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:
[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).

[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.

[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.

[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.

[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.

[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.

[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.

[8]

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

[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.

[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.

[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.

[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.

[13]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[23]

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

[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.

[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.

[26]

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

[27]

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

[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.

[29]

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

[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.

[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.

[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.

[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.

[34]

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

[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.

[36]

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

[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.

[38]

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

[39]

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

[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.

[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.

[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.

[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.

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).

[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.

[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.

[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.

[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.

[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.

[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.

[8]

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

[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.

[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.

[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.

[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.

[13]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[23]

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

[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.

[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.

[26]

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

[27]

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

[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.

[29]

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

[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.

[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.

[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.

[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.

[34]

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

[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.

[36]

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

[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.

[38]

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

[39]

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

[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.

[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.

[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.

[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.

[1]

Hai-Xia Li, Jian-Hua Wu, Yan-Ling Li, Chun-An Liu. Positive solutions to the unstirred chemostat model with Crowley-Martin functional response. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 2951-2966. doi: 10.3934/dcdsb.2017128

[2]

Renji Han, Binxiang Dai, Lin Wang. Delay induced spatiotemporal patterns in a diffusive intraguild predation model with Beddington-DeAngelis functional response. Mathematical Biosciences & Engineering, 2018, 15 (3) : 595-627. doi: 10.3934/mbe.2018027

[3]

Jinliang Wang, Jiying Lang, Xianning Liu. Global dynamics for viral infection model with Beddington-DeAngelis functional response and an eclipse stage of infected cells. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 3215-3233. doi: 10.3934/dcdsb.2015.20.3215

[4]

Sze-Bi Hsu, Shigui Ruan, Ting-Hui Yang. On the dynamics of two-consumers-one-resource competing systems with Beddington-DeAngelis functional response. Discrete & Continuous Dynamical Systems - B, 2013, 18 (9) : 2331-2353. doi: 10.3934/dcdsb.2013.18.2331

[5]

Yu Yang, Shigui Ruan, Dongmei Xiao. Global stability of an age-structured virus dynamics model with Beddington-DeAngelis infection function. Mathematical Biosciences & Engineering, 2015, 12 (4) : 859-877. doi: 10.3934/mbe.2015.12.859

[6]

Haiyin Li, Yasuhiro Takeuchi. Dynamics of the density dependent and nonautonomous predator-prey system with Beddington-DeAngelis functional response. Discrete & Continuous Dynamical Systems - B, 2015, 20 (4) : 1117-1134. doi: 10.3934/dcdsb.2015.20.1117

[7]

Jun Zhou. Qualitative analysis of a modified Leslie-Gower predator-prey model with Crowley-Martin functional responses. Communications on Pure & Applied Analysis, 2015, 14 (3) : 1127-1145. doi: 10.3934/cpaa.2015.14.1127

[8]

Walid Abid, Radouane Yafia, M.A. Aziz-Alaoui, Habib Bouhafa, Azgal Abichou. Global dynamics on a circular domain of a diffusion predator-prey model with modified Leslie-Gower and Beddington-DeAngelis functional type. Evolution Equations & Control Theory, 2015, 4 (2) : 115-129. doi: 10.3934/eect.2015.4.115

[9]

Seong Lee, Inkyung Ahn. Diffusive predator-prey models with stage structure on prey and beddington-deangelis functional responses. Communications on Pure & Applied Analysis, 2017, 16 (2) : 427-442. doi: 10.3934/cpaa.2017022

[10]

Eric Avila-Vales, Gerardo García-Almeida, Erika Rivero-Esquivel. Bifurcation and spatiotemporal patterns in a Bazykin predator-prey model with self and cross diffusion and Beddington-DeAngelis response. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 717-740. doi: 10.3934/dcdsb.2017035

[11]

Zhihua Liu, Pierre Magal, Shigui Ruan. Oscillations in age-structured models of consumer-resource mutualisms. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 537-555. doi: 10.3934/dcdsb.2016.21.537

[12]

Wen-Bin Yang, Yan-Ling Li, Jianhua Wu, Hai-Xia Li. Dynamics of a food chain model with ratio-dependent and modified Leslie-Gower functional responses. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2269-2290. doi: 10.3934/dcdsb.2015.20.2269

[13]

Robert Stephen Cantrell, Chris Cosner, Shigui Ruan. Intraspecific interference and consumer-resource dynamics. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 527-546. doi: 10.3934/dcdsb.2004.4.527

[14]

Benjamin Leard, Catherine Lewis, Jorge Rebaza. Dynamics of ratio-dependent Predator-Prey models with nonconstant harvesting. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 303-315. doi: 10.3934/dcdss.2008.1.303

[15]

Wenjie Zuo, Junping Shi. Traveling wave solutions of a diffusive ratio-dependent Holling-Tanner system with distributed delay. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1179-1200. doi: 10.3934/cpaa.2018057

[16]

C. Connell McCluskey. Global stability of an $SIR$ epidemic model with delay and general nonlinear incidence. Mathematical Biosciences & Engineering, 2010, 7 (4) : 837-850. doi: 10.3934/mbe.2010.7.837

[17]

Andrey V. Melnik, Andrei Korobeinikov. Lyapunov functions and global stability for SIR and SEIR models with age-dependent susceptibility. Mathematical Biosciences & Engineering, 2013, 10 (2) : 369-378. doi: 10.3934/mbe.2013.10.369

[18]

Xinyu Song, Liming Cai, U. Neumann. Ratio-dependent predator-prey system with stage structure for prey. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 747-758. doi: 10.3934/dcdsb.2004.4.747

[19]

Yu Ji, Lan Liu. Global stability of a delayed viral infection model with nonlinear immune response and general incidence rate. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 133-149. doi: 10.3934/dcdsb.2016.21.133

[20]

Jinliang Wang, Lijuan Guan. Global stability for a HIV-1 infection model with cell-mediated immune response and intracellular delay. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 297-302. doi: 10.3934/dcdsb.2012.17.297

2017 Impact Factor: 0.972

Metrics

  • PDF downloads (12)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]