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The state-dependent impulsive control for a general predator-prey model.

Xiaoxiao Zhu1, Huilan Wang1, Zigen Ouyang1

  • 1School of Mathematics and Physics, University of South China, Hengyang, People's Republic of China.

Journal of Biological Dynamics
|June 7, 2021
PubMed
Summary
This summary is machine-generated.

This study analyzes predator-prey models with state-dependent impulses, establishing conditions for periodic solutions using geometric analysis and Poincaré maps. The research also confirms the stability of these solutions through an analogue of the Poincaré Criterion.

Keywords:
34A3734C2537N25Poincaré mapState-dependent impulseorder-1 periodic solutionpredator–prey modelsuccessor function

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Area of Science:

  • Mathematical Biology
  • Dynamical Systems Theory
  • Ecological Modeling

Background:

  • Predator-prey models are fundamental in ecology.
  • State-dependent impulses introduce complex dynamics.
  • Understanding periodic solutions is crucial for ecological stability.

Purpose of the Study:

  • To investigate a general predator-prey model with state-dependent impulses.
  • To establish sufficient conditions for the existence of order-1 periodic solutions.
  • To analyze the stability of these periodic solutions.

Main Methods:

  • Geometric analysis
  • Poincaré map (successor function)
  • Construction of Bendixson domains
  • Analogue of Poincaré Criterion

Main Results:

  • Sufficient conditions for the existence of order-1 periodic solutions were derived.
  • Three typical types of Bendixson domains were constructed.
  • Asymptotic stability of periodic solutions was confirmed.

Conclusions:

  • The study provides a robust framework for analyzing predator-prey dynamics with impulses.
  • The findings contribute to understanding the conditions for stable ecological cycles.
  • Numerical simulations validate the theoretical results.