American Institute of Mathematical Sciences

September  2022, 27(9): 5343-5365. doi: 10.3934/dcdsb.2021277

Dynamic transitions for the S-K-T competition system

 1 School of Sciences, Southwest Petroleum University, Chengdu, Sichuan 610500, China 2 College of Mathematics and Statistics, Chongqing Jiaotong University, Chongqing 400074, China

* Corresponding author: Dongpei Zhang

Received  March 2021 Published  September 2022 Early access  November 2021

Fund Project: The work is supported by the Young Scholars Development Fund of SWPU (Grant No. 201899010079), the scientific research starting project of SWPU (Grant No. 2018QHZ029), Science and Technology Innovation Team of Education Department of Sichuan for Dynamical System and Its Applications (No. 18TD0013), Southwest Petroleum Univ. under Grant 2019CXTD08, Youth Science and Technology Innovation Team of Southwest Petroleum Univ. for Nonlinear Systems (No. 2017CXTD02)

This paper is concerned with dynamical transition for biological competition system modeled by the S-K-T equations. We study the dynamical behaviour of the S-K-T equations with two different boundary conditions. For the system under non-homogeneous Dirichlet boundary condition, we show that the system undergoes a mixed dynamic transition from the homogeneous state to steady state solutions when the bifurcation parameter cross the critical surface. For the system with Neumann boundary condition, we prove that the system undergoes a mixed dynamic transition, a jump transition and a continuous transition when the bifurcation parameter cross the critical number. Finally, two examples are provided to validate the effectiveness of the theoretical results.

Citation: Ruikuan Liu, Dongpei Zhang. Dynamic transitions for the S-K-T competition system. Discrete and Continuous Dynamical Systems - B, 2022, 27 (9) : 5343-5365. doi: 10.3934/dcdsb.2021277
References:
 [1] H. Amann, Dynamic theory of quasilinear parabolic equations, II. Reaction-diffusion systems, Differential Integral Equations, 3 (1990), 13-75. [2] Y. Choi, R. Lui and Y. Yamada, Existence of global solutions for the Shigesada-Kawasaki-Teramoto model with strongly coupled cross-diffusion, Discrete Contin. Dyn. Syst., 10 (2004), 719-730.  doi: 10.3934/dcds.2004.10.719. [3] S. Choo and Y.-H. Kim, Global stability in a discrete Lotka-Volterra competition model, J. Comput. Anal. Appl., 23 (2017), 276-293. [4] D. Deng and R. Liu, Bifurcation solutions of Gross-Pitaevskii equations for spin-1 Bose-Einstein condensates, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 3175-3193.  doi: 10.3934/dcdsb.2018306. [5] P. Du and R. Temam, Weak solutions of the Shigesada-Kawasaki-Teramoto equations and their attractors, Nonlinear Anal., 159 (2017), 339-364.  doi: 10.1016/j.na.2017.01.017. [6] W. Feng, Competitive exclusion and persistence in models of resource and sexual competition, J. Math. Biol., 35 (1997), 683-694.  doi: 10.1007/s002850050071. [7] C.-H. Hsia, C.-S. Lin, T. Ma and S. Wang, Tropical atmospheric circulations with humidity effects, Proc. A., 471 (2015), 1-24.  doi: 10.1098/rspa.2014.0353. [8] Y. Kan-on, Stability of singularly perturbed solutions to nonlinear diffusion systems arising in population dynamics, Hiroshima Math. J., 23 (1993), 509-536. [9] K. Kishimota and H.-F. Weinberger, The spatial homogeneity of stable equilibria of some reaction-diffusion system on convex domain, J. Differential Equations, 58 (1985), 15-21.  doi: 10.1016/0022-0396(85)90020-8. [10] K. Kuto and T. Tsujikawa, Limiting structure of steady-states to the Lotka-Volterra competition model with large diffusion and advection, J. Differential Equations, 258 (2015), 1801-1858.  doi: 10.1016/j.jde.2014.11.016. [11] A. Leung, Equilibria and stabilities for competing-species reaction-diffusion equations with Dirichlet boundary data, J. Math. Anal. Appl., 73 (1980), 204-218.  doi: 10.1016/0022-247X(80)90028-1. [12] A. Leung and D. Clark, Bifurcations and large-time asymptotic behavior for prey-predator reaction-diffusion equations with Dirichlet boundary data, J. Differential Equations, 35 (1980), 113-127.  doi: 10.1016/0022-0396(80)90052-2. [13] L. Li and R. Logan, Positive solutions to general elliptic competition models, Differential Integral Equations, 4 (1991), 817-834. [14] R. Liu and Q. Wang, $S^1$ attractor bifurcation analysis for an electrically conducting fluid flows between two concentric axial cylinders, Phys. D, 392 (2019), 17-33.  doi: 10.1016/j.physd.2019.03.001. [15] Y. Lou and W. Ni, Diffusion self-diffusion and cross-diffusion, J. Differential Equations, 131 (1996), 79-131.  doi: 10.1006/jdeq.1996.0157. [16] Y. Lou, W. Ni and Y. Wu, On the global existence of a cross-diffusion system, Discrete Contin. Dynam. Systems, 4 (1998), 193-203.  doi: 10.3934/dcds.1998.4.193. [17] Y. Lou, W. Ni and S. Yotsutani, On a limiting system in the Lotka-Volterra competition with cross-diffusion, Discrete Contin. Dyn. Syst., 10 (2004), 435-458.  doi: 10.3934/dcds.2004.10.435. [18] C. Lu, Y. Mao, Q. Wang and D. Yan, Hopf bifurcation and transition of three-dimensional wind-driven ocean circulation problem, J. Differential Equations, 267 (2019), 2560-2593.  doi: 10.1016/j.jde.2019.03.021. [19] T. Ma and S. Wang, Bifurcation Theory and Applications, World Scientific Series on Nonlinear Science Series A: Volume 53, World Scientific Publishing Co. Pte. Ltd., 2005. doi: 10.1142/9789812701152. [20] T. Ma and S. Wang, Stability and bifurcation of the Taylor problem, Arch. Ration. Mech. Anal., 181 (2006), 149-176.  doi: 10.1007/s00205-006-0415-8. [21] T. Ma and S. Wang, Dynamic phase transitions for ferromagnetic systems, J. Math. Phys., 49 (2008), 1-18.  doi: 10.1063/1.2913504. [22] T. Ma and S. Wang, Dynamic transition theory for thermohaline circulation, Phys. D, 239 (2010), 167-189.  doi: 10.1016/j.physd.2009.10.014. [23] T. Ma and S. Wang, Phase transitions for Belousov-Zhabotinsky reactions, Math. Methods Appl. Sci., 34 (2011), 1381-1397.  doi: 10.1002/mma.1446. [24] T. Ma and S. Wang, Phase transitions for the Brusselator model, J. Math. Phys., 52 (2011), 1-23.  doi: 10.1063/1.3559120. [25] T. Ma and S. Wang, Dynamic transition theory for thermohaline circulation, Phys. D, 239 (2010), 167-189.  doi: 10.1016/j.physd.2009.10.014. [26] T. Ma and S. Wang, Phase Transition Dynamics, Springer-Verlag, New York, 2014. doi: 10.1007/978-1-4614-8963-4. [27] T. Ma and S. Wang, Dynamic transition and pattern formation for chemotactic systems, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 2809-2835.  doi: 10.3934/dcdsb.2014.19.2809. [28] W. Ni, Y. Wu and Q. Xu, The existence and stability of nontrivial steady states for S-K-T competition model with cross diffusion, Discrete Contin. Dyn. Syst., 34 (2014), 5271-5298.  doi: 10.3934/dcds.2014.34.5271. [29] J. Shi and X. Wang, On global bifurcation for quasilinear elliptic systems on bounded do-mains, J. Differential Equations, 246 (2009), 2788-2812.  doi: 10.1016/j.jde.2008.09.009. [30] N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species, J. Theoret. Biol., 79 (1979), 83-99.  doi: 10.1016/0022-5193(79)90258-3. [31] L. Sun, J. Shi and Y. Wang, Existence and uniqueness of steady state solutions of a nonlocal diffusive logistic equation, Z. Angew. Math. Phys., 64 (2013), 1267-1278.  doi: 10.1007/s00033-012-0286-9. [32] Q.-J. Tan, A free boundary problem describing S-K-T competition ecological model with cross-diffusion, Nonlinear Anal. Real World Appl., 45 (2019), 53-82.  doi: 10.1016/j.nonrwa.2018.06.010. [33] L. Wang, Y. Wu and Q. Xu, Instability of spiky steady states for S-K-T biological competing model with cross-diffusion, Nonlinear Anal., 159 (2017), 424-457.  doi: 10.1016/j.na.2017.02.026. [34] Q. Wang, On the steady states of a shadow system of the SKT competition system, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 2941-2961.  doi: 10.3934/dcdsb.2014.19.2941. [35] Q. Wang, C. Gai and J. Yan, Qualitative analysis of a Lotka-Volterra competition system with advection, Discrete Contin. Dyn. Syst., 35 (2015), 1239-1284.  doi: 10.3934/dcds.2015.35.1239. [36] Y. Wu, The instability of spiky steady states for a competing species model with cross diffusion, J. Differential Equations, 213 (2005), 289-340.  doi: 10.1016/j.jde.2004.08.015. [37] Y. Wu and Q. Xu, The existence and structure of large spiky steady states for S-K-T competition systems with cross-diffusion, Discrete Contin. Dyn. Syst., 29 (2011), 367-385.  doi: 10.3934/dcds.2011.29.367. [38] Y. Wu and X. Zhao, The existence and stability of traveling waves with transition layers for some singular cross-diffuion systems, Phys. D, 200 (2005), 325-358.  doi: 10.1016/j.physd.2004.11.010. [39] A. Yagi, Exponential attractors for competing species model with cross-diffusions, Discrete Contin. Dyn. Syst., 22 (2008), 1091-1120.  doi: 10.3934/dcds.2008.22.1091. [40] Y. Yamada, Global solutions for the Shigesada-Kawasaki-Teramoto model with cross-diffusion, Recent Progress on Reaction-Diffusion Systems and Viscosity Solutions, World Sci. Publ., Hackensack, NJ, (2009), 282–299. doi: 10.1142/9789812834744_0013. [41] D. Zhang and R. Liu, Dynamical transition for S-K-T biological competing model with cross-diffusion, Math. Method Appl. Sci., 41 (2018), 4641-4658.  doi: 10.1002/mma.4919. [42] X. Zhao and P. Zhou, On a Lotka-Volterra competition model: The effects of advection and spatial variation, Calc. Var. Partial Differential Equations, 55 (2016), 1-25.  doi: 10.1007/s00526-016-1021-8.

show all references

References:
 [1] H. Amann, Dynamic theory of quasilinear parabolic equations, II. Reaction-diffusion systems, Differential Integral Equations, 3 (1990), 13-75. [2] Y. Choi, R. Lui and Y. Yamada, Existence of global solutions for the Shigesada-Kawasaki-Teramoto model with strongly coupled cross-diffusion, Discrete Contin. Dyn. Syst., 10 (2004), 719-730.  doi: 10.3934/dcds.2004.10.719. [3] S. Choo and Y.-H. Kim, Global stability in a discrete Lotka-Volterra competition model, J. Comput. Anal. Appl., 23 (2017), 276-293. [4] D. Deng and R. Liu, Bifurcation solutions of Gross-Pitaevskii equations for spin-1 Bose-Einstein condensates, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 3175-3193.  doi: 10.3934/dcdsb.2018306. [5] P. Du and R. Temam, Weak solutions of the Shigesada-Kawasaki-Teramoto equations and their attractors, Nonlinear Anal., 159 (2017), 339-364.  doi: 10.1016/j.na.2017.01.017. [6] W. Feng, Competitive exclusion and persistence in models of resource and sexual competition, J. Math. Biol., 35 (1997), 683-694.  doi: 10.1007/s002850050071. [7] C.-H. Hsia, C.-S. Lin, T. Ma and S. Wang, Tropical atmospheric circulations with humidity effects, Proc. A., 471 (2015), 1-24.  doi: 10.1098/rspa.2014.0353. [8] Y. Kan-on, Stability of singularly perturbed solutions to nonlinear diffusion systems arising in population dynamics, Hiroshima Math. J., 23 (1993), 509-536. [9] K. Kishimota and H.-F. Weinberger, The spatial homogeneity of stable equilibria of some reaction-diffusion system on convex domain, J. Differential Equations, 58 (1985), 15-21.  doi: 10.1016/0022-0396(85)90020-8. [10] K. Kuto and T. Tsujikawa, Limiting structure of steady-states to the Lotka-Volterra competition model with large diffusion and advection, J. Differential Equations, 258 (2015), 1801-1858.  doi: 10.1016/j.jde.2014.11.016. [11] A. Leung, Equilibria and stabilities for competing-species reaction-diffusion equations with Dirichlet boundary data, J. Math. Anal. Appl., 73 (1980), 204-218.  doi: 10.1016/0022-247X(80)90028-1. [12] A. Leung and D. Clark, Bifurcations and large-time asymptotic behavior for prey-predator reaction-diffusion equations with Dirichlet boundary data, J. Differential Equations, 35 (1980), 113-127.  doi: 10.1016/0022-0396(80)90052-2. [13] L. Li and R. Logan, Positive solutions to general elliptic competition models, Differential Integral Equations, 4 (1991), 817-834. [14] R. Liu and Q. Wang, $S^1$ attractor bifurcation analysis for an electrically conducting fluid flows between two concentric axial cylinders, Phys. D, 392 (2019), 17-33.  doi: 10.1016/j.physd.2019.03.001. [15] Y. Lou and W. Ni, Diffusion self-diffusion and cross-diffusion, J. Differential Equations, 131 (1996), 79-131.  doi: 10.1006/jdeq.1996.0157. [16] Y. Lou, W. Ni and Y. Wu, On the global existence of a cross-diffusion system, Discrete Contin. Dynam. Systems, 4 (1998), 193-203.  doi: 10.3934/dcds.1998.4.193. [17] Y. Lou, W. Ni and S. Yotsutani, On a limiting system in the Lotka-Volterra competition with cross-diffusion, Discrete Contin. Dyn. Syst., 10 (2004), 435-458.  doi: 10.3934/dcds.2004.10.435. [18] C. Lu, Y. Mao, Q. Wang and D. Yan, Hopf bifurcation and transition of three-dimensional wind-driven ocean circulation problem, J. Differential Equations, 267 (2019), 2560-2593.  doi: 10.1016/j.jde.2019.03.021. [19] T. Ma and S. Wang, Bifurcation Theory and Applications, World Scientific Series on Nonlinear Science Series A: Volume 53, World Scientific Publishing Co. Pte. Ltd., 2005. doi: 10.1142/9789812701152. [20] T. Ma and S. Wang, Stability and bifurcation of the Taylor problem, Arch. Ration. Mech. Anal., 181 (2006), 149-176.  doi: 10.1007/s00205-006-0415-8. [21] T. Ma and S. Wang, Dynamic phase transitions for ferromagnetic systems, J. Math. Phys., 49 (2008), 1-18.  doi: 10.1063/1.2913504. [22] T. Ma and S. Wang, Dynamic transition theory for thermohaline circulation, Phys. D, 239 (2010), 167-189.  doi: 10.1016/j.physd.2009.10.014. [23] T. Ma and S. Wang, Phase transitions for Belousov-Zhabotinsky reactions, Math. Methods Appl. Sci., 34 (2011), 1381-1397.  doi: 10.1002/mma.1446. [24] T. Ma and S. Wang, Phase transitions for the Brusselator model, J. Math. Phys., 52 (2011), 1-23.  doi: 10.1063/1.3559120. [25] T. Ma and S. Wang, Dynamic transition theory for thermohaline circulation, Phys. D, 239 (2010), 167-189.  doi: 10.1016/j.physd.2009.10.014. [26] T. Ma and S. Wang, Phase Transition Dynamics, Springer-Verlag, New York, 2014. doi: 10.1007/978-1-4614-8963-4. [27] T. Ma and S. Wang, Dynamic transition and pattern formation for chemotactic systems, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 2809-2835.  doi: 10.3934/dcdsb.2014.19.2809. [28] W. Ni, Y. Wu and Q. Xu, The existence and stability of nontrivial steady states for S-K-T competition model with cross diffusion, Discrete Contin. Dyn. Syst., 34 (2014), 5271-5298.  doi: 10.3934/dcds.2014.34.5271. [29] J. Shi and X. Wang, On global bifurcation for quasilinear elliptic systems on bounded do-mains, J. Differential Equations, 246 (2009), 2788-2812.  doi: 10.1016/j.jde.2008.09.009. [30] N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species, J. Theoret. Biol., 79 (1979), 83-99.  doi: 10.1016/0022-5193(79)90258-3. [31] L. Sun, J. Shi and Y. Wang, Existence and uniqueness of steady state solutions of a nonlocal diffusive logistic equation, Z. Angew. Math. Phys., 64 (2013), 1267-1278.  doi: 10.1007/s00033-012-0286-9. [32] Q.-J. Tan, A free boundary problem describing S-K-T competition ecological model with cross-diffusion, Nonlinear Anal. Real World Appl., 45 (2019), 53-82.  doi: 10.1016/j.nonrwa.2018.06.010. [33] L. Wang, Y. Wu and Q. Xu, Instability of spiky steady states for S-K-T biological competing model with cross-diffusion, Nonlinear Anal., 159 (2017), 424-457.  doi: 10.1016/j.na.2017.02.026. [34] Q. Wang, On the steady states of a shadow system of the SKT competition system, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 2941-2961.  doi: 10.3934/dcdsb.2014.19.2941. [35] Q. Wang, C. Gai and J. Yan, Qualitative analysis of a Lotka-Volterra competition system with advection, Discrete Contin. Dyn. Syst., 35 (2015), 1239-1284.  doi: 10.3934/dcds.2015.35.1239. [36] Y. Wu, The instability of spiky steady states for a competing species model with cross diffusion, J. Differential Equations, 213 (2005), 289-340.  doi: 10.1016/j.jde.2004.08.015. [37] Y. Wu and Q. Xu, The existence and structure of large spiky steady states for S-K-T competition systems with cross-diffusion, Discrete Contin. Dyn. Syst., 29 (2011), 367-385.  doi: 10.3934/dcds.2011.29.367. [38] Y. Wu and X. Zhao, The existence and stability of traveling waves with transition layers for some singular cross-diffuion systems, Phys. D, 200 (2005), 325-358.  doi: 10.1016/j.physd.2004.11.010. [39] A. Yagi, Exponential attractors for competing species model with cross-diffusions, Discrete Contin. Dyn. Syst., 22 (2008), 1091-1120.  doi: 10.3934/dcds.2008.22.1091. [40] Y. Yamada, Global solutions for the Shigesada-Kawasaki-Teramoto model with cross-diffusion, Recent Progress on Reaction-Diffusion Systems and Viscosity Solutions, World Sci. Publ., Hackensack, NJ, (2009), 282–299. doi: 10.1142/9789812834744_0013. [41] D. Zhang and R. Liu, Dynamical transition for S-K-T biological competing model with cross-diffusion, Math. Method Appl. Sci., 41 (2018), 4641-4658.  doi: 10.1002/mma.4919. [42] X. Zhao and P. Zhou, On a Lotka-Volterra competition model: The effects of advection and spatial variation, Calc. Var. Partial Differential Equations, 55 (2016), 1-25.  doi: 10.1007/s00526-016-1021-8.
cross-section graph of $\Lambda_k$
The dynamical transitions of $(13)$ when $\int_{\Omega}\psi_{1}^{3}dx \neq 0$
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