Dynamic behaviors of a Lotka\u2013Volterra commensal symbiosis model with density dependent birth rate

Abstract

A Lotka–Volterra commensal symbiosis model with density dependent birth rate that takes the form \t\t\tdxdt=x(b11b12+b13x−b14−a11x+a12y),dydt=y(b21b22+b23y−b24−a22y),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\begin{aligned} &\\frac{dx}{dt}=x \\biggl( \\frac{b_{11}}{b_{12}+b_{13}x}-b_{14}-a_{11}x+a_{12}y \\biggr), \\\\ &\\frac{dy}{dt}=y \\biggl( \\frac{b_{21}}{b_{22}+b_{23}y}-b_{24}-a_{22}y \\biggr), \\end{aligned} $$\\end{document} where b_{ij}, i=1, 2, j=1, 2, 3, 4, a_{11}, a_{12} , and a_{22} are all positive constants, is proposed and studied in this paper. The system may admit four nonnegative equilibria. By constructing some suitable Lyapunov functions, we show that under some suitable assumptions, all of the four equilibria may be globally asymptotically stable, such a property is quite different to the traditional Lotka–Volterra commensalism model. With introduction of the density dependent birth rate, the dynamic behaviors of the commensalism model become complicated.