Stability of Hopf-bifurcating limit cycles in a diffusion-driven prey-predator system with Allee effect and time delay

Kalyan Manna; Malay Banerjee; Kalyan Manna; Malay Banerjee

doi:10.3934/mbe.2019121

Mathematical Biosciences and Engineering

2019, Volume 16, Issue 4: 2411-2446. doi: 10.3934/mbe.2019121

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Stability of Hopf-bifurcating limit cycles in a diffusion-driven prey-predator system with Allee effect and time delay

Kalyan Manna ,
Malay Banerjee ^,

Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur 208016, Uttar Pradesh, India.

Received: 30 December 2018 Accepted: 01 March 2019 Published: 22 March 2019

In this paper, we investigate the effect of the gestation delay on the spatiotemporal pattern formation in a prey-predator system with monotonic functional response with saturation and intraspecific competition among the predator population in presence of the additive Allee effect in prey growth. In this regard, we present rigorous analytical results for determining the delay-induced Hopf bifurcation threshold and the associated properties of the Hopf-bifurcating periodic solutions and verify them with the help of numerical simulations. We derive analytically the delay-induced Hopf bifurcation threshold by employing the linear stability analysis about the unique spatially uniform coexistence steady state. Also, we provide the expressions for determining the direction and stability of the Hopfbifurcating periodic solutions by using the normal form theory and center manifold reduction. The numerical simulation results reveal that the Hopf bifurcation can potentially lead to spatially homogeneous periodic in time distribution of the populations which will eventually settles to chaotic in space and time distribution for sufficiently large value of the time delay. Further, our numerical investigations reveal that the time delay can change one stationary pattern to another through the loss of monotonicity property of the spatially averaged densities.
- prey-predator system,
- Allee effect,
- gestation delay,
- diffusion,
- stationary patterns,
- spatiotemporal chaos
Citation: Kalyan Manna, Malay Banerjee. Stability of Hopf-bifurcating limit cycles in a diffusion-driven prey-predator system with Allee effect and time delay[J]. Mathematical Biosciences and Engineering, 2019, 16(4): 2411-2446. doi: 10.3934/mbe.2019121

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Abstract

In this paper, we investigate the effect of the gestation delay on the spatiotemporal pattern formation in a prey-predator system with monotonic functional response with saturation and intraspecific competition among the predator population in presence of the additive Allee effect in prey growth. In this regard, we present rigorous analytical results for determining the delay-induced Hopf bifurcation threshold and the associated properties of the Hopf-bifurcating periodic solutions and verify them with the help of numerical simulations. We derive analytically the delay-induced Hopf bifurcation threshold by employing the linear stability analysis about the unique spatially uniform coexistence steady state. Also, we provide the expressions for determining the direction and stability of the Hopfbifurcating periodic solutions by using the normal form theory and center manifold reduction. The numerical simulation results reveal that the Hopf bifurcation can potentially lead to spatially homogeneous periodic in time distribution of the populations which will eventually settles to chaotic in space and time distribution for sufficiently large value of the time delay. Further, our numerical investigations reveal that the time delay can change one stationary pattern to another through the loss of monotonicity property of the spatially averaged densities.

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