Article Contents
Article Contents

# Diffusive predator-prey models with stage structure on prey and beddington-deangelis functional responses

• In this paper, we examine a diffusive predator-prey model with Beddington-DeAngelis functional response and stage structure on prey under homogeneous Neumann boundary conditions, where the discrete time delay covers the period from the birth of immature prey to their maturity. We investigate the dynamics of their permanence and the extinction of the predator, and provide sufficient conditions for the global attractiveness and the locally asymptotical stability of the semi-trivial and coexistence equilibria.

Mathematics Subject Classification: 35K40, 35K57, 92D25.

 Citation:

•  [1] R. Arditi and J. Michalski, Nonlinear food web models and their responses to increased basal productivity, in Food Webs: Integration of Patterns and Dynamics (G. A. Polis and K. O. Winemiller eds. ), Chapman and Hall, London, (1995), 122–133. doi: 10.1007/978-1-4615-7007-3_12. [2] U. Brose, R. J. Williams and N. D. Martinez, Comment on "Foraging adaptation and the relationship between food-web complexity and stability", Science, 301 (2003), 918b.doi: 10.1126/science.1085902 . [3] G. Caristi, K. P. Rybakowski and T. Wessolek, Persistence and spatial patterns in a onepredator-two-prey Lotka-Volterra model with diffusion, Annali di Mathematica Pura ed Applicata, 161 (1992), 345–377.doi: 10.1007/BF01759645. [4] W. Chen and M. Wang, Qualitative analysis of predator-prey models with BeddingtonDeAngelis functional response and diffusion, Math. Comp. Modelling, 42 (2005), 31–44.doi: 10.1016/j.mcm.2005.05.013. [5] B. Drossel, P. G. Higgs and A. J. McKane, The influence of predator-prey population dynamics on the long-term evolution of food web structure, J. Theor. Biol., 208 (2001), 91–107.doi: 10.1006/jtbi.2000.2203. [6] Y. Du and Y. Lou, Qualitative behavior of positive solutions of a predator-prey model: effects of saturation, Proc. Roy. Soc. Edinburgh Sect. A, 131 (2001), 321–349.doi: 10.1017/S0308210500000895. [7] J. K. Hale and P. Waltman, Persistence in infinite-dimensional systems, SIAM J. Math. Anal., 20 (1989), 388–395.doi: 10.1137/0520025. [8] E. E. Holmes, M. A. Lewis, J. E. Banks and R. R. Veit, Partial differential equations in ecology: spatial interactions and population dynamics, Ecology, 75 (1994), 17–29.doi: 10.2307/1939378. [9] D. Kesh, A. K. Sarkar and A. B. Roy, Persistence of two prey-one predator system with ratio-dependent predator influence, Math. Meth. Appl. Sci., 23 (2000), 347–356.doi: 10.1002/(SICI)1099-1476(20000310)23:4<347::AID-MMA117>3.0.CO;2-F. [10] W. Ko and I. Ahn, Analysis of ratio-dependent food chain model, J. Math. Anal. Appl., 335 (2007), 498–523.doi: 10.1016/j.jmaa.2007.01.089. [11] W. Ko and I. Ahn, Local stability and bifurcation of a general diffusive consumer-resource model with maturation delay, preprint. [12] W. Ko, S. Liu and I. Ahn, Asymptotical behaviors of a general diffusive consumer-resource model with maturation delay, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1715–1733.doi: 10.3934/dcdsb.2015.20.1715. [13] W. Ko and K. Ryu, Qualitative analysis of a predator-prey model with Holling type Ⅱ functional response incorporating a prey refuge, J. Differential Equations, 231 (2006), 534–550.doi: 10.1016/j.jde.2006.08.001. [14] Z. Lin, Time delayed parabolic system in a two-species competitive model with stage structure, J. Math. Anal. Appl., 315 (2006), 202–215.doi: 10.1016/j.jmaa.2005.06.012 . [15] S. Liu and E. Beretta, A stage-structured predator-prey model with the Beddington-DeAngelis type, SIAM J. Appl. Math., 66 (2006), 1101–1129.doi: 10.1137/050630003. [16] S. Liu and J. Zhang, Coexistence and stability of predator-prey model with BeddingtonDeAngelis functional response and stage structure, J. Math. Anal. Appl., 342 (2008), 446– 460.doi: 10.1016/j.jmaa.2007.12.038. [17] R. M. May, Stability and complexity in model ecosystems, IEEE Transactions on Systems, Man, and Cybernetics, 6 (1976), p887.doi: 10.1109/TSMC.1976.4309488. [18] J. D. Murray, Mathematical Biology Ⅰ: An Introduction, 3rd edition, Interdisciplinary Applied Mathematics, vol. 17, Springer, New York, 2002. doi: 10.1007/b98868. [19] J. D. Murray, Mathematical Biology Ⅱ: Spatial Models and Biomedical Applications, 3rd edition, Interdisciplinary Applied Mathematics, vol. 18, Springer, New York, 2003. doi: 10.1007/b98869. [20] A. Okubo and S. A. Levin, Diffusion and Ecological Problems: Modern Perspectives, 2nd edition, Interdisciplinary Applied Mathematics, Springer, New York, 2001. doi: 10.1007/978-1-4757-4978-6. [21] C. V. Pao, Dynamics of nonlinear parabolic systems with time delays, J. Math. Anal. Appl., 198 (1996), 751–779.doi: 10.1006/jmaa.1996.0111. [22] S. Ruan and X.-Q. Zhao, Persistence and extinction in two species reaction-diffusion systems with delays, J. Differential Equations, 156 (1999), 71–92.doi: 10.1006/jdeq.1998.3599. [23] H. L. Smith, The interaction of steady state and Hopf bifurcations in a two-predator–one-prey competition model, SIAM J. Appl. Math., 42 (1982), 27–43.doi: 10.1137/0142003. [24] D. Xu and X.-Q. Zhao, A nonlocal reaction-diffusion population model with stage structure, Canad. Appl. Math. Quart., 11 (2003), 303–319. [25] R. Xu, Global convergence of a predator-prey model with stage structure and spatio-temporal delay, Discrete Contin. Dyn. Syst. Ser. B, 5 (2011), 273–291.doi: 10.3934/dcdsb.2011.15.273. [26] T. Yi and X. Zou, Map dynamics versus dynamics of associated delay reaction-diffusion equations with a Neumann condition, Proc. R. Soc. A, 466 (2010), 2955–2973.doi: 10.1098/rspa.2009.0650.