Existence of Traveling Waves for Ratio-dependent Predator-prey System with Nonlocal Diffusion
American Journal of Applied Mathematics
Volume 8, Issue 5, October 2020, Pages: 236-246
Received: Feb. 18, 2020;
Accepted: Aug. 14, 2020;
Published: Aug. 25, 2020
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Ke Li, School of Mathematics and Statistics, Shandong Normal University, Jinan, P.R. China
Hongmei Cheng, School of Mathematics and Statistics, Shandong Normal University, Jinan, P.R. China
In this paper, we study the traveling waves for the ratio-dependent predator-prey model with nonlocal diffusion, which is devoted to the existence and nonexistence of traveling wave solution. This model incorporates the ratio-dependent functional response into the Lotka-Volterra type system, and both species obey the logistic growth. Firstly, we construct a nice pair of upper and lower solutions when the wave speed is greater than the minimal wave speed. Then by applying Schauder's fixed point theorem with the help of suitable upper and lower solutions, we can obtain the existence of traveling waves when the wave speed is greater than the minimal wave speed. Moreover, in order to prove the limit behavior of the traveling waves at infinity, we construct a sequence that converges to the coexistence state. Finally, by using the comparison principle, we obtain the nonexistence of the traveling waves when the wave speed is greater than 0 and less than the minimal wave speed. The difficulty of this paper is to construct a suitable upper and lower solution, which is also the novelty of this paper. Under certain restricted condition, this paper concludes the existence and the nonexistence of the traveling waves for the ratio-dependent predator-prey model with nonlocal diffusion.
Existence of Traveling Waves for Ratio-dependent Predator-prey System with Nonlocal Diffusion, American Journal of Applied Mathematics. Special Issue: Application of Nonlinear Analysis.
Vol. 8, No. 5,
2020, pp. 236-246.
P. Bates, P. Fife, X. Ren and X. Wang, Traveling waves in a convolution model for phase transitions, Arch. Ration. Mech. Anal., 1997, 138: 105-136.
J. Carr and A. Chmaj, Uniqueness of travelling waves for nonlocal monostable equations, Proc. Amer. Math. Soc., 2004, 132: 2433-2439.
S.-S. Chen and J.-P. Shi, Global stability in a diffusive Holling-Tanner predator-prey model, Appl. Math. Lett., 2012, 25: 614-618.
X. Chen, Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations, Adv. Differential Equations, 1997, 2: 125-160.
H. Cheng and R. Yuan, The spreading property for a prey-predator reaction-diffusion system with fractional diffusion, Frac. Calc. Appl. Anal., 2015, 18: 565-579.
H. Cheng and R. Yuan, Existence and stability of traveling waves for Leslie-Gower predator-prey system with nonlocal diffusion. Discrete Contin. Dyn. Syst., 2017, 37 (10): 5433-5454.
H. Cheng and R. Yuan, Traveling waves of some Holling-Tanner predator-prey system with nonlocal diffusion. Appl. Math. Comput., 2018, 338 (1): 12-24.
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., 1984, 33: 319-343.
J. Coville and L. Dupaigne, Propagation speed of travelling fronts in non local reaction-diffusion equations, Nonlinear Anal., 2005, 60: 797-819.
J. Coville and L. Dupaigne, On a non-local equation arising in population dynamics, Proc. Roy. Soc. Edinburgh Sect. A, 2007, 137: 727-755.
A. Ducrot, Convergence to generalized transition waves for some Holling-Tanner prey-predator reaction-diffusion system, J. Math. Pures Appl., 2013, 100: 1-15.
S. R. Dunbar, Traveling wave solutions of diffusive Lotka-Volterra equations: a heteroclinic connection in R4, Trans. Amer. Math. Soc., 1984, 286: 557-594.
W. Feng, W.-H. Ruan and X. Lu, On existence of wavefront solutions in mixed monotone reaction-diffusion systems, Discrete Contin. Dyn. Syst. Ser. B, 2016, 21: 815-836.
R. A. Gardner, Existence and stability of travelling wave solutions of competition models: A degree theoretic approach, J. Differential Equations, 1982, 44: 343-364.
C. S. Holling, The functional response of predators to prey density and its role in mimicry and population regulation, Mem. Entomol. Soci. Can., 1965, 97: 5-60.
Y.-L. Huang and G. Lin, Traveling wave solutions in a diffusive system with two preys and one predator, J. Math. Anal. Appl., 2014, 418: 163-184.
J. Huang and X. Zou, Traveling wavefronts in diffusive and cooperative Lotka-Volterra system with delays, J. Math. Anal. Appl., 2002, 271: 455-466. Y. Jin, X.-Q.
Y. Jin, X.-Q. Zhao, Spatial dynamics of a periodic population model with dispersal. Nonlinearity, 2009, 22: 1167-1189.
J. Kanel and L. Zhou, Existence of wave front solutions and estimates of wave speed for a competition-diffusion system, Nonlinear Anal., 1996, 27: 579-587.
G. Lin, Invasion traveling wave solutions of a predator-prey system, Nonlinear Anal., 2014, 96: 47-58.
S.-W. Ma, Traveling wavefronts for delayed reaction-diffusion systems via a fixed point theorem, J. Differential Equations, 2001, 171: 294-314.
M. R. Owen and M. A. Lewis, How predation can slow, stop or reverse a prey invasion, Bull. Math. Biol., 2001, 63: 655-684.
S. Pan, Traveling wave fronts of delayed non-local diffusion systems without quasimonotonicity, J. Math. Anal. Appl., 2008, 346: 415-424.
S. Pan, W.-T. Li and G. Lin, Travelling wave fronts in nonlocal delayed reaction-diffusion systems and applications, Z. Angew. Math. Phys., 2009, 60: 377-392.
R. Peng and M.-X. Wang, Positive steady states of the Holling-Tanner prey-predator model with diffusion, Proc. Roy. Soc. Edinburgh Sect. A, 2005, 135: 149-164.
S. Petrovskii, A. Morozov and B.-L. Li, Regimes of biological invasion in a predator-prey system with the Allee effect, Bull. Math. Biol., 2005, 67: 637-661.
T. Su and G-B. Zhang, Invasion traveling waves for a discrete diffusive ratio-dependent predator-prey model, in press.
J. T. Tanner, The stability and the intrinsic growth rates of prey and predator populations, Ecology, 1975, 855-867.
M. M. Tang and P. C. Fife, Propagating fronts for competing species equations with diffusion, Arch. Rational Mech. Anal., 1980, 73: 69-77.
J. H. Van Vuuren, The existence of travelling plane waves in a general class of competition-diffusion systems, IMA J. Appl. Math., 1995, 55: 135-148.
Z. H. Zhao, R. Li, X. K. Zhao and Z. S. Feng, Traveling wave solutions of a nonlocal dispersal predator-prey model with spatiotemporal delay, Z. Angew. Math. Phys. 69 (2018), no. 6, Paper No. 146, 20 pp.
W. J. Zuo and J. P. Shi, Traveling wave solutions of a diffusive ratio-dependent Holling-Tanner system with distributed delay, Comm. pure Appl. Anal., 2018, 17: 1179-1200.