American Institute of Mathematical Sciences

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

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

Yuanshi Wang and  Hong Wu

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.

Keywords: Persistence, coexistence, extinction, stability, bifurcation.
Mathematics Subject Classification: 34C37, 92D25, 37N25.
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:
[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

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), 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

Figure 1.  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
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  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
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  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
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  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
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Sebastian J. Schreiber. On persistence and extinction for randomly perturbed dynamical systems. Discrete & Continuous Dynamical Systems - B, 2007, 7 (2) : 457-463. doi: 10.3934/dcdsb.2007.7.457
[2]

Kousuke Kuto. Stability and Hopf bifurcation of coexistence steady-states to an SKT model in spatially heterogeneous environment. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 489-509. doi: 10.3934/dcds.2009.24.489
[3]

Georg Hetzer, Tung Nguyen, Wenxian Shen. Coexistence and extinction in the Volterra-Lotka competition model with nonlocal dispersal. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1699-1722. doi: 10.3934/cpaa.2012.11.1699
[4]

Wen Jin, Horst R. Thieme. Persistence and extinction of diffusing populations with two sexes and short reproductive season. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3209-3218. doi: 10.3934/dcdsb.2014.19.3209
[5]

Keng Deng, Yixiang Wu. Extinction and uniform strong persistence of a size-structured population model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 831-840. doi: 10.3934/dcdsb.2017041
[6]

Wen Jin, Horst R. Thieme. An extinction/persistence threshold for sexually reproducing populations: The cone spectral radius. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 447-470. doi: 10.3934/dcdsb.2016.21.447
[7]

Guohong Zhang, Xiaoli Wang. Extinction and coexistence of species for a diffusive intraguild predation model with B-D functional response. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3755-3786. doi: 10.3934/dcdsb.2018076
[8]

Tung Nguyen, Nar Rawal. Coexistence and extinction in Time-Periodic Volterra-Lotka type systems with nonlocal dispersal. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3799-3816. doi: 10.3934/dcdsb.2018080
[9]

Guihong Fan, Yijun Lou, Horst R. Thieme, Jianhong Wu. Stability and persistence in ODE models for populations with many stages. Mathematical Biosciences & Engineering, 2015, 12 (4) : 661-686. doi: 10.3934/mbe.2015.12.661
[10]

Wei Feng, Michael T. Cowen, Xin Lu. Coexistence and asymptotic stability in stage-structured predator-prey models. Mathematical Biosciences & Engineering, 2014, 11 (4) : 823-839. doi: 10.3934/mbe.2014.11.823
[11]

Hal L. Smith, Horst R. Thieme. Persistence and global stability for a class of discrete time structured population models. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4627-4646. doi: 10.3934/dcds.2013.33.4627
[12]

Chunqing Wu, Patricia J.Y. Wong. Global asymptotical stability of the coexistence fixed point of a Ricker-type competitive model. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 3255-3266. doi: 10.3934/dcdsb.2015.20.3255
[13]

Shubo Zhao, Ping Liu, Mingchao Jiang. Stability and bifurcation analysis in a chemotaxis bistable growth system. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1165-1174. doi: 10.3934/dcdss.2017063
[14]

Antoine Perasso. Global stability and uniform persistence for an infection load-structured SI model with exponential growth velocity. Communications on Pure & Applied Analysis, 2019, 18 (1) : 15-32. doi: 10.3934/cpaa.2019002
[15]

Kazuo Yamazaki, Xueying Wang. Global stability and uniform persistence of the reaction-convection-diffusion cholera epidemic model. Mathematical Biosciences & Engineering, 2017, 14 (2) : 559-579. doi: 10.3934/mbe.2017033
[16]

Rui Hu, Yuan Yuan. Stability, bifurcation analysis in a neural network model with delay and diffusion. Conference Publications, 2009, 2009 (Special) : 367-376. doi: 10.3934/proc.2009.2009.367
[17]

Xiaomei Feng, Zhidong Teng, Kai Wang, Fengqin Zhang. Backward bifurcation and global stability in an epidemic model with treatment and vaccination. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 999-1025. doi: 10.3934/dcdsb.2014.19.999
[18]

Fabien Crauste. Global Asymptotic Stability and Hopf Bifurcation for a Blood Cell Production Model. Mathematical Biosciences & Engineering, 2006, 3 (2) : 325-346. doi: 10.3934/mbe.2006.3.325
[19]

Hui Miao, Zhidong Teng, Chengjun Kang. Stability and Hopf bifurcation of an HIV infection model with saturation incidence and two delays. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2365-2387. doi: 10.3934/dcdsb.2017121
[20]

Jungho Park. Bifurcation and stability of the generalized complex Ginzburg--Landau equation. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1237-1253. doi: 10.3934/cpaa.2008.7.1237

2018 Impact Factor: 1.313

Tools

Metrics

  • PDF downloads (20)
  • HTML views (71)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences