# American Institute of Mathematical Sciences

October  2017, 14(5&6): 1463-1475. doi: 10.3934/mbe.2017076

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

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

Received  June 27, 2016 Accepted  October 05, 2016 Published  May 2017

Fund Project: Y. Wang acknowledges support from NSFC of P.R. China (No. 11571382)

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.

Citation: Yuanshi Wang, Hong Wu. Transition of interaction outcomes in a facilitation-competition system of two species. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1463-1475. doi: 10.3934/mbe.2017076
##### References:
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##### References:
 [1] M. Hernandez and I. Barradas, Variation in the outcome of population interactions: Bifurcations and catastrophes, J. Math. Biol., 46 (2003), 571-594. doi: 10.1007/s00285-002-0192-4. Google Scholar [2] J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge, UK, 1998. doi: 10.1017/CBO9781139173179. Google Scholar [3] T. Kawai and M. Tokeshi, Variable modes of facilitation in the upper intertidal: Goose barnacles and mussels, Marine Ecology Progress Series, 272 (2004), 203-213. doi: 10.3354/meps272203. Google Scholar [4] T. Kawai and M. Tokeshi, Asymmetric coexistence: Bidirectional abiotic and biotic effects between goose barnacles and mussels, Journal of Animal Ecology, 75 (2006), 928-941. doi: 10.1111/j.1365-2656.2006.01111.x. Google Scholar [5] T. Kawai and 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. doi: 10.1098/rspb.2007.0871. Google Scholar [6] X. Li, H. Wang and Y. Kuang, Global analysis of a stoichiometric producer-grazer model with Holling type functional responses, J. Math. Biol., 63 (2011), 901-932. doi: 10.1007/s00285-010-0392-2. Google Scholar [7] Z. Liu, P. Magal and S. Ruan, Oscillations in age-structured models of consumer-resource mutualisms, Disc. Cont. Dyna. Systems-B, 21 (2016), 537-555. doi: 10.3934/dcdsb.2016.21.537. Google Scholar [8] C. Neuhauser and J. Fargione, A mutualism-parasitism continuum model and its application to plant-mycorrhizae interactions, Ecological Modelling, 177 (2004), 337-352. doi: 10.1016/j.ecolmodel.2004.02.010. Google Scholar [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), 297-313. doi: 10.1111/brv.12110. Google Scholar [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. doi: 10.1103/PhysRevLett.63.2688. Google Scholar [11] Y. Wang, H. Wu and J. Liang, Dynamics of a lattice gas system of three species, Commun. Non. Sci. Nume. Simu., 39 (2016), 38-57. doi: 10.1016/j.cnsns.2016.02.027. Google Scholar [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), 671-677. doi: 10.4236/oje.2014.411057. Google Scholar
Phase-plane diagrams of (4) with $d_1\ge 1$. Red and blue lines are the isoclines for $x$ and $y$, respectively. Grey arrows display the direction and strength of the vector fields in the phase-plane space. Fix $r_1=r_2=1, d_1=1, d_2=0.1$ and let the facilitation $c$ vary. (a-b) When $c(=6, 10)$ is small, species $X$ goes to extinction and $Y$ approaches its carrying capacity. (c) When $c(=15)$ is large, the species coexist. (d) When $c(=90)$ is very large, the species coexist and species $Y$ approaches a density extremely less than its carrying capacity
Phase-plane diagrams of (4) with $d_1< 1$. Red and blue lines are the isoclines for $x$ and $y$, respectively. Grey arrows display the direction and strength of the vector fields in the phase-plane space. Fix $r_1=r_2=1, d_2=0.1$ and let $d_1$ and $c$ vary. (a-b) When $d_1=0.08, 0.1$ and $c=1$, species $Y$ goes to extinction and $X$ approaches its carrying capacity. (c) When $d_1=0.2$ and $c=1$, species $X$ goes to extinction and $Y$ approaches its carrying capacity. (d) When $d_1=0.2$ and $c=20$, the species coexist
Phase-plane diagrams of (4) when the mortality $d_1$ varies. Red and blue lines are the isoclines for $x$ and $y$, respectively. Grey arrows display the direction and strength of the vector fields in the phase-plane space. Fix $r_1=r_2=1, d_2=0.1, c=20$. (a) When $d_1(=2)$ is large, species $X$ goes to extinction and $Y$ approaches its carrying capacity. (b) When $d_1(=1.1)$ is intermediate, the species coexist. (c) When $d_1(=0.2)$ is small, the species coexist and species $Y$ approaches a density extremely less than its carrying capacity. (d) When $d_1(=0.08)$ is extremely small, species $Y$ goes to extinction and $X$ approaches its carrying capacity
Bifurcation diagram of system (4) on the $d_1-c$ plane. Fix $r_1=r_2=1, d_2=0.2$. Then lines $d_1=0.2, d_1=1.0, c=6.25 (d_1-0.2)$ and $c=5$ divide the first quadrant into 6 regions. In the region $d_1\le 0.2$, the interaction outcome remains amensalism $(0~-)$. In the regions $0.2< d_1< 1.0$, the interaction outcome changes from amensalism $(-~0)$, to competition $(-~-)$, to the other amensalism $(-~0)$ and to parasitism $(+~-)$ in a smooth fashion when the facilitation $c$ increases. Similarly, in the regions $d_1\ge 1.0$, the interaction outcome changes from neutralism $(0~0)$ to parasitism $(+~-)$ in a smooth fashion when the facilitation $c$ increases
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