7 Appendix Robust Uniform Persistence

To obtain the robust persistence result for the system of difference equation of the form

(1)

where $z=(x,y)$ is the state space in ${ℝ}_{+}^{p+q}$ and $\xi \in {ℝ}^l$, where $l\in ℝ$ denotes the vector of parameters. Let $F(t,z(t),\xi )$ denotes the right hand side of (1). Specifically,

$$F(t,z(t),\xi )=(f(t,z,\xi ),A(t,z,\xi )y).$$

Let, $z(t)=\varphi (t+s_0,s_0,z_0,\xi )$ be the non-autonomous solution semiflow generated by (1).
Let $X:=\{(x,y)\in {ℝ}_{+}^p\times {ℝ}_{+}^q|y(t)=0\forall t>0\}$ be the extinction set. Let $P(t+s_0,s_0,z_0,\xi )$ is the solution matrix for the following linear system in $u$, where $u\in {ℝ}_{+}^2$ and $P(s_0,s_0,z_0)=I$,

(2) $$u(t+1)=A_h(t,z(t),\xi )u(t).$$

To prove the robust uniform persistence we need the following assumptions from [salceanu2013robust]:
(H1) There exists a closed set $B$ that absorbs all trajectories corresponding to ${\xi }_0$ (i.e. $\forall s_0\in {\mathbb{Z}}_{+},z_0\in Z,\exists t(s_0,z_0)$ such that $\varphi (t+s_0,s_0,z_0,{\xi }_0)\in B,\forall t\ge t(s_0,z_0)$).
(H2) Let

$$U:=\{\eta \in R_{+}^q||\eta |=1\},$$

$$M:=B\cap X.$$

Then $\forall z_0\in M,\eta \in U,\exists T>0,c>1,s_0\in {\mathbb{Z}}_{+}$ such that $\forall s\ge s_0,\exists T(s)\in (0,T]$ satisfying

$$P|(T(s)+s,s,z_0)\eta |>c,$$

which is equivalent to Proposition 3.8 from [salceanu2013robust].
(H3) $\forall t_0\in {\mathbb{Z}}_{+},z_0\in M,\exists K>0$ such that $|\varphi (t+s_0,s_0,z_0,{\xi }_0)|\le K,\forall s\in {\mathbb{Z}}_{+},t\in [0,t_0].$
(H4) There is a neighbourhood $\widetilde{V_B}$ of $B$ and ${\tilde{N}}_{\xi 0}$ a bounded neighbourhood of ${\xi }_0$ such that for any compact set $\tilde{M}\subset X$, there exists a constant $K>0$ such that

(3) $$\Vert A(t,z_0,{\xi }_0)\Vert \le K,$$

(4) $$|F(t,z_2,\xi )-F(t,z_1,{\xi }_0)|\le (|z_2-z_1|+|\xi -{\xi }_0|),$$

(5) $$\Vert A(t,z_2,\xi )-A(t,z_1,{\xi }_0)\Vert \le K(|z_2-z_1|+|\xi -{\xi }_0|),$$

$\forall t\in {\mathbb{Z}}_{+},z_0,z_1\in \tilde{M},z_2\in \widetilde{V_B},\xi \in N_{\xi 0}.$
Moreover, for every neighborhood $V_B$ of $B$, $V_B\subseteq \widetilde{V_B}$, there exists $N_{\xi 0}$ a neighbourhood of ${\xi }_0$, $N_{\xi 0}\subseteq \widetilde{N_{\xi 0}}$, satisfying the following: $\forall z_0\in Z,s_0\in {\mathbb{Z}}_{+},\xi \in N_{\xi 0},\exists t(z_0,s_0,\xi )\in {\mathbb{Z}}_{+}$ such that

(6) $$\varphi (t+s_0,s_0,z_0,\xi )\in V_B,\forall t\ge t(z_0,s_0,\xi ).$$

(H5) $\{z=(x,y)\in B||y|\le \delta \}$ is bounded (hence compact), for some $\delta >0.$
(D) For every $z_0\in M,\eta \in U\text{and}{t}_0\in {\mathbb{Z}}_{+}$ there exists $C>0$ such that $|P(t_0+s_0,s_0,z_0,{\xi }_0)\eta |\ge C$ for all $s_0\in {\mathbb{Z}}_{+}$.