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 and Continuous Dynamical Systems, 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.

[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.

[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.

[4]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[16]

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

[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.

[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.

[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.

[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.

[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.

[22]

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

[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.

[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.

[25]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[36]

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

[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. 

[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.

[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.

[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.

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.

[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.

[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.

[4]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[16]

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

[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.

[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.

[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.

[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.

[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.

[22]

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

[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.

[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.

[25]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[36]

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

[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. 

[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.

[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.

[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.

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