Permanence conditions for models of population dynamics with switches and delay

Some classes of discrete and continuous generalized Volterra models of population dynamics with parameter switching and constant delay are studied. It is assumed that there are relationships of the type " symbiosis", "compensationism" or "neutralism" between any two species in a biological community. The goal of the work is to obtain sufficient conditions forthe permanence of such models. Original constructions of common Lyapunov-Krasovsky functionals are proposed for families of subsystems corresponding to the switched systems under consideration. Using the constructed functionals, conditions are derived that guarantee permanence for any admissible switching laws and any constant nonnegative delay. These conditions are constructive and are formulated in terms of the existence of a positive solution for an auxiliary system of linear algebraic inequalities. It should be noted that, in the proved theorems, the persistence of the systems is ensured by the positive coefficients of natural growth and the beneficial effect of populations on each other, whereas the ultimate boundedness of species numbers is provided by the intraspecific competition. An example is presented demonstrating the effectiveness of the developed approaches.