A New Lotka-Volterra Model of Competition With Strategic Aggression -- Civil Wars When Strategy Comes Into Play

Abstract

In this monograph, we introduce a new model in population dynamics that describes two species sharing the same environmental resources in a situation of open hostility. The interactions among these populations are described not in terms of random encounters, but via the strategic decisions of one population that can attack the other according to different levels of aggressiveness. This leads to a non-variational model for the two populations at war, taking into account structural parameters such as the relative fit of the two populations with respect to the available resources and the effectiveness of the attack strikes of the aggressive population. The analysis that we perform is rigorous and focuses on the dynamical properties of the system, by detecting and describing all the possible equilibria and their basins of attraction. Moreover, we will analyze the strategies that may lead to the victory of the aggressive population, i.e. the choices of the aggressiveness parameter, in dependence of the structural constants of the system and possibly varying in time in order to optimize the efficacy of the attacks, which take to the extinction in finite time of the defensive population. The model that we present is flexible enough to include also technological competition models of aggressive companies releasing computer viruses to set a rival company out of the market.

Figures (10)

• Figure 1: The figures show a phase portrait for the indicated values of the coefficients. In black, the orbits of the points. The red dots represent the equilibria. The light blue region correspond to $\mathcal{E}$, while the white region to ${\mathcal{B}}$.
• Figure 2: The figure shows the result of a numerical simulation searching a minimizing time strategy $\tilde{a}(t)$ for the problem starting in $(0.5, 0.1875)$ for the parameters $\rho=0.5$, $c=4.0$, $m=0$ and $M=10$. In blue, the value found for $\tilde{a}(t)$; in red, the value of $a_s(t)$ for the corresponding trajectory $(u(t), v(t))$. As one can observe, $\tilde{a}(t)\equiv a_s(t)$ in a long trait. The simulation was done using AMPL-Ipopt on the server NEOS and pictures have been made with Python.
• Figure 3: Trajectories entering ${\mathcal{S}}$.
• Figure 4: Trajectories exiting ${\mathcal{S}}$.
• Figure 5: A possible choice for the function $g(u)$ and intervals $\{I_k\}_{k\in A}$, $\{ K_j \}_{j\in C}$ satisfying the hypothesis of Lemma \ref{['lemma:entrance']}. In blue, the traits for $u\in I_k$ with $k=1,2,3$. In red, the points and traits for $u\in K_j$ with $j=1,2,3$. In green, the outward unit normal vectors; in red, the tangent vectors to the graph of $g$.

Theorems & Definitions (59)

• Remark 3.1
• Theorem 3.2: Dynamics of system \ref{['model']}
• Theorem 3.3
• Theorem 3.4
• Theorem 3.5
• Theorem 3.6
• Theorem 3.7
• Definition 4.1
• Remark 4.2
• Definition 4.3