Multiple spatiotemporal coexistence states and Turing-Hopf bifurcation in a Lotka-Volterra competition system with nonlocal delays

Xianyong Chen; Weihua Jiang

doi:10.3934/dcdsb.2021013

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December 2021, 26(12): 6185-6205. doi: 10.3934/dcdsb.2021013

Multiple spatiotemporal coexistence states and Turing-Hopf bifurcation in a Lotka-Volterra competition system with nonlocal delays

Xianyong Chen and Weihua Jiang ^,

School of Mathematics, Harbin Institute of Technology, Harbin 150001, China

^* Corresponding author: Weihua Jiang

Received October 2020 Revised November 2020 Published December 2021 Early access December 2020

Fund Project: The authors are supported by the National Natural Science Foundation of China (No. 11871176)

We consider a two-species Lotka-Volterra competition system with both local and nonlocal intraspecific and interspecific competitions under the homogeneous Neumann condition. Firstly, we obtain conditions for the existence of Hopf, Turing, Turing-Hopf bifurcations and the necessary and sufficient condition that Turing instability occurs in the weak competition case, and find that the strength of nonlocal intraspecific competitions is the key factor for the stability of coexistence equilibrium. Secondly, we derive explicit formulas of normal forms up to order 3 by applying center manifold theory and normal form method, in which we show the difference compared with system without nonlocal terms in calculating coefficients of normal forms. Thirdly, the existence of complex spatiotemporal phenomena, such as the spatial homogeneous periodic orbit, a pair of stable spatial inhomogeneous steady states and a pair of stable spatial inhomogeneous periodic orbits, is rigorously proved by analyzing the amplitude equations. It is shown that suitably strong nonlocal intraspecific competitions and nonlocal delays can result in various coexistence states for the competition system in the weak competition case. Lastly, these complex spatiotemporal patterns are presented in the numerical results.

Keywords: Lotka-Volterra competition system, nonlocal intraspecific and interspecific competition, Turing-Hopf bifurcation, spatiotemporal patterns.

Mathematics Subject Classification: Primary: 35B32, 35B35; Secondary: 35B36.

Citation: Xianyong Chen, Weihua Jiang. Multiple spatiotemporal coexistence states and Turing-Hopf bifurcation in a Lotka-Volterra competition system with nonlocal delays. Discrete and Continuous Dynamical Systems - B, 2021, 26 (12) : 6185-6205. doi: 10.3934/dcdsb.2021013

References:

[1]	Q. An and W. Jiang, Spatiotemporal attractors generated by the Turing-Hopf bifurcation in a time-delayed reaction-diffusion system, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 487-510. doi: 10.3934/dcdsb.2018183.
[2]	E. Beretta and Y. Tang, Extension of a geometric stability switch criterion, Funkc. Ekvacioj, 46 (2003), 337-361. doi: 10.1619/fesi.46.337.
[3]	N. F. Britton, Spatial structures and periodic travelling waves in an integro-differential reaction-diffusion population model, SIAM J. Appl. Math., 50 (1990), 1663-1688. doi: 10.1137/0150099.
[4]	X. Cao and W. Jiang, Turing-Hopf bifurcation and spatiotemporal patterns in a diffusive predator-prey system with Crowley-Martin functional response, Nonlinear Anal. Real World Appl., 43 (2018), 428-450. doi: 10.1016/j.nonrwa.2018.03.010.
[5]	S. Chen and J. Shi, Global dynamics of the diffusive Lotka-Volterra competition model with stage structure, Calculus of Variations and Partial Differential Equations, 59 (2020), Article number: 33. doi: 10.1007/s00526-019-1693-y.
[6]	S. Chen and J. Shi, Stability and Hopf bifurcation in a diffusive logistic population model with nonlocal delay effect, J. Differ. Equ., 253 (2012), 3440-3470. doi: 10.1016/j.jde.2012.08.031.
[7]	X. Chen and W. Jiang, Turing-Hopf bifurcation and multi-stable spatio-temporal patterns in the Lengyel-Epstein system, Nonlinear Anal. Real World Appl., 49 (2019), 386-404. doi: 10.1016/j.nonrwa.2019.03.013.
[8]	X. Chen, W. Jiang and S. Ruan, Global dynamics and complex patterns in Lotka-Volterra systems: The effects of both local and nonlocal intraspecific and interspecific competitions, To appear.
[9]	J. Dockery, V. Hutson, K. Mischaikow and M. Pernarowski, The evolution of slow dispersal rates: A reaction diffusion model, J. Math. Biol., 37 (1998), 61-83. doi: 10.1007/s002850050120.
[10]	H. I. Freedman and Y. Kuang, Stability switches in linear scalar neutral delay equations, Funkc. Ekvacioj, 34 (1991), 187-209.
[11]	J. Furter and M. Grinfeld, Local vs. non-local interactions in population dynamics, J. Math. Biol., 27 (1989), 65-80. doi: 10.1007/BF00276081.
[12]	S. A. Gourley and N. F. Britton, A predator-prey reaction-diffusion system with nonlocal effects, J. Math. Biol., 34 (1996), 297-333. doi: 10.1007/BF00160498.
[13]	S. A. Gourley and S. Ruan, Convergence and travelling fronts in functional differential equations with nonlocal terms: A competition model, SIAM J. Math. Anal., 35 (2003), 806-822. doi: 10.1137/S003614100139991.
[14]	S. A. Gourley and J. W.-H. So, Dynamics of a food-limited population model incorporating nonlocal delays on a finite domain, J. Math. Biol., 44 (2002), 49-78. doi: 10.1007/s002850100109.
[15]	S. A. Gourley, J. W.-H. So and J. Wu, Nonlocality of reaction-diffusion equations induced by delay: biological modeling and nonlinear dynamics, J. Math. Sci., 124 (2004), 5119-5153. doi: 10.1023/B:JOTH.0000047249.39572.6d.
[16]	J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York, 1990.
[17]	S. Guo and S. Yan, Hopf bifurcation in a diffusive Lotka–Volterra type system with nonlocal delay effect, J. Differ. Equ., 260 (2016), 781-817. doi: 10.1016/j.jde.2015.09.031.
[18]	X. He and W.-M. Ni, Global dynamics of the Lotka-Volterra competition-diffusion system: Diffusion and spatial heterogeneity Ⅰ, Commun. Pure Appl. Math., 69 (2016), 981-1014. doi: 10.1002/cpa.21596.
[19]	R. Hu and Y. Yuan, Spatially nonhomogeneous equilibrium in a reaction–diffusion system with distributed delay, J. Differ. Equ., 250 (2011), 2779-2806. doi: 10.1016/j.jde.2011.01.011.
[20]	W. Jiang, Q. An and J. Shi, Formulation of the normal form of Turing-Hopf bifurcation in partial functional differential equations, J. Differ. Equ., 268 (2019), 6067-6102. doi: 10.1016/j.jde.2019.11.039.
[21]	W. Jiang, H. Wang and X. Cao, Turing instability and Turing-Hopf bifurcation in diffusive schnakenberg systems with gene expression time delay, J. Dyn. Differ. Equ., 31 (2019), 2223-2247. doi: 10.1007/s10884-018-9702-y.
[22]	Y. Kuang and H. L. Smith, Convergence in Lotka-Volterra typediffusive delay systems withoutdominating instantaneous negative feedbacks, J. Austral. Math. Soc. Ser. B, 34 (1993), 471-493. doi: 10.1017/S0334270000009036.
[23]	A. J. Lotka, Elements of Physical Biology, Williams and Wilkins, New York, 1925.
[24]	Y. Lou, On the effects of migration and spatial heterogeneity on single and multiple species, J. Differ. Equ., 223 (2006), 400-426. doi: 10.1016/j.jde.2005.05.010.
[25]	Y. Lou and P. Zhou, Evolution of dispersal in advective homogeneous environment: the effect of boundary conditions, J. Differ. Equ., 259 (2015), 141-171. doi: 10.1016/j.jde.2015.02.004.
[26]	W. Ni, J. Shi and M. Wang, Global stability and pattern formation in a nonlocal diffusive Lotka–Volterra competition model, J. Differ. Equ., 264 (2018), 6891-6932. doi: 10.1016/j.jde.2018.02.002.
[27]	S. Pal, S. Petrovskii, S. Ghorai and M. Banerjee, Spatiotemporal pattern formation in 2d prey-predator system with nonlocal intraspecific competition, Commun. Nonlinear Sci. Numer. Simul., 93 (2021), 105478, 15pp. doi: 10.1016/j.cnsns.2020.105478.
[28]	C. V. Pao, Global asymptotic stability of Lotka-Vlterra competition systems with diffusion and time delays, Nonlinear Anal. Real World Appl., 5 (2004), 91-104. doi: 10.1016/S1468-1218(03)00018-X.
[29]	S. Ruan and J. Wei, On the zeros of transcendental functions with applications to stability of delay differential equations with two delays, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 10 (2003), 863-874.
[30]	V. P. Shukla, Conditions for global stability of two-species population models with discrete time delay, Bull. Math. Biol., 45 (1983), 793-805.
[31]	Y. Song, M. Han and Y. Peng, Stability and Hopf bifurcations in a competitive Lotka-Volterra system with two delays, Chaos Solitons Fract., 22 (2004), 1139-1148. doi: 10.1016/j.chaos.2004.03.026.
[32]	Y. Song, H. Jiang, Q. Liu and Y. Yuan, Spatiotemporal dynamics of the diffusive mussel-algae model near turing-hopf bifurcation, SIAM J. Appl. Dyn. Syst., 16 (2017), 2030-2062. doi: 10.1137/16M1097560.
[33]	Y. Song, T. Zhang and Y. Peng, Turing-Hopf bifurcation in the reaction-diffusion equations and its applications, Commun. Nonlinear Sci. Numer. Simul., 33 (2016), 229-258. doi: 10.1016/j.cnsns.2015.10.002.
[34]	Y. Tang and L. Zhou, Hopf bifurcation and stability of a competition diffusion system with distributed delay, Publ. RIMS, Kyoto Univ., 41 (2005), 579-597. doi: 10.2977/prims/1145475224.
[35]	V. Volterra, Variazionie fluttuazioni del numero d'individui in specie animali conviventi, Mem. Acad. Licei., 2 (1926), 31-113.
[36]	J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4050-1.
[37]	Y. Yamada, Asymptotic stability for some systems of semilinear Volterra diffusion equations, J. Differ. Equ., 52 (1984), 295-326. doi: 10.1016/0022-0396(84)90165-7.
[38]	Y. Yamada, On logistic diffusion equations with nonlocal interaction terms, Nonlinear Anal.-Theory Methods Appl., 118 (2015), 51-62. doi: 10.1016/j.na.2015.01.016.
[39]	J. Zhang, W. Li and X. Yan, Bifurcation and spatiotemporal patterns in a homogeneous diffusion-competition system with delays, Int. J. Biomath., 5 (2012), 1250049, 23pp. doi: 10.1142/S1793524512500490.

show all references

References:

[1]	Q. An and W. Jiang, Spatiotemporal attractors generated by the Turing-Hopf bifurcation in a time-delayed reaction-diffusion system, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 487-510. doi: 10.3934/dcdsb.2018183.
[2]	E. Beretta and Y. Tang, Extension of a geometric stability switch criterion, Funkc. Ekvacioj, 46 (2003), 337-361. doi: 10.1619/fesi.46.337.
[3]	N. F. Britton, Spatial structures and periodic travelling waves in an integro-differential reaction-diffusion population model, SIAM J. Appl. Math., 50 (1990), 1663-1688. doi: 10.1137/0150099.
[4]	X. Cao and W. Jiang, Turing-Hopf bifurcation and spatiotemporal patterns in a diffusive predator-prey system with Crowley-Martin functional response, Nonlinear Anal. Real World Appl., 43 (2018), 428-450. doi: 10.1016/j.nonrwa.2018.03.010.
[5]	S. Chen and J. Shi, Global dynamics of the diffusive Lotka-Volterra competition model with stage structure, Calculus of Variations and Partial Differential Equations, 59 (2020), Article number: 33. doi: 10.1007/s00526-019-1693-y.
[6]	S. Chen and J. Shi, Stability and Hopf bifurcation in a diffusive logistic population model with nonlocal delay effect, J. Differ. Equ., 253 (2012), 3440-3470. doi: 10.1016/j.jde.2012.08.031.
[7]	X. Chen and W. Jiang, Turing-Hopf bifurcation and multi-stable spatio-temporal patterns in the Lengyel-Epstein system, Nonlinear Anal. Real World Appl., 49 (2019), 386-404. doi: 10.1016/j.nonrwa.2019.03.013.
[8]	X. Chen, W. Jiang and S. Ruan, Global dynamics and complex patterns in Lotka-Volterra systems: The effects of both local and nonlocal intraspecific and interspecific competitions, To appear.
[9]	J. Dockery, V. Hutson, K. Mischaikow and M. Pernarowski, The evolution of slow dispersal rates: A reaction diffusion model, J. Math. Biol., 37 (1998), 61-83. doi: 10.1007/s002850050120.
[10]	H. I. Freedman and Y. Kuang, Stability switches in linear scalar neutral delay equations, Funkc. Ekvacioj, 34 (1991), 187-209.
[11]	J. Furter and M. Grinfeld, Local vs. non-local interactions in population dynamics, J. Math. Biol., 27 (1989), 65-80. doi: 10.1007/BF00276081.
[12]	S. A. Gourley and N. F. Britton, A predator-prey reaction-diffusion system with nonlocal effects, J. Math. Biol., 34 (1996), 297-333. doi: 10.1007/BF00160498.
[13]	S. A. Gourley and S. Ruan, Convergence and travelling fronts in functional differential equations with nonlocal terms: A competition model, SIAM J. Math. Anal., 35 (2003), 806-822. doi: 10.1137/S003614100139991.
[14]	S. A. Gourley and J. W.-H. So, Dynamics of a food-limited population model incorporating nonlocal delays on a finite domain, J. Math. Biol., 44 (2002), 49-78. doi: 10.1007/s002850100109.
[15]	S. A. Gourley, J. W.-H. So and J. Wu, Nonlocality of reaction-diffusion equations induced by delay: biological modeling and nonlinear dynamics, J. Math. Sci., 124 (2004), 5119-5153. doi: 10.1023/B:JOTH.0000047249.39572.6d.
[16]	J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York, 1990.
[17]	S. Guo and S. Yan, Hopf bifurcation in a diffusive Lotka–Volterra type system with nonlocal delay effect, J. Differ. Equ., 260 (2016), 781-817. doi: 10.1016/j.jde.2015.09.031.
[18]	X. He and W.-M. Ni, Global dynamics of the Lotka-Volterra competition-diffusion system: Diffusion and spatial heterogeneity Ⅰ, Commun. Pure Appl. Math., 69 (2016), 981-1014. doi: 10.1002/cpa.21596.
[19]	R. Hu and Y. Yuan, Spatially nonhomogeneous equilibrium in a reaction–diffusion system with distributed delay, J. Differ. Equ., 250 (2011), 2779-2806. doi: 10.1016/j.jde.2011.01.011.
[20]	W. Jiang, Q. An and J. Shi, Formulation of the normal form of Turing-Hopf bifurcation in partial functional differential equations, J. Differ. Equ., 268 (2019), 6067-6102. doi: 10.1016/j.jde.2019.11.039.
[21]	W. Jiang, H. Wang and X. Cao, Turing instability and Turing-Hopf bifurcation in diffusive schnakenberg systems with gene expression time delay, J. Dyn. Differ. Equ., 31 (2019), 2223-2247. doi: 10.1007/s10884-018-9702-y.
[22]	Y. Kuang and H. L. Smith, Convergence in Lotka-Volterra typediffusive delay systems withoutdominating instantaneous negative feedbacks, J. Austral. Math. Soc. Ser. B, 34 (1993), 471-493. doi: 10.1017/S0334270000009036.
[23]	A. J. Lotka, Elements of Physical Biology, Williams and Wilkins, New York, 1925.
[24]	Y. Lou, On the effects of migration and spatial heterogeneity on single and multiple species, J. Differ. Equ., 223 (2006), 400-426. doi: 10.1016/j.jde.2005.05.010.
[25]	Y. Lou and P. Zhou, Evolution of dispersal in advective homogeneous environment: the effect of boundary conditions, J. Differ. Equ., 259 (2015), 141-171. doi: 10.1016/j.jde.2015.02.004.
[26]	W. Ni, J. Shi and M. Wang, Global stability and pattern formation in a nonlocal diffusive Lotka–Volterra competition model, J. Differ. Equ., 264 (2018), 6891-6932. doi: 10.1016/j.jde.2018.02.002.
[27]	S. Pal, S. Petrovskii, S. Ghorai and M. Banerjee, Spatiotemporal pattern formation in 2d prey-predator system with nonlocal intraspecific competition, Commun. Nonlinear Sci. Numer. Simul., 93 (2021), 105478, 15pp. doi: 10.1016/j.cnsns.2020.105478.
[28]	C. V. Pao, Global asymptotic stability of Lotka-Vlterra competition systems with diffusion and time delays, Nonlinear Anal. Real World Appl., 5 (2004), 91-104. doi: 10.1016/S1468-1218(03)00018-X.
[29]	S. Ruan and J. Wei, On the zeros of transcendental functions with applications to stability of delay differential equations with two delays, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 10 (2003), 863-874.
[30]	V. P. Shukla, Conditions for global stability of two-species population models with discrete time delay, Bull. Math. Biol., 45 (1983), 793-805.
[31]	Y. Song, M. Han and Y. Peng, Stability and Hopf bifurcations in a competitive Lotka-Volterra system with two delays, Chaos Solitons Fract., 22 (2004), 1139-1148. doi: 10.1016/j.chaos.2004.03.026.
[32]	Y. Song, H. Jiang, Q. Liu and Y. Yuan, Spatiotemporal dynamics of the diffusive mussel-algae model near turing-hopf bifurcation, SIAM J. Appl. Dyn. Syst., 16 (2017), 2030-2062. doi: 10.1137/16M1097560.
[33]	Y. Song, T. Zhang and Y. Peng, Turing-Hopf bifurcation in the reaction-diffusion equations and its applications, Commun. Nonlinear Sci. Numer. Simul., 33 (2016), 229-258. doi: 10.1016/j.cnsns.2015.10.002.
[34]	Y. Tang and L. Zhou, Hopf bifurcation and stability of a competition diffusion system with distributed delay, Publ. RIMS, Kyoto Univ., 41 (2005), 579-597. doi: 10.2977/prims/1145475224.
[35]	V. Volterra, Variazionie fluttuazioni del numero d'individui in specie animali conviventi, Mem. Acad. Licei., 2 (1926), 31-113.
[36]	J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4050-1.
[37]	Y. Yamada, Asymptotic stability for some systems of semilinear Volterra diffusion equations, J. Differ. Equ., 52 (1984), 295-326. doi: 10.1016/0022-0396(84)90165-7.
[38]	Y. Yamada, On logistic diffusion equations with nonlocal interaction terms, Nonlinear Anal.-Theory Methods Appl., 118 (2015), 51-62. doi: 10.1016/j.na.2015.01.016.
[39]	J. Zhang, W. Li and X. Yan, Bifurcation and spatiotemporal patterns in a homogeneous diffusion-competition system with delays, Int. J. Biomath., 5 (2012), 1250049, 23pp. doi: 10.1142/S1793524512500490.

Figure 1. $ w_{*}>0 $ is a positive root of $ F(w) = 0 $

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Figure 2. System parameter values are taken as Table 1 shows. The left figure represents the enlarged figure inside the pink rectangular part of the right figure and the red hollow circles are intersection points of $ \mathcal{L}_{n} $ and $ \mathcal{S}_{n} $. The solid black circles means that the remaining curves $ \mathcal{L}_{n} $ and $ \mathcal{S}_{n} $ are omitted here

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Figure 3. (a), (b) are bifurcation sets and phase portraits respectively

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Figure 4. When the initial values $ u_{1}(t,x) = 0.45-0.01 $, $ u_{2}(t,x) = 0.27-0.01 $, $ t\in[-\tau,0] $ and parameters $ (\tau,d_{1}) = (\tau_{w_{*}},d_{1}^{1}(d_{2}))+(-0.01,0.01)\in \mathscr{D}_{1} $, the coexistence equilibrium is stable

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Figure 5. When the initial values $ u_{1}(t,x) = 0.45-0.01 $, $ u_{2}(t,x) = 0.27-0.01 $, $ t\in[-\tau,0] $ and parameters $ (\tau,d_{1}) = (\tau_{w_{*}},d_{1}^{1}(d_{2}))+(0.01,0.01)\in \mathscr{D}_{2} $, a spatial homogeneous periodic orbit is stable

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Figure 6. For $ (\tau,d_{1}) = (\tau_{w_{*}},d_{1}^{1}(d_{2}))+(0.003,-0.01)\in \mathscr{D}_{4} $, (a), (b) with initial values $ u_{1}(t,x) = 0.45-0.01\cos x $, $ u_{2}(t,x) = 0.27-0.01\cos x $ and (c), (d) with initial values $ u_{1}(t,x) = 0.45+0.01\cos x $, $ u_{2}(t,x) = 0.27+0.01\cos x $, , $ t\in[-\tau,0] $. A pair of spatial inhomogeneous periodic orbits is stable, which indicates $ \mathscr{D}_{4} $ is a bistable region

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Figure 7. For $ (\tau,d_{1}) = (\tau_{w_{*}},d_{1}^{1}(d_{2}))+(-0.01,-0.01)\in \mathscr{D}_{6} $, (a), (b) with initial values $ u_{1}(t,x) = 0.45-0.01\cos x $, $ u_{2}(t,x) = 0.27-0.01\cos x $ and (c), (d) with initial values $ u_{1}(t,x) = 0.45+0.01\cos x $, $ u_{2}(t,x) = 0.27+0.01\cos x $, , $ t\in[-\tau,0] $. a pair of spatial inhomogeneous steady states is stable, which indicates $ \mathscr{D}_{6} $ is a bistable region

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Table 1. Values of system parameters

Parameters	$ r_{1} $	$ r_{2} $	$ a_{11} $	$ a_{12} $	$ a_{21} $	$ a_{22} $	$ b_{11} $	$ b_{12} $	$ b_{21} $	$ b_{22} $
Values	$ 5 $	$ 5 $	$ 4 $	$ 7 $	$ 6 $	$ 5 $	$ 3 $	$ 1 $	$ 0.5 $	$ 4 $

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Parameters	$ r_{1} $	$ r_{2} $	$ a_{11} $	$ a_{12} $	$ a_{21} $	$ a_{22} $	$ b_{11} $	$ b_{12} $	$ b_{21} $	$ b_{22} $
Values	$ 5 $	$ 5 $	$ 4 $	$ 7 $	$ 6 $	$ 5 $	$ 3 $	$ 1 $	$ 0.5 $	$ 4 $

Table 2. Values of system parameters

Parameters	$ d_{2} $	$ r_{1} $	$ r_{2} $	$ a_{11} $	$ a_{12} $	$ a_{21} $	$ a_{22} $	$ b_{11} $	$ b_{12} $	$ b_{21} $	$ b_{22} $
Values	$ 0.6 $	$ 5 $	$ 5 $	$ 4 $	$ 7 $	$ 6 $	$ 5 $	$ 3 $	$ 1 $	$ 0.5 $	$ 4 $

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Parameters	$ d_{2} $	$ r_{1} $	$ r_{2} $	$ a_{11} $	$ a_{12} $	$ a_{21} $	$ a_{22} $	$ b_{11} $	$ b_{12} $	$ b_{21} $	$ b_{22} $
Values	$ 0.6 $	$ 5 $	$ 5 $	$ 4 $	$ 7 $	$ 6 $	$ 5 $	$ 3 $	$ 1 $	$ 0.5 $	$ 4 $

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2021 Impact Factor: 1.497

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American Institute of Mathematical Sciences

Multiple spatiotemporal coexistence states and Turing-Hopf bifurcation in a Lotka-Volterra competition system with nonlocal delays

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American Institute of Mathematical Sciences

Multiple spatiotemporal coexistence states and Turing-Hopf bifurcation in a Lotka-Volterra competition system with nonlocal delays

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