Sect. A, 137 (2007), 727-755. doi: 10.1017/S0308210504000721. Google Scholar

[11]

A. Ducrot, Convergence to generalized transition waves for some Holling-Tanner prey-predator reaction-diffusion system, J. Math. Pures Appl., 100 (2013), 1-15. doi: 10.1016/j.matpur.2012.10.009. Google Scholar

[12]

R. A. Gardner, Existence and stability of travelling wave solutions of competition models: A degree theoretic approach, J. Differential Equations, 44 (1982), 343-364. doi: 10.1016/0022-0396(82)90001-8. Google Scholar

[13]

C. S. Holling, The functional response of predators to prey density and its role in mimicry and population regulation, Mem. Entomol. Soci. Can., 97 (1965), 5-60. doi: 10.4039/entm9745fv. Google Scholar

[14]

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. Google Scholar

[15]

J. Huang and X. Zou, Traveling wavefronts in diffusive and cooperative Lotka-Volterra system with delays, J. Math. Anal. Appl., 271 (2002), 455-466. doi: 10.1016/S0022-247X(02)00135-X. Google Scholar

[16]

V. Hutson and M. Grinfeld, Non-local dispersal and bistability, European J. Appl. Math., 17 (2006), 221-232. doi: 10.1017/S0956792506006462. Google Scholar

[17]

V. HutsonW. Shen and G. Vickers, Spectral theory for nonlocal dispersal with periodic or almost-periodic time dependence, Rocky Mountain J. Math., 38 (2008), 1147-1175. doi: 10.1216/RMJ-2008-38-4-1147. Google Scholar

[18]

J. Kanel and L. Zhou, Existence of wave front solutions and estimates of wave speed for a competition-diffusion system, Nonlinear Anal., 27 (1996), 579-587. doi: 10.1016/0362-546X(95)00221-G. Google Scholar

[19]

A. Korobeinikov, A Lyapunov function for Leslie-Gower predator-prey models, Appl. Math. Lett., 14 (2001), 697-699. doi: 10.1016/S0893-9659(01)80029-X. Google Scholar

[20]

P. Leslie and J. Gower, The properties of a stochastic model for the predator-prey type of interaction between two species, Biometrika, 47 (1960), 219-234. doi: 10.1093/biomet/47.3-4.219. Google Scholar

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R. H. Martin and H. L. Smith, Abstract functional-differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44. doi: 10.2307/2001590. Google Scholar

[22]

R. M. May, Stability and Complexity in Model Ecosystems Princeton University Press, 1973.Google Scholar

[23]

M. Mei and J. W. H. So, Stability of strong travelling waves for a non-local time-delayed reaction-diffusion equation, Proc. Roy. Soc. Edinburgh Sect. A, 138 (2008), 551-568. doi: 10.1017/S0308210506000333. Google Scholar

[24]

M. MeiJ. W. H. SoM. Y. Li and S. S. P. Shen, Asymptotic stability of travelling waves for Nicholson's blowflies equation with diffusion, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 579-594. doi: 10.1017/S0308210500003358. Google Scholar

[25]

J. D. Murray, Mathematical Biology. II Spatial Models and Biomedical Applications Interdisciplinary Applied Mathematics V. 18 Springer-Verlag, New York, 2003. Google Scholar

[26]

A. NindjinM. Aziz-Alaoui and M. Cadivel, Analysis of a predator-prey model with modified Leslie-Gower and Holling-type Ⅱ schemes with time delay, Nonlinear Anal. Real World Appl., 7 (2006), 1104-1118. doi: 10.1016/j.nonrwa.2005.10.003. Google Scholar

[27]

S. Pan, Traveling wave fronts of delayed non-local diffusion systems without quasimonotonicity, J. Math. Anal. Appl., 346 (2008), 415-424. doi: 10.1016/j.jmaa.2008.05.057. Google Scholar

[28]

S. PanW.-T. Li and G. Lin, Travelling wave fronts in nonlocal delayed reaction-diffusion systems and applications, Z. Angew. Math. Phys., 60 (2009), 377-392. doi: 10.1007/s00033-007-7005-y. Google Scholar

[29]

Y. Saito, The necessary and sufficient condition for global stability of a Lotka-Volterra cooperative or competition system with delays, J. Math. Anal. Appl., 268 (2002), 109-124. doi: 10.1006/jmaa.2001.7801. Google Scholar

[30]

Y. SaitoT. Hara and W. Ma, Necessary and sufficient conditions for permanence and global stability of a Lotka-Volterra system with two delays, J. Math. Anal. Appl., 236 (1999), 534-556. doi: 10.1006/jmaa.1999.6464. Google Scholar

[31]

M. M. Tang and P. C. Fife, Propagating fronts for competing species equations with diffusion, Arch. Rational Mech. Anal., 73 (1980), 69-77. doi: 10.1007/BF00283257. Google Scholar

[32]

J. T. Tanner, The stability and the intrinsic growth rates of prey and predator populations, Ecology, 50 (1975), 855-867. doi: 10.2307/1936296. Google Scholar

[33]

J. D. Van Der Waals, On the Continuity of the Gaseous and Liquid States, Translated from the Dutch, Edited and with an introduction by J. S. Rowlinson. Studies in Statistical Mechanics, 1988. Google Scholar

[34]

J. H. Van Vuuren, The existence of travelling plane waves in a general class of competition-diffusion systems, IMA J. Appl. Math., 55 (1995), 135-148. doi: 10.1093/imamat/55.2.135. Google Scholar

[35]

Z.-C. WangW.-T. Li and S. Ruan, Traveling fronts in monostable equations with nonlocal delayed effects, J. Dynam. Differential Equations, 20 (2008), 573-607. doi: 10.1007/s10884-008-9103-8. Google Scholar

[36]

D. V. Widder, The Laplace Transform Princeton University Press, Princeton, NJ, 1941. doi: 10.1142/9781848161016_0007. Google Scholar

[37]

R. YafiaF. El Adnani and H. T. Alaoui, Stability of limit cycle in a predator-prey model with modified Leslie-Gower and Holling-type Ⅱ schemes with time delay, Appl. Math. Sci, 1 (2007), 119-131. Google Scholar

[38]

H. Yagisita, Existence and nonexistence of traveling waves for a nonlocal monostable equation, Publ. Res. Inst. Math. Sci., 45 (2009), 925-953. doi: 10.2977/prims/1260476648. Google Scholar

[39]

H. Yagisita, Existence of traveling waves for a nonlocal monostable equation: An abstract approach, Publ. Res. Inst. Math. Sci., 45 (2009), 955-979. doi: 10.2977/prims/1260476649. Google Scholar

[40]

Z. X. Yu and R. Yuan, Traveling wave solutions in nonlocal reaction-diffusion systems with delays and applications, ANZIAM J., 51 (2009), 49-66. doi: 10.1017/S1446181109000406. Google Scholar

show all references

References:
[1]

M. Aziz-Alaoui and M. D. Okiye, Boundedness and global stability for a predator-prey model with modified Leslie-Gower and Holling-type Ⅱ schemes, Appl. Math. Lett., 16 (2003), 1069-1075. doi: 10.1016/S0893-9659(03)90096-6. Google Scholar

[2]

P. W. BatesP. C. FifeX. Ren and X. Wang, Traveling waves in a convolution model for phase transitions, Arch. Rational Mech. Anal., 138 (1997), 105-136. doi: 10.1007/s002050050037. Google Scholar

[3]

J. Carr and A. Chmaj, Uniqueness of travelling waves for nonlocal monostable equations, Proc. Amer. Math. Soc., 132 (2004), 2433-2439. doi: 10.1090/S0002-9939-04-07432-5. Google Scholar

[4]

X. Chen, Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations, Adv. Differential Equations, 2 (1997), 125-160. Google Scholar

[5]

H. Cheng and R. Yuan, The spreading property for a prey-predator reaction-diffusion system with fractional diffusion, Frac. Calc. Appl. Anal., 18 (2015), 565-579. doi: 10.1515/fca-2015-0035. Google Scholar

[6]

C. Conley and R. Gardner, An application of the generalized Morse index to travelling wave solutions of a competitive reaction-diffusion model, Indiana Univ. Math. J., 33 (1984), 319-343. doi: 10.1512/iumj.1984.33.33018. Google Scholar

[7]

J. Coville, On uniqueness and monotonicity of solutions of non-local reaction diffusion equation, Ann. Mat. Pura Appl., 185 (2006), 461-485. doi: 10.1007/s10231-005-0163-7. Google Scholar

[8]

J. CovilleJ. Dávila and S. Martínez, Nonlocal anisotropic dispersal with monostable nonlinearity, J. Differential Equations, 244 (2008), 3080-3118. doi: 10.1016/j.jde.2007.11.002. Google Scholar

[9]

J. Coville and L. Dupaigne, Propagation speed of travelling fronts in non local reaction-diffusion equations, Nonlinear Anal., 60 (2005), 797-819. doi: 10.1016/j.na.2003.10.030. Google Scholar

[10]

J. Coville and L. Dupaigne, On a non-local equation arising in population dynamics, Proc. Roy. Soc. Edinburgh Sect. A, 137 (2007), 727-755. doi: 10.1017/S0308210504000721. Google Scholar

[11]

A. Ducrot, Convergence to generalized transition waves for some Holling-Tanner prey-predator reaction-diffusion system, J. Math. Pures Appl., 100 (2013), 1-15. doi: 10.1016/j.matpur.2012.10.009. Google Scholar

[12]

R. A. Gardner, Existence and stability of travelling wave solutions of competition models: A degree theoretic approach, J. Differential Equations, 44 (1982), 343-364. doi: 10.1016/0022-0396(82)90001-8. Google Scholar

[13]

C. S. Holling, The functional response of predators to prey density and its role in mimicry and population regulation, Mem. Entomol. Soci. Can., 97 (1965), 5-60. doi: 10.4039/entm9745fv. Google Scholar

[14]

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. Google Scholar

[15]

J. Huang and X. Zou, Traveling wavefronts in diffusive and cooperative Lotka-Volterra system with delays, J. Math. Anal. Appl., 271 (2002), 455-466. doi: 10.1016/S0022-247X(02)00135-X. Google Scholar

[16]

V. Hutson and M. Grinfeld, Non-local dispersal and bistability, European J. Appl. Math., 17 (2006), 221-232. doi: 10.1017/S0956792506006462. Google Scholar

[17]

V. HutsonW. Shen and G. Vickers, Spectral theory for nonlocal dispersal with periodic or almost-periodic time dependence, Rocky Mountain J. Math., 38 (2008), 1147-1175. doi: 10.1216/RMJ-2008-38-4-1147. Google Scholar

[18]

J. Kanel and L. Zhou, Existence of wave front solutions and estimates of wave speed for a competition-diffusion system, Nonlinear Anal., 27 (1996), 579-587. doi: 10.1016/0362-546X(95)00221-G. Google Scholar

[19]

A. Korobeinikov, A Lyapunov function for Leslie-Gower predator-prey models, Appl. Math. Lett., 14 (2001), 697-699. doi: 10.1016/S0893-9659(01)80029-X. Google Scholar

[20]

P. Leslie and J. Gower, The properties of a stochastic model for the predator-prey type of interaction between two species, Biometrika, 47 (1960), 219-234. doi: 10.1093/biomet/47.3-4.219. Google Scholar

[21]

R. H. Martin and H. L. Smith, Abstract functional-differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44. doi: 10.2307/2001590. Google Scholar

[22]

R. M. May, Stability and Complexity in Model Ecosystems Princeton University Press, 1973.Google Scholar

[23]

M. Mei and J. W. H. So, Stability of strong travelling waves for a non-local time-delayed reaction-diffusion equation, Proc. Roy. Soc. Edinburgh Sect. A, 138 (2008), 551-568. doi: 10.1017/S0308210506000333. Google Scholar

[24]

M. MeiJ. W. H. SoM. Y. Li and S. S. P. Shen, Asymptotic stability of travelling waves for Nicholson's blowflies equation with diffusion, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 579-594. doi: 10.1017/S0308210500003358. Google Scholar

[25]

J. D. Murray, Mathematical Biology. II Spatial Models and Biomedical Applications Interdisciplinary Applied Mathematics V. 18 Springer-Verlag, New York, 2003. Google Scholar

[26]

A. NindjinM. Aziz-Alaoui and M. Cadivel, Analysis of a predator-prey model with modified Leslie-Gower and Holling-type Ⅱ schemes with time delay, Nonlinear Anal. Real World Appl., 7 (2006), 1104-1118. doi: 10.1016/j.nonrwa.2005.10.003. Google Scholar

[27]

S. Pan, Traveling wave fronts of delayed non-local diffusion systems without quasimonotonicity, J. Math. Anal. Appl., 346 (2008), 415-424. doi: 10.1016/j.jmaa.2008.05.057. Google Scholar

[28]

S. PanW.-T. Li and G. Lin, Travelling wave fronts in nonlocal delayed reaction-diffusion systems and applications, Z. Angew. Math. Phys., 60 (2009), 377-392. doi: 10.1007/s00033-007-7005-y. Google Scholar

[29]

Y. Saito, The necessary and sufficient condition for global stability of a Lotka-Volterra cooperative or competition system with delays, J. Math. Anal. Appl., 268 (2002), 109-124. doi: 10.1006/jmaa.2001.7801. Google Scholar

[30]

Y. SaitoT. Hara and W. Ma, Necessary and sufficient conditions for permanence and global stability of a Lotka-Volterra system with two delays, J. Math. Anal. Appl., 236 (1999), 534-556. doi: 10.1006/jmaa.1999.6464. Google Scholar

[31]

M. M. Tang and P. C. Fife, Propagating fronts for competing species equations with diffusion, Arch. Rational Mech. Anal., 73 (1980), 69-77. doi: 10.1007/BF00283257. Google Scholar

[32]

J. T. Tanner, The stability and the intrinsic growth rates of prey and predator populations, Ecology, 50 (1975), 855-867. doi: 10.2307/1936296. Google Scholar

[33]

J. D. Van Der Waals, On the Continuity of the Gaseous and Liquid States, Translated from the Dutch, Edited and with an introduction by J. S. Rowlinson. Studies in Statistical Mechanics, 1988. Google Scholar

[34]

J. H. Van Vuuren, The existence of travelling plane waves in a general class of competition-diffusion systems, IMA J. Appl. Math., 55 (1995), 135-148. doi: 10.1093/imamat/55.2.135. Google Scholar

[35]

Z.-C. WangW.-T. Li and S. Ruan, Traveling fronts in monostable equations with nonlocal delayed effects, J. Dynam. Differential Equations, 20 (2008), 573-607. doi: 10.1007/s10884-008-9103-8. Google Scholar

[36]

D. V. Widder, The Laplace Transform Princeton University Press, Princeton, NJ, 1941. doi: 10.1142/9781848161016_0007. Google Scholar

[37]

R. YafiaF. El Adnani and H. T. Alaoui, Stability of limit cycle in a predator-prey model with modified Leslie-Gower and Holling-type Ⅱ schemes with time delay, Appl. Math. Sci, 1 (2007), 119-131. Google Scholar

[38]

H. Yagisita, Existence and nonexistence of traveling waves for a nonlocal monostable equation, Publ. Res. Inst. Math. Sci., 45 (2009), 925-953. doi: 10.2977/prims/1260476648. Google Scholar

[39]

H. Yagisita, Existence of traveling waves for a nonlocal monostable equation: An abstract approach, Publ. Res. Inst. Math. Sci., 45 (2009), 955-979. doi: 10.2977/prims/1260476649. Google Scholar

[40]

Z. X. Yu and R. Yuan, Traveling wave solutions in nonlocal reaction-diffusion systems with delays and applications, ANZIAM J., 51 (2009), 49-66. doi: 10.1017/S1446181109000406. Google Scholar

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