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March  2017, 22(2): 627-645. doi: 10.3934/dcdsb.2017030

Traveling wave solutions in a diffusive producer-scrounger model

Junhao Wen and  Peixuan Weng ,

School of Mathematics, South China Normal University, Guangzhou, Guangdong 510631, China

Corresponding author: Peixuan Weng, Tel:0086-20-85213533

Received  March 2016 Revised  September 2016 Published  December 2016

Fund Project: Supported by the NSF of China (11171120) and the Natural Science Foundation of Guangdong Province (2016A030313426)

This paper looks into the stability of equilibria, existence and non-existence of traveling wave solutions in a diffusive producer-scrounger model. We find that the existence and non-existence of traveling wave solutions are determined by a minimum wave speed $c_{m}$ and a threshold value $R_{0}$. By constructing a suitable invariant convex set $Γ$ and applying Schauder fixed point theorem, the existence for $c>c_{m}, R_{0}>1$ was established. Besides, a Lyapunov function is constructed subtly to explore the asymptotic behaviors of traveling wave solutions. The non-existences of traveling wave solutions for both $c < c_{m}, R_{0}> 1$ and $R_{0}≤1, c > 0$ were obtained by two-sides Laplace transform and reduction method to absurdity.

Keywords: Diffusive producer-scrounger model, linear stability, traveling wave solution, invariant convex set, Lyapunov function.
Mathematics Subject Classification: 35K57, 92D2.
Citation: Junhao Wen, Peixuan Weng. Traveling wave solutions in a diffusive producer-scrounger model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 627-645. doi: 10.3934/dcdsb.2017030
References:
[1]

H. Berestycki and F. Hamel, Quenching and propagation in KPP reaction-diffusion equations with a heat loss, Arch. Ration. Mech. Anal., 178 (2005), 57-80. doi: 10.1007/s00205-005-0367-4. Google Scholar

[2]

C. Cosner and A. L. Nevai, Spatial population dynamics in a producer-scrounger model, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1591-1607. doi: 10.3934/dcdsb.2015.20.1591. Google Scholar

[3]

A. Ducrot and P. Magal, Travelling wave solutions for an infection-age structured epidemic model with external supplies, Nonlinearity, 24 (2011), 2891-2911. doi: 10.1088/0951-7715/24/10/012. Google Scholar

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S. Dunbar, Traveling wave solutions of diffusive Lotka-Volterra equations, J. Math. Biol., 17 (1983), 11-32. doi: 10.1007/BF00276112. Google Scholar

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S. Dunbar, Traveling wave solutions of diffusive Lotka-Volterra equations: A heteroclinic connection in $R^{4}$, Trans. Amer. Math. Soc., 286 (1984), 557-594. doi: 10.2307/1999810. Google Scholar

[6]

S. Dunbar, Traveling waves in diffusive predator-prey equations: Periodic orbits and point-to periodic heteroclinic orbits, SIAM J. Appl. Math, 46 (1986), 1057-1078. doi: 10.1137/0146063. Google Scholar

[7]

Q. T. Gan, R. Xu, Y. L. Li and R. X. Hu, Travelling waves in an infectious disease model with a fixed latent period and a spatio-temporal delay, Math. Comput. Model., 53 (2011), 814-823. doi: 10.1016/j.mcm.2010.10.018. Google Scholar

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W. M. Hirsch, H. Hanisch and J. P. Gabriel, Differential equation models of some parasitic infections: methods for the study of asymptotic behavior, Comm. Pure App. Math., 38 (1985), 733-753. doi: 10.1002/cpa.3160380607. Google Scholar

[9]

K. Hong and P. X. Weng, Stability and traveling waves of a stage-structured predator-prey model with Holling type-Ⅱ functional response and harvesting, Nonlinear Anal. RWA, 14 (2013), 83-103. doi: 10.1016/j.nonrwa.2012.05.004. Google Scholar

[10]

C. H. Hsu, C. R. Yang, T. H. Yang and T. S. Yang, Existence of traveling wave solutions for diffusive predator-prey type systems, J. Diff. Eqns., 252 (2012), 3040-3075. doi: 10.1016/j.jde.2011.11.008. Google Scholar

[11]

J. H. Huang, G. Lu and S. G. Ruan, Existence of traveling wave solutions in a diffusive predator-prey model, J. Math. Biol., 46 (2003), 132-153. doi: 10.1007/s00285-002-0171-9. Google Scholar

[12]

W. Z. Huang, Traveling wave solutions for a class of predator-prey systems, J. Dyn. Diff. Equat., 24 (2012), 633-644. doi: 10.1007/s10884-012-9255-4. Google Scholar

[13]

W. T. Li and S. L. Wu, Traveling waves in a diffusive predator-prey model with holling type-Ⅲ functional response, Chaos, Solitons and Fractals, 37 (2008), 476-486. doi: 10.1016/j.chaos.2006.09.039. Google Scholar

[14]

W. T. Li and F. Y. Yang, Traveling waves for a nonlocal dispersal SIR model with standrad incidence, J. Integral Equ. Appl., 26 (2014), 243-273. doi: 10.1216/JIE-2014-26-2-243. Google Scholar

[15]

Y. Li, W. T. Li and F. Y. Yang, Traveling waves for a nonlocal dispersal SIR model with delay and external supplies, Appl. Math. Comput., 247 (2014), 723-740. doi: 10.1016/j.amc.2014.09.072. Google Scholar

[16]

Y. Li, W. T. Li and G. Lin, Traveling waves of a delayed diffusive SIR epidemic model, Communications on Pure and Applied Analysis, 14 (2015), 1001-1022. doi: 10.3934/cpaa.2015.14.1001. Google Scholar

[17]

G. Lin, W. T. Li and M. Ma, Traveling wave solutions in delayed reaction diffusion systems with applications to multi-species models, Discrete Contin. Dyn. Syst. Ser. B, 13 (2010), 393-414. doi: 10.3934/dcdsb.2010.13.393. Google Scholar

[18]

X. B. Lin, P. X. Weng and C. F. Wu, Traveling wave solutions for a predator-prey system with sigmoidal response function, J. Dyn. Diff. Equat., 23 (2011), 903-921. doi: 10.1007/s10884-011-9220-7. Google Scholar

[19]

Z. C. Wang and J. H. Wu, Travelling waves of a diffusive Kermack-McKendrick epidemic model with non-local delayed transmission, Proc. R. Soc. Ser. A, 466 (2010), 237-264. doi: 10.1098/rspa.2009.0377. Google Scholar

[20]

X. S. Wang, H. Wang and J. H. Wu, Traveling waves of diffusive predator-prey systems: Disease outbreak propagation, Discrete Contin. Dyn. Syst. Ser. B, 32 (2012), 3303-3324. doi: 10.3934/dcds.2012.32.3303. Google Scholar

[21]

Q. R. Wang and K. Zhou, Traveling wave solutions in delayed reaction-diffusion systems with mixed monotonicity, J. Com. Appl. Math., 233 (2010), 2549-2562. doi: 10.1016/j.cam.2009.11.002. Google Scholar

[22]

Z. Q. Xu and P. X. Weng, Traveling waves in a diffusive predator-prey model with general functional response, Electron. J. Differ. Eq., 2012 (2012), 1-13. Google Scholar

[23]

Z. T. Xu, Traveling waves in a Kermack-McKendrick epidemic model with diffusion and latent period, Nonlinear Anal., 111 (2014), 66-81. doi: 10.1016/j.na.2014.08.012. Google Scholar

[24]

F. Y. Yang, Y. Li, W. T. Li and Z. C. Wang, Traveling waves in a nonlocal dispersal Kermack-McKendrick epidemic model, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 1969-1993. doi: 10.3934/dcdsb.2013.18.1969. Google Scholar

[25]

G. B. Zhang, W. T. Li and G. Lin, Traveling waves in delayed predator-prey systems with nonlocal diffusion and stage structure, Math. Comput. Modelling, 49 (2009), 1021-1029. doi: 10.1016/j.mcm.2008.09.007. Google Scholar

show all references

References:
[1]

H. Berestycki and F. Hamel, Quenching and propagation in KPP reaction-diffusion equations with a heat loss, Arch. Ration. Mech. Anal., 178 (2005), 57-80. doi: 10.1007/s00205-005-0367-4. Google Scholar

[2]

C. Cosner and A. L. Nevai, Spatial population dynamics in a producer-scrounger model, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1591-1607. doi: 10.3934/dcdsb.2015.20.1591. Google Scholar

[3]

A. Ducrot and P. Magal, Travelling wave solutions for an infection-age structured epidemic model with external supplies, Nonlinearity, 24 (2011), 2891-2911. doi: 10.1088/0951-7715/24/10/012. Google Scholar

[4]

S. Dunbar, Traveling wave solutions of diffusive Lotka-Volterra equations, J. Math. Biol., 17 (1983), 11-32. doi: 10.1007/BF00276112. Google Scholar

[5]

S. Dunbar, Traveling wave solutions of diffusive Lotka-Volterra equations: A heteroclinic connection in $R^{4}$, Trans. Amer. Math. Soc., 286 (1984), 557-594. doi: 10.2307/1999810. Google Scholar

[6]

S. Dunbar, Traveling waves in diffusive predator-prey equations: Periodic orbits and point-to periodic heteroclinic orbits, SIAM J. Appl. Math, 46 (1986), 1057-1078. doi: 10.1137/0146063. Google Scholar

[7]

Q. T. Gan, R. Xu, Y. L. Li and R. X. Hu, Travelling waves in an infectious disease model with a fixed latent period and a spatio-temporal delay, Math. Comput. Model., 53 (2011), 814-823. doi: 10.1016/j.mcm.2010.10.018. Google Scholar

[8]

W. M. Hirsch, H. Hanisch and J. P. Gabriel, Differential equation models of some parasitic infections: methods for the study of asymptotic behavior, Comm. Pure App. Math., 38 (1985), 733-753. doi: 10.1002/cpa.3160380607. Google Scholar

[9]

K. Hong and P. X. Weng, Stability and traveling waves of a stage-structured predator-prey model with Holling type-Ⅱ functional response and harvesting, Nonlinear Anal. RWA, 14 (2013), 83-103. doi: 10.1016/j.nonrwa.2012.05.004. Google Scholar

[10]

C. H. Hsu, C. R. Yang, T. H. Yang and T. S. Yang, Existence of traveling wave solutions for diffusive predator-prey type systems, J. Diff. Eqns., 252 (2012), 3040-3075. doi: 10.1016/j.jde.2011.11.008. Google Scholar

[11]

J. H. Huang, G. Lu and S. G. Ruan, Existence of traveling wave solutions in a diffusive predator-prey model, J. Math. Biol., 46 (2003), 132-153. doi: 10.1007/s00285-002-0171-9. Google Scholar

[12]

W. Z. Huang, Traveling wave solutions for a class of predator-prey systems, J. Dyn. Diff. Equat., 24 (2012), 633-644. doi: 10.1007/s10884-012-9255-4. Google Scholar

[13]

W. T. Li and S. L. Wu, Traveling waves in a diffusive predator-prey model with holling type-Ⅲ functional response, Chaos, Solitons and Fractals, 37 (2008), 476-486. doi: 10.1016/j.chaos.2006.09.039. Google Scholar

[14]

W. T. Li and F. Y. Yang, Traveling waves for a nonlocal dispersal SIR model with standrad incidence, J. Integral Equ. Appl., 26 (2014), 243-273. doi: 10.1216/JIE-2014-26-2-243. Google Scholar

[15]

Y. Li, W. T. Li and F. Y. Yang, Traveling waves for a nonlocal dispersal SIR model with delay and external supplies, Appl. Math. Comput., 247 (2014), 723-740. doi: 10.1016/j.amc.2014.09.072. Google Scholar

[16]

Y. Li, W. T. Li and G. Lin, Traveling waves of a delayed diffusive SIR epidemic model, Communications on Pure and Applied Analysis, 14 (2015), 1001-1022. doi: 10.3934/cpaa.2015.14.1001. Google Scholar

[17]

G. Lin, W. T. Li and M. Ma, Traveling wave solutions in delayed reaction diffusion systems with applications to multi-species models, Discrete Contin. Dyn. Syst. Ser. B, 13 (2010), 393-414. doi: 10.3934/dcdsb.2010.13.393. Google Scholar

[18]

X. B. Lin, P. X. Weng and C. F. Wu, Traveling wave solutions for a predator-prey system with sigmoidal response function, J. Dyn. Diff. Equat., 23 (2011), 903-921. doi: 10.1007/s10884-011-9220-7. Google Scholar

[19]

Z. C. Wang and J. H. Wu, Travelling waves of a diffusive Kermack-McKendrick epidemic model with non-local delayed transmission, Proc. R. Soc. Ser. A, 466 (2010), 237-264. doi: 10.1098/rspa.2009.0377. Google Scholar

[20]

X. S. Wang, H. Wang and J. H. Wu, Traveling waves of diffusive predator-prey systems: Disease outbreak propagation, Discrete Contin. Dyn. Syst. Ser. B, 32 (2012), 3303-3324. doi: 10.3934/dcds.2012.32.3303. Google Scholar

[21]

Q. R. Wang and K. Zhou, Traveling wave solutions in delayed reaction-diffusion systems with mixed monotonicity, J. Com. Appl. Math., 233 (2010), 2549-2562. doi: 10.1016/j.cam.2009.11.002. Google Scholar

[22]

Z. Q. Xu and P. X. Weng, Traveling waves in a diffusive predator-prey model with general functional response, Electron. J. Differ. Eq., 2012 (2012), 1-13. Google Scholar

[23]

Z. T. Xu, Traveling waves in a Kermack-McKendrick epidemic model with diffusion and latent period, Nonlinear Anal., 111 (2014), 66-81. doi: 10.1016/j.na.2014.08.012. Google Scholar

[24]

F. Y. Yang, Y. Li, W. T. Li and Z. C. Wang, Traveling waves in a nonlocal dispersal Kermack-McKendrick epidemic model, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 1969-1993. doi: 10.3934/dcdsb.2013.18.1969. Google Scholar

[25]

G. B. Zhang, W. T. Li and G. Lin, Traveling waves in delayed predator-prey systems with nonlocal diffusion and stage structure, Math. Comput. Modelling, 49 (2009), 1021-1029. doi: 10.1016/j.mcm.2008.09.007. Google Scholar

Figure 1.  Constant equilibria of model (2)
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