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March
2017, 22(2): 627-645.
doi: 10.3934/dcdsb.2017030
Traveling wave solutions in a diffusive producer-scrounger model
School of Mathematics, South China Normal University, Guangzhou, Guangdong 510631, China |
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 and Continuous Dynamical Systems - B,
2017, 22
(2)
: 627-645.
doi: 10.3934/dcdsb.2017030
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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.
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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.
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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.
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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. |
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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. |
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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. |
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. |
| [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. |
| [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. |
| [4] |
S. Dunbar,
Traveling wave solutions of diffusive Lotka-Volterra equations, J. Math. Biol., 17 (1983), 11-32.
doi: 10.1007/BF00276112. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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.
|
| [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. |
| [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. |
| [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. |
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