Dynamics for a two-species competition–diffusion model with two free boundaries

To understand the spreading and interaction of two-competing species, we study the dynamics for a two-species competition–diffusion model with two free boundaries. Here, the two free boundaries which describe the spreading fronts of two competing species, respectively, may intersect each other. Our result shows there exists a critical value such that the superior competitor always spreads successfully if its territory size is above this constant at some time. Otherwise, the superior competitor can be wiped out by the inferior competitor. Moreover, if the inferior competitor does not spread fast enough such that the superior competitor can catch up with it, the inferior competitor will be wiped out eventually and then a spreading–vanishing trichotomy is established. We also provide some characterization of the spreading–vanishing trichotomy via some parameters of the model. On the other hand, when the superior competitor spreads successfully but with a sufficiently low speed, the inferior competitor can also spread successfully even the superior species is much stronger than the weaker one. It means that the inferior competitor can survive if the superior species cannot catch up with it.

[1]  Mingxin Wang.  On some Free Boundary Problems of the Prey-predator Model , 2013, 1301.2063.

[2]  Zongming Guo,et al.  A diffusive logistic model with a free boundary in time-periodic environment☆ , 2013 .

[3]  Yihong Du,et al.  Spreading speed revisited: Analysis of a free boundary model , 2012, Networks Heterog. Media.

[4]  Yihong Du,et al.  Spreading–vanishing dichotomy in a diffusive logistic model with a free boundary, II☆ , 2011 .

[5]  Bingtuan Li,et al.  Spreading speed and linear determinacy for two-species competition models , 2002, Journal of mathematical biology.

[6]  Mingxin Wang,et al.  Free Boundary Problems for a Lotka–Volterra Competition System , 2014, 1401.0806.

[7]  C. Conley,et al.  An application of the generalized Morse index to travelling wave solutions of a competitive reaction-diffusion model , 1984 .

[8]  Avner Friedman,et al.  A Free Boundary Problem Arising in a Model of Wound Healing , 2000, SIAM J. Math. Anal..

[9]  A. Ōkubo,et al.  On the spatial spread of the grey squirrel in Britain , 1989, Proceedings of the Royal Society of London. B. Biological Sciences.

[10]  Yihong Du,et al.  Spreading and vanishing in nonlinear diffusion problems with free boundaries , 2013, 1301.5373.

[11]  Paul C. Fife,et al.  Propagating fronts for competing species equations with diffusion , 1980 .

[12]  Yihong Du,et al.  Sharp Estimate of the Spreading Speed Determined by Nonlinear Free Boundary Problems , 2014, SIAM J. Math. Anal..

[13]  Yihong Du,et al.  The Stefan problem for the Fisher–KPP equation with unbounded initial range , 2012, Calculus of Variations and Partial Differential Equations.

[14]  Chueh Chang,et al.  Travelling wave solutions of a free boundary problem for a two-species competitive model , 2012 .

[15]  Yukio Kan-On,et al.  Parameter dependence of propagation speed of travelling waves for competition-diffusion equations , 1995 .

[16]  Alimardon Elmurodov,et al.  Free boundary problem for predator-prey model , 2023, E3S Web of Conferences.

[17]  Yukio Kan-on,et al.  Fisher wave fronts for the Lotka-Volterra competition model with diffusion , 1997 .

[18]  C. Cosner,et al.  Spatial Ecology via Reaction-Diffusion Equations , 2003 .

[19]  Zhigui Lin,et al.  The diffusive competition model with a free boundary: Invasion of a superior or inferior competitor , 2013, 1303.0454.

[20]  Xiao-Qiang Zhao,et al.  The diffusive logistic model with a free boundary and seasonalsuccession , 2012 .

[21]  Toyohiko Aiki,et al.  A free-boundary problem for concrete carbonation : rigorous justification of the $\sqrt t$-law of propagation , 2011, 1102.3452.

[22]  Y. Hosono,et al.  The minimal speed of traveling fronts for a diffusive Lotka-Volterra competition model , 1998 .

[23]  Masayasu Mimura,et al.  A free boundary problem in ecology , 1985 .

[24]  Yihong Du,et al.  Spreading-Vanishing Dichotomy in the Diffusive Logistic Model with a Free Boundary , 2010, SIAM J. Math. Anal..

[25]  Bingtuan Li,et al.  Cooperative , 1952, The Fairchild Books Dictionary of Fashion.

[26]  Y. Kaneko.  Spreading and vanishing behaviors for radially symmetric solutions of free boundary problems for reaction–diffusion equations , 2014 .

[27]  Robert A. Gardner.  Existence and stability of travelling wave solutions of competition models: A degree theoretic approach , 1982 .

[28]  O. Ladyženskaja.  Linear and Quasilinear Equations of Parabolic Type , 1968 .

[29]  Bei Hu.  Blow-up Theories for Semilinear Parabolic Equations , 2011 .

[30]  Spreading speed and traveling waves for a two-species weak competition system with free boundary , 2013 .

[31]  Jong-Shenq Guo,et al.  On a Free Boundary Problem for a Two-Species Weak Competition System , 2012 .