Journal of Mathematical Biology

, Volume 78, Issue 6, pp 1605–1636 | Cite as

Dynamics of a consumer–resource reaction–diffusion model

Homogeneous versus heterogeneous environments
  • Xiaoqing He
  • King-Yeung LamEmail author
  • Yuan Lou
  • Wei-Ming Ni


We study the dynamics of a consumer–resource reaction–diffusion model, proposed recently by Zhang et al. (Ecol Lett 20(9):1118–1128, 2017), in both homogeneous and heterogeneous environments. For homogeneous environments we establish the global stability of constant steady states. For heterogeneous environments we study the existence and stability of positive steady states and the persistence of time-dependent solutions. Our results illustrate that for heterogeneous environments there are some parameter regions in which the resources are only partially limited in space, a unique feature which does not occur in homogeneous environments. Such difference between homogeneous and heterogeneous environments seems to be closely connected with a recent finding by Zhang et al. (2017), which says that in consumer–resource models, homogeneously distributed resources could support higher population abundance than heterogeneously distributed resources. This is opposite to the prediction by Lou (J Differ Equ 223(2):400–426, 2006. for logistic-type models. For both small and high yield rates, we also show that when a consumer exists in a region with a heterogeneously distributed input of exploitable renewed limiting resources, the total population abundance at equilibrium can reach a greater abundance when it diffuses than when it does not. In contrast, such phenomenon may fail for intermediate yield rates.


Spatial heterogeneity Global asymptotic stability Consumer–resource model Reaction–diffusion equations 

Mathematics Subject Classification

92D25 92D40 35K57 35B40 



We sincerely thank the anonymous referees for their comments which help improve the presentation of the paper. The research of X. He is supported in part by NSFC (11601155), Recruitment Program of Global Experts in China, and Science and Technology Commission of Shanghai Municipality (No. 18dz2271000); the research of K.-Y. Lam and Y. Lou is partially supported by NSF Grant DMS-1411476; the research of W.-M. Ni is partially supported by the Presidential Chair Fund at CUHK (SZ), NSF Grants DMS-1210400 and DMS-1714487, and NSFC Grant Nos. 11571363, 11431005. Part of the research was carried out while K.-Y. Lam was visiting the Center for PDE, ECNU and while Y. Lou was visiting the NCTS at the National Taiwan University.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Center for PDE, School of Mathematical Sciences and Shanghai Key Laboratory of PMMPEast China Normal UniversityShanghaiChina
  2. 2.Department of MathematicsThe Ohio State UniversityColumbusUSA
  3. 3.Chinese University of Hong Kong - ShenzhenShenzhenChina
  4. 4.School of MathematicsUniversity of MinnesotaMinneapolisUSA

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