Global stability and oscillations in classical Lotka-Volterra loops
The classical Lotka-Volterra model \[ \dot N_ i=N_ i(b_ i+\sum^{n}_{j=1}a_{ij}N_ j);\quad N_ i(0)>0;\quad i,j=1,...,n;\quad n\geq 3, \] is investigated under the assumptions \(a_{ii}<0\), \(a_{ij}a_{ji}<0\), \(i\neq j\), and \(a_{ij}=0\) if the remainder classes of i and j mod n are not neighbouring. This last assumption expresses that the system is an n-loop. It is called cyclic, if sign \(a_{i,i+1}=sign a_{n1}\), for all \(i=1,2,...,n-1\); otherwise it is called acyclic. Sufficient conditions are provided for the existence of a nonnegative and stable equilibrium. Boundedness of the solutions is proved for acyclic loops and sufficient conditions are given for the case of cyclic loops. The entire parameter space is studied for a 3-species cyclic loop.
sufficient conditions for global asymptotic stability
scientific article; zbMATH DE number 3987110