Robust uniform persistence in discrete and continuous nonautonomous systems
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.
scientific article; zbMATH DE number 6117534