Traveling Wave Solutions for Some Three-Species Predator-Prey Systems

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Jong-Shenq Guo

Abstract




In this paper, we present some recent developments on the application of Schauder’s fixed point theorem to the existence of traveling waves for some three-species predator-prey systems. The existence of traveling waves of predator-prey systems is closely related to the invasion phenomenon of some alien species to the habitat of aboriginal species. Three different three-species predator-prey models with different invaded and invading states are presented. In this paper, we focus on the methodology of deriving the convergence of stale tail of wave profiles.




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How to Cite
Guo, J.-S. (2021). Traveling Wave Solutions for Some Three-Species Predator-Prey Systems. Tamkang Journal of Mathematics, 52(1), 25–36. https://doi.org/10.5556/j.tkjm.52.2021.4029
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Papers

References

Z. Bi and S. Pan, Dynamics of a predator-prey system with three species, Bound. Value. Probl. (2018) 2018:162.

Y.-S. Chen and J.-S. Guo, Traveling wave solutions for a three-species predator-prey model with two aborigine preys, Japan J. Industrial and Applied Mathematics, https://doi.org/10.1007/s13160-020- 00445-9.

Y.-S. Chen, T. Giletti and J.-S. Guo, Persistence of preys in a diffusive three species predator-prey system with a pair of strong-weak competing preys, arXiv:2008.11939.

Y.-Y. Chen, J.-S. Guo and C.-H. Yao, Traveling wave solutions for a continuous and discrete diffusive predator-prey model, J. Math. Anal. Appl. 445 (2017), 212-239.

Y. Du and S.-B. Hsu, A diffusive predator-prey model in heterogeneous environment, J. Differential Equations 203 (2004), 331-364.

Y. Du and R. Xu, Traveling wave solutions in a three-species food-chain model with diffusion and delays, Int. J. Biomath. 5 (2012), 1250002, 17 pp.

A. Ducrot, T. Giletti, J.-S. Guo and M. Shimojo, Asymptotic spreading speeds for a predator-prey system with two predators and one prey, arXiv:2007.02568.

A. Ducrot, T. Giletti and H. Matano, Spreading speeds for multidimensional reaction-diffusion systems of the prey-predator type, Calc. Var. Partial Differential Equations 58 (2019), no. 4, Paper No. 137, 34 pp.

A. Ducrot and J.-S. Guo, Asymptotic behavior of solutions to a class of diffusive predator-prey systems, J. Evolution Equations 18 (2018), 755-775.

J.-S. Guo, K.-I. Nakamura, T. Ogiwara and C.-C. Wu, Traveling wave solutions for a predator-prey system with two predators and one prey, Nonlinear Analysis: Real World Applications 54 (2020), 103111, 13 pp.

J. Huang, G. Lu and S. Ruan, Existence of traveling wave solutions in a diffusive predator-prey model, J. Math. Biol. 46 (2003), 132-152.

W. Huang, Traveling wave solutions for a class of predator-prey systems, J. Dynam. Differential Equa- tions 24 (2012), 633-644.

Y.L. Huang and G. Lin, Traveling wave solutions in a diffusive system with two preys and one predator, J. Math. Anal. Appl. 418 (2014), 163-184.

J. Huang and X. Zou, Existence of traveling wave fronts of delayed reaction-diffusion systems without monotonicity, Disc. Cont. Dyn. Systems 9 (2003), 925-936.

W.T. Li, G. Lin and S. Ruan, Existence of traveling wave solutions in delayed reaction-diffusion systems with applications to diffusion-competition systems, Nonlinearity 19 (2006), 1253-1273.

G. Lin, Invasion traveling wave solutions of a predator-prey system, Nonlinear Anal. 96 (2014), 47-58.

G. Lin, W.T. Li and M. Ma, Traveling wave solutions in delayed reaction diffusion systems with applications to multi-species models, Disc. Cont. Dyn. Systems, Ser. B 13 (2010), 393-414.

G. Lin and S. Ruan, Traveling wave solutions for delayed reaction-diffusion systems and applications to diffusive Lotka-Volterra competition models with distributed delays, J. Dyn. Diff. Equat., 26 (2014), 583-605.

J.-J. Lin, W. Wang, C. Zhao and T.-H. Yang, Global dynamics and traveling wave solutions of two predators-one prey models, Discrete and Continuous Dynamical System, Series B, 20 (2015), 1135-1154.

J.-J. Lin and T.-H. Yang, Traveling wave solutions for a diffusive three-species intraguild predation model, Int. J. Biomath. 11 (2018), 1850022, 27 pp.

S. Ma, Traveling wavefronts for delayed reaction-diffusion systems via a fixed point theorem, J. Differential Equations 171 (2001), 294-314.

J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with delay, J. Dynam. Differential Equations 13 (2001), 651-687.

T. Zhang, Minimal wave speed for a class of non-cooperative reaction-diffusion systems of three equations, J. Differential Equations 262 (2017), 4724-4770.

T. Zhang and Y. Jin, Traveling waves for a reaction-diffusion-advection predator-prey model, Nonlinear Analysis: Real World Applications 36 (2017), 203-232.