Invasion of open space by two competitors: spreading properties of monostable two‐species competition‐diffusion systems

Abstract

This paper is concerned with some spreading properties of monostable Lotka–Volterra two‐species competition‐diffusion systems when the initial values are null or exponentially decaying in a right half line. Thanks to a careful construction of super‐solutions and sub‐solutions, we improve previously known results and settle open questions. In particular, we show that if the weaker competitor is also the faster one, then it is able to evade the stronger and slower competitor by invading first into unoccupied territories. The pair of speeds depends on the initial values. If these are null in a right half‐line, then the first speed is the KPP speed of the fastest competitor and the second speed is given by an exact formula describing the possibility of nonlocal pulling. Furthermore, the unbounded set of pairs of speeds achievable with exponentially decaying initial values is characterized, up to a negligible set.

Asymptotic spreading of interacting species with multiple fronts Ⅰ: A geometric optics approach

“…They showed that the second speed c 2 can be determined by the linear instability of the zero solution of a single equation with space-time inhomogeneous coefficient. For coupled systems, the case 0 < a < 1 < b was treated in a recently appeared paper of Girardin and the third author [26]. By deriving an explicit formula for c 2 , it is observed that c 2 can sometimes be strictly greater than the minimal speed of traveling wave connecting E 1 and E 2 , and that it depends on the first speed c 1 in a non-increasing manner.…”

“…By deriving an explicit formula for c 2 , it is observed that c 2 can sometimes be strictly greater than the minimal speed of traveling wave connecting E 1 and E 2 , and that it depends on the first speed c 1 in a non-increasing manner. The proof in [26] is based on a delicate construction of (piecewise smooth) super-and sub-solutions for the parabolic system. In our previous paper [41], we showed that in the weak competition case 0 < a, b < 1 the formula for c 2 is exactly the same as the one in [26] but with a novel strategy of proof based on obtaining large deviation estimates via analyzing the Hamilton-Jacobi equations obtained in the thin-front limit.…”

“…The proof in [26] is based on a delicate construction of (piecewise smooth) super-and sub-solutions for the parabolic system. In our previous paper [41], we showed that in the weak competition case 0 < a, b < 1 the formula for c 2 is exactly the same as the one in [26] but with a novel strategy of proof based on obtaining large deviation estimates via analyzing the Hamilton-Jacobi equations obtained in the thin-front limit. We also mention that coupled parabolic systems were also treated in [18,23] based on the large deviations approach, but in these papers all components spread with a single spreading speed.…”

“…1 2 QIAN LIU, SHUANG LIU AND KING-YEUNG LAM ASYMPTOTIC SPREADING OF INTERACTING SPECIES 3 Nevertheless, the exact formula of c 2 remained open.In the bistable case a, b > 1 with appropriate initial conditions, the spreading problem was studied by Carrère [9], who showed that solutions of (1) exhibit two moving interfaces connecting, starting from the right, (0, 0) to (0, 1) to (1, 0). The first interface moves with the expected speed of c 1 = 2 √ dr, whereas the second interface moves with speed c 2 which is the speed of the unique traveling wave solution connecting (0, 1) to (1, 0).The monostable case 0 < a < 1 < b, which is closely related to our problem, was considered by Girardin and the last author [18]. By a delicate construction of piecewise smooth super-and sub-solutions for (1), it was shown that while the faster species spreads at the expected speed c 1 = 2 √ dr, the spreading speed c 2 of the slower species depends on c 1 in a non-trivial way, and is a nonlocally determined quantity in general (see Subsection 1.1 for details).…”

“…This was accomplished in [18] for the case a < 1 < b by a delicate construction of global super-and sub-solutions, in the sense that they are defined and respect the differential inequalities for (t, x) ∈ (0, ∞) × R.…”