Journal of Mathematical Biology

, Volume 78, Issue 3, pp 683–710 | Cite as

Global dynamics of a mutualism–competition model with one resource and multiple consumers

  • Yuanshi Wang
  • Hong WuEmail author
  • Donald L. DeAngelis


Recent simulation modeling has shown that species can coevolve toward clusters of coexisting consumers exploiting the same limiting resource or resources, with nearly identical ratios of coefficients related to growth and mortality. This paper provides a mathematical basis for such as situation; a full analysis of the global dynamics of a new model for such a class of n-dimensional consumer–resource system, in which a set of consumers with identical growth to mortality ratios compete for the same resource and in which each consumer is mutualistic with the resource. First, we study the system of one resource and two consumers. By theoretical analysis, we demonstrate the expected result that competitive exclusion of one of the consumers can occur when the growth to mortality ratios differ. However, when these ratios are identical, the outcomes are complex. Either equilibrium coexistence or mutual extinction can occur, depending on initial conditions. When there is coexistence, interaction outcomes between the consumers can transition between effective mutualism, parasitism, competition, amensalism and neutralism. We generalize to the global dynamics of a system of one resource and multiple consumers. Changes in one factor, either a parameter or initial density, can determine whether all of the consumers either coexist or go to extinction together. New results are presented showing that multiple competing consumers can coexist on a single resource when they have coevolved toward identical growth to mortality ratios. This coexistence can occur because of feedbacks created by all of the consumers providing a mutualistic service to the resource. This is biologically relevant to the persistence of pollination–mutualisms.


Principle of competitive exclusion Cooperation Global stability Bifurcation Coexistence 

Mathematics Subject Classification

34C12 37N25 34C28 37G20 



We would like to thank the anonymous reviewers for their helpful comments on the manuscript. Yuanshi Wang and Hong Wu were supported by NSF of People’s Republic of China (11571382). D. L. DeAngelis acknowledges the support of the US Geological Survey’s Greater Everglades Priority Ecosystem Sciences program. Any use of trade, firm, or product names is for descriptive purposes only and does not imply endorsement by the US Government.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of MathematicsSun Yat-sen UniversityGuangzhouPeople’s Republic of China
  2. 2.U.S. Geological Survey, Wetland and Aquatic Research CenterGainesvilleUSA

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