August  2015, 20(6): 1715-1733. doi: 10.3934/dcdsb.2015.20.1715

Asymptotical behaviors of a general diffusive consumer-resource model with maturation delay

1. 

Department of Mathematics, Korea University, 2511, Sejong-Ro, Sejong, 339-700, South Korea, South Korea

2. 

The Academy of Fundamental and Interdisciplinary Science, Harbin Institute of Technology, Nan-Gang District, Harbin, 150080, China

Received  September 2013 Revised  August 2014 Published  June 2015

In this paper, we examine the asymptotic behaviors of a diffusive delayed consumer-resource model subject to homogeneous Neumann boundary conditions, where the discrete time delay covers the period from the birth of juvenile consumers to their maturity, and the predation is of a general type of functional response. We construct the threshold dynamics of the persistence and extinction of the consumer. Moreover, we establish the sufficient conditions for the global attractivity of the semitrivial and coexistence equilibria. Finally, we apply our results to the specific consumer-resource models with Beddington-DeAngelis, Crowley-Martin, and ratio-dependent type of functional responses.
Citation: Wonlyul Ko, Inkyung Ahn, Shengqiang Liu. Asymptotical behaviors of a general diffusive consumer-resource model with maturation delay. Discrete & Continuous Dynamical Systems - B, 2015, 20 (6) : 1715-1733. doi: 10.3934/dcdsb.2015.20.1715
References:
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Z. Artstein, Limiting equations and stability of nonautonomous ordinary differential equations, Appendix to J. P. LaSalle, the stability of dynamical systems,, in CBMS, (1976). Google Scholar

[2]

E. Beretta and Y. Kuang, Geometric stability switch criteria in delay differential systems with delay dependent parameters,, SIAM J. Appl. Math., 33 (2002), 1144. doi: 10.1137/S0036141000376086. Google Scholar

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W. Chen and M. Wang, Qualitative analysis of predator-prey models with Beddington-DeAngelis functional response and diffusion,, Math. Comp. Modelling, 42 (2005), 31. doi: 10.1016/j.mcm.2005.05.013. Google Scholar

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S. Chen, J. Shi and J. Wei, Global stability and Hopf bifurcation in a delayed diffusive Leslie-Gower predator-prey system,, Int. J. Bifurcation and Chaos, 22 (2012). doi: 10.1142/S0218127412500617. Google Scholar

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S. Chen, J. Shi and J. Wei, The effect of delay on a diffusive predator-prey system with Holling type-II predator functional response,, Comm. on Pure and Appl. Analy., 12 (2013), 481. doi: 10.3934/cpaa.2013.12.481. Google Scholar

[7]

M. G. Crandall and P. H. Rabinowitz, The Hopf bifurcation theorem in infinite dimensions,, Arch. Rational Mech. Anal., 67 (1977), 53. doi: 10.1007/BF00280827. Google Scholar

[8]

D. L. DeAngelis, R. A. Goldstein and R. Neill, A model for trophic interaction,, Ecology, 56 (1975), 881. Google Scholar

[9]

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. doi: 10.1017/S0308210500000895. Google Scholar

[10]

H. I. Freedman and X.-Q. Zhao, Global asymptotics in some quasimonotone reaction-diffusion systems with delays,, J. Differential Equations, 137 (1997), 340. doi: 10.1006/jdeq.1997.3264. Google Scholar

[11]

S. A. Gourley and Y. Kuang, A delay reaction-diffusion model of the spread of bacteriophage infection,, SIAM J. Appl. Math., 65 (2005), 550. doi: 10.1137/S0036139903436613. Google Scholar

[12]

S. A. Gourley and S. Ruan, Dynamics of the diffusive Nicholson's blowflies equation with distributed delay,, Proc. R. Soc. A, 130 (2000), 1275. doi: 10.1017/S0308210500000688. Google Scholar

[13]

J. K. Hale and P. Waltman, Persistence in infinite-dimensional systems,, SIAM J. Math. Anal., 20 (1989), 388. doi: 10.1137/0520025. Google Scholar

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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. doi: 10.2307/1939378. Google Scholar

[15]

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. doi: 10.1002/(SICI)1099-1476(20000310)23:4<347::AID-MMA117>3.0.CO;2-F. Google Scholar

[16]

W. Ko and I. Ahn, Analysis of ratio-dependent food chain model,, J. Math. Anal. Appl., 335 (2007), 498. doi: 10.1016/j.jmaa.2007.01.089. Google Scholar

[17]

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. doi: 10.1016/j.jde.2006.08.001. Google Scholar

[18]

A. Korobeinikov, Stability of ecosystem: Global properties of a general predator-prey model,, Mathematical Medicine and Biology, 26 (2009), 309. doi: 10.1093/imammb/dqp009. Google Scholar

[19]

Z. Lin, Time delayed parabolic system in a two-species competitive model with stage structure,, J. Math. Anal. Appl., 315 (2006), 202. doi: 10.1016/j.jmaa.2005.06.012. Google Scholar

[20]

S. Liu and E. Beretta, Stage-structured Predator-prey Model with the Beddington-DeAngelis functional response,, SIAM J. Appl. Math., 66 (2006), 1101. doi: 10.1137/050630003. Google Scholar

[21]

S. Liu and J. Zhang, Coexistence and stability of predator-prey model with Beddington-DeAngelis functional response and stage structure,, J. Math. Anal. Appl., 342 (2008), 446. doi: 10.1016/j.jmaa.2007.12.038. Google Scholar

[22]

R. H. Martin and H. L. Smith, Reaction-diffusion systems with time delays: Monotonicity, invariance, comparison and convergence,, J. Reine Angew. Math., 413 (1991), 1. Google Scholar

[23]

R. M. May, Stability and complexity in model ecosystems,, in IEEE Transactions on Systems, SMC-6 (1976). doi: 10.1109/TSMC.1976.4309488. Google Scholar

[24]

K. Mischaikow, H. Smith and H. R Thieme, Asympotically autonomous semiflows: Chain recurrence and Lyapunov functions,, Trans. AMS., 347 (1995), 1669. doi: 10.1090/S0002-9947-1995-1290727-7. Google Scholar

[25]

P. De Mottoni and F. Rothe, Convergence to homogeneous equilibrium state for generalized Volterra-Lotka systems with diffusion,, SIAM J. Appl. Math., 37 (1979), 648. doi: 10.1137/0137048. Google Scholar

[26]

J. D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications,, Third edition, (2003). Google Scholar

[27]

C. V. Pao, Nonlinear Parabolic and Elliptic Equations,, Plenum Press, (1992). Google Scholar

[28]

C. V. Pao, Dynamics of nonlinear parabolic systems with time delays,, J. Math. Anal. Appl., 198 (1996), 751. doi: 10.1006/jmaa.1996.0111. Google Scholar

[29]

C. V. Pao, Coupled nonlinear parabolic systems with time delays,, J. Math. Anal. Appl., 196 (1995), 237. doi: 10.1006/jmaa.1995.1408. Google Scholar

[30]

R. Peng and S. Liu, Global stability of the steady states of an SIS epidemic reaction-diffusion models,, Nonlin. Analy., 71 (2009), 239. doi: 10.1016/j.na.2008.10.043. Google Scholar

[31]

S. Ruan and X.-Q. Zhao, Persistence and extinction in two species reaction-diffusion systems with delays,, J. Differential Equations, 156 (1999), 71. doi: 10.1006/jdeq.1998.3599. Google Scholar

[32]

G. T. Skalski and J. F. Gilliam, Functional responses with predator interference: Viable alternatives to the Holling type II model,, Ecology, 82 (2001), 3083. Google Scholar

[33]

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. doi: 10.1137/0142003. Google Scholar

[34]

J. W.-H. So and X. Q Zhao, A Threshold Phenomenon in a Reaction-Diffusion Equation with Temporal Delays, Note,, 1997., (). Google Scholar

[35]

Y. Su, J. Wei and J. Shi, Hopf bifurcations in a reaction-diffusion population model with delay effect,, J. Differential Equations, 247 (2009), 1156. doi: 10.1016/j.jde.2009.04.017. Google Scholar

[36]

Y. Takeuchi, Global Dynamical Properties of Lotka-Volterra Systems,, World Scientific Publishing Co. Pte. Ltd, (1996). doi: 10.1142/9789812830548. Google Scholar

[37]

M. Wang and Peter Y. H. Pang, Qualitative analysis of a diffusive variable-territory prey-predator model,, Discrete Contin. Dyn. Syst., 23 (2009), 1061. doi: 10.3934/dcds.2009.23.1061. Google Scholar

[38]

J. Wu, Theory and Applications of Partial Functional Differential Equations,, Applied Mathematical Sciences, (1996). doi: 10.1007/978-1-4612-4050-1. Google Scholar

[39]

D. Xu and X.-Q. Zhao, A nonlocal reaction-diffusion population model with stage structure,, Canad. Appl. Math. Quart., 11 (2003), 303. Google Scholar

[40]

R. Xu, Global Convergence of a predator-prey model with stage structure and spatio-temporal delay,, Discrete Contin. Dyn. Syst. Ser. B, 15 (2011), 273. doi: 10.3934/dcdsb.2011.15.273. Google Scholar

[41]

F. Yi, J. Wei and J. Shi, Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system,, J. Differential Equations, 246 (2009), 1944. doi: 10.1016/j.jde.2008.10.024. Google Scholar

[42]

T. Yi and X. Zou, Map dynamics versus dynamics of associated delay reaction-diffusion equations with a Neumann condition,, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 466 (2010), 2955. doi: 10.1098/rspa.2009.0650. Google Scholar

[43]

T. Yi and X. Zou, Global attractivity of the diffusive Nicholson blowflies equation with Neumann boundary condition: A non-monotone case,, J. Differential equations, 245 (2008), 3376. doi: 10.1016/j.jde.2008.03.007. Google Scholar

show all references

References:
[1]

Z. Artstein, Limiting equations and stability of nonautonomous ordinary differential equations, Appendix to J. P. LaSalle, the stability of dynamical systems,, in CBMS, (1976). Google Scholar

[2]

E. Beretta and Y. Kuang, Geometric stability switch criteria in delay differential systems with delay dependent parameters,, SIAM J. Appl. Math., 33 (2002), 1144. doi: 10.1137/S0036141000376086. Google Scholar

[3]

G. Caristi, K. Rybakowski and T. Wessolek, Persistence and spatial patterns in a one-predator-two-prey Lotka-Volterra model with diffusion,, Annali di Mathematica pura ed applicata, 161 (1992), 345. doi: 10.1007/BF01759645. Google Scholar

[4]

W. Chen and M. Wang, Qualitative analysis of predator-prey models with Beddington-DeAngelis functional response and diffusion,, Math. Comp. Modelling, 42 (2005), 31. doi: 10.1016/j.mcm.2005.05.013. Google Scholar

[5]

S. Chen, J. Shi and J. Wei, Global stability and Hopf bifurcation in a delayed diffusive Leslie-Gower predator-prey system,, Int. J. Bifurcation and Chaos, 22 (2012). doi: 10.1142/S0218127412500617. Google Scholar

[6]

S. Chen, J. Shi and J. Wei, The effect of delay on a diffusive predator-prey system with Holling type-II predator functional response,, Comm. on Pure and Appl. Analy., 12 (2013), 481. doi: 10.3934/cpaa.2013.12.481. Google Scholar

[7]

M. G. Crandall and P. H. Rabinowitz, The Hopf bifurcation theorem in infinite dimensions,, Arch. Rational Mech. Anal., 67 (1977), 53. doi: 10.1007/BF00280827. Google Scholar

[8]

D. L. DeAngelis, R. A. Goldstein and R. Neill, A model for trophic interaction,, Ecology, 56 (1975), 881. Google Scholar

[9]

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. doi: 10.1017/S0308210500000895. Google Scholar

[10]

H. I. Freedman and X.-Q. Zhao, Global asymptotics in some quasimonotone reaction-diffusion systems with delays,, J. Differential Equations, 137 (1997), 340. doi: 10.1006/jdeq.1997.3264. Google Scholar

[11]

S. A. Gourley and Y. Kuang, A delay reaction-diffusion model of the spread of bacteriophage infection,, SIAM J. Appl. Math., 65 (2005), 550. doi: 10.1137/S0036139903436613. Google Scholar

[12]

S. A. Gourley and S. Ruan, Dynamics of the diffusive Nicholson's blowflies equation with distributed delay,, Proc. R. Soc. A, 130 (2000), 1275. doi: 10.1017/S0308210500000688. Google Scholar

[13]

J. K. Hale and P. Waltman, Persistence in infinite-dimensional systems,, SIAM J. Math. Anal., 20 (1989), 388. doi: 10.1137/0520025. Google Scholar

[14]

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. doi: 10.2307/1939378. Google Scholar

[15]

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. doi: 10.1002/(SICI)1099-1476(20000310)23:4<347::AID-MMA117>3.0.CO;2-F. Google Scholar

[16]

W. Ko and I. Ahn, Analysis of ratio-dependent food chain model,, J. Math. Anal. Appl., 335 (2007), 498. doi: 10.1016/j.jmaa.2007.01.089. Google Scholar

[17]

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. doi: 10.1016/j.jde.2006.08.001. Google Scholar

[18]

A. Korobeinikov, Stability of ecosystem: Global properties of a general predator-prey model,, Mathematical Medicine and Biology, 26 (2009), 309. doi: 10.1093/imammb/dqp009. Google Scholar

[19]

Z. Lin, Time delayed parabolic system in a two-species competitive model with stage structure,, J. Math. Anal. Appl., 315 (2006), 202. doi: 10.1016/j.jmaa.2005.06.012. Google Scholar

[20]

S. Liu and E. Beretta, Stage-structured Predator-prey Model with the Beddington-DeAngelis functional response,, SIAM J. Appl. Math., 66 (2006), 1101. doi: 10.1137/050630003. Google Scholar

[21]

S. Liu and J. Zhang, Coexistence and stability of predator-prey model with Beddington-DeAngelis functional response and stage structure,, J. Math. Anal. Appl., 342 (2008), 446. doi: 10.1016/j.jmaa.2007.12.038. Google Scholar

[22]

R. H. Martin and H. L. Smith, Reaction-diffusion systems with time delays: Monotonicity, invariance, comparison and convergence,, J. Reine Angew. Math., 413 (1991), 1. Google Scholar

[23]

R. M. May, Stability and complexity in model ecosystems,, in IEEE Transactions on Systems, SMC-6 (1976). doi: 10.1109/TSMC.1976.4309488. Google Scholar

[24]

K. Mischaikow, H. Smith and H. R Thieme, Asympotically autonomous semiflows: Chain recurrence and Lyapunov functions,, Trans. AMS., 347 (1995), 1669. doi: 10.1090/S0002-9947-1995-1290727-7. Google Scholar

[25]

P. De Mottoni and F. Rothe, Convergence to homogeneous equilibrium state for generalized Volterra-Lotka systems with diffusion,, SIAM J. Appl. Math., 37 (1979), 648. doi: 10.1137/0137048. Google Scholar

[26]

J. D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications,, Third edition, (2003). Google Scholar

[27]

C. V. Pao, Nonlinear Parabolic and Elliptic Equations,, Plenum Press, (1992). Google Scholar

[28]

C. V. Pao, Dynamics of nonlinear parabolic systems with time delays,, J. Math. Anal. Appl., 198 (1996), 751. doi: 10.1006/jmaa.1996.0111. Google Scholar

[29]

C. V. Pao, Coupled nonlinear parabolic systems with time delays,, J. Math. Anal. Appl., 196 (1995), 237. doi: 10.1006/jmaa.1995.1408. Google Scholar

[30]

R. Peng and S. Liu, Global stability of the steady states of an SIS epidemic reaction-diffusion models,, Nonlin. Analy., 71 (2009), 239. doi: 10.1016/j.na.2008.10.043. Google Scholar

[31]

S. Ruan and X.-Q. Zhao, Persistence and extinction in two species reaction-diffusion systems with delays,, J. Differential Equations, 156 (1999), 71. doi: 10.1006/jdeq.1998.3599. Google Scholar

[32]

G. T. Skalski and J. F. Gilliam, Functional responses with predator interference: Viable alternatives to the Holling type II model,, Ecology, 82 (2001), 3083. Google Scholar

[33]

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. doi: 10.1137/0142003. Google Scholar

[34]

J. W.-H. So and X. Q Zhao, A Threshold Phenomenon in a Reaction-Diffusion Equation with Temporal Delays, Note,, 1997., (). Google Scholar

[35]

Y. Su, J. Wei and J. Shi, Hopf bifurcations in a reaction-diffusion population model with delay effect,, J. Differential Equations, 247 (2009), 1156. doi: 10.1016/j.jde.2009.04.017. Google Scholar

[36]

Y. Takeuchi, Global Dynamical Properties of Lotka-Volterra Systems,, World Scientific Publishing Co. Pte. Ltd, (1996). doi: 10.1142/9789812830548. Google Scholar

[37]

M. Wang and Peter Y. H. Pang, Qualitative analysis of a diffusive variable-territory prey-predator model,, Discrete Contin. Dyn. Syst., 23 (2009), 1061. doi: 10.3934/dcds.2009.23.1061. Google Scholar

[38]

J. Wu, Theory and Applications of Partial Functional Differential Equations,, Applied Mathematical Sciences, (1996). doi: 10.1007/978-1-4612-4050-1. Google Scholar

[39]

D. Xu and X.-Q. Zhao, A nonlocal reaction-diffusion population model with stage structure,, Canad. Appl. Math. Quart., 11 (2003), 303. Google Scholar

[40]

R. Xu, Global Convergence of a predator-prey model with stage structure and spatio-temporal delay,, Discrete Contin. Dyn. Syst. Ser. B, 15 (2011), 273. doi: 10.3934/dcdsb.2011.15.273. Google Scholar

[41]

F. Yi, J. Wei and J. Shi, Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system,, J. Differential Equations, 246 (2009), 1944. doi: 10.1016/j.jde.2008.10.024. Google Scholar

[42]

T. Yi and X. Zou, Map dynamics versus dynamics of associated delay reaction-diffusion equations with a Neumann condition,, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 466 (2010), 2955. doi: 10.1098/rspa.2009.0650. Google Scholar

[43]

T. Yi and X. Zou, Global attractivity of the diffusive Nicholson blowflies equation with Neumann boundary condition: A non-monotone case,, J. Differential equations, 245 (2008), 3376. doi: 10.1016/j.jde.2008.03.007. Google Scholar

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