Global dynamics of anti-competitive systems in the plane (Q2849212)

An anticompetitive system of scalar equations in the plane of the form NEWLINE\[NEWLINEx_{n+1}=T_1(x_n,y_n),\quad y_{n+1}=T_2(x_n,y_n)\tag{1}NEWLINE\]NEWLINE is considered. The anticompetitiveness means that the continuous functions \(T_1\) and \(T_2\) are monotone with \(T_1(x,y)\) being non-increasing in \(x\) and non-decreasing in \(y\), while \(T_2(x,y)\) being non-decreasing in \(x\) and non-increasing in \(y\). The dynamics of System (1) is equivalent to that of the discrete map \(F:=(F_1,F_2)\) of the plane into itself. The second iteration of the map \(F, F\circ F=F(F(\cdot))=G(\cdot)=(G_1, G_2),\) is dynamically equivalent to the system in the plane NEWLINE\[NEWLINEx_{n+1}=G_1(x_n,y_n),\quad y_{n+1}=G_2(x_n,y_n),\tag{2}NEWLINE\]NEWLINE for which the continuous functions \(G_1\) and \(G_2\) have certain monotonicity properties as well: \(G_1(x,y)\) is non-decreasing in \(x\) and non-increasing in \(y\), while \(G_2(x,y)\) is non-increasing in \(x\) and non-decreasing in \(y\). Such systems (2) are called competitive, and many of their dynamical properties are well studied.NEWLINENEWLINEThe authors use their own results, as well as those of others, to derive some properties of system (1) based on the known properties of system (2). The obtained results are applied to the model system NEWLINE\[NEWLINEx_{n+1}=\frac{\alpha_1+\gamma_1 y_n}{A_1+x_n},\quad y_{n+1}=\frac{\alpha_2+\beta_2 x_n}{A_2+y_n}. \tag{3}NEWLINE\]NEWLINE For the latter several specific results are derived: conditions for the global asymptotic stability of a unique equilibrium are established; the dynamics are described in the case when the unique equilibrium is a saddle; conditions are given when system (3) has infinitely many period-two solutions.