# American Institute of Mathematical Sciences

December  2020, 19(12): 5609-5626. doi: 10.3934/cpaa.2020256

## On competition models under allee effect: Asymptotic behavior and traveling waves

 Department of Mathematics and Statistics, University of North Carolina at Wilmington, Wilmington, NC 28403

*Corresponding author

Received  February 2020 Revised  June 2020 Published  December 2020 Early access  October 2020

In this article, we study a reaction-diffusion model on infinite spatial domain for two competing biological species ($u$ and $v$). Under one-side Allee effect on $u$-species, the model demonstrates complexity on its coexistence and $u$-dominance steady states. The conditions for persistence, permanence and competitive exclusion of the species are obtained through analysis on asymptotic behavior of the solutions and stability of the steady states, including the attraction regions and convergent rates depending on the biological parameters. When the Allee effect constant $K$ is large relative to other biological parameters, the asymptotic stability of the $v$-dominance state $(0,\:1)$ indicates the competitive exclusion of the $u$-species. Applying upper-lower solution method, we further prove that for a family of wave speeds with specific minimum wave speed determined by several biological parameters (including the magnitude of the $u$-dominance states), there exist traveling wave solutions flowing from the $u$-dominance states to the $v$-dominance state. The asymptotic rates of the traveling waves at $\xi \rightarrow \mp \infty$ are also explicitly calculated. Finally, numerical simulations are presented to illustrate the theoretical results and population dynamics of coexistence or dominance-shifting.

Citation: Wei Feng, Michael Freeze, Xin Lu. On competition models under allee effect: Asymptotic behavior and traveling waves. Communications on Pure and Applied Analysis, 2020, 19 (12) : 5609-5626. doi: 10.3934/cpaa.2020256
##### References:
 [1] S. Ahmad and A. Lazer, An elementary approach to traveling front solutions to a system of N competition-diffusion equations, Nonlinear Anal., 16 (1991), 893-901.  doi: 10.1016/0362-546X(91)90152-Q. [2] P. Ashwin, M. V. Bartuccelli, T. J. Bridges and S. A. Gourley, Travelling fronts for the KPP equation with spatio-temporal delay, Z. Angew. math. phys., 53 (2002), 103-122.  doi: 10.1007/s00033-002-8145-8. [3] J. Blat and K. J. Brown, Bifurcation of steady-state solutions in predator-prey and competition systems, P. Roy. Soc. Edinb. A, 97 (1984), 21-34.  doi: 10.1017/S0308210500031802. [4] A. Boumenir and V. Nguyen, Perron theorem in the monotone iteration for traveling waves in delayed in delayed reaction diffusion equations, J. Differ. Equ., 244 (2008), 1151-1570.  doi: 10.1016/j.jde.2008.01.004. [5] C. Conley and R. Gardner, An application of the generalized Mores index to traveling wave solutions of a competition reaction diffusion model, Indiana U. Math. J., 44 (1984), 319-343.  doi: 10.1512/iumj.1984.33.33018. [6] S. Dunbar, Traveling wave solutions of diffusive Lotka-Volterra equations: a. heteroclinic connection in $R^{4}$, T. Am. Math. Soc., 286 (1984), 557-594.  doi: 10.2307/1999810. [7] W. Feng, Competitive exclusion and persistence in models of resource and sexual competition, J. Math. Biol., 35 (1997), 683-694.  doi: 10.1007/s002850050071. [8] W. Feng and X. Lu, Harmless delays for permanence in a class of population models with diffusion effects, J. Math. Anal. Appl., 206 (1997), 547-566.  doi: 10.1006/jmaa.1997.5265. [9] W. Feng and X. Lu, Traveling waves and competitive exclusion in models of resource competition and mating interference, J. Math. Anal. Appl., 424 (2015), 542-562.  doi: 10.1016/j.jmaa.2014.11.027. [10] W. Feng, W. Ruan and X. Lu, On existence of wavefront solutions in mixed monotone reaction-diffusion systems, Discrete Cont. Dyn. B, 21 (2016), 815-836.  doi: 10.3934/dcdsb.2016.21.815. [11] R. Gardner, Existence and stability of traveling wave solutions of competition models: a degree theoretic approach, J. Differ. Equ., 44 (1982), 343-364.  doi: 10.1016/0022-0396(82)90001-8. [12] X. Hou and W. Feng, Traveling waves and their stability in a coupled reaction diffusion system, Commun. Pure Appl. Anal., 10 (2011), 141-160.  doi: 10.3934/cpaa.2011.10.141. [13] J. Huang and X. Zou, Traveling wavefronts in diffusive and cooperative Lotka-Voltera system with delays, J. Math. Anal. Appl, 271 (2002), 455-466.  doi: 10.1016/S0022-247X(02)00135-X. [14] J. I. Kanel and L. Zhou, Existence of wave front solutions and estimates of wave speed for a competition-diffusion system, Nonlinear Analysis, Theory, Method and Application, 27 (1996), 579-587.  doi: 10.1016/0362-546X(95)00221-G. [15] Y. Kan-on, Parameter dependence of propagation speed of traveling waves for competition-diffusion equations, SIAM J. Math. Anal., 26 (1995), 340-363.  doi: 10.1137/S0036141093244556. [16] A. Leung, Nonlinear Systems of Partial Differential Equations: Applications to Life and Physical Sciences, World Scientific, Singapore, 2009. doi: 10.1142/9789814277709. [17] A. W. Leung, X. Hou and W. Feng, Traveling wave solutions for Lotka-Volterra system re-visited, Discrete Cont. Dyn. B, 15 (2011), 171-196.  doi: 10.3934/dcdsb.2011.15.171. [18] Mark A. Lewis, Bi ngtuan Li and Hans F. Weinberger, Spreading speed and linear determinacy for two-species competition models, J. Math. Biol., 45 (2002), 219-233.  doi: 10.1007/s002850200144. [19] X. Lu, Monotone method and convergence acceleration for finite-difference solutions of parabolic problems with time delays, Numerical Methods for Partial Differential Equations, 11 (1995), 591-602.  doi: 10.1002/num.1690110605. [20] C. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, N.Y., 1992. [21] C. V. Pao, Numerical methods for semilinear parabolic equations, SIAM J. Numer. Anal., 24 (1987), 24-35.  doi: 10.1137/0724003. [22] C. V. Pao and X. Lu, Block monotone iterative methods for numerical solutions of nonlinear parabolic equations, SIAM J. Numer. Anal., 47 (2010), 4581-4606.  doi: 10.1137/090748706. [23] W. Ruan, W. Feng and X. Lu, On traveling wave solutions in general reaction-diffusion systems with time delays, J. Math. Anal. Appl., 448 (2017), 376-400.  doi: 10.1016/j.jmaa.2016.10.070. [24] W. Ruan, W. Feng and X. Lu, Wavefront solutions of quasilinear reaction-diffusion systems with mixed quasimonotonicity, Appl. Anal., 98 (2019), 934-968.  doi: 10.1080/00036811.2017.1408077. [25] D. H. Sattinger, On the stability of waves of nonlinear parabolic systems, Adv. Math., 22 (1976), 312-355.  doi: 10.1016/0001-8708(76)90098-0. [26] G. Wang, X. Liang and F. Wang, The competitive dynamics of populations subject to an Allee effect, Ecol. Modell., 124 (1999), 183-192. [27] J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with delay, J. Dyn. Differ. Equ., 13 (2001), 651-687.  doi: 10.1023/A:1016690424892. [28] L. Zhou and C. V. Pao, Asymptotic behavior of a competition-diffusion system in population dynamics, Nonlinear Anal., 6 (1982), 1163-1184.  doi: 10.1016/0362-546X(82)90028-1. [29] S. Zhou, C. Liu and G. Wang, The competitive dynamics of metapopulationss subject to the Allee-like effect, Theor. Popul. Biol., 65 (2004), 29-37. [30] S. Zhou, Y. Liu and G. Wang, The stability of predator-prey systems subject to the Allee effects, Theor. Popul. Biol., 67 (2005), 23-31. [31] W. J. Bo, G. Lin and S. Ruan, Traveling wave solutions for time periodic reaction-diffusion systems, Discrete Cont. Dyn. A, 38 (2018), 4329-4351.  doi: 10.3934/dcds.2018189. [32] Ma saharu Taniguchi, Axisymmetric traveling fronts in balanced bistable reaction-diffusion equations, Discrete Cont. Dyn. A, 40 (2020), 3981-3995.  doi: 10.1016/j.anihpc.2019.05.001. [33] X. Bao, W. T. Li and Z. C. Wang, Uniqueness and stability of time-periodic pyramidal fronts for a periodic competition-diffusion system, Commun. Pure Appl. Anal., 19 (2020), 253-277.  doi: 10.3934/cpaa.2020014. [34] Y. Yang, Y. R. Yang and X. J. Jiao, Traveling waves for a nonlocal dispersal SIR model equipped delay and generalized incidence, Electron. Res. Arc., 28 (2020), 1-13.

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##### References:
 [1] S. Ahmad and A. Lazer, An elementary approach to traveling front solutions to a system of N competition-diffusion equations, Nonlinear Anal., 16 (1991), 893-901.  doi: 10.1016/0362-546X(91)90152-Q. [2] P. Ashwin, M. V. Bartuccelli, T. J. Bridges and S. A. Gourley, Travelling fronts for the KPP equation with spatio-temporal delay, Z. Angew. math. phys., 53 (2002), 103-122.  doi: 10.1007/s00033-002-8145-8. [3] J. Blat and K. J. Brown, Bifurcation of steady-state solutions in predator-prey and competition systems, P. Roy. Soc. Edinb. A, 97 (1984), 21-34.  doi: 10.1017/S0308210500031802. [4] A. Boumenir and V. Nguyen, Perron theorem in the monotone iteration for traveling waves in delayed in delayed reaction diffusion equations, J. Differ. Equ., 244 (2008), 1151-1570.  doi: 10.1016/j.jde.2008.01.004. [5] C. Conley and R. Gardner, An application of the generalized Mores index to traveling wave solutions of a competition reaction diffusion model, Indiana U. Math. J., 44 (1984), 319-343.  doi: 10.1512/iumj.1984.33.33018. [6] S. Dunbar, Traveling wave solutions of diffusive Lotka-Volterra equations: a. heteroclinic connection in $R^{4}$, T. Am. Math. Soc., 286 (1984), 557-594.  doi: 10.2307/1999810. [7] W. Feng, Competitive exclusion and persistence in models of resource and sexual competition, J. Math. Biol., 35 (1997), 683-694.  doi: 10.1007/s002850050071. [8] W. Feng and X. Lu, Harmless delays for permanence in a class of population models with diffusion effects, J. Math. Anal. Appl., 206 (1997), 547-566.  doi: 10.1006/jmaa.1997.5265. [9] W. Feng and X. Lu, Traveling waves and competitive exclusion in models of resource competition and mating interference, J. Math. Anal. Appl., 424 (2015), 542-562.  doi: 10.1016/j.jmaa.2014.11.027. [10] W. Feng, W. Ruan and X. Lu, On existence of wavefront solutions in mixed monotone reaction-diffusion systems, Discrete Cont. Dyn. B, 21 (2016), 815-836.  doi: 10.3934/dcdsb.2016.21.815. [11] R. Gardner, Existence and stability of traveling wave solutions of competition models: a degree theoretic approach, J. Differ. Equ., 44 (1982), 343-364.  doi: 10.1016/0022-0396(82)90001-8. [12] X. Hou and W. Feng, Traveling waves and their stability in a coupled reaction diffusion system, Commun. Pure Appl. Anal., 10 (2011), 141-160.  doi: 10.3934/cpaa.2011.10.141. [13] J. Huang and X. Zou, Traveling wavefronts in diffusive and cooperative Lotka-Voltera system with delays, J. Math. Anal. Appl, 271 (2002), 455-466.  doi: 10.1016/S0022-247X(02)00135-X. [14] J. I. Kanel and L. Zhou, Existence of wave front solutions and estimates of wave speed for a competition-diffusion system, Nonlinear Analysis, Theory, Method and Application, 27 (1996), 579-587.  doi: 10.1016/0362-546X(95)00221-G. [15] Y. Kan-on, Parameter dependence of propagation speed of traveling waves for competition-diffusion equations, SIAM J. Math. Anal., 26 (1995), 340-363.  doi: 10.1137/S0036141093244556. [16] A. Leung, Nonlinear Systems of Partial Differential Equations: Applications to Life and Physical Sciences, World Scientific, Singapore, 2009. doi: 10.1142/9789814277709. [17] A. W. Leung, X. Hou and W. Feng, Traveling wave solutions for Lotka-Volterra system re-visited, Discrete Cont. Dyn. B, 15 (2011), 171-196.  doi: 10.3934/dcdsb.2011.15.171. [18] Mark A. Lewis, Bi ngtuan Li and Hans F. Weinberger, Spreading speed and linear determinacy for two-species competition models, J. Math. Biol., 45 (2002), 219-233.  doi: 10.1007/s002850200144. [19] X. Lu, Monotone method and convergence acceleration for finite-difference solutions of parabolic problems with time delays, Numerical Methods for Partial Differential Equations, 11 (1995), 591-602.  doi: 10.1002/num.1690110605. [20] C. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, N.Y., 1992. [21] C. V. Pao, Numerical methods for semilinear parabolic equations, SIAM J. Numer. Anal., 24 (1987), 24-35.  doi: 10.1137/0724003. [22] C. V. Pao and X. Lu, Block monotone iterative methods for numerical solutions of nonlinear parabolic equations, SIAM J. Numer. Anal., 47 (2010), 4581-4606.  doi: 10.1137/090748706. [23] W. Ruan, W. Feng and X. Lu, On traveling wave solutions in general reaction-diffusion systems with time delays, J. Math. Anal. Appl., 448 (2017), 376-400.  doi: 10.1016/j.jmaa.2016.10.070. [24] W. Ruan, W. Feng and X. Lu, Wavefront solutions of quasilinear reaction-diffusion systems with mixed quasimonotonicity, Appl. Anal., 98 (2019), 934-968.  doi: 10.1080/00036811.2017.1408077. [25] D. H. Sattinger, On the stability of waves of nonlinear parabolic systems, Adv. Math., 22 (1976), 312-355.  doi: 10.1016/0001-8708(76)90098-0. [26] G. Wang, X. Liang and F. Wang, The competitive dynamics of populations subject to an Allee effect, Ecol. Modell., 124 (1999), 183-192. [27] J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with delay, J. Dyn. Differ. Equ., 13 (2001), 651-687.  doi: 10.1023/A:1016690424892. [28] L. Zhou and C. V. Pao, Asymptotic behavior of a competition-diffusion system in population dynamics, Nonlinear Anal., 6 (1982), 1163-1184.  doi: 10.1016/0362-546X(82)90028-1. [29] S. Zhou, C. Liu and G. Wang, The competitive dynamics of metapopulationss subject to the Allee-like effect, Theor. Popul. Biol., 65 (2004), 29-37. [30] S. Zhou, Y. Liu and G. Wang, The stability of predator-prey systems subject to the Allee effects, Theor. Popul. Biol., 67 (2005), 23-31. [31] W. J. Bo, G. Lin and S. Ruan, Traveling wave solutions for time periodic reaction-diffusion systems, Discrete Cont. Dyn. A, 38 (2018), 4329-4351.  doi: 10.3934/dcds.2018189. [32] Ma saharu Taniguchi, Axisymmetric traveling fronts in balanced bistable reaction-diffusion equations, Discrete Cont. Dyn. A, 40 (2020), 3981-3995.  doi: 10.1016/j.anihpc.2019.05.001. [33] X. Bao, W. T. Li and Z. C. Wang, Uniqueness and stability of time-periodic pyramidal fronts for a periodic competition-diffusion system, Commun. Pure Appl. Anal., 19 (2020), 253-277.  doi: 10.3934/cpaa.2020014. [34] Y. Yang, Y. R. Yang and X. J. Jiao, Traveling waves for a nonlocal dispersal SIR model equipped delay and generalized incidence, Electron. Res. Arc., 28 (2020), 1-13.
Permanence in the competition model under Allee effect
Asymptotic stability of the steady state $(0,1)$, $v$ species dominance
Traveling wave front connecting $(u_s^{(1)},0)$ to $(0,1)$, competitive exclusion of $u$-species
Traveling wave front connecting $(u_s^{(2)},0)$ to $(0,1)$, competitive exclusion of $u$-species
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