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Persistence of preys in a diffusive three species predator-prey system with a pair of strong-weak competing preys. (English) Zbl 1465.35274

Summary: We investigate the traveling wave solutions of a three-species system involving a single predator and a pair of strong-weak competing preys. Our results show how the predation may affect this dynamics. More precisely, we describe several situations where the environment is initially inhabited by the predator and by either one of the two preys. When the weak competing prey is an aboriginal species, we show that there exist traveling waves where the strong prey invades the environment and either replaces its weak counterpart, or more surprisingly the three species eventually co-exist. Furthermore, depending on the parameters, we can also construct traveling waves where the weaker prey actually invades the environment initially inhabited by its strong competitor and the predator. In all those situations, we find the infimum of the set of admissible wave speeds; these results are sharp at least when the three species diffusive at the same speed.

MSC:

35K45 Initial value problems for second-order parabolic systems
35K57 Reaction-diffusion equations
34B40 Boundary value problems on infinite intervals for ordinary differential equations
35C07 Traveling wave solutions
92D25 Population dynamics (general)
35K58 Semilinear parabolic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
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References:

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