ON THE UNION OF VOLTERRA-TYPE POPULATIONS

Ray Redheffer
2006 Demonstratio Mathematica  
where i -1,2,..., m, each u l (0) > 0, the e, and p l j are real constants and the e^ are so chosen that the system has a positive stationary solution («¿) = (pi). The system is called globally asymptotically stable if every solution («¿) tends to (pt) as t -» oo. Suppose we have another system like this one, with n unknowns V{ instead of Ui and positive stationary solution ((ft) instead of (pi). We form a system inm+n unknowns (u, v) by joining some of the vertices of the graph G(p) to some of
more » ... aph G(p) to some of those of G(q). If the original systems are globally asymptotically stable, what additional conditions ensure global asymptotic stability of the larger system so obtained? That is the question with which this paper is concerned. A system of Volterra type is a system of the form Here ei, pij and pi are real constants, Ui are real-valued continuous functions of t for t > 0, differentiate for t > 0, and iii = dui/dt. Condition (lb) ensures that there is a positive stationary point (Pi). Such systems were introduced in the study of populations by Volterra around 1930 and independently by Lotka at about the same time. However Volterra's researches were much more comprehensive [3], a fact which influences our title. The solutions of (1) are called globally asymptotically stable if they all have one and the same finite limit as t oo, independent of the initial Abstract. A system of Volterra type is a system of the form m Unauthenticated Download Date
doi:10.1515/dema-2006-0117 fatcat:igd54dmelbho7pw4dv4d4hgtci