TY - JOUR
T1 - The complex dynamics of a diffusive prey–predator model with an Allee effect in prey
AU - Rao, Feng
AU - Kang, Yun
N1 - Funding Information:
This research of F.R. is supported by Jiangsu Provincial Natural Science Foundation (BK 20140927) and Tianyuan Fund for Mathematics of NSFC (11426132). The research of Y.K. is partially supported by NSF-DMS (1313312), Simons Collaboration Grants for Mathematicians (208902), NSF-IOS/DMS (1558127), The James S. McDonnell Foundation-UHC Scholar Award (220020472), and the research scholarship from College of Integrative Sciences and Arts, ASU.
Funding Information:
This research of F.R. is supported by Jiangsu Provincial Natural Science Foundation ( BK 20140927 ) and Tianyuan Fund for Mathematics of NSFC ( 11426132 ). The research of Y.K. is partially supported by NSF-DMS ( 1313312 ), Simons Collaboration Grants for Mathematicians ( 208902 ), NSF-IOS/DMS (1558127), The James S. McDonnell Foundation-UHC Scholar Award (220020472), and the research scholarship from College of Integrative Sciences and Arts, ASU.
Publisher Copyright:
© 2016 Elsevier B.V.
PY - 2016/12/1
Y1 - 2016/12/1
N2 - 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.
AB - 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.
KW - Allee effect
KW - Diffusion
KW - Non-constant positive solution
KW - Pattern formation
KW - Turing instability
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U2 - 10.1016/j.ecocom.2016.07.001
DO - 10.1016/j.ecocom.2016.07.001
M3 - Article
AN - SCOPUS:84994376888
SN - 1476-945X
VL - 28
SP - 123
EP - 144
JO - Ecological Complexity
JF - Ecological Complexity
ER -