• Previous Article
    On sufficiency issues, first integrals and exact solutions of Uzawa-Lucas model with unskilled labor
  • DCDS-S Home
  • This Issue
  • Next Article
    From conservative to dissipative non-linear differential systems. An application to the cardio-respiratory regulation
doi: 10.3934/dcdss.2020129

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

1. 

Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China

2. 

Department of Mathematics, Anqing Normal University, Anqing 246133, China

* Corresponding author: Hongyong Zhao

Received  November 2018 Revised  March 2019 Published  November 2019

Fund Project: 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)

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.

Citation: Hongyong Zhao, Daiyong Wu. Point to point traveling wave and periodic traveling wave induced by Hopf bifurcation for a diffusive predator-prey system. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020129
References:
[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.  Google Scholar

[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.  Google Scholar

[3] S. CantrellC. Cosner and S. G. Ruan, Spatial Ecology, Mathematical and Computational Biology Series, Chapman & Hall/CRC Mathematical and Computational Biology Series, CRC Press, Boca Raton, FL, 2010.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[6]

S. R. Dunbar, Travelling wave solutions of diffusive lotka-volterra equations, J. Math. Biol., 17 (1983), 11-32.  doi: 10.1007/BF00276112.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

J. E. Marsden and M. McCracken, The Hopf Bifurcation and Its Applications, Applied Mathematical Sciences, Vol. 19. Springer-Verlag, New York, 1976.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[25] N. Shigesada and K. Kawasaki, Biology Invasions: Theory and Practice, Oxford University Press, Oxford, 1997.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[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.  Google Scholar

[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.  Google Scholar

[3] S. CantrellC. Cosner and S. G. Ruan, Spatial Ecology, Mathematical and Computational Biology Series, Chapman & Hall/CRC Mathematical and Computational Biology Series, CRC Press, Boca Raton, FL, 2010.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[6]

S. R. Dunbar, Travelling wave solutions of diffusive lotka-volterra equations, J. Math. Biol., 17 (1983), 11-32.  doi: 10.1007/BF00276112.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

J. E. Marsden and M. McCracken, The Hopf Bifurcation and Its Applications, Applied Mathematical Sciences, Vol. 19. Springer-Verlag, New York, 1976.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[25] N. Shigesada and K. Kawasaki, Biology Invasions: Theory and Practice, Oxford University Press, Oxford, 1997.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

Na Min, Mingxin Wang. Hopf bifurcation and steady-state bifurcation for a Leslie-Gower prey-predator model with strong Allee effect in prey. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 1071-1099. doi: 10.3934/dcds.2019045

[2]

Qizhen Xiao, Binxiang Dai. Heteroclinic bifurcation for a general predator-prey model with Allee effect and state feedback impulsive control strategy. Mathematical Biosciences & Engineering, 2015, 12 (5) : 1065-1081. doi: 10.3934/mbe.2015.12.1065

[3]

Miljana JovanoviĆ, Marija KrstiĆ. Extinction in stochastic predator-prey population model with Allee effect on prey. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2651-2667. doi: 10.3934/dcdsb.2017129

[4]

Zuolin Shen, Junjie Wei. Hopf bifurcation analysis in a diffusive predator-prey system with delay and surplus killing effect. Mathematical Biosciences & Engineering, 2018, 15 (3) : 693-715. doi: 10.3934/mbe.2018031

[5]

Xiaoling Zou, Dejun Fan, Ke Wang. Stationary distribution and stochastic Hopf bifurcation for a predator-prey system with noises. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1507-1519. doi: 10.3934/dcdsb.2013.18.1507

[6]

Yujing Gao, Bingtuan Li. Dynamics of a ratio-dependent predator-prey system with a strong Allee effect. Discrete & Continuous Dynamical Systems - B, 2013, 18 (9) : 2283-2313. doi: 10.3934/dcdsb.2013.18.2283

[7]

Yuying Liu, Yuxiao Guo, Junjie Wei. Dynamics in a diffusive predator-prey system with stage structure and strong allee effect. Communications on Pure & Applied Analysis, 2020, 19 (2) : 883-910. doi: 10.3934/cpaa.2020040

[8]

Jiang Liu, Xiaohui Shang, Zengji Du. Traveling wave solutions of a reaction-diffusion predator-prey model. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1063-1078. doi: 10.3934/dcdss.2017057

[9]

Xiaoyuan Chang, Junjie Wei. Stability and Hopf bifurcation in a diffusive predator-prey system incorporating a prey refuge. Mathematical Biosciences & Engineering, 2013, 10 (4) : 979-996. doi: 10.3934/mbe.2013.10.979

[10]

Eduardo González-Olivares, Betsabé González-Yañez, Jaime Mena-Lorca, José D. Flores. Uniqueness of limit cycles and multiple attractors in a Gause-type predator-prey model with nonmonotonic functional response and Allee effect on prey. Mathematical Biosciences & Engineering, 2013, 10 (2) : 345-367. doi: 10.3934/mbe.2013.10.345

[11]

Yun Kang, Sourav Kumar Sasmal, Amiya Ranjan Bhowmick, Joydev Chattopadhyay. Dynamics of a predator-prey system with prey subject to Allee effects and disease. Mathematical Biosciences & Engineering, 2014, 11 (4) : 877-918. doi: 10.3934/mbe.2014.11.877

[12]

Pengmiao Hao, Xuechen Wang, Junjie Wei. Global Hopf bifurcation of a population model with stage structure and strong Allee effect. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 973-993. doi: 10.3934/dcdss.2017051

[13]

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

[14]

Shanshan Chen, Jianshe Yu. Stability and bifurcation on predator-prey systems with nonlocal prey competition. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 43-62. doi: 10.3934/dcds.2018002

[15]

Na Min, Mingxin Wang. Dynamics of a diffusive prey-predator system with strong Allee effect growth rate and a protection zone for the prey. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1721-1737. doi: 10.3934/dcdsb.2018073

[16]

Wenjie Ni, Mingxin Wang. Dynamical properties of a Leslie-Gower prey-predator model with strong Allee effect in prey. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3409-3420. doi: 10.3934/dcdsb.2017172

[17]

Moitri Sen, Malay Banerjee, Yasuhiro Takeuchi. Influence of Allee effect in prey populations on the dynamics of two-prey-one-predator model. Mathematical Biosciences & Engineering, 2018, 15 (4) : 883-904. doi: 10.3934/mbe.2018040

[18]

Xiang-Sheng Wang, Haiyan Wang, Jianhong Wu. Traveling waves of diffusive predator-prey systems: Disease outbreak propagation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (9) : 3303-3324. doi: 10.3934/dcds.2012.32.3303

[19]

Bing Zeng, Shengfu Deng, Pei Yu. Bogdanov-Takens bifurcation in predator-prey systems. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020130

[20]

Shanshan Chen, Junping Shi, Junjie Wei. The effect of delay on a diffusive predator-prey system with Holling Type-II predator functional response. Communications on Pure & Applied Analysis, 2013, 12 (1) : 481-501. doi: 10.3934/cpaa.2013.12.481

2018 Impact Factor: 0.545

Metrics

  • PDF downloads (14)
  • HTML views (28)
  • Cited by (0)

Other articles
by authors

[Back to Top]