Assume that every system here is of monostable structure and its solution operator is order preserved. H. F. Weinberger [SIAM J. Math. Anal. 13, 353–396 (1982; Zbl 0529.92010)] established the spreading speed and its coincidence with the minimal wave speed for a scalar discrete recursion model on a one-dimensional habitat \(\mathcal{H}\) which maybe either discrete or continuous. The space chosen by Weinberger is \(C(\mathcal{H}, \mathbb{R})\) and the recursion map is assumed to be compact under the compact open topology. The compactness plays an important role in the proof of the existence of traveling waves. Weiberger’s result was developed by R. Lui [Math. Biosci. 93, No. 2, 269–295 (1989; Zbl 0706.92014)] to systems of recursion on \(C(\mathcal{H}, \mathbb{R}^l)\), and by X. Liang and X.-Q. Zhao [Commun. Pure Appl. Math. 60, No. 1, 1–40 (2007; Zbl 1106.76008)] to compact semiflows \(C(\mathcal{H}, C(M, \mathbb{R}^l))\), where \(M\) is a compact metric space. X. Liang and X.-Q. Zhao [J. Funct. Anal. 259, No. 4, 857–903 (2010; Zbl 1201.35068)] further weakened the compactness assumption to be “interval-\(\alpha\)-contraction” under the abstract setting, where the kinetic phase space is an appropriate subset of a Banach lattice \(X\).
The aim of this article is to weaken the compactness of “interval-\(\alpha\)-contraction to “point-\(\alpha\)-contraction”. The authors chose the phase space to be a set of monotone functions from \(\mathbb{R}\) to the special Banach space lattice \(X=C(\Omega, \mathbb{R}^l)\), rather than continuous functions from \(\mathbb{R}\) to a general Banach space lattice \(X\), where \(\Omega\) is a compact metric space with metric \(d\). They discussed the difference of these two compactness assumptions in detail, and overcame the difficulty due to the lack of compactness. At last, they applied the developed theory to discuss the existence of traveling waves for three prototypical noncompact systems: a partially degenerate reaction-diffusion system, a nonlocal dispersal equation with time delay, and a two species competition model with nonlocal dispersal.