On a Lotka-Volterra Competition Diffusion Model with Advection

Chinese Annals of Mathematics, Series B - Tập 42 - Trang 891-908 - 2021
Qi Wang1
1College of Science, University of Shanghai for Science and Technology, Shanghai, China

Tóm tắt

In this paper, the author focuses on the joint effects of diffusion and advection on the dynamics of a classical two species Lotka-Volterra competition-diffusion-advection system, where the ratio of diffusion and advection rates are supposed to be a positive constant. For comparison purposes, the two species are assumed to have identical competition abilities throughout this paper. The results explore the condition on the diffusion and advection rates for the stability of former species. Meanwhile, an asymptotic behavior of the stable coexistence steady states is obtained.

Tài liệu tham khảo

Averill, I., Lam, K.-Y. and Lou, Y., The role of advection in a two-species competition model: A bifurcation approach, Mem. Amer. Math. Soc., 245(1161), 2017, v+ 117 pp. Cantrell, R. S. and Cosner, C., Spatial Ecology Via Reaction-Diffusion Equations, Series in Mathematical and Computational Biology, Wiley, Chichester, UK, 2003. Cantrell, R. S., Cosner, C. and Lou, Y., Multiple reversals of competitive dominance in ecological reserves via external habitat degradation, J. Dynam. Differential Equations, 16, 2004, 973–1010. Cantrell, R. S., Cosner, C. and Lou, Y., Movement toward better environments and the evolution of rapid diffusion, Math. Biosci., 204, 2006, 199–214. Cantrell, R. S., Cosner, C. and Lou, Y., Advection-mediated coexistence of competing species, Proc. Roy. Soc. Edinburgh Sect. A, 137, 2007, 497–518. Chen, X. F., Hambrock, R. and Lou, Y., Evolution of conditional dispersal: A reaction-diffusion-advection model, J. Math. Biol., 57, 2008, 361–386. Cosner, C. and Lou, Y., When does movement toward better environment benefit a population? J. Math. Anal. Appl., 277, 2003, 489–503. Du, Y., Effects of a degeneracy in the competition model, Part I, Classical and generalized steady-state solutions, J. Differential Equations, 181, 2002, 92–132. Du, Y., Effects of a degeneracy in the competition model, Part II, Perturbation and dynamical behavior, J. Differential Equations, 181, 2002, 133–164. Du, Y., Realization of prescribed patterns in the competition model, J. Differential Equations, 193, 2003, 147–179. Evans, L. C., Partial Differential Equations, Graduate Studies in Mathematics, 19, American Mathematical Society, USA, 1999. López-Gómez, J., Coexistence and meta-coexistence for competing species, Houston J. Math., 29(2), 2003, 483–536. Hambrock, R. and Lou, Y., The evolution of conditional dispersal strategies in spatially heterogeneous habitats, Bull. Math. Biol., 71, 2009, 1793–1817. He, X. and Ni, W.-M., Global dynamics of the Lotka-Volterra competition-diffusion system: Diffusion and spatial heterogeneity I, Comm. Pure Appl. Math., 69, 2016, 981–1014. He, X. and Ni, W.-M., Global dynamics of the Lotka-Volterra competition-diffusion system with equal amount of total resources, II, Calc. Var. Partial Differential Equations, 55, 2016, 25, 20 pp. He, X., and Ni W.-M., Global dynamics of the Lotka-Volterra competition-diffusion system with equal amount of total resources, III, Calc. Var. Partial Differential Equations, 56, 2017, 132, 26 pp. Hess, P., Periodic-parabolic Boundary Value Problems and Positivity, Pitman Research Notes in Mathematics, 247. Longman Sci. Tech., Harlow, 1991. Hutson, V., Lou, Y. and Mischaikow, K., Spatial heterogeneity of resources versus Lotka-Volterra dynamics, J. Differential Equations, 185, 2002, 97–136. Hutson, V., Lou, Y. and Mischaikow, K., Convergence in competition models with small diffusion coeffcients, J. Differential Equations, 211, 2005, 135–161. Hutson, V., Lou, Y. and Mischaikow, K., Poláčik, P., Competing species near the degenerate limit, SIAM J. Math. Anal., 35, 2003, 453–491. Hutson, V., Martinez, S., Mischaikow, K. and Vicker, G. T., The evolution of dispersal, J. Math. Biol., 47, 2003, 483–517. Hutson, V., Mischaikow, K. and Poláčik, P., The evolution of dispersal rates in a heterogeneous time-periodic environment, J. Math. Biol., 43, 2001, 501–533. Lam, K.-Y., Concentration phenomena of a semilinear elliptic equation with large advection in an ecological model, J. Differential Equations, 250, 2011, 161–181. Lam, K.-Y., Limiting profiles of semilinear elliptic equations with large advection in population dynamics II, SIAM J. Math. Anal., 44, 2012, 1808–1830. Lou, Y., On the effects of migration and spatial heterogeneity on single and multiple species, J. Differential Equations, 223, 2006, 400–426. Lou, Y., Martinez, S. and Poláčik, P., Loops and branches of coexistence states in a Lotka-Volterra competition model, J. Differential Equations, 230, 2006, 720–742. Protter, M. H. and Weinberger, H. F., Maximum Principles in Differential Equations, 2nd ed., Springer-Verlag, New York, 1984. Saut, J. C. and Scheurer, B., Remarks on a nonlinear equation arising in population genetics, Commun. Part. Differ. Eq., 23, 1978, 907–931. Sattinger, D. H., Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J., 21, 1972, 979–1000. Wang, Q., On steady state of some Lotka-Volterra competition-diffusion-advection model, Discrete Contin. Dyn. Syst. Ser. B, 25, 2020, 859–875. Zhou, P. and Xiao, D., Global dynamics of a classical Lotka-Volterra competition-diffusion-advection system, J. Funct. Anal., 275, 2018, 356–380.