March  2020, 19(3): 1205-1232. doi: 10.3934/cpaa.2020056

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

* Corresponding author

Received  July 2017 Revised  August 2019 Published  November 2019

Fund Project: The first author is supported by Jiangxi Science and Technology Project (Grant No. GJJ170844), and the second author is supported by the National Natural Science Foundation of China (Grants Nos. 11671123, 11801089, 11901110).

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.

Citation: Li Ma, Shangjiang Guo. Bifurcation and stability of a two-species diffusive Lotka-Volterra model. Communications on Pure and Applied Analysis, 2020, 19 (3) : 1205-1232. doi: 10.3934/cpaa.2020056
References:
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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.

[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.

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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.

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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.

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H. Kierstead and L. B. Slobodkin, The size of water masses containing plankton bloom, J. Mar. Res., 12 (1953), 141–147.

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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.

[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.

[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.

[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.

[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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