Lyapunov exponents and persistence in discrete dynamical systems

Competitive exclusion and coexistence in ann-species Ricker model

“…As above, it can be shown that λ 2 (x) > 0 for all x ∈ (X 2 ∩ B 3 ) and again, from Proposition 1 and Theorem 2.3 in [10] we obtain that there exist ε 2 > 0 such that Equation (30) holds with ε 2 and with min i=1,2,3 {x i (t, x)} replaced by min i=2,3 {x i (t, x)}.…”

“…Although the proof of Lemma 2.3 is very similar to the proof of Lemma 4.1 in [10], we give it here, for completeness, in the appendix. Lemma 2.4 is the discrete-time version of Proposition 2 in [4].…”

“…If we show that λ i (x) > 0 for all x ∈ {0} ∪ M i , i = 1, 2, then Proposition 1 and Theorem 2.3 in [10] imply that Equation (20) holds.…”

“…If we show that λ 3 (x) > 0 for all x ∈ (X 3 ), then Proposition 1 and Theorem 2.3 in [10] imply that Equation (30) holds with min i=1,2,3 {x i (t, x)} replaced by x 3 (t, x). Now, for any x ∈ X 3 ,…”

“…Although, for our purposes, a weaker version of this lemma would suffice, we state it here in a more general form. The set-up and idea of proof follows [9,10]. Thus, consider a difference equation of the form…”