The influence of commensalism on a Lotka\u2013Volterra commensal symbiosis model with Michaelis\u2013Menten type harvesting

Abstract

In this paper, we study the following Lotka–Volterra commensal symbiosis model of two populations with Michaelis–Menten type harvesting for the first species: \t\t\tdxdt=r1x(1−xK1+αyK1)−qExm1E+m2x,dydt=r2y(1−yK2),\\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}$$ \\begin{gathered} \\frac{dx}{dt} = r_{1}x \\biggl(1- \\frac{x}{K_{1}}+\\alpha \\frac{y}{K_{1}} \\biggr)- \\frac{qEx}{m_{1}E+m_{2}x}, \\\\ \\frac{dy}{dt} = r_{2}y \\biggl(1- \\frac{y}{K_{2}} \\biggr), \\end{gathered} $$\\end{document} where r_{1}, r_{2}, K_{1}, K_{2}, α, q, E, m_{1} and m_{2} are all positive constants. The local and global dynamic behaviors of the system are investigated, respectively. For the limited harvesting case (i.e., q is small enough), we show that the system admits a unique globally stable positive equilibrium. For the over harvesting case, if the cooperate intensity of the both species (α) and the capacity of the second species (K_{2}) are large enough, the two species could coexist in a stable state; otherwise, the first species will be driven to extinction. Numeric simulations are carried out to show the feasibility of the main results.