March  2017, 16(2): 427-442. doi: 10.3934/cpaa.2017022

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

Department of Mathematics, Korea University, 2511, Sejong-Ro, Sejong, 30019, Korea

ahnik@korea.ac.kr

Received  February 2016 Revised  November 2016 Published  January 2017

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.

Citation: 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
References:
[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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J. K. Hale and P. Waltman, Persistence in infinite-dimensional systems, SIAM J. Math. Anal., 20 (1989), 388–395. 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), 17–29. doi: 10.2307/1939378.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

W. Ko and I. Ahn, Local stability and bifurcation of a general diffusive consumer-resource model with maturation delay, preprint. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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 .  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

J. D. Murray, Mathematical Biology Ⅰ: An Introduction, 3rd edition, Interdisciplinary Applied Mathematics, vol. 17, Springer, New York, 2002. doi: 10.1007/b98868.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

D. Xu and X.-Q. Zhao, A nonlocal reaction-diffusion population model with stage structure, Canad. Appl. Math. Quart., 11 (2003), 303–319.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[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.  Google Scholar

[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 .  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

J. K. Hale and P. Waltman, Persistence in infinite-dimensional systems, SIAM J. Math. Anal., 20 (1989), 388–395. doi: 10.1137/0520025.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

W. Ko and I. Ahn, Local stability and bifurcation of a general diffusive consumer-resource model with maturation delay, preprint. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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 .  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

J. D. Murray, Mathematical Biology Ⅰ: An Introduction, 3rd edition, Interdisciplinary Applied Mathematics, vol. 17, Springer, New York, 2002. doi: 10.1007/b98868.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

D. Xu and X.-Q. Zhao, A nonlocal reaction-diffusion population model with stage structure, Canad. Appl. Math. Quart., 11 (2003), 303–319.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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