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.

Original languageEnglish (US)
Pages (from-to)1177-1214
Number of pages38
JournalJournal of Mathematical Analysis and Applications
Volume461
Issue number2
DOIs
StatePublished - May 15 2018

Fingerprint

Prey-predator Model
Reaction-diffusion Model
Prey
Interior
Hopf bifurcation
Predator
Time delay
Hopf Bifurcation
Time Delay
Prey-predator
Asymptotic stability
Predator-prey
Complex Dynamics
Interaction
Extinction
Asymptotic Stability
Attractor
Periodic Solution
Computer simulation
Numerical Simulation

Keywords

  • Diffusion–reaction
  • Hopf bifurcation
  • Predator–prey system
  • Time delay
  • Turing instability

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

Cite this

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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",
year = "2018",
month = "5",
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",
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TY - JOUR

T1 - Dynamics of a diffusion reaction prey–predator model with delay in prey

T2 - Effects of delay and spatial components

AU - Rao, Feng

AU - Castillo-Chavez, Carlos

AU - Kang, Yun

PY - 2018/5/15

Y1 - 2018/5/15

N2 - 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.

AB - 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.

KW - Diffusion–reaction

KW - Hopf bifurcation

KW - Predator–prey system

KW - Time delay

KW - Turing instability

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JF - Journal of Mathematical Analysis and Applications

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