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

- 194 Downloads

## Abstract

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.

## Keywords

Principle of competitive exclusion Cooperation Global stability Bifurcation Coexistence## Mathematics Subject Classification

34C12 37N25 34C28 37G20## Notes

### Acknowledgements

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.

## References

- Albrecht M, Padrn B, Bartomeus I, Traveset A (2014) Consequences of plant invasions on compartmentalization and species’ roles in plant–pollinator networks. Proc R Soc Lond B 281:20140773CrossRefGoogle Scholar
- Bascompte J, Jordano P, Melián CJ, Olesen JM (2003) The nested assembly of plant–animal mutualistic networks. Proc Natl Acad Sci 100:9383–987CrossRefGoogle Scholar
- Bronstein JL (1994) Conditional outcomes in mutualistic interactions. Trends Ecol Evol 9:214–217CrossRefGoogle Scholar
- Campbell C, Yang S, Albert R, Shea K (2015) Plant–pollinator community network response to species invasion depends on both invader and community characteristics. Oikos 124:406–413CrossRefGoogle Scholar
- Cantrell RS, Cosner C, Ruan S (2004) Intraspecific interference and consumer–resource dynamics. Discrete Cont Dyn Syst B 4:527–546MathSciNetCrossRefzbMATHGoogle Scholar
- Fagan WF, Bewick S, Cantrell S, Cosner C, Varassin IG, Inouye DW (2014) Phenologically explicitmodelsforstudyingplant–pollinator interactions under climate change. Theor Ecol 7:289–297CrossRefGoogle Scholar
- Freedman HI, Waltman P (1984) Persistence in models of three interacting predator–prey populations. Math Bios 68:213–231MathSciNetCrossRefzbMATHGoogle Scholar
- Gilbert LE (1980) Food web organization and conservation of neotropical diversity. In: Soule Michael, Wilcox Bruce A (eds) Conservation biology. An evolutionary-ecological perspective. Sinauer Associates, Sunderland, pp 11–34Google Scholar
- Hale JK (1969) Ordinary differential equations. Wiley, New YorkzbMATHGoogle Scholar
- Hofbauer J, Sigmund K (1998) Evolutionary games and population dynamics. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
- Hubbell SP (2001) The unified neutral theory of biodiversity and biogeography. Princeton University PressGoogle Scholar
- Jang SR-J (2002) Dynamics of herbivore–plant–pollinator models. J Math Biol 44:129–149MathSciNetCrossRefzbMATHGoogle Scholar
- Li B, Smith HL (2007) Global dynamics of microbial competition for two resources with internal storage competition model. J Math Biol 55:481–511MathSciNetCrossRefzbMATHGoogle Scholar
- Llibre J, Xiao D (2014) Global dynamics of a Lotka–Volterra model with two predators competing for one prey. SIAM J Appl Math 74:434–453MathSciNetCrossRefzbMATHGoogle Scholar
- Lundberg S, Ingvarsson P (1998) Population dynamics of resource limited plants and their pollinators. Theor Popul Biol 54:44–49CrossRefzbMATHGoogle Scholar
- Memmott J, Waser NM, Price MV (2004) Tolerance of pollination networks to species extinctions. Proc R Soc Lond B 271:2605–2611CrossRefGoogle Scholar
- Nguyen DH, Yin G (2017) Coexistence and exclusion of stochastic competitive Lotka–Volterra models. J Differ Equ 262:1192–1225MathSciNetCrossRefzbMATHGoogle Scholar
- Oleson JM, Bascompte J, Dupont YL, Jordano P (2007) The modularity of pollination networks. Proc Natl Acad Sci 104:19891–19896CrossRefGoogle Scholar
- Perko L (2001) Differential equations and dynamical systems. Springer, New YorkCrossRefzbMATHGoogle Scholar
- Revilla T (2015) Numerical responses in resource-based mutualisms: a time scale approach. J Theor Biol 378:39–46MathSciNetCrossRefzbMATHGoogle Scholar
- Sakavara A, Tsirtsis G, Roelke DL, Nancy R, Spatharis S (2018) Lumpy species coexistence arises robustly in fluctuating resource environments. Proc Natl Acad Sci 115:738–743CrossRefGoogle Scholar
- Scheffer AM, van Nes EH (2006) Self-organized similarity, the evolutionary emergence of groups of similar species. Proc Natl Acad Sci 103:6230–6235CrossRefGoogle Scholar
- Scheffer M, van Nes EH, Vergnon R (2018) Toward a unifying theory of biodiversity. Proc Natl Acad Sci 115:639–641CrossRefGoogle Scholar
- Terry I (2001) Thrips and weevils as dual, specialist pollinators of the Australian cycad
*Macrozamia communis*(Zamiaceae). Int J Plant Sci 162:1293–1305CrossRefGoogle Scholar - Tilman D (1982) Resource competition and community structure. Princeton University Press, PrincetonGoogle Scholar
- Traveset A, Richardson DM (2006) Biological invasions as disrupters of plant reproductive mutualisms. Trends Ecol Evol 21:208–215CrossRefGoogle Scholar
- Vanbergen AJ (2013) Threats to an ecosystem service: pressures on pollinators. Front Ecol Environ 11:251–259CrossRefGoogle Scholar
- Vanbergen AJ, Woodcock BA, Heard MS, Chapman DS (2017) Network size, structure and mutualisms dependence affect the propensity for plant–pollinator extinction cascades. Funct Ecol. https://doi.org/10.1111/1365-2435.12823 Google Scholar
- Wang Y, DeAngelis DL (2016) Stability of an intraguild predation system with mutual predation. Commun Nonlinear Sci Numer Simul 33:141–159MathSciNetCrossRefGoogle Scholar