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Positive solutions of iterative functional differential equations and application to mixed-type functional differential equations
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 |
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.
Mathematics Subject Classification:
Primary: 35J57, 35B09, 35B32, 35B35; Secondary: 92B05.
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
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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. Li, Y. 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. Li, P. 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. Li, J. 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. Peng, M. 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. Wei, J. 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. Li, Y. 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. Li, P. 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. Li, J. 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. Peng, M. 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. Wei, J. 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 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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