Introduction

In this chapter, we describe the content of this book, in which we will introduce a new mathematical model to analyze the situation of two biological populations competing for the same resources, in a mutual conflict caused by an aggressive population which, depending on the parameters of the system, may attack the other. We employ “civil war” as a stylized expression that we employ to resume the two main features of our model: aggressive behavior from one hand and competition for the resources from the other. The main questions that we deal with in this monograph are the characterization of the initial conditions for which there exists a winning strategy, the success of the constant strategies, compared to all possible strategies, the construction of a winning strategy for a given initial datum, and the existence of a single winning strategy independently of the initial datum. Our analysis will characterize the equilibria of the system and their features in terms of the different parameters of the model (such as relative fitness to the environment, aggressiveness, and effectiveness of strikes). Moreover, we will study the initial configurations which may lead to the victory of the aggressive population, discussing the possible strategies to achieve the victory. The analysis of the optimal strategies is complex, also because there exist initial configurations for which the aggressive population cannot obtain the victory of the war, no matter what strategy is adopted. Different scenarios will arise in dependence of the different parameters of the system. For example, for populations with the same fit to the environment, the constant strategies suffice for the aggressive population to possibly achieve the victory, but for populations with different fit to the environment the constant strategies do not exhaust all the possible winning strategies, but it is still enough to consider strategies with at most one jump discontinuity. The utility of the strategy is also subject to different possible parameters, such as the duration of the war. For example, we show that among all the winning strategies, jump discontinuous strategies may not be optimal.