The main objective of this article is to investigate the dynamical transition for a 3-component Lotka-Volterra model with diffusion. Based on the spectral analysis, the principle of exchange of stability conditions for eigenvalues is obtained. In addition, when $ \delta_0 < \delta_1 $, the first eigenvalues are complex, and we show that the system undergoes a continuous or jump transition. In the small oscillation frequency limit, the transition is always continuous and the time periodic rolls are stable after the transition. In the case where $ \delta_0 > \delta_1 $, the first eigenvalue is real. Generically, the first eigenvalue is simple and all three types of transition are possible. In particular, the transition is mixed if $ \int_{\Omega}e_{k_0}^3dx\neq 0 $, and is continuous or jump in the case where $ \int_{\Omega}e_{k_0}^3dx = 0 $. In this case we also show that the system bifurcates to two saddle points on $ \delta < \delta_1 $ as $ \tilde{\theta} > 0 $, and bifurcates to two stable singular points on $ \delta > \delta_1 $ as $ \tilde{\theta} < 0 $ where $ \tilde{\theta} $ depends on the system parameters.
Citation: Ruili Wu, Limei Li, Junyan Li. Dynamical transition for a 3-component Lotka-Volterra model with diffusion[J]. AIMS Mathematics, 2021, 6(5): 4345-4369. doi: 10.3934/math.2021258
The main objective of this article is to investigate the dynamical transition for a 3-component Lotka-Volterra model with diffusion. Based on the spectral analysis, the principle of exchange of stability conditions for eigenvalues is obtained. In addition, when $ \delta_0 < \delta_1 $, the first eigenvalues are complex, and we show that the system undergoes a continuous or jump transition. In the small oscillation frequency limit, the transition is always continuous and the time periodic rolls are stable after the transition. In the case where $ \delta_0 > \delta_1 $, the first eigenvalue is real. Generically, the first eigenvalue is simple and all three types of transition are possible. In particular, the transition is mixed if $ \int_{\Omega}e_{k_0}^3dx\neq 0 $, and is continuous or jump in the case where $ \int_{\Omega}e_{k_0}^3dx = 0 $. In this case we also show that the system bifurcates to two saddle points on $ \delta < \delta_1 $ as $ \tilde{\theta} > 0 $, and bifurcates to two stable singular points on $ \delta > \delta_1 $ as $ \tilde{\theta} < 0 $ where $ \tilde{\theta} $ depends on the system parameters.
| [1] |
K. Kuto, T. Tsujikawa, Limiting structure of steady-states to the Lotka-Volterra competition model with large diffusion and advection, J. Differ. Equations, 258 (2015), 1801–1858. doi: 10.1016/j.jde.2014.11.016
|
| [2] |
J. C. Eilbeck, J. E. Furter, J. Lopezgomez, Coexistence in the competition model with diffusion, J. Differ. Equations, 107 (1994), 96–139. doi: 10.1006/jdeq.1994.1005
|
| [3] | L. Lou, W. M. Ni, S. Yotsutani, On a limiting system in the Lotka-Volterra competition with cross-diffusion, Discrete Contin. Dyn. Syst., 10 (2004), 435–458. |
| [4] |
Y. Wu, The instability of spiky steady states for a competing species model with cross diffusion, J. Differ. Equations, 213 (2005), 289–340. doi: 10.1016/j.jde.2004.08.015
|
| [5] |
W. M. Ni, Y. Wu, 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
|
| [6] |
L. Lou, W. M. Ni, S. Yotsutani, Pattern formation in a cross-diffusion system, Discrete Contin. Dyn. Syst., 35 (2015), 1589–1607. doi: 10.3934/dcds.2015.35.1589
|
| [7] | W. M. Ni, M. Salom$\acute{e}$, Y. Lou, On $3\times 3$ Lotka-Volterra competition systems with cross-diffusion, Discrete Contin. Dyn. Syst., 6 (1999), 175–190. |
| [8] |
P. Y. H. Pang, M. Wang, Strategy and stationary pattern in a three-species predator-prey model, J. Differ. Equations, 200 (2004), 245–273. doi: 10.1016/j.jde.2004.01.004
|
| [9] |
N. Ali, M. Haque, E. Venturino, S. Chakravarty, Dynamics of a three species ratio-dependent food chain model with intra-specific competition within the top predator, Comp. Biol. Med., 85 (2017), 63–74. doi: 10.1016/j.compbiomed.2017.04.007
|
| [10] |
T. Ikeda, M. Mimura, An interfacial approach to regional segregation of two competing species mediated by a predator, J. Math. Biol., 31 (1993), 215–240. doi: 10.1007/BF00166143
|
| [11] |
Y. Kan-on, M. Mimura, Singular perturbation approach to a 3-component reaction-diffusion system arising in population dynamics, SIAM J. Math. Anal., 29 (1998), 1519–1536. doi: 10.1137/S0036141097318328
|
| [12] |
Y. Kan-on, Existence and instability of Neumann layer solutions for a 3-component Lotka-Volterra model with diffusion, J. Math. Anal. Appl., 243 (2000), 357–372. doi: 10.1006/jmaa.1999.6676
|
| [13] |
M. X. Wang, Stationary patterns of strongly coupled prey-predator models, J. Math. Anal. Appl., 292 (2004), 484–505. doi: 10.1016/j.jmaa.2003.12.027
|
| [14] |
W. Chen, R. Peng, Stationary patterns created by cross-diffusion for the competitor-competitor-mutualist model, J. Math. Anal. Appl., 291 (2004), 550–564. doi: 10.1016/j.jmaa.2003.11.015
|
| [15] |
W. Ko, K. Ryu, I. Ahn, Coexistence of Three Competing Species with Non-negative Cross-diffusion rate, J. Dynam. Contr. Syst., 20 (2014), 229–240. doi: 10.1007/s10883-014-9219-6
|
| [16] |
K. Ryu, I. Ahn, Coexistence theorem of steady states for nonlinear self-cross diffusion systems with competitive dynamics, J. Math. Anal. Appl., 283 (2003), 46–65. doi: 10.1016/S0022-247X(03)00162-8
|
| [17] |
L. Li, Coexistence Theorems of Steady States for Predator-Prey Interacting Systems, Trans. Am. Math. Soc., 305 (1988), 143–166. doi: 10.1090/S0002-9947-1988-0920151-1
|
| [18] | T. Ma, S. H. Wang, Bifurcation Theory and Applications, World Scientific, Singapore, 2005. |
| [19] | T. Ma, S. H. Wang, Stability and Bifurcation of Nonlinear Evolutions Equations, Science Press, Beijing, 2007. |
| [20] | T. Ma, S. H. Wang, Phase Transition Dynamics, New York: Springer-Verlag, 2014. |
| [21] |
T. Ma, S. H. Wang, Phase transitions for Belousov-Zhabotinsky reactions, Math. Methods. Appl. Sci., 34 (2011), 1381–1397. doi: 10.1002/mma.1446
|
| [22] |
C. H. Hsia, T. Ma, S. H. Wang, Rotating Boussinesq equations: dynamic stability and transitions, Discrete Contin. Dyn. Syst., 28 (2010), 99–130. doi: 10.3934/dcds.2010.28.99
|
| [23] | T. Ma, S. H. Wang, Dynamic transition and pattern formation for chemotactic systems, Discrete Contin. Dyn. Syst. Ser B., 19 (2014), 2809–2835. |
| [24] |
Z. G. Pan, T. Sengul, Q. Wang, On the viscous instabilities and transitions of two-layer model with a layered topography, Commun. Nonlinear Sci. Numer. Simul., 80 (2020), 104978. doi: 10.1016/j.cnsns.2019.104978
|
| [25] |
C. H. Lu, Y. Mao, Q Wang, D. M. Yan, Hopf bifurcation and transition of three-dimensional wind-driven ocean circulation problem, J. Differ. Equations, 267 (2019), 2560–2593. doi: 10.1016/j.jde.2019.03.021
|
| [26] |
R. Liu, Q. Wang, $S^1$ attractor bifurcation analysis for an electrically conducting fluid flow between two rotating cylinders, Phys. D., 392 (2019), 17–33. doi: 10.1016/j.physd.2019.03.001
|
| [27] | T. Ma, S. H. Wang, Dynamic transition theory for thermohaline circulation, Phys. D. Nonlinear Phenomena, 239 (2009), 167–189. |
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Ruili Wu, Limei Li, Junyan Li. Dynamical transition for a 3-component Lotka-Volterra model with diffusion[J]. AIMS Mathematics, 2021, 6(5): 4345-4369. doi: 10.3934/math.2021258
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