Global asymptotic stability of the periodic Lotka-Volterra system with two predators and one prey
The authors consider the three species Lotka-Volterra periodic model with two predators and one prey \[ \dot x_i= x_i[\delta_i(b_i(t)- a_{i1}(t))x_1- a_{i2}(t) x_2- a_{i3}(t) x_3],\tag{1} \] where \(\delta_1= 1= -\delta_2= -\delta_3\), \(a_{23}(t)= a_{32}(t)= 0\), \(i= 1, 2, 3\). Suppose that the functions \(b_i(t)\), \(a_{ij}(t)\), \(i,j= 1, 2, 3\) are continuous, nonnegative and periodic functions with a common period \(T> 0\) and \(b_1(t)\), \(a_{ii}(t)\), \(i= 1, 2, 3\) are strictly positive. By using Brouwer's fixed point theorem, the authors show that the system (1) has at least one strictly positive \(T\)-periodic solution, if the coefficients of (1) satisfy some conditions. The authors also give some sufficient conditions to guarantee that a unique strictly positive solution of (1) is globally asymptotically stable.
three species Lotka-Volterra periodic model with two predators and one prey
scientific article; zbMATH DE number 840329