Convergence and ultimate bounds of solutions of the nonautonomous Volterra-Lotka competition equations

On almost periodic solutions of the competing species problems

“…Then, for -rij < to, we have u2nj(-rij) -£ < û(-rij) and v(-rij) < k2 = v2rlj(-rij). It follows from our earlier comparison result (see [1]) that u2n. (t) < u(t) and v{t) < v2rij (t) if -rij < t < 00.…”

“…In [1] the author showed that if n -2 and the functions o¿j(¿), 1 < i,j < 2, and 6¿(¿) are merely assumed to be continuous and bounded above and below by positive constants on an interval [in,oo), then conditions (Gi) imply that the differences of the corresponding components of two solutions of (S), both of whose components are positive at £o, tend to zero as t -* oo. To prove the above claim, it is therefore sufficient to show that if n = 2 and the functions are almost periodic and bounded above and below by positive constants, then conditions (Gi) imply the existence of an almost periodic solution both of whose components are bounded below by positive constants.…”

“…Given a function g(t), we let gL and #m denote mî-O0<t<O0 g{t) and sup_00<t<00 g(t), respectively. Throughout this paper we assume the inequalities a¿ > caíí¿m//l and di > eMO,M¡bh-As in [1], let £,/ci,/c2 be numbers satisfying the inequalities 0 < e < rci,fc2, aM/bL < fci, dM/fL<k2, Ûl -e\iki -}mE > 0, aL -6m£ -c«^ > 0.…”

“…In [1] it was shown that if col(tti(i), t>i(f)) and co\(u2(t): v2(t)) are two solutions with ui(io) = ^li ^lUo) -£■, ^2(io) = £, and ^2(^0) -k2 for some ¿o m (-00,00), then e < u2(t) < ui(i) < Aci and e < vi(t) < v2(t) < k2 for all t > t0.…”

Extinction in nonautonomous competitive Lotka-Volterra systems

“…Let u 1 (t) be a solution of the nonautonomous logistic equation (1.2) such that u 1 (t 0 ) ≥ x 1 (t 0 ). Then u 1 (t) > x 1 (t) for all t > t 0 (see Ahmad [1], Lemma 2.8 or Tineo and Alvarez [12], Proposition 2.1), and u 1 (t) is bounded (Lemmas 1.1 and 1.2). We now follow a technique similar to that used in the proof of the previous theorem to compare u 1 (t) and x 1 (t) as t → ∞:…”