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The role of noise in a predator–prey model with Allee effect

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Abstract

The existence and implications of alternative stable states in ecological systems have been investigated extensively within deterministic models. However, it is known that natural systems are undeniably subject to random fluctuations, arising from either environmental variability or internal effects. Thus, in this paper, we study the role of noise on the pattern formation of a spatial predator–prey model with Allee effect. The obtained results show that the spatially extended system exhibits rich dynamic behavior. More specifically, the stationary pattern can be induced to be a stable target wave when the noise intensity is small. As the noise intensity is increased, patchy invasion emerges. These results indicate that the dynamic behavior of predator–prey models may be partly due to stochastic factors instead of deterministic factors, which may also help us to understand the effects arising from the undeniable susceptibility to random fluctuations of real ecosystems.

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Acknowledgements

This work is supported by the National Natural Science Foundation of China under Grant No. 60771026, Program for New Century Excellent Talents in University (NCET050271), the Natural Science Foundation of Shan’Xi Province Grant No. 2006011009, the Special Scientific Research Foundation for the Subjects of Doctors in University (20060110005), and Graduate Students’ Excellent Innovative Item of Shanxi Province No. 20081018.

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Authors and Affiliations

  1. National Key Laboratory for Electronic Measurement Technology, North University of China, Taiyuan, 030051, Shanxi, People’s Republic of China

    Gui-Quan Sun

  2. School of Mechatronic Engineering, North University of China, Taiyuan, 030051, Shanxi, People’s Republic of China

    Gui-Quan Sun & Zhen Jin

  3. Department of Mathematics, North University of China, Taiyuan, 030051, Shanxi, People’s Republic of China

    Gui-Quan Sun, Zhen Jin, Li Li & Quan-Xing Liu

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  1. Gui-Quan Sun
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  2. Zhen Jin
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Correspondence to Zhen Jin.

Appendix: More detail about noise term η(r, t)

Appendix: More detail about noise term η(r, t)

The Ornstein–Uhlenbeck process obeys the following stochastic partial differential equation:

$$ \frac{\partial \eta(r,t)}{\partial t}=-\frac{1}{\tau}\eta(r,t)+\frac{1}{\tau}\xi(r,t), \label{tur1} $$
(10)

where ξ(r,t) is a Gaussian white noise with zero mean and correlation,

$$ \langle \xi(r,t) \rangle=0, $$
(11a)
$$ \left\langle \xi(r,t)\xi\left(r',t'\right) \right\rangle=2\varepsilon \delta \left(r-r'\right)\delta \left(t-t'\right). $$
(11b)

As a result, the colored noise η(r,t), which is temporally correlated and white in space, satisfies

$$ \left\langle \eta(r,t)\eta\left(r',t'\right) \right\rangle=\frac{\varepsilon}{\tau}\exp\left(-\frac{|t-t'|}{\tau}\right)\delta\left(r-r'\right), $$
(12)

where τ controls the temporal correlation, and ε measures the noise intensity.

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Sun, GQ., Jin, Z., Li, L. et al. The role of noise in a predator–prey model with Allee effect. J Biol Phys 35, 185–196 (2009). https://doi.org/10.1007/s10867-009-9139-y

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