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May 2019, 18(3): 1483-1508. doi: 10.3934/cpaa.2019071

## Existence and non-monotonicity of traveling wave solutions for general diffusive predator-prey models

 1 Department of Mathematics, National Central University, Zhongli District, Taoyuan City 32001, Taiwan 2 General Education Center, National Taipei University of Technology, Taipei 10608, Taiwan

* Corresponding author

Received  October 2017 Revised  June 2018 Published  November 2018

Fund Project: The first author is partially supported by the NCTS and MOST of Taiwan, and the second author is partially supported by the MOST of Taiwan

This paper is concerned with the existence and non-monotonicity of traveling wave solutions for general diffusive predator-prey models. By using Schauder's fixed point theorem and the existence of contracting rectangles, we obtain the existence result. Then we investigate the asymptotic behavior of positive monotone traveling wave solutions by using the modified Ikehara's Theorem. With the help of their asymptotic behavior, we provide a sufficient condition which guarantee that all positive traveling wave solutions of the system are non-monotone. Furthermore, to illustrate our main results, the existence and non-monotonicity of traveling wave solutions of Lotka-Volterra predator-prey model and modified Leslie-Gower predator-prey models with different kinds of functional responses are also discussed.

Citation: Cheng-Hsiung Hsu, Jian-Jhong Lin. Existence and non-monotonicity of traveling wave solutions for general diffusive predator-prey models. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1483-1508. doi: 10.3934/cpaa.2019071
##### References:
 [1] S. Ai, Y. Du and R. Peng, Traveling waves for a generalized Holling-Tanner predator-prey model, J. Differential Equations, 263 (2017), 7782-7814. doi: 10.1016/j.jde.2017.08.021. [2] M. A. Aziz-Alaoui and M. D. Okiye, Boundedness and global stability for a predator-prey model with modified Leslie-Gower and Holling-type II schemes, Applied Mathematics Letters, 16 (2003), 1069-1075. doi: 10.1016/S0893-9659(03)90096-6. [3] J. R. Beddington, Mutual interference between parasites or predators and it's effect on searching efficiency, J. Anim. Ecol., 44 (1975), 331-340. [4] A. Boumenir and V. Nguyen, Erron Theorem in the monotone iteration method for traveling waves in delayed reaction-diffusion equations, J. Differential Equations, 244 (2008), 1551-1570. doi: 10.1016/j.jde.2008.01.004. [5] J. B. Conway, Functions of One Complex Variable, $2^{nd}$ edition, Springer-Verlag, New York, 1978. [6] W. Ding and W. Huang, Traveling wave solutions for some classes of diffusive predator-prey models, Journal of Dynamics and Differential Equations, 28 (2016), 1293-1308. doi: 10.1007/s10884-015-9472-8. [7] Y. H. Du and S. B. Hsu, A diffusive predator-prey model in heterogeneous environment, J. Differential Equations, 203 (2004), 331-364. doi: 10.1016/j.jde.2004.05.010. [8] Y. H. Du and M. X. Wang, Asymptotic behaviour of positive steady states to a predator-prey model, Proc. Roy. Soc. Edinburgh Sect. A, 136 (2006), 759-778. doi: 10.1017/S0308210500004704. [9] S. R. Dubar, Travelling wave solutions of diffusive Lotka-Volterra equations, Journal of Mathematical Biology, 17 (1983), 11-32. doi: 10.1007/BF00276112. [10] S. R. Dubar, Traveling wave solutions of diffusive Lotka-Volterra equations: a heteroclinic connection in $R^4$, Transactions of American Mathematical Society, 286 (1984), 557-594. doi: 10.2307/1999810. [11] S. R. Dubar, Traveling waves in diffusive predator-prey equations: periodic orbits and point-to-periodic heteroclinic orbits, SIAM Journal on Applied Mathematics, 46 (1986), 1057-1078. doi: 10.1137/0146063. [12] W. Ding and W. Huang, Traveling wave solutions for some classes of diffusive predator-prey models, Journal of Dynamics and Differential Equations, 28 (2016), 1293-1308. doi: 10.1007/s10884-015-9472-8. [13] W. Ellison and F. Ellison, Prime Numbers, A Wiley-Interscience Publication, John Wiley & Sons Inc., New York, 1985. [14] R. Gardner, Existence of traveling wave solutions of predator-prey system via the connection index, SIAM Journal on Applied Mathematics, 44 (1984), 56-79. doi: 10.1137/0144006. [15] C.-H. Hsu, C.-R. Yang, T.-H. Yang and T.-S. Yang, Existence of traveling wave solutions for diffusive predator-prayer type model, J. of Differential Equations, 252 (2012), 3040-3075. doi: 10.1016/j.jde.2011.11.008. [16] Y. L. Huang and G. Lin, Traveling wave solutions in a diffusive system with two preys and one predator, Journal of Mathematical Analysis and Applications, 41 (2014), 163-184. doi: 10.1016/j.jmaa.2014.03.085. [17] J. Huang and X. Zou, Existence of traveling wavefronts of delayed reaction-diffusion systems without monotonicity, Discrete and Continuous Dynamical System, 9 (2003), 925-936. doi: 10.3934/dcds.2003.9.925. [18] J. Huang, G. Lu and S. Ruan, Existence of traveling wave solutions in diffusive predator-prey model, Journal of Mathematical Biology, 46 (2003), 132-152. doi: 10.1007/s00285-002-0171-9. [19] W. Huang, Traveling wave solutions for a class of predator-prey system, Journal of Dynamics and Differential Equations, 24 (2012), 633-644. doi: 10.1007/s10884-012-9255-4. [20] W. Huang, A geometric approach in the study of traveling waves for some classes of non-monotone reaction-diffusion systems, J. Differential Equations, 260 (2016), 2190-2224. doi: 10.1016/j.jde.2015.09.060. [21] W. Khellaf and N. Hamri, Boundedness and global stability for a predator-prey system with the Beddington-DeAngelis functional response, Differ. Equ. Nonlinear Mech., 2010 (2010), Article ID 813289. doi: 10.1155/2010/813289. [22] 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. doi: 10.1088/0951-7715/19/6/003. [23] W. T. Li and S. L. Wu, Traveling waves in a diffusive predator-prey model with holling type-Ⅲ functional response, Chaos Soliton Fractals, 37 (2008), 476-486. doi: 10.1016/j.chaos.2006.09.039. [24] G. Lin, Invasion traveling wave solutions of a predator-prey system, Nonlinear Anal., 96 (2014), 4-58. doi: 10.1016/j.na.2013.10.024. [25] G. Lin, W. T. Li and M. Ma, Traveling wave solutions in delayed reaction diffusion systems with applications to multi-species models, Discrete and Continuous Dynamical System-Series B, 13 (2010), 393-414. doi: 10.3934/dcdsb.2010.13.393. [26] X. Lin, C. Wu and P. Weng, Traveling wave solutions for a predator-prey system with sigmoidal response function, Journal of Dynamics and Differential Equations, 23 (2011), 903-921. doi: 10.1007/s10884-011-9220-7. [27] D. Liang, P. Weng and J. Wu, Travelling wave solutions in a delayed predator-prey diffusion PDE system point-to-periodic and point-to-point waves, IMA Journal of Applied Mathematics, 77 (2012), 516-545. doi: 10.1093/imamat/hxr031. [28] G. Lin and S. Ruan, Traveling wave solutions for delayed reaction-diffusion systems and applications to diffusive Lotka-Volterra competition models with distributed delays, Journal of Dynamics and Differential Equations, 23 (2014), 583-605. doi: 10.1007/s10884-014-9355-4. [29] 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. doi: 10.3934/dcdsb.2015.20.1135. [30] S. Ma, Traveling wavefronts for delayed reaction-diffusion system via a fixed point theorem, J. Differential Equations, 171 (2001), 294-314. doi: 10.1006/jdeq.2000.3846. [31] S. Pan, Convergence and traveling wave solutions for a predator-prey system with distributed delays, Mediterr. J. Math., 14 (2017). doi: 10.1007/s00009-017-0905-y. [32] S. Pan, Invasion speed of a predator-prey system, Appl. Math. Lett., 74 (2017), 4-51. doi: 10.1016/j.aml.2017.05.014. [33] H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, AMS, Providence, 1995. [34] E. Trafimchuk, M. Pinto and S. Trafimchuk, Traveling waves for a model of the Belousov-Zhabotinsky reaction, J. of Differential Equations, 254 (2013), 3690-3714. doi: 10.1016/j.jde.2013.02.005. [35] X. S. Wang, H. Wang and J. Wu, Traveling waves of diffusive predator-prey systems: disease outbreak propagation, Discrete and Continuous Dynamical System-Series A, 32 (2012), 3303-3324. doi: 10.3934/dcds.2012.32.3303. [36] D. V. Widder, The Laplace Transform, Princeton University Press, Princeton, NJ, 1941. [37] Q. Ye, Z, Li, M. X. Wang and Y. Wu, Introduction to Reaction-Diffusion Equations, $2^{nd}$ edition, Science Press, Beijing, 2011. [38] J. Zhou, Positive solutions of a diffusive predator-prey model with modified Leslie-Gower and Holling-type Ⅱ schemes, Journal of Mathematical Analysis and Applications, 389 (2012), 1380-1393. doi: 10.1016/j.jmaa.2012.01.013.

show all references

##### References:
 [1] S. Ai, Y. Du and R. Peng, Traveling waves for a generalized Holling-Tanner predator-prey model, J. Differential Equations, 263 (2017), 7782-7814. doi: 10.1016/j.jde.2017.08.021. [2] M. A. Aziz-Alaoui and M. D. Okiye, Boundedness and global stability for a predator-prey model with modified Leslie-Gower and Holling-type II schemes, Applied Mathematics Letters, 16 (2003), 1069-1075. doi: 10.1016/S0893-9659(03)90096-6. [3] J. R. Beddington, Mutual interference between parasites or predators and it's effect on searching efficiency, J. Anim. Ecol., 44 (1975), 331-340. [4] A. Boumenir and V. Nguyen, Erron Theorem in the monotone iteration method for traveling waves in delayed reaction-diffusion equations, J. Differential Equations, 244 (2008), 1551-1570. doi: 10.1016/j.jde.2008.01.004. [5] J. B. Conway, Functions of One Complex Variable, $2^{nd}$ edition, Springer-Verlag, New York, 1978. [6] W. Ding and W. Huang, Traveling wave solutions for some classes of diffusive predator-prey models, Journal of Dynamics and Differential Equations, 28 (2016), 1293-1308. doi: 10.1007/s10884-015-9472-8. [7] Y. H. Du and S. B. Hsu, A diffusive predator-prey model in heterogeneous environment, J. Differential Equations, 203 (2004), 331-364. doi: 10.1016/j.jde.2004.05.010. [8] Y. H. Du and M. X. Wang, Asymptotic behaviour of positive steady states to a predator-prey model, Proc. Roy. Soc. Edinburgh Sect. A, 136 (2006), 759-778. doi: 10.1017/S0308210500004704. [9] S. R. Dubar, Travelling wave solutions of diffusive Lotka-Volterra equations, Journal of Mathematical Biology, 17 (1983), 11-32. doi: 10.1007/BF00276112. [10] S. R. Dubar, Traveling wave solutions of diffusive Lotka-Volterra equations: a heteroclinic connection in $R^4$, Transactions of American Mathematical Society, 286 (1984), 557-594. doi: 10.2307/1999810. [11] S. R. Dubar, Traveling waves in diffusive predator-prey equations: periodic orbits and point-to-periodic heteroclinic orbits, SIAM Journal on Applied Mathematics, 46 (1986), 1057-1078. doi: 10.1137/0146063. [12] W. Ding and W. Huang, Traveling wave solutions for some classes of diffusive predator-prey models, Journal of Dynamics and Differential Equations, 28 (2016), 1293-1308. doi: 10.1007/s10884-015-9472-8. [13] W. Ellison and F. Ellison, Prime Numbers, A Wiley-Interscience Publication, John Wiley & Sons Inc., New York, 1985. [14] R. Gardner, Existence of traveling wave solutions of predator-prey system via the connection index, SIAM Journal on Applied Mathematics, 44 (1984), 56-79. doi: 10.1137/0144006. [15] C.-H. Hsu, C.-R. Yang, T.-H. Yang and T.-S. Yang, Existence of traveling wave solutions for diffusive predator-prayer type model, J. of Differential Equations, 252 (2012), 3040-3075. doi: 10.1016/j.jde.2011.11.008. [16] Y. L. Huang and G. Lin, Traveling wave solutions in a diffusive system with two preys and one predator, Journal of Mathematical Analysis and Applications, 41 (2014), 163-184. doi: 10.1016/j.jmaa.2014.03.085. [17] J. Huang and X. Zou, Existence of traveling wavefronts of delayed reaction-diffusion systems without monotonicity, Discrete and Continuous Dynamical System, 9 (2003), 925-936. doi: 10.3934/dcds.2003.9.925. [18] J. Huang, G. Lu and S. Ruan, Existence of traveling wave solutions in diffusive predator-prey model, Journal of Mathematical Biology, 46 (2003), 132-152. doi: 10.1007/s00285-002-0171-9. [19] W. Huang, Traveling wave solutions for a class of predator-prey system, Journal of Dynamics and Differential Equations, 24 (2012), 633-644. doi: 10.1007/s10884-012-9255-4. [20] W. Huang, A geometric approach in the study of traveling waves for some classes of non-monotone reaction-diffusion systems, J. Differential Equations, 260 (2016), 2190-2224. doi: 10.1016/j.jde.2015.09.060. [21] W. Khellaf and N. Hamri, Boundedness and global stability for a predator-prey system with the Beddington-DeAngelis functional response, Differ. Equ. Nonlinear Mech., 2010 (2010), Article ID 813289. doi: 10.1155/2010/813289. [22] 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. doi: 10.1088/0951-7715/19/6/003. [23] W. T. Li and S. L. Wu, Traveling waves in a diffusive predator-prey model with holling type-Ⅲ functional response, Chaos Soliton Fractals, 37 (2008), 476-486. doi: 10.1016/j.chaos.2006.09.039. [24] G. Lin, Invasion traveling wave solutions of a predator-prey system, Nonlinear Anal., 96 (2014), 4-58. doi: 10.1016/j.na.2013.10.024. [25] G. Lin, W. T. Li and M. Ma, Traveling wave solutions in delayed reaction diffusion systems with applications to multi-species models, Discrete and Continuous Dynamical System-Series B, 13 (2010), 393-414. doi: 10.3934/dcdsb.2010.13.393. [26] X. Lin, C. Wu and P. Weng, Traveling wave solutions for a predator-prey system with sigmoidal response function, Journal of Dynamics and Differential Equations, 23 (2011), 903-921. doi: 10.1007/s10884-011-9220-7. [27] D. Liang, P. Weng and J. Wu, Travelling wave solutions in a delayed predator-prey diffusion PDE system point-to-periodic and point-to-point waves, IMA Journal of Applied Mathematics, 77 (2012), 516-545. doi: 10.1093/imamat/hxr031. [28] G. Lin and S. Ruan, Traveling wave solutions for delayed reaction-diffusion systems and applications to diffusive Lotka-Volterra competition models with distributed delays, Journal of Dynamics and Differential Equations, 23 (2014), 583-605. doi: 10.1007/s10884-014-9355-4. [29] 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. doi: 10.3934/dcdsb.2015.20.1135. [30] S. Ma, Traveling wavefronts for delayed reaction-diffusion system via a fixed point theorem, J. Differential Equations, 171 (2001), 294-314. doi: 10.1006/jdeq.2000.3846. [31] S. Pan, Convergence and traveling wave solutions for a predator-prey system with distributed delays, Mediterr. J. Math., 14 (2017). doi: 10.1007/s00009-017-0905-y. [32] S. Pan, Invasion speed of a predator-prey system, Appl. Math. Lett., 74 (2017), 4-51. doi: 10.1016/j.aml.2017.05.014. [33] H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, AMS, Providence, 1995. [34] E. Trafimchuk, M. Pinto and S. Trafimchuk, Traveling waves for a model of the Belousov-Zhabotinsky reaction, J. of Differential Equations, 254 (2013), 3690-3714. doi: 10.1016/j.jde.2013.02.005. [35] X. S. Wang, H. Wang and J. Wu, Traveling waves of diffusive predator-prey systems: disease outbreak propagation, Discrete and Continuous Dynamical System-Series A, 32 (2012), 3303-3324. doi: 10.3934/dcds.2012.32.3303. [36] D. V. Widder, The Laplace Transform, Princeton University Press, Princeton, NJ, 1941. [37] Q. Ye, Z, Li, M. X. Wang and Y. Wu, Introduction to Reaction-Diffusion Equations, $2^{nd}$ edition, Science Press, Beijing, 2011. [38] J. Zhou, Positive solutions of a diffusive predator-prey model with modified Leslie-Gower and Holling-type Ⅱ schemes, Journal of Mathematical Analysis and Applications, 389 (2012), 1380-1393. doi: 10.1016/j.jmaa.2012.01.013.
graphs of functions $g_1(\cdot)$ and $g_2(\cdot).$
Graphs of $\overline{\phi}_n(\xi)$ and $\underline{\phi}_n(\xi)$ with $n = 1, 2$.
The regions of $\Omega_1, \Omega_2$, line segments $L_1, L_2$ and tangent line $L_{2T}$.
The regions of $\Omega_3, \Omega_4$, line segments $L_3, L_4$ and tangent line $L_{4T}$.
The strictly contracting rectangle $[{\bf{a}}(s), {\bf{b}}(s)]$ with $s \in [0, 1]$.
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