The complex dynamics of a diffusive prey-predator model with an Allee effect in prey

Feng Rao, Yun Kang

Research output: Contribution to journalArticle

11 Citations (Scopus)

Abstract

This paper investigates complex dynamics of a predator-prey interaction model that incorporates: (a) an Allee effect in prey; (b) the Michaelis-Menten type functional response between prey and predator; and (c) diffusion in both prey and predator. We provide rigorous mathematical results of the proposed model including: (1) the stability of non-negative constant steady states; (2) sufficient conditions that lead to Hopf/Turing bifurcations; (3) a prior estimates of positive steady states; (4) the non-existence and existence of non-constant positive steady states when the model is under zero-flux boundary condition. We also perform completed analysis of the corresponding ODE model to obtain a better understanding on effects of diffusion on the stability. Our analytical results show that the small values of the ratio of the prey's diffusion rate to the predator's diffusion rate are more likely to destabilize the system, thus generate Hopf-bifurcation and Turing instability that can lead to different spatial patterns. Through numerical simulations, we observe that our model, with or without Allee effect, can exhibit extremely rich pattern formations that include but not limit to strips, spotted patterns, symmetric patterns. In addition, the strength of Allee effects also plays an important role in generating distinct spatial patterns.

Original languageEnglish (US)
JournalEcological Complexity
DOIs
StateAccepted/In press - Dec 10 2015

Fingerprint

Allee effect
predator
predators
bifurcation
predator-prey interaction
functional response
predator-prey relationships
strength (mechanics)
boundary condition
simulation

Keywords

  • Allee effect
  • Diffusion
  • Non-constant positive solution
  • Pattern formation
  • Turing instability

ASJC Scopus subject areas

  • Ecology, Evolution, Behavior and Systematics
  • Ecological Modeling

Cite this

The complex dynamics of a diffusive prey-predator model with an Allee effect in prey. / Rao, Feng; Kang, Yun.

In: Ecological Complexity, 10.12.2015.

Research output: Contribution to journalArticle

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