@article{ad166c0c597e44dd9e3a42f74525dc0d,
title = "Dynamics of a diffusion reaction prey–predator model with delay in prey: Effects of delay and spatial components",
abstract = "We study the complex dynamics of a Monod–Haldane-type predator–prey interaction model that incorporates: (1) A constant time delay in the prey growth; and (2) diffusion in both prey and predator. We provide the rigorous results of our system including the asymptotic stability of a positive equilibrium; Hopf bifurcation; and the direction of Hopf bifurcation and the stability of bifurcated periodic solutions. We also perform numerical simulations on the effects of diffusion or/and delay when the corresponding ODE model has either a unique interior equilibrium with two interior attractors or two interior equilibria. Our theoretical and numerical results show that diffusion can either stabilize or destabilize the system; large delay could destabilize the system; and the combination of diffusion and delay could intensify the instability of the system. Moreover, when the corresponding ODE system has two interior equilibria, diffusion or time delay in prey or both could lead to the extinction of predator. Our results may provide us useful biological insights on population managements for prey–predator interaction systems.",
keywords = "Diffusion–reaction, Hopf bifurcation, Predator–prey system, Time delay, Turing instability",
author = "Feng Rao and Carlos Castillo-Chavez and Yun Kang",
note = "Funding Information: This research is partially supported by the National Natural Science Foundation of China (Grant Nos. 11601226 , 11426132 ), the Natural Science Foundation of Jiangsu Province of China (Grant No. BK20140927 ), and the research funds from Nanjing Tech University and Jiangsu Government Scholarship for Overseas Studies . The work is also partially supported by NSF -DMS ( 1313312 & 1716802 ); NSF -IOS/DMS ( 1558127 ), and The James S. McDonnell Foundation 21st Century Science Initiative in Studying Complex Systems Scholar Award (UHC Scholar Award 220020472 ). This project has been partially supported by grants from the National Science Foundation ( DMS-1263374 and DMS-1261211 ), and the Offices of the President and the Provost of Arizona State University . Funding Information: This research is partially supported by the National Natural Science Foundation of China (Grant Nos. 11601226, 11426132), the Natural Science Foundation of Jiangsu Province of China (Grant No. BK20140927), and the research funds from Nanjing Tech University and Jiangsu Government Scholarship for Overseas Studies. The work is also partially supported by NSF-DMS (1313312 & 1716802); NSF-IOS/DMS (1558127), and The James S. McDonnell Foundation 21st Century Science Initiative in Studying Complex Systems Scholar Award (UHC Scholar Award 220020472). This project has been partially supported by grants from the National Science Foundation (DMS-1263374 and DMS-1261211), and the Offices of the President and the Provost of Arizona State University. Publisher Copyright: {\textcopyright} 2018 The Author(s)",
year = "2018",
month = may,
day = "15",
doi = "10.1016/j.jmaa.2018.01.046",
language = "English (US)",
volume = "461",
pages = "1177--1214",
journal = "Journal of Mathematical Analysis and Applications",
issn = "0022-247X",
publisher = "Academic Press Inc.",
number = "2",
}