May  2020, 25(5): 1959-1983. doi: 10.3934/dcdsb.2020011

Traveling waves for nonlocal Lotka-Volterra competition systems

†. 

School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu, 730000, China

§. 

School of Mathematics, Southwest Jiaotong University, Chengdu, Sichuan, 611756, China

‡. 

School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, Jiangsu, 221116, China

* Corresponding author: Zhi-Cheng Wang

Received  February 2019 Revised  May 2019 Published  December 2019

In this paper, we study the traveling wave solutions of a Lotka-Volterra diffusion competition system with nonlocal terms. We prove that there exists traveling wave solutions of the system connecting equilibrium $ (0, 0) $ to some unknown positive steady state for wave speed $ c>c^* = \max\left\{2, 2\sqrt{dr}\right\} $ and there is no such traveling wave solutions for $ c<c^* $, where $ d $ and $ r $ respectively corresponds to the diffusion coefficients and intrinsic rate of an competition species. Furthermore, we also demonstrate the unknown steady state just is the positive equilibrium of the system when the nonlocal delays only appears in the interspecific competition term, which implies that the nonlocal delay appearing in the interspecific competition terms does not affect the existence of traveling wave solutions. Finally, for a specific kernel function, some numerical simulations are given to show that the traveling wave solutions may connect the zero equilibrium to a periodic steady state.

Citation: Bang-Sheng Han, Zhi-Cheng Wang, Zengji Du. Traveling waves for nonlocal Lotka-Volterra competition systems. Discrete & Continuous Dynamical Systems - B, 2020, 25 (5) : 1959-1983. doi: 10.3934/dcdsb.2020011
References:
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J. Fang and J. H. Wu, Monotone traveling waves for delayed Lotka-Volterra competition systems, Discrete Contin. Dyn. Syst., 32 (2012), 3043-3058.  doi: 10.3934/dcds.2012.32.3043.  Google Scholar

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J. Fang and X.-Q. Zhao, Monotone wave fronts of the nonlocal Fisher-KPP equation, Nonlinearity, 24 (2011), 3043-3054.  doi: 10.1088/0951-7715/24/11/002.  Google Scholar

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A. Gomez and S. Trofimchuk, Monotone traveling wavefronts of the KPP-Fisher delayed equation, J. Differential Equations, 250 (2011), 1767-1787.  doi: 10.1016/j.jde.2010.11.011.  Google Scholar

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J.-S. Guo and X. Liang, The minimal speed of traveling fronts for the Lotka-Volterra competition system, J. Dynam. Differential Equations, 23 (2011), 353-363.  doi: 10.1007/s10884-011-9214-5.  Google Scholar

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K. HasikJ. KopfováP. Nábělková and S. Trofimchuk, Traveling waves in the nonlocal KPP-Fisher equation: Different roles of the right and the left interactions, J. Differential Equations, 260 (2016), 6130-6175.  doi: 10.1016/j.jde.2015.12.035.  Google Scholar

[21]

K. Hasik and S. Trofimchuk, Slowly oscillating wavefronts of the Fisher-KPP delayed equation, Discrete Contin. Dyn. Syst., 34 (2014), 3511-3533.  doi: 10.3934/dcds.2014.34.3511.  Google Scholar

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[23]

Y. Hosono, The minimal spread of traveling fronts for a diffusive Lotka-Volterra competition model, Bull. Math. Biol., 66 (1998), 435-448.  doi: 10.1006/bulm.1997.0008.  Google Scholar

[24]

A. Huang and P. X. Weng, Traveling wavefronts for a Lotka-Volterra system of type-$K$ with delays, Nonlinear Anal. Real World Appl., 14 (2013), 1114-1129.  doi: 10.1016/j.nonrwa.2012.09.002.  Google Scholar

[25]

J. H. Huang and X. F. 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

[26]

J. H. Huang and X. F. Zou, Existence of traveling wavefronts of delayed reaction-diffusion systems without monotonicity, Discrete Cont. Dyn. Syst., 9 (2003), 925-936.  doi: 10.3934/dcds.2003.9.925.  Google Scholar

[27]

J. H. Huang and X. F. Zou, Travelling wave solutions in delayed reaction diffusion systems with partial monotonicity, Acta Math. Appl. Sin. Engl. Ser., 22 (2006), 243-256.  doi: 10.1007/s10255-006-0300-0.  Google Scholar

[28]

J. I. 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

[29]

Y. Kan-on, Parameter dependence of propagation speed of traveling waves for competition-diffusion equations, SIAM J. Math. Anal., 26 (1995), 340-363.  doi: 10.1137/S0036141093244556.  Google Scholar

[30]

M. K. Kwong and C. H. Ou, Existence and nonexistence of monotone traveling waves for the delayed Fisher equation, J. Differential Equations, 249 (2010), 728-745.  doi: 10.1016/j.jde.2010.04.017.  Google Scholar

[31]

K. Li and X. Li, Traveling wave solutions in a delayed diffusive competition system, Nonlinear Anal., 75 (2012), 3705-3722.  doi: 10.1016/j.na.2012.01.024.  Google Scholar

[32]

W.-T. LiG. Lin and S. G. Ruan, Existence of travelling wave solutions in delayed reaction-diffusion systems with applications to diffusion-competition systems, Nonlinearity, 19 (2006), 1253-1273.  doi: 10.1088/0951-7715/19/6/003.  Google Scholar

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G. Lin and W.-T. Li, Bistable wavefronts in a diffusive and competitive Lotka-Volterra type system with nonlocal delays, J. Differential Equations, 244 (2008), 487-513.  doi: 10.1016/j.jde.2007.10.019.  Google Scholar

[34]

G. Lin and S. G. Ruan, Traveling wave solutions for delayed reaction-diffusion systems and applications to diffusive Lotka-Volterra competition models with distributed delays, J. Dynam. Differential Equations, 26 (2014), 583-605.  doi: 10.1007/s10884-014-9355-4.  Google Scholar

[35]

G.-Y. Lv and M. X. Wang, Traveling wave front in diffusive and competitive Lotka-Volterra system with delays, Nonlinear Anal. Real World Appl., 11 (2010), 1323-1329.  doi: 10.1016/j.nonrwa.2009.02.020.  Google Scholar

[36]

G.-Y. Lv and M. Wang, Traveling wave front and stability as planar wave of reaction diffusion equations with nonlocal delays, Z. Angew. Math. Phys., 64 (2013), 1005-1023.   Google Scholar

[37]

S. W. 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

[38]

A. OkuboP. K. MainiM. H. Williamson and J. D. Murray, On the spatial spread of grey squrrel in Britatin, Proc. R. Soc. Lond. B, 238 (1989), 113-125.  doi: 10.1098/rspb.1989.0070.  Google Scholar

[39]

C. H. Ou and J. H. Wu, Traveling wavefronts in a delayed food-limited population model, SIAM J. Math. Anal., 39 (2007), 103-125.  doi: 10.1137/050638011.  Google Scholar

[40]

S. X. Pan, Traveling wave solutions in delayed diffusion systems via a cross iteration scheme, Nonlinear Anal. Real World Appl., 10 (2009), 2807-2818.  doi: 10.1016/j.nonrwa.2008.08.007.  Google Scholar

[41]

K. W. Schaaf, Asymptotic behavior and traveling wave solutions for parabolic functional differential equations, Trans. Amer. Math. Soc., 302 (1987), 587-615.  doi: 10.2307/2000859.  Google Scholar

[42]

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

[43]

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

[44]

X. P. Yang and Y. F. Wang, Travelling wave and global attractivity in a competition-diffusion system with nonlocal delays, Comput. Math. Appl., 59 (2010), 3338-3350.  doi: 10.1016/j.camwa.2010.03.020.  Google Scholar

[45]

L.-H. YaoZ.-X. Yu and R. Yuan, Spreading speed and traveling waves for a nonmonotone reaction-diffusion model with distributed delay and nonlocal effect, Appl. Math. Model., 35 (2011), 2916-2929.  doi: 10.1016/j.apm.2010.12.011.  Google Scholar

[46]

Z.-X. Yu and R. Yuan, Traveling waves of delayed reaction-diffusion systems with applications, Nonlinear Anal. Real World Appl., 12 (2011), 2475-2488.  doi: 10.1016/j.nonrwa.2011.02.005.  Google Scholar

[47]

Z.-C. WangW.-T. Li and S. G. Ruan, Travelling wave fronts in reaction-diffusion systems with spatio-temporal delays, J. Differential Equations, 222 (2006), 185-232.  doi: 10.1016/j.jde.2005.08.010.  Google Scholar

[48]

J. H. Wu and X. F. Zou, Traveling wave fronts of reaction-diffusion systems with delay, J. Dynam. Differential Equations, 13 (2001), 651-687.  doi: 10.1023/A:1016690424892.  Google Scholar

[49]

X. F. Zou and J. H. Wu, Existence of traveling wave fronts in delayed reaction-diffusion systems via the monotone iteration method, Proc. Amer. Math. Soc., 125 (1997), 2589-2598.  doi: 10.1090/S0002-9939-97-04080-X.  Google Scholar

show all references

References:
[1]

M. Alfaro and J. Coville, Rapid traveling waves in the nonlocal Fisher equation connect two unstable states, Appl. Math. Lett., 25 (2012), 2095-2099.  doi: 10.1016/j.aml.2012.05.006.  Google Scholar

[2]

M. AlfaroJ. Coville and G. Raoul, Traveling waves in a nonlocal reaction-diffusion equation as a model for a population structured by a space variable and a phenotypic trait, Comm. Partial Differential Equations, 38 (2013), 2126-2154.  doi: 10.1080/03605302.2013.828069.  Google Scholar

[3]

H. BerestyckiG. NadinB. Perthame and L. Ryzhik, The non-local Fisher-KPP equation: Travelling waves and steady states, Nonlinearity, 22 (2009), 2813-2844.  doi: 10.1088/0951-7715/22/12/002.  Google Scholar

[4]

O. BonnefonJ. GarnierF. Hamel and L. Roques, Inside deyanics of delayed traveling waves, Math. Model. Nat. Phenom., 8 (2013), 42-59.  doi: 10.1051/mmnp/20138305.  Google Scholar

[5]

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

[6]

J. Fang and J. H. Wu, Monotone traveling waves for delayed Lotka-Volterra competition systems, Discrete Contin. Dyn. Syst., 32 (2012), 3043-3058.  doi: 10.3934/dcds.2012.32.3043.  Google Scholar

[7]

J. Fang and X.-Q. Zhao, Monotone wave fronts of the nonlocal Fisher-KPP equation, Nonlinearity, 24 (2011), 3043-3054.  doi: 10.1088/0951-7715/24/11/002.  Google Scholar

[8]

G. Faye and M. Holzer, Modulated traveling fronts for a nonlocal Fisher-KPP equation: A dynamical systems approach, J. Differential Equations, 258 (2015), 2257-2289.  doi: 10.1016/j.jde.2014.12.006.  Google Scholar

[9]

R. A. Gardner, Existence and stability of traveling 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

[10]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer-Verlag, Berlin, 2001.  Google Scholar

[11]

A. Gomez and S. Trofimchuk, Monotone traveling wavefronts of the KPP-Fisher delayed equation, J. Differential Equations, 250 (2011), 1767-1787.  doi: 10.1016/j.jde.2010.11.011.  Google Scholar

[12]

S. A. Gourley and S. G. Ruan, Convergence and traveling fronts in functional differential equations with nonlocal terms: A competition model, SIAM J. Math. Anal., 35 (2003), 806-822.  doi: 10.1137/S003614100139991.  Google Scholar

[13]

J.-S. Guo and X. Liang, The minimal speed of traveling fronts for the Lotka-Volterra competition system, J. Dynam. Differential Equations, 23 (2011), 353-363.  doi: 10.1007/s10884-011-9214-5.  Google Scholar

[14]

S.-J. Guo and J. Zimmer, Stability of travelling wavefronts in discrete reaction-diffusion equations with nonlocal delay effects, Nonlinearity, 28 (2015), 463-492.  doi: 10.1088/0951-7715/28/2/463.  Google Scholar

[15]

J.-S. Guo and C.-H. Wu, Recent developments on wave propagation in 2-species competition systems, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2713-2724.  doi: 10.3934/dcdsb.2012.17.2713.  Google Scholar

[16]

F. Hamel and L. Ryzhik, On the nonlocal Fisher-KPP equation: Steady states, spreading speed and global bounds, Nonlinearity, 27 (2014), 2735-2753.  doi: 10.1088/0951-7715/27/11/2735.  Google Scholar

[17]

B.-S. Han and Z.-C. Wang, Traveling wave solutions in a nonlocal reaction-diffusion population model, Commun. Pure Appl. Anal., 15 (2016), 1057-1076.  doi: 10.3934/cpaa.2016.15.1057.  Google Scholar

[18]

B.-S. Han and Z.-C. Wang, Traveling waves for the nonlocal diffusive single species model with allee effect, J. Math. Anal. Appl., 443 (2016), 243-264.  doi: 10.1016/j.jmaa.2016.05.031.  Google Scholar

[19]

B.-S. Han and Y. H. Yang, An integro-PDE model with variable motility, Nonlinear Anal. Real World Appl., 45 (2019), 186-199.  doi: 10.1016/j.nonrwa.2018.07.004.  Google Scholar

[20]

K. HasikJ. KopfováP. Nábělková and S. Trofimchuk, Traveling waves in the nonlocal KPP-Fisher equation: Different roles of the right and the left interactions, J. Differential Equations, 260 (2016), 6130-6175.  doi: 10.1016/j.jde.2015.12.035.  Google Scholar

[21]

K. Hasik and S. Trofimchuk, Slowly oscillating wavefronts of the Fisher-KPP delayed equation, Discrete Contin. Dyn. Syst., 34 (2014), 3511-3533.  doi: 10.3934/dcds.2014.34.3511.  Google Scholar

[22]

Y. Hosono, Singular perturbation analysis of traveling waves for diffusive Lotak-Volterra competitive models, Numerical and Applied Mathemections, IMACS Ann. Comput. Appl. Math., IMACS Trans. Sci. Comput. '88, Baltzer, Basel, 1 (1988), 687-692.   Google Scholar

[23]

Y. Hosono, The minimal spread of traveling fronts for a diffusive Lotka-Volterra competition model, Bull. Math. Biol., 66 (1998), 435-448.  doi: 10.1006/bulm.1997.0008.  Google Scholar

[24]

A. Huang and P. X. Weng, Traveling wavefronts for a Lotka-Volterra system of type-$K$ with delays, Nonlinear Anal. Real World Appl., 14 (2013), 1114-1129.  doi: 10.1016/j.nonrwa.2012.09.002.  Google Scholar

[25]

J. H. Huang and X. F. 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

[26]

J. H. Huang and X. F. Zou, Existence of traveling wavefronts of delayed reaction-diffusion systems without monotonicity, Discrete Cont. Dyn. Syst., 9 (2003), 925-936.  doi: 10.3934/dcds.2003.9.925.  Google Scholar

[27]

J. H. Huang and X. F. Zou, Travelling wave solutions in delayed reaction diffusion systems with partial monotonicity, Acta Math. Appl. Sin. Engl. Ser., 22 (2006), 243-256.  doi: 10.1007/s10255-006-0300-0.  Google Scholar

[28]

J. I. 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

[29]

Y. Kan-on, Parameter dependence of propagation speed of traveling waves for competition-diffusion equations, SIAM J. Math. Anal., 26 (1995), 340-363.  doi: 10.1137/S0036141093244556.  Google Scholar

[30]

M. K. Kwong and C. H. Ou, Existence and nonexistence of monotone traveling waves for the delayed Fisher equation, J. Differential Equations, 249 (2010), 728-745.  doi: 10.1016/j.jde.2010.04.017.  Google Scholar

[31]

K. Li and X. Li, Traveling wave solutions in a delayed diffusive competition system, Nonlinear Anal., 75 (2012), 3705-3722.  doi: 10.1016/j.na.2012.01.024.  Google Scholar

[32]

W.-T. LiG. Lin and S. G. Ruan, Existence of travelling wave solutions in delayed reaction-diffusion systems with applications to diffusion-competition systems, Nonlinearity, 19 (2006), 1253-1273.  doi: 10.1088/0951-7715/19/6/003.  Google Scholar

[33]

G. Lin and W.-T. Li, Bistable wavefronts in a diffusive and competitive Lotka-Volterra type system with nonlocal delays, J. Differential Equations, 244 (2008), 487-513.  doi: 10.1016/j.jde.2007.10.019.  Google Scholar

[34]

G. Lin and S. G. Ruan, Traveling wave solutions for delayed reaction-diffusion systems and applications to diffusive Lotka-Volterra competition models with distributed delays, J. Dynam. Differential Equations, 26 (2014), 583-605.  doi: 10.1007/s10884-014-9355-4.  Google Scholar

[35]

G.-Y. Lv and M. X. Wang, Traveling wave front in diffusive and competitive Lotka-Volterra system with delays, Nonlinear Anal. Real World Appl., 11 (2010), 1323-1329.  doi: 10.1016/j.nonrwa.2009.02.020.  Google Scholar

[36]

G.-Y. Lv and M. Wang, Traveling wave front and stability as planar wave of reaction diffusion equations with nonlocal delays, Z. Angew. Math. Phys., 64 (2013), 1005-1023.   Google Scholar

[37]

S. W. 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

[38]

A. OkuboP. K. MainiM. H. Williamson and J. D. Murray, On the spatial spread of grey squrrel in Britatin, Proc. R. Soc. Lond. B, 238 (1989), 113-125.  doi: 10.1098/rspb.1989.0070.  Google Scholar

[39]

C. H. Ou and J. H. Wu, Traveling wavefronts in a delayed food-limited population model, SIAM J. Math. Anal., 39 (2007), 103-125.  doi: 10.1137/050638011.  Google Scholar

[40]

S. X. Pan, Traveling wave solutions in delayed diffusion systems via a cross iteration scheme, Nonlinear Anal. Real World Appl., 10 (2009), 2807-2818.  doi: 10.1016/j.nonrwa.2008.08.007.  Google Scholar

[41]

K. W. Schaaf, Asymptotic behavior and traveling wave solutions for parabolic functional differential equations, Trans. Amer. Math. Soc., 302 (1987), 587-615.  doi: 10.2307/2000859.  Google Scholar

[42]

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

[43]

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

[44]

X. P. Yang and Y. F. Wang, Travelling wave and global attractivity in a competition-diffusion system with nonlocal delays, Comput. Math. Appl., 59 (2010), 3338-3350.  doi: 10.1016/j.camwa.2010.03.020.  Google Scholar

[45]

L.-H. YaoZ.-X. Yu and R. Yuan, Spreading speed and traveling waves for a nonmonotone reaction-diffusion model with distributed delay and nonlocal effect, Appl. Math. Model., 35 (2011), 2916-2929.  doi: 10.1016/j.apm.2010.12.011.  Google Scholar

[46]

Z.-X. Yu and R. Yuan, Traveling waves of delayed reaction-diffusion systems with applications, Nonlinear Anal. Real World Appl., 12 (2011), 2475-2488.  doi: 10.1016/j.nonrwa.2011.02.005.  Google Scholar

[47]

Z.-C. WangW.-T. Li and S. G. Ruan, Travelling wave fronts in reaction-diffusion systems with spatio-temporal delays, J. Differential Equations, 222 (2006), 185-232.  doi: 10.1016/j.jde.2005.08.010.  Google Scholar

[48]

J. H. Wu and X. F. Zou, Traveling wave fronts of reaction-diffusion systems with delay, J. Dynam. Differential Equations, 13 (2001), 651-687.  doi: 10.1023/A:1016690424892.  Google Scholar

[49]

X. F. Zou and J. H. Wu, Existence of traveling wave fronts in delayed reaction-diffusion systems via the monotone iteration method, Proc. Amer. Math. Soc., 125 (1997), 2589-2598.  doi: 10.1090/S0002-9939-97-04080-X.  Google Scholar

Figure 1.  The time and space evolution of $ u(x,t) $ in nonlocal equation (36) with kernel $ \phi_{\sigma}(x) = \frac{3a}{2\sigma}e^{-\frac{a}{\sigma}|x|}-\frac{1}{\sigma}e^{-\frac{|x|}{\sigma}} $. Our computational domain is $ x\in [0,85], t \in [0,30] $. The corresponding parameter values are: $ L_0 = 15,\ a = 0.7,\ a_1 = 0.4,\ a_2 = 0.5,\ d = 0.1,\ r = 2 $ and $ \sigma $ follows by 0.3, 0.6, 1.2, 1.5
Figure 2.  The time and space evolution of $ v(x,t) $ in nonlocal equation (36) with kernel $ \phi_{\sigma}(x) = \frac{3a}{2\sigma}e^{-\frac{a}{\sigma}|x|}-\frac{1}{\sigma}e^{-\frac{|x|}{\sigma}} $. Our computational domain is $ x\in [0,85],k ∈ [0,30] $. The corresponding parameter values are: $ L_0 = 15,\ a = 0.7,\ a_1 = 0.4,\ a_2 = 0.5,\ d = 0.1,\ r = 2 $ and $ \sigma $ follows by 0.3, 0.6, 1.2, 1.5
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