Traveling waves for a periodic Lotka–Volterra predator–prey system

ABSTRACT

This paper is concerned with the time periodic traveling wave solutions for a periodic Lotka–Volterra predator–prey system, which formulates that both species synchronously invade a new habitat. We first establish the existence of periodic traveling wave solutions by combining the upper and lower solutions with contracting mapping principle and Schauder’s fixed point theorem. The asymptotic behavior of nontrivial solution is given precisely by the stability of the corresponding kinetic system that has been widely investigated. Then, the nonexistence of periodic traveling wave solutions is confirmed by applying the theory of asymptotic spreading. We show the conclusion for all positive wave speed and obtain the minimal wave speed.

KEYWORDS:

• Minimal wave speed
• upper and lower solutions
• analytic semigroup
• asymptotic spreading

AMS SUBJECT CLASSIFICATIONS:

• 35K57
• 35B40
• 92D40

Acknowledgements

The authors would like to thank the anonymous reviewer for his/her helpful comments and careful reading.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by NSF of China [grant number 11471149], [grant number 11731005]; Fundamental Research Funds for the Central Universities.