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, Minimal (left) and maximal (right) total effort ratio to get into the basin of 0 (in days) for various values of (? E , ?), the minimum and maximum being taken with respect to (T, ?), with a period and an entrance time shown in parentheses. The total effort ratio is defined as the total number of released male mosquitoes divided by the initial (wild) male mosquito population, vol.9

, 23133) 317) 667 (1, 420) 752 (1, 474) 826 (1, 521) 896 (1, 565) 528 (1, 629) 661 (1, 749) 735 (1, 833) 803 (1, 909) 868 (1, 982) 534 (1, 1012) 642 (1, 1179) 708 (1, 1300) 771 (1, 1414) 830 (1, 1522), vol.7155, p.470, 13603.

E. Bubbles-chapter, MATHEMATICAL PERSPECTIVES under (too) specific conditions on the nonlinearities in [53] (see the introduction of the cited article for a historical and synthetic presentation of the techniques of proof)

. First, 3) simultaneously. By classical results on competitive systems (see the discussion in Section 4.3.3), there exists a traveling wave ? := (? 1 , ? 2 ) connecting 0 at ?? to E + at +? and traveling at speed c ? R. Using this particular solution and the comparison principle yields: Proposition 13.1. Assume c = 0, vol.13

, has a solution then for all L > L * it also has a solution

, ) be a solution to (13.2) for some L > 0

?. K-o-n, By the symmetric construction (extending by E + on R\(?L, L)), we can show that any solution to (13.3) gives rise to a "bubble" super-solution, which prevents E + from being the invading state and imposes c > 0 if c = 0. In particular, assuming c = 0 implies that for any L 1 , L 2 > 0, the existence of a solution to (13.2) with L = L 1 and to (13.3) with L = L 2 are incompatible, whence the first part of the result. Then, the sub-and super-solution method exposed in Chapter 4, Proposition 4.5 for scalar elliptic equations extends to systems (by the same process of building monotone and bounded sequences of sub-and super-solutions), This inequality is preserved by the time-dynamics, and in particular 0 cannot be the invading state: if c = 0 then c < 0

, Under the additional assumptions of Proposition 7.8, we know that there is exactly one bubble of radius L * > 0 and two bubbles of radius L for all L > L *, particular Theorem 7.1 and Section, vol.7

, or D 1 and D 2 ), we can indeed deduce from Proposition 13.1 that generically, (13.2) and (13.3) cannot be solved simultaneously. As noted in Section 4.3.3, the sign of c is usually not simple to determine. With the bubble viewpoint, it amounts to checking which Dirichlet problem has solutions among (13.2) and (13.3). Numerical simulations lead us to the following conjecture, which may hold under specific addi

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