October 2018, 23(8): 2951-2966. doi: 10.3934/dcdsb.2017128

Positive solutions to the unstirred chemostat model with Crowley-Martin functional response

1. 

Institute of Mathematics and Information Sciences, Baoji University of Arts and Sciences, Baoji, Shaanxi 721013, China,

2. 

College of Mathematics and Information Science, Shaanxi Normal University, Xi'an, Shaanxi 710062, China,

3. 

Institute of Mathematics and Information Sciences, Baoji University of Arts and Sciences, Baoji, Shaanxi 721013, China

* Corresponding author

Received  March 2016 Revised  February 2017 Published  April 2017

Fund Project: The work is supported by the Natural Science Foundation of China (61672021,11401356,11671 243), the Natural Science Basic Research Plan in Shaanxi Province of China (2015JM1008), the Foundations of Shaanxi Educational Committee (16JK1046), the Postdoctoral Science Foundation of China (2016M602767) and the Special Fund of Education Department of Shaanxi Province (16JK1710).

A food-chain model with Crowley-Martin functional response in the unstirred chemostat is considered. First, the global framework of coexistence solutions is discussed by the maximum principle and bifurcation theory. We obtain the sufficient and necessary conditions for coexistence of steady-state. Second, the stability and uniqueness of coexistence solutions are investigated by means of the combination of the perturbation theory and fixed point index theory. Our results indicate that if the magnitude of interference among predator is sufficiently large, the model has only one unique linearly stable coexistence solution when the maximal growth rate of predator belongs to certain range. Finally, some numerical simulations are carried out to verify and complement the theoretical results.

Citation: 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
References:
[1]

P. H. Crowley and E. K. Martin, Functional responses and interference within and between year classes of a dragonfly population, Journal of the North American Benthological Society, 8 (1989), 211-221. doi: 10.2307/1467324.

[2]

E. N. Dancer, On the indices of fixed points of mapping in cones and applications, Journal of Mathematical Analysis and Applications, 91 (1983), 131-151. doi: 10.1016/0022-247X(83)90098-7.

[3]

Y. Y. DongS. B. Li and Y. L. Li, Multiplicity and uniqueness of positive solutions for a predator-prey model with C-M functional response, Acta Applicandae Mathematicae, 139 (2015), 187-206. doi: 10.1007/s10440-014-9985-x.

[4]

Y. H. Du and Y. Lou, Some uniqueness and exact multiplicity results for a predator-prey model, Transactions of The American Mathematical Society, 349 (1997), 2443-2475. doi: 10.1090/S0002-9947-97-01842-4.

[5]

L. Dung and H. L. Smith, A parabolic system modeling microbial competition in an unmixed bio-reactor, Journal of Differential Equations, 130 (1996), 59-91. doi: 10.1006/jdeq.1996.0132.

[6]

D. G. Figueiredo and J. P. Gossez, Strict monotonicity of eigenvalues and unique continuation, Communications in Partial Differential Equations, 17 (1992), 339-346. doi: 10.1080/03605309208820844.

[7]

G. H. GuoJ. H. Wu and Y. E. Wang, Bifurcation from a double eigenvalue in the unstirred chemostat, Applicable Analysis, 92 (2013), 1449-1461. doi: 10.1080/00036811.2012.683786.

[8]

H. J. Guo and S. N. Zheng, A competition model for two resources in un-stirred chemostat, Applied Mathematics and Computation, 217 (2011), 6934-6949. doi: 10.1016/j.amc.2011.01.102.

[9]

S. B. HsuJ. F. Jiang and F. B. Wang, On a system of reaction-diffusion equations arising from competition with internal storage in an unstirred chemostat, Journal of Differential Equations, 248 (2010), 2470-2496. doi: 10.1016/j.jde.2009.12.014.

[10]

S. B. HsuJ. P. Shi and F. B. Wang, Further studies of a reaction-diffusion system for an unstirred chemostat with internal storage, Discrete and Continuous Dynamical Systems-Series B, 19 (2014), 3169-3189. doi: 10.3934/dcdsb.2014.19.3169.

[11]

S. B. Hsu and P. Waltman, On a system of reaction-diffusion equations arising from competition in an unstirred chemostat, SIAM Journal on Applied Mathematics, 53 (1993), 1026-1044. doi: 10.1137/0153051.

[12]

H. X. Li, Asymptotic behavior and multiplicity for a diffusive Leslie-Gower predator-prey system with Crowley-Martin functional response, Computers and Mathematics with Applications, 68 (2014), 693-705. doi: 10.1016/j.camwa.2014.07.018.

[13]

H. X. Li, Existence and multiplicity of positive solutions for an unstirred chemostat model with B-D functional response, Chinese Journal of Engineering Methematics(China), 32 (2015), 369-380.

[14]

S. B. Li and J. H. Wu, Qualitative analysis of a predator-prey model with predator saturation and competition, Acta Applicandae Mathematicae, 141 (2016), 165-185. doi: 10.1007/s10440-015-0009-2.

[15]

S. B. LiJ. H. Wu and Y. Y. Dong, Uniqueness and stability of a predator-prey model with C-M functional response, Computers and Mathematics with Applications, 69 (2015), 1080-1095. doi: 10.1016/j.camwa.2015.03.007.

[16]

J. Liu and S. N. Zheng, A reaction-diffusion system arising from food chain in an unstirred Chemostat, Journal of Biomathematics, 17 (2002), 263-272.

[17]

Y. Z. LuJ. LiuX. T. Chen and S. F. Xu, Coexistence of steady states to an annular model in an un-stirred chemostat, International Journal of Information and Systems Sciences, 5 (2009), 359-368.

[18]

H. NieN. Liu and J. H. Wu, Coexistence solutions and their stability of an unstirred chemostat model with toxins, Nonlinear Analysis: Real World Applications, 20 (2014), 36-51. doi: 10.1016/j.nonrwa.2014.04.002.

[19]

H. Nie and J. H. Wu, A system of reaction-diffusion equations in the unstirred chemostat with an inhibitor, International Journal of Bifurcation and Chaos, 16 (2006), 989-1009. doi: 10.1142/S0218127406015246.

[20]

H. Nie and J. H. Wu, Asymptotic behaviour of an unstirred chemostat with internal inhibitor, Journal of Mathematical Analysis and Applications, 334 (2007), 889-908. doi: 10.1016/j.jmaa.2007.01.014.

[21]

H. Nie and J. H. Wu, Positive solutions of a competition model for two resources in the unstirred chemostat, Journal of Mathematical Analysis and Applications, 355 (2009), 231-242. doi: 10.1016/j.jmaa.2009.01.045.

[22]

H. Nie and J. H. Wu, Coexistence of an unstirred chemostat model with Beddington-DeAngelis functional response and inhibitor, Nonlinear Analysis: Real World Applications, 11 (2010), 3639-3652. doi: 10.1016/j.nonrwa.2010.01.010.

[23]

H. Nie and J. H. Wu, Exact multiplicity of coexistence solutions to the unstirred chemostat model with the plasmid and internal inhibitor, Science in China: A, 41 (2011), 497-516.

[24]

H. Nie and J. H. Wu, The effect of toxins on the plasmid-bearing and plasmid-free model in the unstirred chemostat, Discrete and Continuous Dynamical systems, 32 (2012), 303-329. doi: 10.3934/dcds.2012.32.303.

[25]

H. Nie and J. H. Wu, Multiplicity results for the unstirred chemostat model with general response functions, Science in China: A, 56 (2013), 2035-2050. doi: 10.1007/s11425-012-4550-4.

[26]

H. Nie and J. H. Wu, Multiple coexistence solutions to the unstirred chemostat model with plasmid and toxin, European Journal of Applied Mathematics, 25 (2014), 481-510. doi: 10.1017/S0956792514000096.

[27]

H. NieH. W. Zhang and J. H. Wu, Characterization of positive solutions of the unstirred chemostat with an inhibitor, Nonlinear Analysis: Real World Applications, 9 (2008), 1078-1089. doi: 10.1016/j.nonrwa.2007.01.018.

[28]

P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, Journal of Functional Analysis, 7 (1971), 487-513. doi: 10.1016/0022-1236(71)90030-9.

[29]

X. Y. ShiX. Y. Zhou and X. Y. Song, Analysis of a stage-structured predator-prey model with Crowley-Martin function, Journal of Applied Mathematics and Computing, 36 (2011), 459-472. doi: 10.1007/s12190-010-0413-8.

[30]

J. Smoller, Shock Waves and Reaction-Diffusion Equations 2nd edition, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0.

[31]

M. X. Wang and W. Qiang, Positive solutions of a prey-predator model with predator saturation and competition, Journal of Mathematical Analysis and Applications, 345 (2008), 708-718. doi: 10.1016/j.jmaa.2008.04.054.

[32]

Y. F. Wang and J. X. Yin, Predator-prey in an unstirred chemostat with periodical input and washout, Nonlinear Analysis: Real World Applications, 3 (2002), 597-610. doi: 10.1016/S1468-1218(01)00051-7.

[33]

M. H. WeiJ. H. Wu and G. H. Guo, The effect of predator competition on positive solutions for a predator-prey model with diffusion, Nonlinear Analysis, 75 (2012), 5053-5068. doi: 10.1016/j.na.2012.04.021.

[34]

J. H. Wu, Global bifurcation of coexistence state for the competition model in the chemostat, Nonlinear Analysis, 39 (2000), 817-835. doi: 10.1016/S0362-546X(98)00250-8.

[35]

J. H. WuH. Nie and G. S. K. Wolkowicz, A mathematical model of competition for two essential resources in the unstirred chemostat, SIAM Journal on Applied Mathematics, 65 (2004), 209-229. doi: 10.1137/S0036139903423285.

[36]

J. H. WuH. Nie and G. S. K. Wolkowicz, The effect of inhibitor on the plasmid-bearing and plasmid-free model in the unstirred chemostat, SIAM Journal on Applied Mathematics, 38 (2007), 1860-1885. doi: 10.1137/050627514.

[37]

J. H. Wu and G. S. K. Wolkowicz, A system of resource-based growth models with two resources in the unstirred chemostat, Journal of Differential Equations, 172 (2001), 300-332. doi: 10.1006/jdeq.2000.3870.

[38]

S. N. Zheng and J. Liu, Coexistence solutions for a reaction-diffusion system of un-stirred chemostat model, Applied Mathematics and Computation, 145 (2003), 579-590. doi: 10.1016/S0096-3003(02)00732-4.

show all references

References:
[1]

P. H. Crowley and E. K. Martin, Functional responses and interference within and between year classes of a dragonfly population, Journal of the North American Benthological Society, 8 (1989), 211-221. doi: 10.2307/1467324.

[2]

E. N. Dancer, On the indices of fixed points of mapping in cones and applications, Journal of Mathematical Analysis and Applications, 91 (1983), 131-151. doi: 10.1016/0022-247X(83)90098-7.

[3]

Y. Y. DongS. B. Li and Y. L. Li, Multiplicity and uniqueness of positive solutions for a predator-prey model with C-M functional response, Acta Applicandae Mathematicae, 139 (2015), 187-206. doi: 10.1007/s10440-014-9985-x.

[4]

Y. H. Du and Y. Lou, Some uniqueness and exact multiplicity results for a predator-prey model, Transactions of The American Mathematical Society, 349 (1997), 2443-2475. doi: 10.1090/S0002-9947-97-01842-4.

[5]

L. Dung and H. L. Smith, A parabolic system modeling microbial competition in an unmixed bio-reactor, Journal of Differential Equations, 130 (1996), 59-91. doi: 10.1006/jdeq.1996.0132.

[6]

D. G. Figueiredo and J. P. Gossez, Strict monotonicity of eigenvalues and unique continuation, Communications in Partial Differential Equations, 17 (1992), 339-346. doi: 10.1080/03605309208820844.

[7]

G. H. GuoJ. H. Wu and Y. E. Wang, Bifurcation from a double eigenvalue in the unstirred chemostat, Applicable Analysis, 92 (2013), 1449-1461. doi: 10.1080/00036811.2012.683786.

[8]

H. J. Guo and S. N. Zheng, A competition model for two resources in un-stirred chemostat, Applied Mathematics and Computation, 217 (2011), 6934-6949. doi: 10.1016/j.amc.2011.01.102.

[9]

S. B. HsuJ. F. Jiang and F. B. Wang, On a system of reaction-diffusion equations arising from competition with internal storage in an unstirred chemostat, Journal of Differential Equations, 248 (2010), 2470-2496. doi: 10.1016/j.jde.2009.12.014.

[10]

S. B. HsuJ. P. Shi and F. B. Wang, Further studies of a reaction-diffusion system for an unstirred chemostat with internal storage, Discrete and Continuous Dynamical Systems-Series B, 19 (2014), 3169-3189. doi: 10.3934/dcdsb.2014.19.3169.

[11]

S. B. Hsu and P. Waltman, On a system of reaction-diffusion equations arising from competition in an unstirred chemostat, SIAM Journal on Applied Mathematics, 53 (1993), 1026-1044. doi: 10.1137/0153051.

[12]

H. X. Li, Asymptotic behavior and multiplicity for a diffusive Leslie-Gower predator-prey system with Crowley-Martin functional response, Computers and Mathematics with Applications, 68 (2014), 693-705. doi: 10.1016/j.camwa.2014.07.018.

[13]

H. X. Li, Existence and multiplicity of positive solutions for an unstirred chemostat model with B-D functional response, Chinese Journal of Engineering Methematics(China), 32 (2015), 369-380.

[14]

S. B. Li and J. H. Wu, Qualitative analysis of a predator-prey model with predator saturation and competition, Acta Applicandae Mathematicae, 141 (2016), 165-185. doi: 10.1007/s10440-015-0009-2.

[15]

S. B. LiJ. H. Wu and Y. Y. Dong, Uniqueness and stability of a predator-prey model with C-M functional response, Computers and Mathematics with Applications, 69 (2015), 1080-1095. doi: 10.1016/j.camwa.2015.03.007.

[16]

J. Liu and S. N. Zheng, A reaction-diffusion system arising from food chain in an unstirred Chemostat, Journal of Biomathematics, 17 (2002), 263-272.

[17]

Y. Z. LuJ. LiuX. T. Chen and S. F. Xu, Coexistence of steady states to an annular model in an un-stirred chemostat, International Journal of Information and Systems Sciences, 5 (2009), 359-368.

[18]

H. NieN. Liu and J. H. Wu, Coexistence solutions and their stability of an unstirred chemostat model with toxins, Nonlinear Analysis: Real World Applications, 20 (2014), 36-51. doi: 10.1016/j.nonrwa.2014.04.002.

[19]

H. Nie and J. H. Wu, A system of reaction-diffusion equations in the unstirred chemostat with an inhibitor, International Journal of Bifurcation and Chaos, 16 (2006), 989-1009. doi: 10.1142/S0218127406015246.

[20]

H. Nie and J. H. Wu, Asymptotic behaviour of an unstirred chemostat with internal inhibitor, Journal of Mathematical Analysis and Applications, 334 (2007), 889-908. doi: 10.1016/j.jmaa.2007.01.014.

[21]

H. Nie and J. H. Wu, Positive solutions of a competition model for two resources in the unstirred chemostat, Journal of Mathematical Analysis and Applications, 355 (2009), 231-242. doi: 10.1016/j.jmaa.2009.01.045.

[22]

H. Nie and J. H. Wu, Coexistence of an unstirred chemostat model with Beddington-DeAngelis functional response and inhibitor, Nonlinear Analysis: Real World Applications, 11 (2010), 3639-3652. doi: 10.1016/j.nonrwa.2010.01.010.

[23]

H. Nie and J. H. Wu, Exact multiplicity of coexistence solutions to the unstirred chemostat model with the plasmid and internal inhibitor, Science in China: A, 41 (2011), 497-516.

[24]

H. Nie and J. H. Wu, The effect of toxins on the plasmid-bearing and plasmid-free model in the unstirred chemostat, Discrete and Continuous Dynamical systems, 32 (2012), 303-329. doi: 10.3934/dcds.2012.32.303.

[25]

H. Nie and J. H. Wu, Multiplicity results for the unstirred chemostat model with general response functions, Science in China: A, 56 (2013), 2035-2050. doi: 10.1007/s11425-012-4550-4.

[26]

H. Nie and J. H. Wu, Multiple coexistence solutions to the unstirred chemostat model with plasmid and toxin, European Journal of Applied Mathematics, 25 (2014), 481-510. doi: 10.1017/S0956792514000096.

[27]

H. NieH. W. Zhang and J. H. Wu, Characterization of positive solutions of the unstirred chemostat with an inhibitor, Nonlinear Analysis: Real World Applications, 9 (2008), 1078-1089. doi: 10.1016/j.nonrwa.2007.01.018.

[28]

P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, Journal of Functional Analysis, 7 (1971), 487-513. doi: 10.1016/0022-1236(71)90030-9.

[29]

X. Y. ShiX. Y. Zhou and X. Y. Song, Analysis of a stage-structured predator-prey model with Crowley-Martin function, Journal of Applied Mathematics and Computing, 36 (2011), 459-472. doi: 10.1007/s12190-010-0413-8.

[30]

J. Smoller, Shock Waves and Reaction-Diffusion Equations 2nd edition, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0.

[31]

M. X. Wang and W. Qiang, Positive solutions of a prey-predator model with predator saturation and competition, Journal of Mathematical Analysis and Applications, 345 (2008), 708-718. doi: 10.1016/j.jmaa.2008.04.054.

[32]

Y. F. Wang and J. X. Yin, Predator-prey in an unstirred chemostat with periodical input and washout, Nonlinear Analysis: Real World Applications, 3 (2002), 597-610. doi: 10.1016/S1468-1218(01)00051-7.

[33]

M. H. WeiJ. H. Wu and G. H. Guo, The effect of predator competition on positive solutions for a predator-prey model with diffusion, Nonlinear Analysis, 75 (2012), 5053-5068. doi: 10.1016/j.na.2012.04.021.

[34]

J. H. Wu, Global bifurcation of coexistence state for the competition model in the chemostat, Nonlinear Analysis, 39 (2000), 817-835. doi: 10.1016/S0362-546X(98)00250-8.

[35]

J. H. WuH. Nie and G. S. K. Wolkowicz, A mathematical model of competition for two essential resources in the unstirred chemostat, SIAM Journal on Applied Mathematics, 65 (2004), 209-229. doi: 10.1137/S0036139903423285.

[36]

J. H. WuH. Nie and G. S. K. Wolkowicz, The effect of inhibitor on the plasmid-bearing and plasmid-free model in the unstirred chemostat, SIAM Journal on Applied Mathematics, 38 (2007), 1860-1885. doi: 10.1137/050627514.

[37]

J. H. Wu and G. S. K. Wolkowicz, A system of resource-based growth models with two resources in the unstirred chemostat, Journal of Differential Equations, 172 (2001), 300-332. doi: 10.1006/jdeq.2000.3870.

[38]

S. N. Zheng and J. Liu, Coexistence solutions for a reaction-diffusion system of un-stirred chemostat model, Applied Mathematics and Computation, 145 (2003), 579-590. doi: 10.1016/S0096-3003(02)00732-4.

Figure 1.  Existence and non-existence of coexistence states. In (a), $a=5,b=10.$ In (b) and (c), $a=8,b=5$ and $a=15,b=5$ respectively. In (d), $a=5,b=1.$.
Figure 2.  Different values of the parameters $k_{1},k_{2},m_{1}$. In (a) and (b), $k_{2}=1,m_{1}=2,k_{1}=1,2$ respectively.
Figure 3.  Different values of the parameter $m_{2}$. In (a)-(d), $m_{2}=1,2,200,1000$ respectively. In (e) and (f), $m_{2}=500$.
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