# American Institute of Mathematical Sciences

April  2021, 26(4): 2239-2255. doi: 10.3934/dcdsb.2021007

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

 Department of Mathematical Sciences, National Chengchi University, 64, S-2 Zhi-Nan Road, Taipei 116, Taiwan

* Corresponding author: Sheng-Chen Fu

Dedicated to Professor Sze-Bi Hsu

Received  September 2020 Revised  November 2020 Published  December 2020

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

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.

Citation: Ching-Hui Wang, Sheng-Chen Fu. Traveling wave solutions to diffusive Holling-Tanner predator-prey models. Discrete & Continuous Dynamical Systems - B, 2021, 26 (4) : 2239-2255. doi: 10.3934/dcdsb.2021007
##### References:
 [1] S. Ai, Y. 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.  Google Scholar [2] I. Barbălat, Systèmes d'équations différentielles d'oscillations non Linéaires, Rev. Math. Pures Appl., 4 (1959), 267-270.   Google Scholar [3] 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.  doi: 10.1016/j.jmaa.2016.07.071.  Google Scholar [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.  Google Scholar [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. Google Scholar [6] J. K. Hale, Ordinary Differential Equations, R. E. Krieger Publ., 1980.  Google Scholar

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##### References:
 [1] S. Ai, Y. 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.  Google Scholar [2] I. Barbălat, Systèmes d'équations différentielles d'oscillations non Linéaires, Rev. Math. Pures Appl., 4 (1959), 267-270.   Google Scholar [3] 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.  doi: 10.1016/j.jmaa.2016.07.071.  Google Scholar [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.  Google Scholar [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. Google Scholar [6] J. K. Hale, Ordinary Differential Equations, R. E. Krieger Publ., 1980.  Google Scholar
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 $s = 0.6$
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 $s = 0.5$
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 $s = 0.6$
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 $s = 0.5$
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