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Steady-state bifurcations of a diffusive–advective predator–prey system with hostile boundary conditions and spatial heterogeneity

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Abstract

In this paper, we consider a diffusive–advective predator–prey system in a spatially heterogeneous environment subject to a hostile boundary condition, where the interaction term is governed by a Holling type II functional response. We investigate the existence and global attractivity of both trivial and semi-trivial steady-state solutions and the existence and local stability of coexistence steady-state solutions, depending on the size of a key principal eigenvalue. In addition, we show that the effect of advection on the principal eigenvalue is monotonic for small advection rates, depending on the concavity of the resource distribution. For arbitrary advection rates, we consider two explicit resource distributions for which we can say precisely the behaviour of the principal eigenvalue as it depends on advection, highlighting that advection can either improve or impair a population’s ability to persist, depending on the characteristics of the resource distribution. We present some numerical simulations to demonstrate the outcomes as they depend on the advection rates for the full predator–prey system. These insights highlight the intimate relationship between environmental heterogeneity, directed movement, and the hostile boundary. The methods employed include upper and lower solution techniques, bifurcation theory, spectral analysis, and the comparison principle.

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Acknowledgements

The work of the first author and the fourth author was partially supported by the National Natural Science Foundation of China (No. 12371165). The work of the second author was supported by an NSERC Postdoctoral Fellowship (NSERC Grant PDF-578181-2023). The work of the third author was partially supported by the Natural Sciences and Engineering Research Council of Canada (Individual Discovery Grant RGPIN-2020-03911 and Discovery Accelerator Supplement Award RGPAS-2020-00090).

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LD and JWH wrote the main manuscript text; SY and WH prepared figures 1, 2, 3 , examples and discussion. All authors reviewed the manuscript.

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Correspondence to Weihua Jiang.

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Liu, D., Salmaniw, Y., Wang, H. et al. Steady-state bifurcations of a diffusive–advective predator–prey system with hostile boundary conditions and spatial heterogeneity. Z. Angew. Math. Phys. 75, 124 (2024). https://doi.org/10.1007/s00033-024-02267-y

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