Traveling wavefronts of a prey-predator diffusion system with stage-structure and harvesting. (English) Zbl 1207.92047

Summary: From a biological point of view, we consider a prey-predator-type free diffusion fishery model with stage-structure and harvesting. First, we study the stability of the non-negative constant equilibria. In particular, the effect of harvesting on the stability of equilibria is discussed and supported with numerical simulations. Then, employing the upper and lower solution method, we show that when the wave speed is large enough there exists a traveling wavefront connecting the zero solution to the positive equilibrium of the system. Numerical simulation is also carried out to illustrate the main result.


92D40 Ecology
35Q92 PDEs in connection with biology, chemistry and other natural sciences
91B76 Environmental economics (natural resource models, harvesting, pollution, etc.)


Full Text: DOI


[1] Azar, C.; Holmberg, J.; Lindgren, K., Stability analysis of harvesting in a predator-prey model, J. Theoret. Biol., 174, 13-19 (1995)
[2] Brauer, F.; Soudack, A. C., Stability regions and transition phenomena for harvested predator-prey systems, J. Math. Biol., 7, 319-337 (1979) · Zbl 0397.92019
[3] Brauer, F.; Soudack, A. C., Coexistence properties of some predator-prey systems under constant rate harvesting and stocking, J. Math. Biol., 12, 101-114 (1981) · Zbl 0482.92015
[4] Dai, G.; Tang, M., Coexistence region and global dynamics of a harvested predator-prey system, SIAM J. Appl. Math., 58, 193-210 (1998) · Zbl 0916.34034
[5] Gao, S.; Chen, L., The effect of seasonal harvesting on a single-species discrete population model with stage structure and birth pulses, Chaos Solitons Fractals, 24, 1013-1023 (2005) · Zbl 1061.92059
[6] He, Z. R., Optimal harvesting of two competing species with age dependence, Nonlinear Anal. RWA, 7, 769-788 (2006) · Zbl 1105.35303
[7] da Silveira Costa, M. I., Harvesting induced fluctuations: insights from a threshold management policy, Math. Biosci., 205, 77-82 (2007) · Zbl 1106.92069
[8] Kar, T. K., Modelling and analysis of a harvested prey-predator system incorporating a prey refuge, J. Comput. Appl. Math., 185, 19-33 (2006) · Zbl 1071.92041
[9] Liao, X. Y.; Cheng, S. S., Convergent and divergent solutions of a discrete nonautonomous Lotka-Volterra model, Tamkang J. Math., 36, 337-344 (2005) · Zbl 1103.39012
[10] Lu, Z.; Chi, X.; Chen, L., Global attractivity of nonautonomous stage-structured population models with dispersal and harvest, J. Comput. Appl. Math., 166, 411-425 (2004) · Zbl 1061.34033
[11] Luo, Z.; Li, W. T.; Wang, M., Optimal harvesting control problem for linear periodic age-dependent population dynamics, Appl. Math. Comput., 151, 789-800 (2004) · Zbl 1043.92039
[12] Martin, A.; Ruan, S., Predator-prey models with delay and prey harvesting, J. Math. Biol., 43, 247-267 (2001) · Zbl 1008.34066
[13] Ngom, D.; Iggidr, A.; Guiro, A.; Ouahbi, A., An observer for a nonlinear age-structured model of a harvested fish population, Math. Biosci. Eng., 5, 337-354 (2008) · Zbl 1159.92318
[14] Song, X.; Chen, L., Optimal harvesting and stability for a two-species competitive system with stage structure, Math. Biosci., 170, 173-186 (2001) · Zbl 1028.34049
[15] Song, X.; Chen, L., Global asymptotic stability of a two species competitive system with stage structure and harvesting, Commun. Nonlinear Sci. Numer. Simul., 6, 81-87 (2001) · Zbl 0994.34066
[16] Tang, S.; Chen, L., The effect of seasonal harvesting on stage-structured population models, J. Math. Biol., 48, 357-374 (2004) · Zbl 1058.92051
[17] Xiao, D.; Jennings, L. S., Bifurcations of a ratio-dependent predator-prey system with constant rate harvesting, SIAM J. Appl. Math., 65, 737-753 (2005) · Zbl 1094.34024
[18] Xiao, D.; Li, W.; Han, M., Dynamics in a ratio-dependent predator-prey model with predator harvesting, J. Math. Anal. Appl., 324, 14-29 (2006) · Zbl 1122.34035
[19] Fahmy, E. S., Travelling wave solutions for some time-delayed equations through factorizations, Chaos Solitons Fractals, 38, 1209-1216 (2008) · Zbl 1152.35438
[20] Ge, Z.; He, Y., Traveling wavefronts for a two-species predator-prey system with diffusion terms and stage structure, Appl. Math. Model., 33, 1356-1365 (2009) · Zbl 1396.35034
[21] Hou, X.; Leung, A. W., Traveling wave solutions for a competitive reaction-diffusion system and their asymptotics, Nonlinear Anal. RWA, 9, 2196-2213 (2008) · Zbl 1156.35405
[22] Hou, X.; Li, Y., Local stability of traveling-wave solutions of nonlinear reaction-diffusion equations, Discrete Contin. Dyn. Syst., 15, 681-701 (2006) · Zbl 1114.35018
[23] Huang, J.; Zou, X., Traveling wavefronts in diffusive and cooperative Lotka-Volterra system with delays, J. Math. Anal. Appl., 271, 455-466 (2002) · Zbl 1017.35116
[24] Li, B.; Weinberger, H. F.; Lewis, M. A., Spreading speeds as slowest wave speeds for cooperative systems, Math. Biosci., 196, 82-98 (2005) · Zbl 1075.92043
[25] Lin, G.; Hong, Y., Travelling wave fronts in a vector disease model with delay, Appl. Math. Model., 32, 2831-2838 (2008) · Zbl 1167.35392
[26] Ma, S., Traveling wavefronts for delayed reaction-diffusion systems via a fixed point theorem, J. Differential Equations, 171, 294-314 (2001) · Zbl 0988.34053
[27] Smith, H. L.; Zhao, X. Q., Global asymptotic stability of traveling waves in delayed reaction-diffusion equations, SIAM J. Math. Anal., 31, 514-534 (2000) · Zbl 0941.35125
[28] So, J. W.H.; Zou, X., Traveling waves for the diffusive Nicholson’s blowflies equation, Appl. Math. Comput., 122, 385-392 (2001) · Zbl 1027.35051
[29] Wang, Z. C.; Li, W. T.; Ruan, S., Existence and stability of traveling wave fronts in reaction advection diffusion equations with nonlocal delay, J. Differential Equations, 238, 153-200 (2007) · Zbl 1124.35089
[30] Weng, P.; Zhao, X. Q., Spreading speed and traveling waves for a multi-type SIS epidemic model, J. Differential Equations, 229, 270-296 (2006) · Zbl 1126.35080
[31] Wu, J.; Zou, X., Traveling wave fronts of reaction-diffusion systems with delay, J. Dynam. Differential Equations, 13, 651-687 (2001) · Zbl 0996.34053
[32] Zou, X., Delay induced traveling wave fronts in reaction diffusion equations of KPP-Fisher type, J. Comput. Appl. Math., 146, 309-321 (2002) · Zbl 1058.35114
[33] Canosa, J., On a nonlinear diffusion equation describing population growth, IBM J. Res. Dev., 17, 307-313 (1973) · Zbl 0266.65080
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