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July  2022, 27(7): 3625-3641. doi: 10.3934/dcdsb.2021199

Positive solutions of a diffusive two competitive species model with saturation

Department of Mathematics, University of Mandalay, Mandalay 05032, Myanmar

* Corresponding author: Aung Zaw Myint

Received  April 2021 Published  July 2022 Early access  August 2021

In this paper, the positive solutions of a diffusive two competitive species model with Bazykin functional response are investigated. We give the a priori estimates and compute the fixed point indices of trivial and semi-trivial solutions. And obtain the existence of solution and demonstrate the bifurcation of a coexistence state emanating from semi-trivial solutions. Finally, multiplicity and stability results are presented.

Citation: Aung Zaw Myint. Positive solutions of a diffusive two competitive species model with saturation. Discrete and Continuous Dynamical Systems - B, 2022, 27 (7) : 3625-3641. doi: 10.3934/dcdsb.2021199
References:
[1]

A. D. Bazykin, Nonlinear Dynamics of Interacting Populations, World Scientific Series on Nonlinear Science Series A, London, 1998. doi: 10.1142/9789812798725.

[2]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis, 8 (1971), 321-340.  doi: 10.1016/0022-1236(71)90015-2.

[3]

E. N. Dancer, On the indices of fixed points of mappings in cones and applications, J. Math. Anal. Appl., 91 (1983), 131-151.  doi: 10.1016/0022-247X(83)90098-7.

[4]

E. N. Dancer, On positive solutions of some pairs of differential equations, Trans. Amer. Math. Soc., 284 (1984), 729-743.  doi: 10.1090/S0002-9947-1984-0743741-4.

[5]

E. N. Dancer and Z. T. Zhang, Dynamics of Lotka-Volterra competition systems with large interaction, J. Differential Equations, 182 (2002), 470-489.  doi: 10.1006/jdeq.2001.4102.

[6]

Y. Du, Realization of prescribed patterns in the competition model, J. Differential Equations, 193 (2003), 147-179.  doi: 10.1016/S0022-0396(03)00056-1.

[7]

M. Kamenskiĭ, Measures of noncompactness and the perturbation theory of linear operators, Tartu Riikl. Ül. Toimetised, 430 (1977), 112-122. 

[8]

T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, 1995.

[9]

H. LiY. Li and W. Yang, Existence and asymptotic behavior of positive solutions for a one-prey and two-competing-predators system with diffusion, Nonlinear Anal. Real World Appl., 27 (2016), 261-282.  doi: 10.1016/j.nonrwa.2015.07.010.

[10]

H. LiP. Y. H. Pang and M. X. Wang, Qualitative analysis of a diffusive prey-predator model with trophic interactions of three levels, Discrete Contin. Dyn. Syst. Ser. B., 17 (2012), 127-152.  doi: 10.3934/dcdsb.2012.17.127.

[11]

L. Li, Coexistence theorems of steady states for predator-prey interacting systems, Trans. Amer. Math. Soc., 305 (1988), 143-166.  doi: 10.1090/S0002-9947-1988-0920151-1.

[12]

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

[13]

W. Ni and M. X. Wang, Dynamics and patterns of a diffusive Leslie-Gower prey-predator model with strong Allee effect in prey, J. Differential Equations, 261 (2016), 4244-4274.  doi: 10.1016/j.jde.2016.06.022.

[14]

P. Y. H. Pang and M. X. Wang, Non-constant positive steady states of a predator-prey system with non-monotonic functional response and diffusion, Proc. London Math. Soc., 88 (2004), 135-157.  doi: 10.1112/S0024611503014321.

[15]

R. Peng and M. X. Wang, On multiplicity and stability of positive solutions of a diffusive prey-predator model, J. Math. Anal. Appl., 316 (2006), 256-268.  doi: 10.1016/j.jmaa.2005.04.033.

[16]

R. PengM. X. Wang and W. Chen, Positive steady states of a prey-predator model with diffusion and non-monotone conversion rate, Acta Math. Sin. Engl. Ser., 23 (2007), 749-760.  doi: 10.1007/s10114-005-0789-9.

[17]

K. Ryu and I. Ahn, Positive solutions for ratio-dependent predator-prey interaction systems, J. Differential Equations, 218 (2005), 117-135.  doi: 10.1016/j.jde.2005.06.020.

[18] M. X. Wang, Nonlinear Partial Differential Equations of Parabolic Type, Science Press, Beijing, (in Chinese), 1993. 
[19]

M. X. Wang and Q. Wu, Positive solutions of a prey-predator model with predator saturation and competition, J. Math. Anal. Appl., 345 (2008), 708-718.  doi: 10.1016/j.jmaa.2008.04.054.

[20]

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

[21]

Y. Yamada, Positive solutions for Lotka-Volterra systems with cross-diffusion, Handbook of Differential Equations: Stationary Partial Differential Equations, 6 (2008), 411-501.  doi: 10.1016/S1874-5733(08)80023-X.

[22]

J. Zhou, Positive solutions of a diffusive Leslie-Gower predator-prey model with Bazykin functional response, Z. Angew. Math. Phys., 65 (2014), 1-18.  doi: 10.1007/s00033-013-0315-3.

show all references

References:
[1]

A. D. Bazykin, Nonlinear Dynamics of Interacting Populations, World Scientific Series on Nonlinear Science Series A, London, 1998. doi: 10.1142/9789812798725.

[2]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis, 8 (1971), 321-340.  doi: 10.1016/0022-1236(71)90015-2.

[3]

E. N. Dancer, On the indices of fixed points of mappings in cones and applications, J. Math. Anal. Appl., 91 (1983), 131-151.  doi: 10.1016/0022-247X(83)90098-7.

[4]

E. N. Dancer, On positive solutions of some pairs of differential equations, Trans. Amer. Math. Soc., 284 (1984), 729-743.  doi: 10.1090/S0002-9947-1984-0743741-4.

[5]

E. N. Dancer and Z. T. Zhang, Dynamics of Lotka-Volterra competition systems with large interaction, J. Differential Equations, 182 (2002), 470-489.  doi: 10.1006/jdeq.2001.4102.

[6]

Y. Du, Realization of prescribed patterns in the competition model, J. Differential Equations, 193 (2003), 147-179.  doi: 10.1016/S0022-0396(03)00056-1.

[7]

M. Kamenskiĭ, Measures of noncompactness and the perturbation theory of linear operators, Tartu Riikl. Ül. Toimetised, 430 (1977), 112-122. 

[8]

T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, 1995.

[9]

H. LiY. Li and W. Yang, Existence and asymptotic behavior of positive solutions for a one-prey and two-competing-predators system with diffusion, Nonlinear Anal. Real World Appl., 27 (2016), 261-282.  doi: 10.1016/j.nonrwa.2015.07.010.

[10]

H. LiP. Y. H. Pang and M. X. Wang, Qualitative analysis of a diffusive prey-predator model with trophic interactions of three levels, Discrete Contin. Dyn. Syst. Ser. B., 17 (2012), 127-152.  doi: 10.3934/dcdsb.2012.17.127.

[11]

L. Li, Coexistence theorems of steady states for predator-prey interacting systems, Trans. Amer. Math. Soc., 305 (1988), 143-166.  doi: 10.1090/S0002-9947-1988-0920151-1.

[12]

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

[13]

W. Ni and M. X. Wang, Dynamics and patterns of a diffusive Leslie-Gower prey-predator model with strong Allee effect in prey, J. Differential Equations, 261 (2016), 4244-4274.  doi: 10.1016/j.jde.2016.06.022.

[14]

P. Y. H. Pang and M. X. Wang, Non-constant positive steady states of a predator-prey system with non-monotonic functional response and diffusion, Proc. London Math. Soc., 88 (2004), 135-157.  doi: 10.1112/S0024611503014321.

[15]

R. Peng and M. X. Wang, On multiplicity and stability of positive solutions of a diffusive prey-predator model, J. Math. Anal. Appl., 316 (2006), 256-268.  doi: 10.1016/j.jmaa.2005.04.033.

[16]

R. PengM. X. Wang and W. Chen, Positive steady states of a prey-predator model with diffusion and non-monotone conversion rate, Acta Math. Sin. Engl. Ser., 23 (2007), 749-760.  doi: 10.1007/s10114-005-0789-9.

[17]

K. Ryu and I. Ahn, Positive solutions for ratio-dependent predator-prey interaction systems, J. Differential Equations, 218 (2005), 117-135.  doi: 10.1016/j.jde.2005.06.020.

[18] M. X. Wang, Nonlinear Partial Differential Equations of Parabolic Type, Science Press, Beijing, (in Chinese), 1993. 
[19]

M. X. Wang and Q. Wu, Positive solutions of a prey-predator model with predator saturation and competition, J. Math. Anal. Appl., 345 (2008), 708-718.  doi: 10.1016/j.jmaa.2008.04.054.

[20]

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

[21]

Y. Yamada, Positive solutions for Lotka-Volterra systems with cross-diffusion, Handbook of Differential Equations: Stationary Partial Differential Equations, 6 (2008), 411-501.  doi: 10.1016/S1874-5733(08)80023-X.

[22]

J. Zhou, Positive solutions of a diffusive Leslie-Gower predator-prey model with Bazykin functional response, Z. Angew. Math. Phys., 65 (2014), 1-18.  doi: 10.1007/s00033-013-0315-3.

Figure 1.  The existence of coexistence states and bifurcation lines
Figure 2.  For $ \alpha\gg 1 $, the existence and multiplicity of coexistence states
Figure 3.  For $ \beta\gg 1 $, the existence and multiplicity of coexistence states
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