Subharmonic solutions for degenerate periodic systems of Lotka-Volterra type with impulsive effects

Yinyin Wu; Fanfan Chen; Qingchi Ma; Dingbian Qian; Yinyin Wu; Fanfan Chen; Qingchi Ma; Dingbian Qian

doi:10.3934/math.20231023

AIMS Mathematics

2023, Volume 8, Issue 9: 20080-20096. doi: 10.3934/math.20231023

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Subharmonic solutions for degenerate periodic systems of Lotka-Volterra type with impulsive effects

1.
Department of Fundamental Courses, Wuxi Institute of Technology, Wuxi 214121, China
2.
Department of Mathematics, School of Science, Zhejiang Sci-Tech University, Hangzhou, 310018, China
3.
Suzhou High School affiliated to Xian Jiaotong University, Suzhou 215006, China
4.
School of Mathematical Sciences, Soochow University, Suzhou 215006, China

Received: 17 March 2023 Revised: 07 June 2023 Accepted: 12 June 2023 Published: 19 June 2023
MSC : 34A34, 34C25, 37E40

In this paper, we are concerned with the existence of subharmonic solutions for the degenerate periodic systems of Lotka-Volterra type with impulsive effects. In our degenerate model, the variation of the predator and prey populations may vanish on a time interval, which imitates the (real) possibility that the predation is seasonally absent. Our proof is based on the Poincaré-Birkhoff theorem. By using phase plane analysis, we can find the large gap in the rotation numbers between the "small" solutions and the "large" solutions, which guarantees a suitable twist property. By applying the Poincaré-Birkhoff theorem, we then obtain the existence of subharmonic solutions. Our main theorem extends the associated results by J. López-Gómez et al.
- periodic systems of Lotka-Volterra type,
- subharmonic solutions,
- degenerate systems,
- impulsive effects,
- Poincaré-Birkhoff theorem
Citation: Yinyin Wu, Fanfan Chen, Qingchi Ma, Dingbian Qian. Subharmonic solutions for degenerate periodic systems of Lotka-Volterra type with impulsive effects[J]. AIMS Mathematics, 2023, 8(9): 20080-20096. doi: 10.3934/math.20231023

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Abstract

In this paper, we are concerned with the existence of subharmonic solutions for the degenerate periodic systems of Lotka-Volterra type with impulsive effects. In our degenerate model, the variation of the predator and prey populations may vanish on a time interval, which imitates the (real) possibility that the predation is seasonally absent. Our proof is based on the Poincaré-Birkhoff theorem. By using phase plane analysis, we can find the large gap in the rotation numbers between the "small" solutions and the "large" solutions, which guarantees a suitable twist property. By applying the Poincaré-Birkhoff theorem, we then obtain the existence of subharmonic solutions. Our main theorem extends the associated results by J. López-Gómez et al.

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