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 & Applied Analysis, 2020, 19 (3) : 1205-1232. doi: 10.3934/cpaa.2020056
References:
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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.  Google Scholar

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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.  Google Scholar

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R. A. Fisher, The wave of advance of advantageous genes, Ann. Eugenics, 7 (1937), 355–369. Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

J. G. Skellam, Random dispersal in theoritical populations, Biometrika, 38 (1951), 196–218. doi: 10.1093/biomet/38.1-2.196.  Google Scholar

[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.  Google Scholar

[36]

J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer, New York, 1983.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[41]

A. Turing, The chemical basis of morphogenesis, Phil. Trans. Royal Soc. Lond, 237 (1952), 37–72.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[46]

Q. X. Ye and Z. Y. Li, Introduction to reaction-Diffusion Equations (in Chinese), Beijing, Science Press, 1990.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

T. Kato, Perturbation Theory for Linear Operators, Second edition, Springer-Verlag, Berlin, New York, 1976.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

J. G. Skellam, Random dispersal in theoritical populations, Biometrika, 38 (1951), 196–218. doi: 10.1093/biomet/38.1-2.196.  Google Scholar

[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.  Google Scholar

[36]

J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer, New York, 1983.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[41]

A. Turing, The chemical basis of morphogenesis, Phil. Trans. Royal Soc. Lond, 237 (1952), 37–72.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[46]

Q. X. Ye and Z. Y. Li, Introduction to reaction-Diffusion Equations (in Chinese), Beijing, Science Press, 1990.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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