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Point to point traveling wave and periodic traveling wave induced by Hopf bifurcation for a diffusive predator-prey system

  • * Corresponding author: Hongyong Zhao

    * Corresponding author: Hongyong Zhao 

The work is partially supported by the National Natural Science Foundation of China (Nos 11571170, 31570417); the Natural Science Foundation of Anhui Province of China (No 1608085 MA14); the Key Project of Natural Science Research of Anhui Higher Education Institutions of China (No KJ2018A0365)

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  • In this paper, we consider a diffusive Leslie-Gower predator-prey system with prey subject to Allee effect. First, taking into account the diffusion of both species, we obtain the existence of traveling wave solution connecting predator-free constant steady state and coexistence steady state by using the upper and lower solutions method. However, due to the singularity in the predator equation, we need construct a positive suitable lower solution for the prey density. Such a traveling wave solution can model the spatial-temporal process where the predator invades the territory of the prey and they eventually coexist. Second, taking into account two cases: the diffusion of both species and the diffusion of prey-only, we prove the existence of small amplitude periodic traveling wave train solutions by using the Hopf bifurcation theory. Such traveling wave solutions show that the predator invasion leads to the periodic population densities in the coexistence domain.

    Mathematics Subject Classification: Primary: 35K57, 35C07; Secondary: 34C23.


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  • [1] D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. Math., 30 (1978), 33-76.  doi: 10.1016/0001-8708(78)90130-5.
    [2] M. A. Aziz-Alaoui and M. Daher 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.
    [3] S. CantrellC. Cosner and  S. G. RuanSpatial Ecology, Mathematical and Computational Biology Series, Chapman & Hall/CRC Mathematical and Computational Biology Series, CRC Press, Boca Raton, FL, 2010. 
    [4] Y.-Y. ChenJ.-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.  doi: 10.1016/j.jmaa.2016.07.071.
    [5] J. B. Collings, The effect of the functional response on the bifurcation behavior of a mite predator-prey interaction model, J. Math. Biol., 36 (1997), 149-168.  doi: 10.1007/s002850050095.
    [6] S. R. Dunbar, Travelling wave solutions of diffusive lotka-volterra equations, J. Math. Biol., 17 (1983), 11-32.  doi: 10.1007/BF00276112.
    [7] S. R. Dunbar, Traveling wave solutions of diffusive lotka-volterra equations: a heteroclinic connection in $\mathbb{R}^{4}$, Trans. Amer. Math. Soc., 286 (1984), 557-594.  doi: 10.2307/1999810.
    [8] C.-H. HsuC.-R. YangT.-H. Yang and T.-S. Yang, Existence of traveling wave solutions for diffusive predator-prey type systems, J. Differ. Equations, 252 (2012), 3040-3075.  doi: 10.1016/j.jde.2011.11.008.
    [9] W. Z. Huang, Traveling wave solutions for a class of predator-prey systems, J. Dyn. Differ. Equ., 24 (2012), 633-644.  doi: 10.1007/s10884-012-9255-4.
    [10] J. H. HuangG. Lu and S. G. Ruan, Existence of traveling wave solutions in a diffusive predator-prey model, J. Math. Biol., 46 (2003), 132-152.  doi: 10.1007/s00285-002-0171-9.
    [11] W. Z. Huang, A geometric approach in the study of traveling waves for some classes of non-monotone reaction-diffusion systems, J. Differ. Equations, 260 (2016), 2190-2224.  doi: 10.1016/j.jde.2015.09.060.
    [12] Y. H. Huang and P. X. Weng, Traveling waves for a diffusive predator-prey system with general functional response, Nonlinear Anal. Real World Appl., 14 (2013), 940-959.  doi: 10.1016/j.nonrwa.2012.08.007.
    [13] Y.-L. Huang and G. Lin, Traveling wave solutions in a diffusion system with two preys and one predator, J. Math. Anal. Appl., 418 (2014), 163-184.  doi: 10.1016/j.jmaa.2014.03.085.
    [14] W. Z. Huang and M. A. Han, Non-linear determinacy of minimum wave speed for a Lotka-Volterra competition model, J. Differ. Equations, 251 (2011), 1549-1561.  doi: 10.1016/j.jde.2011.05.012.
    [15] 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.
    [16] M. A. LewisB. T. Li and H. F. Weinberger, Spreading speed and linear determinacy for two-species competition models, J. Math. Biol., 45 (2002), 219-233.  doi: 10.1007/s002850200144.
    [17] B. T. LiH. F. Weinberger and M. A. Lewis, Spreading speeds as slowest wave speeds for cooperative systems, Math. Biosci., 196 (2005), 82-98.  doi: 10.1016/j.mbs.2005.03.008.
    [18] X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math., 60 (2007), 1-40.  doi: 10.1002/cpa.20154.
    [19] G. Lin, Invasion traveling wave solutions of a predator-prey system, Nonlinear Anal., 96 (2014), 47-58.  doi: 10.1016/j.na.2013.10.024.
    [20] J. E. Marsden and M. McCracken, The Hopf Bifurcation and Its Applications, Applied Mathematical Sciences, Vol. 19. Springer-Verlag, New York, 1976.
    [21] W. J. Ni and M. X. Wang, Dynamical properties of a Leslie-Gower prey-predator model with strong Allee effect in prey, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 3409-3420.  doi: 10.3934/dcdsb.2017172.
    [22] W. J. Ni and M. X. Wang, Dynamics and patterns of a diffusive Leslie-Gower prey-predator model with strong Allee effect in prey, J. Differ. Equations, 261 (2016), 4244-4274.  doi: 10.1016/j.jde.2016.06.022.
    [23] S. E. Riechert, Spiders as Representative 'Sit-and-Wait' Predators, Natural Enemies: The Population Biology of Predators, Parasites and Diseases, John Wiley & Sons, 2009.
    [24] H. M. SafuanI. N. TowersZ. Jovanoski and H. S. Sidhu, On travelling wave solutions of the diffusive Leslie-Gower model, Appl. Math. Comput., 274 (2016), 362-371.  doi: 10.1016/j.amc.2015.10.088.
    [25] N. Shigesada and  K. KawasakiBiology Invasions: Theory and Practice, Oxford University Press, Oxford, 1997. 
    [26] A. I. Volpert, V. A. Volpert and V. A. Volpert, Traveling Wave Solutions of Parabolic Systems, Translations of Mathematical Monographs, 140. American Mathematical Society, Providence, RI, 1994.
    [27] J. H. Wu and X. F. Zou, Traveling wave fronts of reaction-diffusion systems with delay, J. Dyn. Differ. Equ., 13 (2001), 651-687.  doi: 10.1023/A:1016690424892.
    [28] X. B. Zhang and H. L. Zhu, Dynamics and pattern formation in homogeneous diffusive predator-prey systems with predator interference or foraging facilitation, Nonlinear Anal. Real World Appl., 48 (2019), 267-287.  doi: 10.1016/j.nonrwa.2019.01.016.
    [29] W. Zuo and J. Shi, Traveling wave solutions of a diffusive reatio-dependent Holling-tanner system with distributed delay, Comm. Pure Appl. Math., 17 (2018), 1179-1200.  doi: 10.3934/cpaa.2018057.
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