ENDOGENOUS CONTROL IN A TERNARY LOTKA-VOLTERRA MODEL AND ITS APPLICATIONS - Piccinini, Livio Clemente; Lepellere, Maria Antonietta; Chang, Ting Fa Margherita; Iseppi, Luca (Italian Journal of Pure and Applied Mathematics, 2015)
This paper aims at highlighting the role of endogenous controls in Lotka-Volterra predator-prey models. Unlike other studies in which the core lies in expanding the number of conflicting species, here the stress is laid upon a control variable that does not need to be of the same nature as the species involved in the conflict. As in the case of the logistic equation Lotka-Volterra models have proved highly fruitful also outside the pure biological frame, clarifying many socio-economical and psychological evolution phenomena. A main problem lies in the change of the structural parameters, that can alter the evolution. Rather than an exterior action a model is most satisfactory in presence of an endogenous change, that should happen without any external intervention. The critical model is described in Section 2 under the name of tripartite antagonist model, where there are three actors, and three predator-prey equations involving, two at each time, the actors. The structure is that each actor once is predator and once prey in a circular scheme. In some natural cases the three variables are all biological, but in most meaningful cases the control has a different nature, usually social or behavioral. The paper highlights the particular case where the initial existence of a control, albeit at a very low level, is essential to allow the development towards equilibrium, while its absence leads to the destruction of the prey. A similar case arises when predator and prey are in temporary equilibrium, but a hidden control variable operates until a sudden change of equilibrium bursts out. Section 1 is devoted to recall some preliminary knowledge of system theory and of Lotka.Volterra classical equations. Section 3 is devoted to illustrate some actual cases where tripartite antagonist system can explain social, psychological, economic developments, even with reference to the academic world. The long lasting competition between Stoicism and Epicurism, between hedonism and Aristotle's eudaimonia fully enters in this scheme.