Bulletin of Mathematical Biology

, Volume 81, Issue 6, pp 1665–1686 | Cite as

Asymptotic State of a Two-Patch System with Infinite Diffusion

  • Yuanshi WangEmail author


Mathematical theory has predicted that populations diffusing in heterogeneous environments can reach larger total size than when not diffusing. This prediction was tested in a recent experiment, which leads to extension of the previous theory to consumer-resource systems with external resource input. This paper studies a two-patch model with diffusion that characterizes the experiment. Solutions of the model are shown to be nonnegative and bounded, and global dynamics of the subsystems are completely exhibited. It is shown that there exist stable positive equilibria as the diffusion rate is large, and the equilibria converge to a unique positive point as the diffusion tends to infinity. Rigorous analysis on the model demonstrates that homogeneously distributed resources support larger carrying capacity than heterogeneously distributed resources with or without diffusion, which coincides with experimental observations but refutes previous theory. It is shown that spatial diffusion increases total equilibrium population abundance in heterogeneous environments, which coincides with real data and previous theory while a new insight is exhibited. A novel prediction of this work is that these results hold even with source–sink populations and increasing diffusion rate of consumer could change its persistence to extinction in the same-resource environments.


Consumer-resource model Spatially distributed population Diffusion Uniform persistence Liapunov stability 

Mathematics Subject Classification

34C12 37N25 34C28 37G20 



I would like to thank the anonymous reviewer for the helpful comments on the manuscript. This work was supported by NSF of P.R. China (11571382).


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Copyright information

© Society for Mathematical Biology 2019

Authors and Affiliations

  1. 1.School of MathematicsSun Yat-sen UniversityGuangzhouPeople’s Republic of China

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