Hopf bifurcation analysis in a reaction-diffusion-advection model with strong Allee effect and delay

Yuying Liu; Xin Wei; Yuying Liu; Xin Wei

doi:10.3934/math.2026476

AIMS Mathematics

2026, Volume 11, Issue 4: 11559-11579. doi: 10.3934/math.2026476

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Hopf bifurcation analysis in a reaction-diffusion-advection model with strong Allee effect and delay

Yuying Liu ^{1,2
,
,},
Xin Wei ³

1.
School of Mathematics, China University of Mining and Technology, Xuzhou, Jiangsu 221116, China
2.
Jiangsu Center for Applied Mathematics at CUMT, Xuzhou, Jiangsu 221116, China
3.
School of Mathematical Sciences, Heilongjiang University, Harbin, Heilongjiang 150001, China

Received: 27 February 2026 Revised: 02 April 2026 Accepted: 10 April 2026 Published: 27 April 2026
MSC : 35K57, 37G15

This paper investigates a delayed population model incorporating advection and strong Allee effect. First, we prove the well-posedness of the solutions in the model. The effect of the advection rate on the dynamics of the population is examined. Analysis indicates that under the given conditions, a larger advection rate can stabilize the equilibrium of the model. Second, by adopting delay as the varying parameter, the Hopf bifurcation of the model is studied. Third, the normal form in the vicinity of the Hopf bifurcation singularity is calculated by adopting a weighted inner product. The reliability of the conclusion is then verified by means of the numerical simulations. Research shows that under specific conditions, there exists a sequence of Hopf bifurcation singularities in the system.
- Allee effect,
- reaction-diffusion-advection,
- normal form,
- Hopf bifurcation
Citation: Yuying Liu, Xin Wei. Hopf bifurcation analysis in a reaction-diffusion-advection model with strong Allee effect and delay[J]. AIMS Mathematics, 2026, 11(4): 11559-11579. doi: 10.3934/math.2026476

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Abstract

This paper investigates a delayed population model incorporating advection and strong Allee effect. First, we prove the well-posedness of the solutions in the model. The effect of the advection rate on the dynamics of the population is examined. Analysis indicates that under the given conditions, a larger advection rate can stabilize the equilibrium of the model. Second, by adopting delay as the varying parameter, the Hopf bifurcation of the model is studied. Third, the normal form in the vicinity of the Hopf bifurcation singularity is calculated by adopting a weighted inner product. The reliability of the conclusion is then verified by means of the numerical simulations. Research shows that under specific conditions, there exists a sequence of Hopf bifurcation singularities in the system.

References

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