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doi: 10.3934/dcdsb.2021287
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Dynamics of a delayed Lotka-Volterra model with two predators competing for one prey

1. 

Business School, Hunan University, Changsha, Hunan 410082, China

2. 

School of Mathematics and Physics, China University of Geosciences, Wuhan, Hubei 430074, China

*Corresponding author: Shangjiang Guo

Received  June 2021 Revised  July 2021 Early access December 2021

Fund Project: The second author is supported by the National Natural Science Foundation of P.R. China (Grant No. 12071446), and by the Fundamental Research Funds for the Central Universities, People's Republic of China, China University of Geosciences (Wuhan) (Grant No. CUGST2)

In this paper, we study the local dynamics of a class of 3-dimensional Lotka-Volterra systems with a discrete delay. This system describes two predators competing for one prey. Firstly, linear stability and Hopf bifurcation are investigated. Then some regions of attraction for the positive steady state are obtained by means of Liapunov functional in a restricted region. Finally, sufficient and necessary conditions for the principle of competitive exclusion are obtained.

Citation: Minzhen Xu, Shangjiang Guo. Dynamics of a delayed Lotka-Volterra model with two predators competing for one prey. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021287
References:
[1]

R. A. Armstrong and R. McGehee, Competitive exclusion, Amer. Natur., 115 (1980), 151-170.  doi: 10.1086/283553.  Google Scholar

[2]

E. Beretta and Y. Kuang, Convergence results in a well-known delayed predator-prey system, J. Math. Anal. Appl., 204 (1996), 840-853.  doi: 10.1006/jmaa.1996.0471.  Google Scholar

[3]

F. Brauer and C. Castillo-Chávez, Mathematical Models in Population Biology and Epidemiology, Springer-Verlag, New York, 2001.  Google Scholar

[4]

E. ChauvetJ. E. PaulletJ. P. Previte and Z. Walls, A Lotka-Volterra three-species food chain, Math. Mag., 75 (2002), 243-255.  doi: 10.2307/3219158.  Google Scholar

[5]

J. P. Gao and S. J. Guo, Global dynamics and spatio-temporal patterns in a two-species chemotaxis system with two chemicals, Z. Angew. Math. Phys., 72 (2021), 25, 28pp. doi: 10.1007/s00033-020-01449-8.  Google Scholar

[6]

S. J. Guo, Bifurcation in a reaction-diffusion model with nonlocal delay effect and nonlinear boundary condition, J. Differential Equations, 289 (2021), 236-278.  doi: 10.1016/j.jde.2021.04.021.  Google Scholar

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S. J. Guo and S. Z. Li, On the stability of reaction-diffusion models with nonlocal delay effect and nonlinear boundary condition, Appl. Math. Lett., 103 (2020), 106197, 7pp. doi: 10.1016/j.aml.2019.106197.  Google Scholar

[8]

S. J. Guo, S. Z. Li and B. Sounvoravong, Oscillatory and stationary patterns in a diffusive model with delay effect, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 31 (2021), 2150035, 21pp. doi: 10.1142/S0218127421500358.  Google Scholar

[9]

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

[10]

G. Hardin, The competitive exclusion principle, Science, 131 (1960), 1292-1297.  doi: 10.1126/science.131.3409.1292.  Google Scholar

[11]

A. Korobeinikov and G. C. Wake, Global properties of the three-dimensional predator-prey Lotka-Volterra systems, J. Appl. Math. Decis. Sci., 3 (1999), 155-162.  doi: 10.1155/S1173912699000085.  Google Scholar

[12] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, Inc., Boston, 1993.   Google Scholar
[13]

Y. Kuang, Global stability in delay differential systems without dominating instantaneous negative feedbacks, J. Differential Equations, 119 (1995), 503-532.  doi: 10.1006/jdeq.1995.1100.  Google Scholar

[14]

Y. Kuang and H. L. Smith, Global stability for infinite delay Lotka-Volterra type systems, J. Differential Equations, 103 (1993), 221-246.  doi: 10.1006/jdeq.1993.1048.  Google Scholar

[15]

J. P. LaSalle, The Stability of Dynamical Systems, Reg. Conf. Ser. Appl. Math., SIMA, Philadelphia, 1976.  Google Scholar

[16]

B. Li and H. L. Smith, Global dynamics of microbial competition for two resources with internal storage competition model, J. Math. Biol., 55 (2007), 481-515.  doi: 10.1007/s00285-007-0092-8.  Google Scholar

[17]

S. Z. Li and S. J. Guo, Permanence and extinction of a stochastic prey-predator model with a general functional response, Math. Comput. Simulation, 187 (2021), 308-336.  doi: 10.1016/j.matcom.2021.02.025.  Google Scholar

[18]

S. Z. Li and S. J. Guo, Dynamics of a stage-structured population model with a state-dependent delay, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 3523-3551.  doi: 10.3934/dcdsb.2020071.  Google Scholar

[19]

C. F. Liu and S. J. Guo, Steady states of Lotka-Volterra competition models with nonlinear cross-diffusion, J. Differential Equations, 292 (2021), 247-286.  doi: 10.1016/j.jde.2021.05.014.  Google Scholar

[20]

J. Llibre and D. Xiao, Global dynamics of a Lotka-Volterra model with two predators competing for one prey, SIAM J. Appl. Math., 74 (2014), 434-453.  doi: 10.1137/130923907.  Google Scholar

[21]

A. J. Lotka, Elements of Physical Biology, Dover Publications, Inc., New York, 1958.  Google Scholar

[22]

L. Ma and S. J. Guo, Bifurcation and stability of a two-species diffusive Lotka-Volterra model, Commun. Pure Appl. Anal., 19 (2020), 1205-1232.  doi: 10.3934/cpaa.2020056.  Google Scholar

[23]

L. Ma and S. J. Guo, Bifurcation and stability of a two-species reaction-diffusion-advection competition model, Nonlinear Anal. Real World Appl., 59 (2021), 103241.  doi: 10.1016/j.nonrwa.2020.103241.  Google Scholar

[24]

L. Ma and S. J. Guo, Positive solutions in the competitive Lotka-Volterra reaction-diffusion model with advection terms, Proc. Amer. Math. Soc., 149 (2021), 3013-3019.  doi: 10.1090/proc/15443.  Google Scholar

[25]

M. R. May, Limit cycles in predator-prey communities, Science, 177 (1972), 900-902.  doi: 10.1126/science.177.4052.900.  Google Scholar

[26]

R. McGehee and R. A. Armstrong, Some mathmatical problems concerning the principle of comprtitive exclusion, J. Differential Equation, 23 (1977), 30-52.  doi: 10.1016/0022-0396(77)90135-8.  Google Scholar

[27]

H. Qiu and S. J. Guo, Global existence and stability in a two-species chemotaxis system, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 1569-1587.  doi: 10.3934/dcdsb.2018220.  Google Scholar

[28] H. L. Smith and P. Waltman, The Theory of the Chemostat,, Cambridge University Press, Cambridge, UK, 1995.  doi: 10.1017/CBO9780511530043.  Google Scholar
[29]

V. Volterra, Variations and fluctuations of the number of individuals in animal species living together, ICES Journal of Marine Science, 3 (1928), 3-51.  doi: 10.1093/icesjms/3.1.3.  Google Scholar

[30]

H. Y. Wang, S. J. Guo and S. Z. Li, Stationary solutions of advective Lotka-Volterra models with a weak Allee effect and large diffusion, Nonlinear Anal. Real World Appl., 56 (2020), 103171, 23pp. doi: 10.1016/j.nonrwa.2020.103171.  Google Scholar

[31]

Y. Z. Wang and S. J. Guo, Global existence and asymptotic behavior of a two-species competitive Keller-Segel system on $\mathbb{R}^N$, Nonlinear Anal. Real World Appl., 61 (2021), 103342, 41pp. doi: 10.1016/j.nonrwa.2021.103342.  Google Scholar

[32]

Y. Z. Wang and S. J. Guo, Dynamics for a two-species competitive Keller-Segel chemotaxis system with a free boundary, J. Math. Anal. Appl., 502 (2021), 125259, 39pp. doi: 10.1016/j.jmaa.2021.125259.  Google Scholar

[33]

D. Wei and S. J. Guo, Qualitative analysis of a Lotka-Volterra competition-diffusion-advection system, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 2599-2623.  doi: 10.3934/dcdsb.2020197.  Google Scholar

[34]

G. S. K. Wolkowicz and Z. Lu, Global dynamics of a mathematical model of competition in the chemostat: General response function and differential death rates, SIAM J. Appl. Math., 52 (1992), 222-233.  doi: 10.1137/0152012.  Google Scholar

[35]

S. L. Yan and S. J. Guo, Stability analysis of a stage-structure model with spatial heterogeneity, Math. Meth. Appl. Sci., 44 (2021), 10993-11005.  doi: 10.1002/mma.7464.  Google Scholar

[36]

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

[37]

T. ZhaoY. Kuang and H. L. Smith, Global existence of periodic solutions in a class of delayed Gause-type predator-prey systems, Nonlinear Anal., 28 (1997), 1373-1394.  doi: 10.1016/0362-546X(95)00230-S.  Google Scholar

show all references

References:
[1]

R. A. Armstrong and R. McGehee, Competitive exclusion, Amer. Natur., 115 (1980), 151-170.  doi: 10.1086/283553.  Google Scholar

[2]

E. Beretta and Y. Kuang, Convergence results in a well-known delayed predator-prey system, J. Math. Anal. Appl., 204 (1996), 840-853.  doi: 10.1006/jmaa.1996.0471.  Google Scholar

[3]

F. Brauer and C. Castillo-Chávez, Mathematical Models in Population Biology and Epidemiology, Springer-Verlag, New York, 2001.  Google Scholar

[4]

E. ChauvetJ. E. PaulletJ. P. Previte and Z. Walls, A Lotka-Volterra three-species food chain, Math. Mag., 75 (2002), 243-255.  doi: 10.2307/3219158.  Google Scholar

[5]

J. P. Gao and S. J. Guo, Global dynamics and spatio-temporal patterns in a two-species chemotaxis system with two chemicals, Z. Angew. Math. Phys., 72 (2021), 25, 28pp. doi: 10.1007/s00033-020-01449-8.  Google Scholar

[6]

S. J. Guo, Bifurcation in a reaction-diffusion model with nonlocal delay effect and nonlinear boundary condition, J. Differential Equations, 289 (2021), 236-278.  doi: 10.1016/j.jde.2021.04.021.  Google Scholar

[7]

S. J. Guo and S. Z. Li, On the stability of reaction-diffusion models with nonlocal delay effect and nonlinear boundary condition, Appl. Math. Lett., 103 (2020), 106197, 7pp. doi: 10.1016/j.aml.2019.106197.  Google Scholar

[8]

S. J. Guo, S. Z. Li and B. Sounvoravong, Oscillatory and stationary patterns in a diffusive model with delay effect, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 31 (2021), 2150035, 21pp. doi: 10.1142/S0218127421500358.  Google Scholar

[9]

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

[10]

G. Hardin, The competitive exclusion principle, Science, 131 (1960), 1292-1297.  doi: 10.1126/science.131.3409.1292.  Google Scholar

[11]

A. Korobeinikov and G. C. Wake, Global properties of the three-dimensional predator-prey Lotka-Volterra systems, J. Appl. Math. Decis. Sci., 3 (1999), 155-162.  doi: 10.1155/S1173912699000085.  Google Scholar

[12] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, Inc., Boston, 1993.   Google Scholar
[13]

Y. Kuang, Global stability in delay differential systems without dominating instantaneous negative feedbacks, J. Differential Equations, 119 (1995), 503-532.  doi: 10.1006/jdeq.1995.1100.  Google Scholar

[14]

Y. Kuang and H. L. Smith, Global stability for infinite delay Lotka-Volterra type systems, J. Differential Equations, 103 (1993), 221-246.  doi: 10.1006/jdeq.1993.1048.  Google Scholar

[15]

J. P. LaSalle, The Stability of Dynamical Systems, Reg. Conf. Ser. Appl. Math., SIMA, Philadelphia, 1976.  Google Scholar

[16]

B. Li and H. L. Smith, Global dynamics of microbial competition for two resources with internal storage competition model, J. Math. Biol., 55 (2007), 481-515.  doi: 10.1007/s00285-007-0092-8.  Google Scholar

[17]

S. Z. Li and S. J. Guo, Permanence and extinction of a stochastic prey-predator model with a general functional response, Math. Comput. Simulation, 187 (2021), 308-336.  doi: 10.1016/j.matcom.2021.02.025.  Google Scholar

[18]

S. Z. Li and S. J. Guo, Dynamics of a stage-structured population model with a state-dependent delay, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 3523-3551.  doi: 10.3934/dcdsb.2020071.  Google Scholar

[19]

C. F. Liu and S. J. Guo, Steady states of Lotka-Volterra competition models with nonlinear cross-diffusion, J. Differential Equations, 292 (2021), 247-286.  doi: 10.1016/j.jde.2021.05.014.  Google Scholar

[20]

J. Llibre and D. Xiao, Global dynamics of a Lotka-Volterra model with two predators competing for one prey, SIAM J. Appl. Math., 74 (2014), 434-453.  doi: 10.1137/130923907.  Google Scholar

[21]

A. J. Lotka, Elements of Physical Biology, Dover Publications, Inc., New York, 1958.  Google Scholar

[22]

L. Ma and S. J. Guo, Bifurcation and stability of a two-species diffusive Lotka-Volterra model, Commun. Pure Appl. Anal., 19 (2020), 1205-1232.  doi: 10.3934/cpaa.2020056.  Google Scholar

[23]

L. Ma and S. J. Guo, Bifurcation and stability of a two-species reaction-diffusion-advection competition model, Nonlinear Anal. Real World Appl., 59 (2021), 103241.  doi: 10.1016/j.nonrwa.2020.103241.  Google Scholar

[24]

L. Ma and S. J. Guo, Positive solutions in the competitive Lotka-Volterra reaction-diffusion model with advection terms, Proc. Amer. Math. Soc., 149 (2021), 3013-3019.  doi: 10.1090/proc/15443.  Google Scholar

[25]

M. R. May, Limit cycles in predator-prey communities, Science, 177 (1972), 900-902.  doi: 10.1126/science.177.4052.900.  Google Scholar

[26]

R. McGehee and R. A. Armstrong, Some mathmatical problems concerning the principle of comprtitive exclusion, J. Differential Equation, 23 (1977), 30-52.  doi: 10.1016/0022-0396(77)90135-8.  Google Scholar

[27]

H. Qiu and S. J. Guo, Global existence and stability in a two-species chemotaxis system, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 1569-1587.  doi: 10.3934/dcdsb.2018220.  Google Scholar

[28] H. L. Smith and P. Waltman, The Theory of the Chemostat,, Cambridge University Press, Cambridge, UK, 1995.  doi: 10.1017/CBO9780511530043.  Google Scholar
[29]

V. Volterra, Variations and fluctuations of the number of individuals in animal species living together, ICES Journal of Marine Science, 3 (1928), 3-51.  doi: 10.1093/icesjms/3.1.3.  Google Scholar

[30]

H. Y. Wang, S. J. Guo and S. Z. Li, Stationary solutions of advective Lotka-Volterra models with a weak Allee effect and large diffusion, Nonlinear Anal. Real World Appl., 56 (2020), 103171, 23pp. doi: 10.1016/j.nonrwa.2020.103171.  Google Scholar

[31]

Y. Z. Wang and S. J. Guo, Global existence and asymptotic behavior of a two-species competitive Keller-Segel system on $\mathbb{R}^N$, Nonlinear Anal. Real World Appl., 61 (2021), 103342, 41pp. doi: 10.1016/j.nonrwa.2021.103342.  Google Scholar

[32]

Y. Z. Wang and S. J. Guo, Dynamics for a two-species competitive Keller-Segel chemotaxis system with a free boundary, J. Math. Anal. Appl., 502 (2021), 125259, 39pp. doi: 10.1016/j.jmaa.2021.125259.  Google Scholar

[33]

D. Wei and S. J. Guo, Qualitative analysis of a Lotka-Volterra competition-diffusion-advection system, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 2599-2623.  doi: 10.3934/dcdsb.2020197.  Google Scholar

[34]

G. S. K. Wolkowicz and Z. Lu, Global dynamics of a mathematical model of competition in the chemostat: General response function and differential death rates, SIAM J. Appl. Math., 52 (1992), 222-233.  doi: 10.1137/0152012.  Google Scholar

[35]

S. L. Yan and S. J. Guo, Stability analysis of a stage-structure model with spatial heterogeneity, Math. Meth. Appl. Sci., 44 (2021), 10993-11005.  doi: 10.1002/mma.7464.  Google Scholar

[36]

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

[37]

T. ZhaoY. Kuang and H. L. Smith, Global existence of periodic solutions in a class of delayed Gause-type predator-prey systems, Nonlinear Anal., 28 (1997), 1373-1394.  doi: 10.1016/0362-546X(95)00230-S.  Google Scholar

Figure 1.  Numerical simulations of system (46) with $ \tau = 0.00177 $ shows that the solution of the system with initial values (a) (2.026, 1.013) and (b) (1.974, 0.987) tends to the positive equilibrium $ E_+ $
Figure 2.  Numerical simulations of system (46) with $ \tau = 0.00177 $ shows that the solution of the system with initial value (a) (0.5, 3) and (b) (5, 0.2) tends to the positive equilibrium $ E_+ $
Figure 3.  Numerical simulations of the solution of system (46) with initial value (2.01, 1.01), (a) $ \tau = 0.212 $, and (b) $ \tau = 0.220 $
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