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Hindrances to bistable front propagation: application to Wolbachia invasion. (English) Zbl 1411.34039
The authors study the biological situation when an invading population propagates and replaces an existing population with different characteristics. They aim at quantifying the propagules and the invasive power. They rigorously show that a heterogeneous environment inducing a strong enough population gradient can stop an invading front, which will converge in this case to a stable front. They characterize the critical population jump, and also prove the existence of unstable fronts above the stable (blocking) fronts. They are particularly interested in the case of artificial Wolbachia infection, used as a tool to fight arboviruses.
The main results are the characterization of the asymptotic behavior of p in two settings: when $$N$$ (total population density) only depends on $$x$$ and when $$\partial x \log N$$ is equal to a constant time the characteristic function of an interval. Overall, two possible sets of asymptotic behaviors appear. Also they demand a numerical conjecture that for generic bistable function, there exists exactly two barriers.

##### MSC:
 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations 35K57 Reaction-diffusion equations 35B40 Asymptotic behavior of solutions to PDEs 92D25 Population dynamics (general) 35C07 Traveling wave solutions
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