The criteria of ultimate boundedness for nonautonomous Lotka-Volterra tree systems (Q5938944)

The authors study the nonautonomous Lotka-Volterra tree system NEWLINE\[NEWLINE \dot x_i(t)=x_i(t)(b_i(t)-\sum_{j=1}^n a_{ij}(t)x_j(t)),\quad i\in N,\tag{1}NEWLINE\]NEWLINE with \(N=\{1,2,\dots ,n\}\), \(n\) is the species number and \(x_i(t)\) represents the density of species \(i\) at time \(t\). Functions \(a_{ij}, b_i\) are supposed to be bounded on \(\mathbb{R}\). By introducing a concept called the degree of species, the authors obtain a set of easily verifiable sufficient conditions for the ultimate boundedness of solutions to (1). As a consequence, they give criteria of the existence of a globally stable equilibrium point for an autonomous Lotka-Volterra tree system.