Abstract
This paper deals with the minimal wave speed of delayed lattice dynamical systems without monotonicity in the sense of standard partial ordering in R2. By constructing upper and lower solutions appealing to the exponential ordering, we prove the existence of traveling wave solutions if the wave speed is not smaller than some threshold. The nonexistence of traveling wave solutions is obtained when the wave speed is smaller than the threshold. Therefore, we confirm the threshold is the minimal wave speed, which completes the known results.
Highlights
Propagation thresholds including minimal wave speed of traveling wave solutions and asymptotic speed of spreading in population models have attracted much attention; see [1, 2] for their biological backgrounds
By constructing upper and lower solutions appealing to the exponential ordering, we prove the existence of traveling wave solutions if the wave speed is not smaller than some threshold
To obtain the existence and asymptotic behavior of traveling wave solutions of nonmonotone systems is not easy; at least the oscillation of traveling wave solutions has been observed in some delayed equations without quasimonotonicity [26]
Summary
Propagation thresholds including minimal wave speed of traveling wave solutions and asymptotic speed of spreading in population models have attracted much attention; see [1, 2] for their biological backgrounds. If a noncooperative system admits comparison principle in the sense of standard partial ordering in Rn, there are some results on propagation thresholds, e.g., predator-prey systems [5,6,7] and competitive system [8, 9]. By showing the existence or nonexistence of traveling wave solutions with small wave speed, they completed the conclusions in Lin and Li [22, Example 5.1] if b1, b2 ∈ [0, 1). The purpose of this paper is to confirm the existence or nonexistence of traveling wave solutions if the wave speed is positive when b1, b2 ∈ [0, 1). We establish the existence of traveling wave solutions by an abstract result in Lin and Li [22, Theorem 4.5] with the help of exponential ordering [25], which will be finished by constructing proper upper and lower solutions if the wave speed is large. It should be noted that our discussion includes all positive wave speed, so we obtain a minimal wave speed and complete the conclusion in [22]
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
