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Bifurcation and stability of a two-species diffusive Lotka-Volterra model
1. | Key Laboratory of Jiangxi Province for Numerical Simulation and Emulation Techniques, College of Mathematics and Computer Science, Gannan Normal University, Ganzhou, Jiangxi 341000, China |
2. | School of Mathematics and Physics, China University of Geosciences, Wuhan, Hubei 430074, China |
This paper is devoted to a two-species Lotka-Volterra model with general functional response. The existence, local and global stability of boundary (including trivial and semi-trivial) steady-state solutions are analyzed by means of the signs of the associated principal eigenvalues. Moreover, the nonexistence and steady-state bifurcation of coexistence steady-state solutions at each of the boundary steady states are investigated. In particular, the coincidence of bifurcating coexistence steady-state solution branches is also described. It should be pointed out that the methods we applied here are mainly based on spectral analysis, perturbation theory, comparison principle, monotone theory, Lyapunov-Schmidt reduction, and bifurcation theory.
References:
[1] |
R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-diffusion Equations, John Wiley & Sons, 2003.
doi: 10.1002/0470871296. |
[2] |
N. Chafee and E. F. Infante, A bifurcation problem for a nonlinear partial differential equation of parabolic type, Applicable Analysis, 4 (1974), 17–37.
doi: 10.1080/00036817408839081. |
[3] |
M. G. Crandall and P. H. Rabinowtiz, Bifurcation from simple eigenvalues, J. Functional Analysis, 8 (1971), 321–340.
doi: 10.1016/0022-1236(71)90015-2. |
[4] |
M. G. Crandall and P. H. Rabinowtiz, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rational Mech. Anal, 52 (1973), 161–180.
doi: 10.1007/BF00282325. |
[5] |
Y. B. Deng, S. J. Peng and S. S. Yan, Positive soliton solutions for generalized quasilinear Schr$\ddot{o}$dinger equations with critical growth, J. Differential Equations, 258 (2015), 115–147.
doi: 10.1016/j.jde.2014.09.006. |
[6] |
T. Faria, Stability and bifurcation for a delayed predator-prey model and the effect of diffusion, J. Math. Anal. Appl., 254 (2001), 433–463.
doi: 10.1006/jmaa.2000.7182. |
[7] |
R. A. Fisher, The wave of advance of advantageous genes, Ann. Eugenics, 7 (1937), 355–369. Google Scholar |
[8] |
G. H. Guo, J. H. Wu and Y. E Wang, Bifurcation from a double eigenvalue in the unstirred chemostat, Applicable Analysis, 92 (2013), 1449–1461.
doi: 10.1080/00036811.2012.683786. |
[9] |
S. J. Guo, Bifurcation and spatio-temporal patterns in a diffusive predator-prey system, Nonlinear Anal. Real World Appl., 42 (2018), 448–477.
doi: 10.1016/j.nonrwa.2018.01.011. |
[10] |
S. J. Guo, Patterns in a nonlocal time-delayed reaction-diffusion equation, Zeitschrift Fur Angewandte Mathematik Und Physik, 69 (2018), 10.
doi: 10.1007/s00033-017-0904-7. |
[11] |
S. J. Guo, Spatio-temporal patterns in a diffusive model with non-local delay effect, IMA Journal of Applied Mathematics, 82 (2017), 864–908.
doi: 10.1093/imamat/hxx018. |
[12] |
S. J. Guo, Stability and bifurcation in a reaction-diffusion model with nonlocal delay effect, J. Differential Equations, 259 (2015), 1409–1448.
doi: 10.1016/j.jde.2015.03.006. |
[13] |
S. J. Guo and S. L. Yan, Hopf bifurcation in a diffusive Lotka-Volterra type system with nonlocal delay effect, J. Differential Equations, 260 (2016), 781–817.
doi: 10.1016/j.jde.2015.09.031. |
[14] |
S. J. Guo and L. Ma, Stability and bifurcation in a delayed reaction-diffusion equation with Dirichlet boundary condition, J. Nonlinear Science, 26 (2016), 545–580.
doi: 10.1007/s00332-016-9285-x. |
[15] |
X. Q. He and W. M. Ni, The effects of diffusion and spatial variation in Lotka-Volterra competition-diffusion system Ⅰ: Heterogeneity vs. homogeneity, Journal of Differential Equations, 254 (2013), 528–546.
doi: 10.1016/j.jde.2012.08.032. |
[16] |
H. J. Hu and X. F. Zou, Existence of an extinction wave in the fisher equation with a shifting habitat, Proceedings of the American Mathematical Society, 145 (2017), 4763–4771.
doi: 10.1090/proc/13687. |
[17] |
H. J. Hu, Y. X. Tan and J. H. Huang, Hopf bifurcation analysis on a delayed reaction-diffusion system modelling the spatial spread of bacterial and viral diseases, Chaos, Solitons & Fractals, 125 (2019), 152–162.
doi: 10.1016/j.chaos.2019.05.002. |
[18] |
M. Ito, Global aspect of steady-states for competitive-diffusive systems with homogeneous Dirichlet conditions, Phys. D, 14 (1984), 1–28.
doi: 10.1016/0167-2789(84)90002-2. |
[19] |
T. Kato, Perturbation Theory for Linear Operators, Second edition, Springer-Verlag, Berlin, New York, 1976. |
[20] |
H. Kierstead and L. B. Slobodkin, The size of water masses containing plankton bloom, J. Mar. Res., 12 (1953), 141–147. Google Scholar |
[21] |
A. Kolmogoroff, I. Petrovsky and N. Piscounoff, Study of the diffusion equation with growth of the quantity of matter and its application to a biological problem, Moscow Univ. Bull. Math., 1 (2012), 105 (French).
doi: 10.1016/0022-0396(85)90137-8. |
[22] |
M. G. Krein and M. A. Rutman, Linear operators leaving invariant a cone in Banach space, Uspekhi Mat. Nauk (N.S.), 3 (1948), 3–95. |
[23] |
K. Kuto and T. Tsujikawa, Limiting structure of steady-states to the Lotka-Volterra compttition model with large diffusion and advection, J. Differential Equations, 258 (2015), 1801–1858.
doi: 10.1016/j.jde.2014.11.016. |
[24] |
F. Li and H. W. Li, Hopf bifurcation of a predator-prey model with time delay and stage structure for the prey, Mathematical and Computer Modelling, 55 (2012), 672–679.
doi: 10.1016/j.mcm.2011.08.041. |
[25] |
S. B. Li, J. H. Wu and Y. Y. Dong, Bifurcation and Stability for the Unstirred Chemostat Model with Beddington-DeAngelis Functional Response, J. Taiwanese Journal of Mathematics, 20 (2016), 849–870.
doi: 10.11650/tjm.20.2016.5482. |
[26] |
S. Z. Li, S. J. Guo, Stability and Hopf bifurcation in a Hutchinson model, Appl. Math. Lett. 101 (2020), 106066.
doi: 10.1016/j.aml.2019.106066. |
[27] |
L. Ma, S. J. Guo and T. Chen, Dynamics of a nonlocal dispersal model with a nonlocal reaction term. Internat, J. Bifur. Chaos Appl. Sci. Engrg., 28 (2018), 1850033.
doi: 10.1142/S0218127418500335. |
[28] |
L. Ma and S. J. Guo, Stability and bifurcation in a diffusive Lotka-Volterra system with delay, Computers and Mathematics with Applications, 72 (2016), 147–177.
doi: 10.1016/j.camwa.2016.04.049. |
[29] |
H. Nie and J. H. Wu, Uniqueness and stability for coexistence solutions of the unstirred chemostat model, App. Anal., 89 (2010), 1141–1159.
doi: 10.1080/00036811003717954. |
[30] |
C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992. |
[31] |
M. H. Protter and H. F. Weinberger, Maximum Principle in Differential Equations, Springer, New York, 1984.
doi: 10.1007/978-1-4612-5282-5. |
[32] |
H. H. Qiu and S. J. Guo, Steady-states of a Leslie-Gower model with diffusion and advection, Appl. Math. Comput., 346 (2019), 695–709.
doi: 10.1016/j.amc.2018.10.002. |
[33] |
K. Ryu and I. Ann, Positive solutions for ratio-dependent predator-prey interaction system, J. Differential Equations, 218 (2005), 117–135.
doi: 10.1016/j.jde.2005.06.020. |
[34] |
J. G. Skellam, Random dispersal in theoritical populations, Biometrika, 38 (1951), 196–218.
doi: 10.1093/biomet/38.1-2.196. |
[35] |
H. Smith, Monotone Dynamical Systems, An Introduction to the Theory of Competitive and Cooperative Syatems, Mth. Surveys Monogr. 41, American Mathematical Society, Providence, RI, 1995. |
[36] |
J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer, New York, 1983. |
[37] |
J. W.-H. So and P. Waltman, A nonlinear boundary value problem arising from competition in the chemostat, Appl. Math. Comput., 32 (1989), 169–183.
doi: 10.1016/0096-3003(89)90092-1. |
[38] |
M. E. Solomon, The natural control of animal populations, Journal of Animal Ecology, 18 (1949), 1–35. Google Scholar |
[39] |
B. Sounvoravong, J. P. Gao and S. J. Guo, Bifurcation analysis of a diffusive predator-prey system with nonmonotonic functional response, Nonlinear Dyn, 94 (2018), 2901–2918.
doi: 10.1007/s11071-018-4533-2. |
[40] |
Y. X. Tan, C. X. Huang, B. Sun and T. Wang, Dynamics of a class of delayed reaction-diffusion systems with Neumann boundary condition, Journal of Mathematical Analysis and Applications, 458 (2018), 1115–1130.
doi: 10.1016/j.jmaa.2017.09.045. |
[41] |
A. Turing, The chemical basis of morphogenesis, Phil. Trans. Royal Soc. Lond, 237 (1952), 37–72. |
[42] |
J. H. Wu, Global bifurcation of coexistence state for the competition model in the chemostat, Nonlinear Anal. Ser. A: Theory Methods, 39 (2000), 817–835.
doi: 10.1016/S0362-546X(98)00250-8. |
[43] |
Y. Yamada, Stability of steady states for prey-predator diffusion equations with homogeneous Dirichlet conditions, SIAM J. Math. Anal., 21 (1990), 327–345.
doi: 10.1137/0521018. |
[44] |
S. L. Yan and S. J. Guo, Bifurcation phenomena in a Lotka-Volterra model with cross-diffusion and delay effect, International Journal of Bifurcation and Chaos, 27 (2017), 1750105.
doi: 10.1142/S021812741750105X. |
[45] |
S. L. Yan and S. J. Guo, Dynamics of a Lotka-Volterra competition-diffusion model with stage structure and spatial heterogeneity, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 1559–1579.
doi: 10.3934/dcdsb.2018059. |
[46] |
Q. X. Ye and Z. Y. Li, Introduction to reaction-Diffusion Equations (in Chinese), Beijing, Science Press, 1990. |
[47] |
F. Q. Yi, J. J. Wei and J. P. Shi, Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system, J. Differential Equations, 246 (2009), 1944–1977.
doi: 10.1016/j.jde.2008.10.024. |
[48] |
F. Q. Yi, H. Zhang, A. Cherif A and W. Zhang, Spatiotemporal patterns of a homogeneous diffusive system modeling hair growth: global asymptotic behavior and multiple bifurcation analysis, Communications on Pure and Applied Analysis, 13 (2014), 347–369.
doi: 10.3934/cpaa.2014.13.347. |
[49] |
X. Y. Zhong, S. J. Guo and M. F. Peng, Stability of stochastic SIRS epidemic models with saturated incidence rates and delay, Stochastic Analysis and Applications, 35 (2017), 1–26.
doi: 10.1080/07362994.2016.1244644. |
[50] |
R. Zou and S. J. Guo, Dynamics in a diffusive predator-prey system with ratio-dependent predator influence, Comput. Math. Appl., 75 (2018), 1237–1258.
doi: 10.1016/j.camwa.2017.11.002. |
[51] |
R. Zou and S. J. Guo, Bifurcation of reaction cross-diffusion systems, International Journal of Bifurcation and Chaos, 27 (2017), 1750049.
doi: 10.1142/S0218127417500493. |
show all references
References:
[1] |
R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-diffusion Equations, John Wiley & Sons, 2003.
doi: 10.1002/0470871296. |
[2] |
N. Chafee and E. F. Infante, A bifurcation problem for a nonlinear partial differential equation of parabolic type, Applicable Analysis, 4 (1974), 17–37.
doi: 10.1080/00036817408839081. |
[3] |
M. G. Crandall and P. H. Rabinowtiz, Bifurcation from simple eigenvalues, J. Functional Analysis, 8 (1971), 321–340.
doi: 10.1016/0022-1236(71)90015-2. |
[4] |
M. G. Crandall and P. H. Rabinowtiz, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rational Mech. Anal, 52 (1973), 161–180.
doi: 10.1007/BF00282325. |
[5] |
Y. B. Deng, S. J. Peng and S. S. Yan, Positive soliton solutions for generalized quasilinear Schr$\ddot{o}$dinger equations with critical growth, J. Differential Equations, 258 (2015), 115–147.
doi: 10.1016/j.jde.2014.09.006. |
[6] |
T. Faria, Stability and bifurcation for a delayed predator-prey model and the effect of diffusion, J. Math. Anal. Appl., 254 (2001), 433–463.
doi: 10.1006/jmaa.2000.7182. |
[7] |
R. A. Fisher, The wave of advance of advantageous genes, Ann. Eugenics, 7 (1937), 355–369. Google Scholar |
[8] |
G. H. Guo, J. H. Wu and Y. E Wang, Bifurcation from a double eigenvalue in the unstirred chemostat, Applicable Analysis, 92 (2013), 1449–1461.
doi: 10.1080/00036811.2012.683786. |
[9] |
S. J. Guo, Bifurcation and spatio-temporal patterns in a diffusive predator-prey system, Nonlinear Anal. Real World Appl., 42 (2018), 448–477.
doi: 10.1016/j.nonrwa.2018.01.011. |
[10] |
S. J. Guo, Patterns in a nonlocal time-delayed reaction-diffusion equation, Zeitschrift Fur Angewandte Mathematik Und Physik, 69 (2018), 10.
doi: 10.1007/s00033-017-0904-7. |
[11] |
S. J. Guo, Spatio-temporal patterns in a diffusive model with non-local delay effect, IMA Journal of Applied Mathematics, 82 (2017), 864–908.
doi: 10.1093/imamat/hxx018. |
[12] |
S. J. Guo, Stability and bifurcation in a reaction-diffusion model with nonlocal delay effect, J. Differential Equations, 259 (2015), 1409–1448.
doi: 10.1016/j.jde.2015.03.006. |
[13] |
S. J. Guo and S. L. Yan, Hopf bifurcation in a diffusive Lotka-Volterra type system with nonlocal delay effect, J. Differential Equations, 260 (2016), 781–817.
doi: 10.1016/j.jde.2015.09.031. |
[14] |
S. J. Guo and L. Ma, Stability and bifurcation in a delayed reaction-diffusion equation with Dirichlet boundary condition, J. Nonlinear Science, 26 (2016), 545–580.
doi: 10.1007/s00332-016-9285-x. |
[15] |
X. Q. He and W. M. Ni, The effects of diffusion and spatial variation in Lotka-Volterra competition-diffusion system Ⅰ: Heterogeneity vs. homogeneity, Journal of Differential Equations, 254 (2013), 528–546.
doi: 10.1016/j.jde.2012.08.032. |
[16] |
H. J. Hu and X. F. Zou, Existence of an extinction wave in the fisher equation with a shifting habitat, Proceedings of the American Mathematical Society, 145 (2017), 4763–4771.
doi: 10.1090/proc/13687. |
[17] |
H. J. Hu, Y. X. Tan and J. H. Huang, Hopf bifurcation analysis on a delayed reaction-diffusion system modelling the spatial spread of bacterial and viral diseases, Chaos, Solitons & Fractals, 125 (2019), 152–162.
doi: 10.1016/j.chaos.2019.05.002. |
[18] |
M. Ito, Global aspect of steady-states for competitive-diffusive systems with homogeneous Dirichlet conditions, Phys. D, 14 (1984), 1–28.
doi: 10.1016/0167-2789(84)90002-2. |
[19] |
T. Kato, Perturbation Theory for Linear Operators, Second edition, Springer-Verlag, Berlin, New York, 1976. |
[20] |
H. Kierstead and L. B. Slobodkin, The size of water masses containing plankton bloom, J. Mar. Res., 12 (1953), 141–147. Google Scholar |
[21] |
A. Kolmogoroff, I. Petrovsky and N. Piscounoff, Study of the diffusion equation with growth of the quantity of matter and its application to a biological problem, Moscow Univ. Bull. Math., 1 (2012), 105 (French).
doi: 10.1016/0022-0396(85)90137-8. |
[22] |
M. G. Krein and M. A. Rutman, Linear operators leaving invariant a cone in Banach space, Uspekhi Mat. Nauk (N.S.), 3 (1948), 3–95. |
[23] |
K. Kuto and T. Tsujikawa, Limiting structure of steady-states to the Lotka-Volterra compttition model with large diffusion and advection, J. Differential Equations, 258 (2015), 1801–1858.
doi: 10.1016/j.jde.2014.11.016. |
[24] |
F. Li and H. W. Li, Hopf bifurcation of a predator-prey model with time delay and stage structure for the prey, Mathematical and Computer Modelling, 55 (2012), 672–679.
doi: 10.1016/j.mcm.2011.08.041. |
[25] |
S. B. Li, J. H. Wu and Y. Y. Dong, Bifurcation and Stability for the Unstirred Chemostat Model with Beddington-DeAngelis Functional Response, J. Taiwanese Journal of Mathematics, 20 (2016), 849–870.
doi: 10.11650/tjm.20.2016.5482. |
[26] |
S. Z. Li, S. J. Guo, Stability and Hopf bifurcation in a Hutchinson model, Appl. Math. Lett. 101 (2020), 106066.
doi: 10.1016/j.aml.2019.106066. |
[27] |
L. Ma, S. J. Guo and T. Chen, Dynamics of a nonlocal dispersal model with a nonlocal reaction term. Internat, J. Bifur. Chaos Appl. Sci. Engrg., 28 (2018), 1850033.
doi: 10.1142/S0218127418500335. |
[28] |
L. Ma and S. J. Guo, Stability and bifurcation in a diffusive Lotka-Volterra system with delay, Computers and Mathematics with Applications, 72 (2016), 147–177.
doi: 10.1016/j.camwa.2016.04.049. |
[29] |
H. Nie and J. H. Wu, Uniqueness and stability for coexistence solutions of the unstirred chemostat model, App. Anal., 89 (2010), 1141–1159.
doi: 10.1080/00036811003717954. |
[30] |
C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992. |
[31] |
M. H. Protter and H. F. Weinberger, Maximum Principle in Differential Equations, Springer, New York, 1984.
doi: 10.1007/978-1-4612-5282-5. |
[32] |
H. H. Qiu and S. J. Guo, Steady-states of a Leslie-Gower model with diffusion and advection, Appl. Math. Comput., 346 (2019), 695–709.
doi: 10.1016/j.amc.2018.10.002. |
[33] |
K. Ryu and I. Ann, Positive solutions for ratio-dependent predator-prey interaction system, J. Differential Equations, 218 (2005), 117–135.
doi: 10.1016/j.jde.2005.06.020. |
[34] |
J. G. Skellam, Random dispersal in theoritical populations, Biometrika, 38 (1951), 196–218.
doi: 10.1093/biomet/38.1-2.196. |
[35] |
H. Smith, Monotone Dynamical Systems, An Introduction to the Theory of Competitive and Cooperative Syatems, Mth. Surveys Monogr. 41, American Mathematical Society, Providence, RI, 1995. |
[36] |
J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer, New York, 1983. |
[37] |
J. W.-H. So and P. Waltman, A nonlinear boundary value problem arising from competition in the chemostat, Appl. Math. Comput., 32 (1989), 169–183.
doi: 10.1016/0096-3003(89)90092-1. |
[38] |
M. E. Solomon, The natural control of animal populations, Journal of Animal Ecology, 18 (1949), 1–35. Google Scholar |
[39] |
B. Sounvoravong, J. P. Gao and S. J. Guo, Bifurcation analysis of a diffusive predator-prey system with nonmonotonic functional response, Nonlinear Dyn, 94 (2018), 2901–2918.
doi: 10.1007/s11071-018-4533-2. |
[40] |
Y. X. Tan, C. X. Huang, B. Sun and T. Wang, Dynamics of a class of delayed reaction-diffusion systems with Neumann boundary condition, Journal of Mathematical Analysis and Applications, 458 (2018), 1115–1130.
doi: 10.1016/j.jmaa.2017.09.045. |
[41] |
A. Turing, The chemical basis of morphogenesis, Phil. Trans. Royal Soc. Lond, 237 (1952), 37–72. |
[42] |
J. H. Wu, Global bifurcation of coexistence state for the competition model in the chemostat, Nonlinear Anal. Ser. A: Theory Methods, 39 (2000), 817–835.
doi: 10.1016/S0362-546X(98)00250-8. |
[43] |
Y. Yamada, Stability of steady states for prey-predator diffusion equations with homogeneous Dirichlet conditions, SIAM J. Math. Anal., 21 (1990), 327–345.
doi: 10.1137/0521018. |
[44] |
S. L. Yan and S. J. Guo, Bifurcation phenomena in a Lotka-Volterra model with cross-diffusion and delay effect, International Journal of Bifurcation and Chaos, 27 (2017), 1750105.
doi: 10.1142/S021812741750105X. |
[45] |
S. L. Yan and S. J. Guo, Dynamics of a Lotka-Volterra competition-diffusion model with stage structure and spatial heterogeneity, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 1559–1579.
doi: 10.3934/dcdsb.2018059. |
[46] |
Q. X. Ye and Z. Y. Li, Introduction to reaction-Diffusion Equations (in Chinese), Beijing, Science Press, 1990. |
[47] |
F. Q. Yi, J. J. Wei and J. P. Shi, Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system, J. Differential Equations, 246 (2009), 1944–1977.
doi: 10.1016/j.jde.2008.10.024. |
[48] |
F. Q. Yi, H. Zhang, A. Cherif A and W. Zhang, Spatiotemporal patterns of a homogeneous diffusive system modeling hair growth: global asymptotic behavior and multiple bifurcation analysis, Communications on Pure and Applied Analysis, 13 (2014), 347–369.
doi: 10.3934/cpaa.2014.13.347. |
[49] |
X. Y. Zhong, S. J. Guo and M. F. Peng, Stability of stochastic SIRS epidemic models with saturated incidence rates and delay, Stochastic Analysis and Applications, 35 (2017), 1–26.
doi: 10.1080/07362994.2016.1244644. |
[50] |
R. Zou and S. J. Guo, Dynamics in a diffusive predator-prey system with ratio-dependent predator influence, Comput. Math. Appl., 75 (2018), 1237–1258.
doi: 10.1016/j.camwa.2017.11.002. |
[51] |
R. Zou and S. J. Guo, Bifurcation of reaction cross-diffusion systems, International Journal of Bifurcation and Chaos, 27 (2017), 1750049.
doi: 10.1142/S0218127417500493. |
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