Effects of a protection zone in a reaction-advection-diffusion model with strong Allee effect

Davis Henderson; Bingtuan Li; Davis Henderson; Bingtuan Li

doi:10.3934/mbe.2026057

Mathematical Biosciences and Engineering

2026, Volume 23, Issue 6: 1572-1595. doi: 10.3934/mbe.2026057

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Effects of a protection zone in a reaction-advection-diffusion model with strong Allee effect

Davis Henderson ¹,
Bingtuan Li ^{2,
,
,}

1.
Department of Mathematics, Indiana University, Bloomington, IN 47404, USA
2.
Department of Mathematics, University of Louisville, Louisville, KY 40292, USA
Dedicated to Professor Yang Kuang on the occasion of his 60th birthday

Received: 31 August 2025 Revised: 07 February 2026 Accepted: 09 February 2026 Published: 21 April 2026

We consider a reaction-advection-diffusion equation that models a population in a bounded habitat with a strong Allee effect and a protection zone. We assume that the population growth function exhibits the strong Allee effect and has a positive integral within the protection zone (indicating population persistence) and a negative integral in the surrounding patches (indicating population decay). We prove that the existence of positive steady-state solutions to this system depends on the length of the protection zone. It is demonstrated that there exists a threshold value $ H^* $ such that, for a protection zone of size $ H^* $, there exists one positive steady-state solution and, for a larger protection zone, there exist multiple positive steady-state solutions. For smaller protection zones, we prove there exists no positive steady-state solution. The dynamics of the equation are further examined via numerical simulations.
- reaction-advection-diffusion equation,
- protection zone,
- Allee effect,
- steady-state solution,
- persistence
Citation: Davis Henderson, Bingtuan Li. Effects of a protection zone in a reaction-advection-diffusion model with strong Allee effect[J]. Mathematical Biosciences and Engineering, 2026, 23(6): 1572-1595. doi: 10.3934/mbe.2026057

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Abstract

We consider a reaction-advection-diffusion equation that models a population in a bounded habitat with a strong Allee effect and a protection zone. We assume that the population growth function exhibits the strong Allee effect and has a positive integral within the protection zone (indicating population persistence) and a negative integral in the surrounding patches (indicating population decay). We prove that the existence of positive steady-state solutions to this system depends on the length of the protection zone. It is demonstrated that there exists a threshold value $ H^* $ such that, for a protection zone of size $ H^* $, there exists one positive steady-state solution and, for a larger protection zone, there exist multiple positive steady-state solutions. For smaller protection zones, we prove there exists no positive steady-state solution. The dynamics of the equation are further examined via numerical simulations.

References

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