Robust uniform persistence in discrete and continuous nonautonomous systems (Q695102)

The paper extends earlier results of the author on nonautonomous equations. In detail, it addresses difference equations \[ x_{n+1}=f_n(x_n,y_n),\quad y_{n+1}=A(x_n,y_n)y_n, \] as well as ordinary differential equations \[ \dot x=f(t,x,y),\quad \dot y=A(t,x,y)y \] in the positive cone of \({\mathbb R}^p\times{\mathbb R}^q\) which exhibit a positively invariant boundary hyperplane \(X\). It is shown that robust uniform persistence for the dynamics in \(({\mathbb R}^p\times{\mathbb R}^q)\setminus X\) is obtained, provided a compact subset of \(X\) attracting all solutions in \(X\) is a robust uniform weak repellent. Here, the nonautonomous nature of the equations requires some additional uniformity assumptions. The time-periodic case is treated in detail. Finally, the results are illustrated using a periodic discrete-time epidemic model and a continuous-time SIRS model with \(n\) infection strains. It has to be pointed out that convergence or persistence statements are always understood in the forward rather than the pullback sense.