doi: 10.3934/dcdsb.2021199
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

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 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 & Continuous Dynamical Systems - B, 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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
[1]

Yukio Kan-On. Bifurcation structures of positive stationary solutions for a Lotka-Volterra competition model with diffusion II: Global structure. Discrete & Continuous Dynamical Systems, 2006, 14 (1) : 135-148. doi: 10.3934/dcds.2006.14.135

[2]

Tai-Chia Lin, Tsung-Fang Wu. Existence and multiplicity of positive solutions for two coupled nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems, 2013, 33 (7) : 2911-2938. doi: 10.3934/dcds.2013.33.2911

[3]

Jiafeng Liao, Peng Zhang, Jiu Liu, Chunlei Tang. Existence and multiplicity of positive solutions for a class of Kirchhoff type problems at resonance. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1959-1974. doi: 10.3934/dcdss.2016080

[4]

Leonelo Iturriaga, Eugenio Massa. Existence, nonexistence and multiplicity of positive solutions for the poly-Laplacian and nonlinearities with zeros. Discrete & Continuous Dynamical Systems, 2018, 38 (8) : 3831-3850. doi: 10.3934/dcds.2018166

[5]

Yukio Kan-On. Global bifurcation structure of stationary solutions for a Lotka-Volterra competition model. Discrete & Continuous Dynamical Systems, 2002, 8 (1) : 147-162. doi: 10.3934/dcds.2002.8.147

[6]

Shao-Yuan Huang. Exact multiplicity and bifurcation curves of positive solutions of a one-dimensional Minkowski-curvature problem and its application. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1271-1294. doi: 10.3934/cpaa.2018061

[7]

Michael E. Filippakis, Nikolaos S. Papageorgiou. Existence and multiplicity of positive solutions for nonlinear boundary value problems driven by the scalar $p$-Laplacian. Communications on Pure & Applied Analysis, 2004, 3 (4) : 729-756. doi: 10.3934/cpaa.2004.3.729

[8]

Shao-Yuan Huang. Global bifurcation and exact multiplicity of positive solutions for the one-dimensional Minkowski-curvature problem with sign-changing nonlinearity. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3267-3284. doi: 10.3934/cpaa.2019147

[9]

Xin Xu. Existence of monotone positive solutions of a neighbour based chemotaxis model and aggregation phenomenon. Communications on Pure & Applied Analysis, 2020, 19 (9) : 4327-4348. doi: 10.3934/cpaa.2020195

[10]

Xudong Shang, Jihui Zhang. Multiplicity and concentration of positive solutions for fractional nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2239-2259. doi: 10.3934/cpaa.2018107

[11]

Yi-hsin Cheng, Tsung-Fang Wu. Multiplicity and concentration of positive solutions for semilinear elliptic equations with steep potential. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2457-2473. doi: 10.3934/cpaa.2016044

[12]

Xiyou Cheng, Zhaosheng Feng, Zhitao Zhang. Multiplicity of positive solutions to nonlinear systems of Hammerstein integral equations with weighted functions. Communications on Pure & Applied Analysis, 2020, 19 (1) : 221-240. doi: 10.3934/cpaa.2020012

[13]

Lynnyngs Kelly Arruda, Francisco Odair de Paiva, Ilma Marques. A remark on multiplicity of positive solutions for a class of quasilinear elliptic systems. Conference Publications, 2011, 2011 (Special) : 112-116. doi: 10.3934/proc.2011.2011.112

[14]

Masataka Shibata. Multiplicity of positive solutions to semi-linear elliptic problems on metric graphs. Communications on Pure & Applied Analysis, 2021, 20 (12) : 4107-4126. doi: 10.3934/cpaa.2021147

[15]

Chin-Chin Wu. Existence of traveling wavefront for discrete bistable competition model. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 973-984. doi: 10.3934/dcdsb.2011.16.973

[16]

Ziqing Yuan, Jianshe Yu. Existence and multiplicity of positive solutions for a class of quasilinear Schrödinger equations in $ \mathbb R^N $$ ^\diamondsuit $. Discrete & Continuous Dynamical Systems - S, 2021, 14 (9) : 3285-3303. doi: 10.3934/dcdss.2020281

[17]

Meili Li, Maoan Han, Chunhai Kou. The existence of positive periodic solutions of a generalized. Mathematical Biosciences & Engineering, 2008, 5 (4) : 803-812. doi: 10.3934/mbe.2008.5.803

[18]

Feliz Minhós, João Fialho. Existence and multiplicity of solutions in fourth order BVPs with unbounded nonlinearities. Conference Publications, 2013, 2013 (special) : 555-564. doi: 10.3934/proc.2013.2013.555

[19]

Qi-Lin Xie, Xing-Ping Wu, Chun-Lei Tang. Existence and multiplicity of solutions for Kirchhoff type problem with critical exponent. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2773-2786. doi: 10.3934/cpaa.2013.12.2773

[20]

Ziqing Yuana, Jianshe Yu. Existence and multiplicity of nontrivial solutions of biharmonic equations via differential inclusion. Communications on Pure & Applied Analysis, 2020, 19 (1) : 391-405. doi: 10.3934/cpaa.2020020

2020 Impact Factor: 1.327

Article outline

Figures and Tables

[Back to Top]