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Transition of interaction outcomes in a facilitationcompetition system of two species
School of Mathematics, Sun Yatsen University, Guangzhou 510275, China 
A facilitationcompetition 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 facilitationcompetition 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 facilitationcompetition 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.
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K. Tainaka, Stationary pattern of vortices or strings in biological systems: lattice version of the LotkaVolterra model, Physical Review Letters, 63 (1989), 26882691. doi: 10.1103/PhysRevLett.63.2688. 
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H. Yokoi, T. Uehara, T. Kawai, Y. Tateoka and K. Tainaka, Lattice and lattice gas models for commensalism: Two shellfishes in intertidal Zone, Scientific Research Publishing, Open Journal of Ecology, 4 (2014), 671677. doi: 10.4236/oje.2014.411057. 
show all references
References:
[1] 
M. Hernandez and I. Barradas, Variation in the outcome of population interactions: Bifurcations and catastrophes, J. Math. Biol., 46 (2003), 571594. doi: 10.1007/s0028500201924. 
[2] 
J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge, UK, 1998. doi: 10.1017/CBO9781139173179. 
[3] 
T. Kawai and M. Tokeshi, Variable modes of facilitation in the upper intertidal: Goose barnacles and mussels, Marine Ecology Progress Series, 272 (2004), 203213. doi: 10.3354/meps272203. 
[4] 
T. Kawai and M. Tokeshi, Asymmetric coexistence: Bidirectional abiotic and biotic effects between goose barnacles and mussels, Journal of Animal Ecology, 75 (2006), 928941. doi: 10.1111/j.13652656.2006.01111.x. 
[5] 
T. Kawai and M. Tokeshi, Testing the facilitationcompetition paradigm under the stressgradient hypothesis: Decoupling multiple stress factors, Proceedings of the Royal Society B: Biological Sciences, 274 (2007), 25032508. doi: 10.1098/rspb.2007.0871. 
[6] 
X. Li, H. Wang and Y. Kuang, Global analysis of a stoichiometric producergrazer model with Holling type functional responses, J. Math. Biol., 63 (2011), 901932. doi: 10.1007/s0028501003922. 
[7] 
Z. Liu, P. Magal and S. Ruan, Oscillations in agestructured models of consumerresource mutualisms, Disc. Cont. Dyna. SystemsB, 21 (2016), 537555. doi: 10.3934/dcdsb.2016.21.537. 
[8] 
C. Neuhauser and J. Fargione, A mutualismparasitism continuum model and its application to plantmycorrhizae interactions, Ecological Modelling, 177 (2004), 337352. doi: 10.1016/j.ecolmodel.2004.02.010. 
[9] 
S. Soliveres, C. Smit and 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), 297313. doi: 10.1111/brv.12110. 
[10] 
K. Tainaka, Stationary pattern of vortices or strings in biological systems: lattice version of the LotkaVolterra model, Physical Review Letters, 63 (1989), 26882691. doi: 10.1103/PhysRevLett.63.2688. 
[11] 
Y. Wang, H. Wu and J. Liang, Dynamics of a lattice gas system of three species, Commun. Non. Sci. Nume. Simu., 39 (2016), 3857. doi: 10.1016/j.cnsns.2016.02.027. 
[12] 
H. Yokoi, T. Uehara, T. Kawai, Y. Tateoka and K. Tainaka, Lattice and lattice gas models for commensalism: Two shellfishes in intertidal Zone, Scientific Research Publishing, Open Journal of Ecology, 4 (2014), 671677. doi: 10.4236/oje.2014.411057. 
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