Transition of interaction outcomes in a facilitation-competition system of two species

Yuanshi Wang; Hong Wu; Yuanshi Wang; Hong Wu

doi:10.3934/mbe.2017076

Mathematical Biosciences and Engineering

2017, Volume 14, Issue 5&6: 1463-1475. doi: 10.3934/mbe.2017076

Previous Article Next Article

Transition of interaction outcomes in a facilitation-competition system of two species

Yuanshi Wang ,
Hong Wu

School of Mathematics, Sun Yat-sen University, Guangzhou 510275, China

Received: 27 June 2016 Accepted: 05 October 2016 Published: 01 October 2017
MSC : 34C37, 92D25, 37N25

A facilitation-competition system of two species is considered, where one species has a facilitation effect on the other but there is spatial competition between them. Our aim is to show mechanism by which the facilitation promotes coexistence of the species. A lattice gas model describing the facilitation-competition system is analyzed, in which nonexistence of periodic solution is shown and previous results are extended. Global dynamics of the model demonstrate essential features of the facilitation-competition system. When a species cannot survive alone, a strong facilitation from the other would lead to its survival. However, if the facilitation is extremely strong, both species go extinct. When a species can survive alone and its mortality rate is not larger than that of the other species, it would drive the other one into extinction. When a species can survive alone and its mortality rate is larger than that of the other species, it would be driven into extinction if the facilitation from the other is weak, while it would coexist with the other if the facilitation is strong. Moreover, an extremely strong facilitation from the other would lead to extinction of species. Bifurcation diagram of the system exhibits that interaction outcome between the species can transition between competition, amensalism, neutralism and parasitism in a smooth fashion. A novel result of this paper is the rigorous and thorough analysis, which displays transparency of dynamics in the system. Numerical simulations validate the results.
- Persistence,
- coexistence,
- extinction,
- stability,
- bifurcation
Citation: Yuanshi Wang, Hong Wu. Transition of interaction outcomes in a facilitation-competition system of two species[J]. Mathematical Biosciences and Engineering, 2017, 14(5&6): 1463-1475. doi: 10.3934/mbe.2017076

Related Papers:

Abstract

A facilitation-competition system of two species is considered, where one species has a facilitation effect on the other but there is spatial competition between them. Our aim is to show mechanism by which the facilitation promotes coexistence of the species. A lattice gas model describing the facilitation-competition system is analyzed, in which nonexistence of periodic solution is shown and previous results are extended. Global dynamics of the model demonstrate essential features of the facilitation-competition system. When a species cannot survive alone, a strong facilitation from the other would lead to its survival. However, if the facilitation is extremely strong, both species go extinct. When a species can survive alone and its mortality rate is not larger than that of the other species, it would drive the other one into extinction. When a species can survive alone and its mortality rate is larger than that of the other species, it would be driven into extinction if the facilitation from the other is weak, while it would coexist with the other if the facilitation is strong. Moreover, an extremely strong facilitation from the other would lead to extinction of species. Bifurcation diagram of the system exhibits that interaction outcome between the species can transition between competition, amensalism, neutralism and parasitism in a smooth fashion. A novel result of this paper is the rigorous and thorough analysis, which displays transparency of dynamics in the system. Numerical simulations validate the results.

References

[1]	[ M. Hernandez,I. Barradas, Variation in the outcome of population interactions: Bifurcations and catastrophes, J. Math. Biol., 46 (2003): 571-594.
[2]	[ J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge, UK, 1998.
[3]	[ T. Kawai,M. Tokeshi, Variable modes of facilitation in the upper intertidal: Goose barnacles and mussels, Marine Ecology Progress Series, 272 (2004): 203-213.
[4]	[ T. Kawai,M. Tokeshi, Asymmetric coexistence: Bidirectional abiotic and biotic effects between goose barnacles and mussels, Journal of Animal Ecology, 75 (2006): 928-941.
[5]	[ T. Kawai,M. Tokeshi, Testing the facilitation-competition paradigm under the stress-gradient hypothesis: Decoupling multiple stress factors, Proceedings of the Royal Society B: Biological Sciences, 274 (2007): 2503-2508.
[6]	[ X. Li,H. Wang,Y. Kuang, Global analysis of a stoichiometric producer-grazer model with Holling type functional responses, J. Math. Biol., 63 (2011): 901-932.
[7]	[ Z. Liu,P. Magal,S. Ruan, Oscillations in age-structured models of consumer-resource mutualisms, Disc. Cont. Dyna. Systems-B, 21 (2016): 537-555.
[8]	[ C. Neuhauser,J. Fargione, A mutualism-parasitism continuum model and its application to plant-mycorrhizae interactions, Ecological Modelling, 177 (2004): 337-352.
[9]	[ S. Soliveres,C. Smit,F. T. Maestre, Moving forward on facilitation research: Response to changing environments and effects on the diversity, functioning and evolution of plant communities, Biological Reviews., 90 (2015): 297-313.
[10]	[ K. Tainaka, Stationary pattern of vortices or strings in biological systems: lattice version of the Lotka-Volterra model, Physical Review Letters, 63 (1989): 2688-2691.
[11]	[ Y. Wang,H. Wu,J. Liang, Dynamics of a lattice gas system of three species, Commun. Non. Sci. Nume. Simu., 39 (2016): 38-57.
[12]	[ H. Yokoi,T. Uehara,T. Kawai,Y. Tateoka,K. Tainaka, Lattice and lattice gas models for commensalism: Two shellfishes in intertidal Zone, Scientific Research Publishing, Open Journal of Ecology, 4 (2014): 671-677.

Reader Comments

Your name:*

Email:*
© 2017 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)