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Traveling wave solutions to diffusive Holling-Tanner predator-prey models

  • * Corresponding author: Sheng-Chen Fu

    * Corresponding author: Sheng-Chen Fu

Dedicated to Professor Sze-Bi Hsu

The second author is supported by MOST grant 109-2115-M-004-004

Abstract / Introduction Full Text(HTML) Figure(4) Related Papers Cited by
  • In this paper, we first establish the existence of semi-traveling wave solutions to a diffusive generalized Holling-Tanner predator-prey model in which the functional response may depend on both the predator and prey populations. Then, by constructing the Lyapunov function, we apply the obtained result to show the existence of traveling wave solutions to the diffusive Holling-Tanner predator-prey models with various functional responses, including the Lotka-Volterra type functional response, the Holling type Ⅱ functional response and the Beddington-DeAngelis functional response.

    Mathematics Subject Classification: Primary: 35c07, 35K57; Secondary: 35K40.

    Citation:

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  • Figure 1.  The solution as a function of the spatial variable x is plotted at t = 0, t = 10, t = 20 and t = 30. The initial data $ (u_0,v_0) $ is chosen so that $ u_0 = 1 $ and $ v_0 = 0.05*(1+sign(51-x))*(1+sign(x-49))/4 $. The parameter values are $ k = 1.4 $, $ b = e = 1 $, $ d = 1 $, $ r = 4 $ and $ s = 0.6 $

    Figure 2.  The solution as a function of the spatial variable x is plotted at t = 0, t = 5, t = 10 and t = 20. The initial data $ (u_0,v_0) $ is chosen so that $ u_0 = 1 $ and $ v_0 = 0.05*(1+sign(51-x))*(1+sign(x-49))/4 $. The parameter values are $ k = 4 $, $ b = e = 0 $, $ d = 1 $, $ r = 2 $ and $ s = 0.5 $

    Figure 3.  The solution as a function of the spatial variable x is plotted at t = 0, t = 10, t = 20 and t = 30. The initial data $ (u_0,v_0) $ is chosen so that $ u_0 = 1 $ and $ v_0 = 0.05*(1+sign(51-x))*(1+sign(x-49))/4 $. The parameter values are $ k = 10 $, $ b = 5 $, $ e = 1 $, $ d = 1 $, $ r = 4 $ and $ s = 0.6 $

    Figure 4.  The solution as a function of the spatial variable x is plotted at t = 0, t = 5, t = 10 and t = 20. The initial data $ (u_0,v_0) $ is chosen so that $ u_0 = 1 $ and $ v_0 = 0.05*(1+sign(51-x))*(1+sign(x-49))/4 $. The parameter values are $ k = 10 $, $ b = e = 0 $, $ d = 1 $, $ r = 2 $ and $ s = 0.5 $

  • [1] S. AiY. Du and R. Peng, Traveling waves for a generalized Holling-Tanner predator-prey model, J. Diff. Eqns., 263 (2017), 7782-7814.  doi: 10.1016/j.jde.2017.08.021.
    [2] I. Barbălat, Systèmes d'équations différentielles d'oscillations non Linéaires, Rev. Math. Pures Appl., 4 (1959), 267-270. 
    [3] 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.
    [4] Y. Du and S.-B. Hsu, A diffusive predator-prey model in heterogeneous environment, J. Diff. Eqns., 203 (2004), 331-364.  doi: 10.1016/j.jde.2004.05.010.
    [5] S.-C. Fu, M. Mimura and J.-C. Tsai, Traveling waves in a hybrid model of demic and cultural diffusions in Neolithic transition, submitted.
    [6] J. K. Hale, Ordinary Differential Equations, R. E. Krieger Publ., 1980.
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