• Previous Article
    Optimal convergence rates of the magnetohydrodynamic model for quantum plasmas with potential force
  • DCDS-B Home
  • This Issue
  • Next Article
    A modified May–Holling–Tanner predator-prey model with multiple Allee effects on the prey and an alternative food source for the predator
February  2021, 26(2): 963-985. doi: 10.3934/dcdsb.2020149

Asymmetric diffusion in a two-patch mutualism system characterizing exchange of resource for resource

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

Received  May 2019 Revised  February 2020 Published  May 2020

Fund Project: The author is supported by NSF of P.R. China (11571382)

This paper considers a two-patch mutualism system derived from exchange of resource for resource, where the obligate mutualist can diffuse asymmetrically between patches. First, we give a complete analysis on dynamics of the system without diffusion, which exhibit how resource production of the obligate mutualist leads to its survival/extinction. Using monotone dynamics theory, we show global stability of a positive equilibrium in the three-dimensional system with diffusion. A novel finding of this work is that the obligate species' final abundance is explicitly expressed as a function of the diffusion rate and asymmetry, which demonstrates precise mechanisms by which the diffusion and asymmetry lead to the abundance higher than if non-diffusing, even though the facultative species declines. It is shown that for a fixed diffusion rate, intermediate asymmetry is favorable while extremely large asymmetry is unfavorable; For a fixed asymmetry, small diffusion is favorable while extremely large asymmetry is unfavorable. Initial densities of the species are also shown to be important in species' persistence and abundance. Numerical simulations confirm and extend our results.

Citation: Yuanshi Wang. Asymmetric diffusion in a two-patch mutualism system characterizing exchange of resource for resource. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 963-985. doi: 10.3934/dcdsb.2020149
References:
[1]

R. ArditiC. Lobry and T. Sari, Is dispersal always beneficial to carrying capacity? New insights from the multi-patch logistic equation, Theor. Popul. Biol., 106 (2015), 45-59.  doi: 10.1016/j.tpb.2015.10.001.  Google Scholar

[2]

R. ArditiC. Lobry and T. Sari, Asymmetric dispersal in the multi-patch logistic equation, Theor. Popul. Biol., 120 (2018), 11-15.  doi: 10.1016/j.tpb.2017.12.006.  Google Scholar

[3]

J. $\hat{A}$str$\ddot{o}$m and T. P$\ddot{a}$rt, Negative and matrix-dependent effects of dispersal corri- dors in an experimental metacommunity, Ecology, 94 (2013), 1939-1970.   Google Scholar

[4]

C. J. Briggs and M. F. Hoopes, Stabilizing effects in spatial parasitoid-host and predator-prey models: A review, Theor. Popul. Biol., 65 (2004), 299-315.  doi: 10.1016/j.tpb.2003.11.001.  Google Scholar

[5]

G. J. ButlerH. I. Freedman and P. Waltman, Uniformly persistent systems, Proc. Amer. Math. Sco., 96 (1986), 425-430.  doi: 10.1090/S0002-9939-1986-0822433-4.  Google Scholar

[6]

L. Fahrig, Effect of habitat fragmentation on the extinction threshold: A synthesis, Ecol. Appl., 12 (2002), 346-353.   Google Scholar

[7]

H. I. Freedman and D. Waltman, Mathematical models of population interactions with dispersal. I. Stability of two habitats with and without a predator, SIAM J Appl Math., 32 (1977), 631-648.  doi: 10.1137/0132052.  Google Scholar

[8] J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge, UK, 1998.  doi: 10.1017/CBO9781139173179.  Google Scholar
[9]

R. D. Holt, Population dynamics in two-patch environments: Some anomalous consequences of an optimal habitat distribution, Theor. Popul. Biol., 28 (1985), 181-208.  doi: 10.1016/0040-5809(85)90027-9.  Google Scholar

[10]

J. N. Holland and D. L. DeAngelis, A consumer-resource approach to the density-dependent population dynamics of mutualism, Ecology, 91 (2010), 1286-1295.   Google Scholar

[11]

V. HutsonY. Lou and K. Mischaikow, Convergence in competition models with small diffusion coefficients, J. Diff. Equa., 211 (2005), 135-161.  doi: 10.1016/j.jde.2004.06.003.  Google Scholar

[12]

V. A. A. Jansen, The dynamics of two diffusively coupled predator-prey populations, Theor. Popul. Biol., 59 (2001), 119-131.  doi: 10.1006/tpbi.2000.1506.  Google Scholar

[13]

J. Jiang, Three- and four-dimensional cooperative systems with every equilibrium stable, J. Math. Anal. Appl., 188 (1994), 92-100.  doi: 10.1006/jmaa.1994.1413.  Google Scholar

[14]

Y. Lou, On the effects of migration and spatial heterogeneity on single and multiple species, J. Diff. Equa., 223 (2006), 400-426.  doi: 10.1016/j.jde.2005.05.010.  Google Scholar

[15]

T. A. Revilla, Numerical responses in resource-based mutualisms: A time scale approach, J. Theor. Biol., 378 (2015), 39-46.  doi: 10.1016/j.jtbi.2015.04.012.  Google Scholar

[16]

S. Rinaldi and M. Scheffer, Geometric analysis of ecological models with slow and fast processes, Ecosystems, 3 (2000), 507-521.  doi: 10.1007/s100210000045.  Google Scholar

[17]

A. Ruiz-Herrera and P. J. Torres, Effects of diffusion on total biomass in simple metacommunities, J. Theor. Biol., 447 (2018), 12-24.  doi: 10.1016/j.jtbi.2018.03.018.  Google Scholar

[18] H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Amer. Math. Soci. Press, New York, USA, 1995.   Google Scholar
[19] H. L. Smith and P. Waltman, The Theory of the Chemostat, New York: Cambridge University Press, 1995.  doi: 10.1017/CBO9780511530043.  Google Scholar
[20]

G. Takimoto and K. Suzuki, Global stability of obligate mutualism in community modules with facultative mutualists, OIKOS, 125 (2015), 535-540.  doi: 10.1111/oik.02741.  Google Scholar

[21]

J. J. Tewksbury et al., Corridors affect plants, animals, and their interactions in fragmented landscapes, Proc. Natl. Acad. Sci. U.S.A., 99 (2002), 12923-12926. Google Scholar

[22]

Y. Wang and D. L. DeAngelis, Comparison of effects of diffusion in heterogeneous and homogeneous with the same total carrying capacity on total realized population size, Theor. Popul. Biol., 125 (2019), 30-37.   Google Scholar

[23]

Y. WangH. Wu and D. L. DeAngelis, Global dynamics of a mutualism-competition model with one resource and multiple consumers, J. Math. Biol., 78 (2019), 683-710.  doi: 10.1007/s00285-018-1288-9.  Google Scholar

[24]

B. ZhangK. AlexM. L. KeenanZ. LuL. R. ArrixW.-M. NiD. L. DeAngelis and J. D. Dyken, Carrying capacity in a heterogeneous environment with habitat connectivity, Ecology Letters, 20 (2017), 1118-1128.  doi: 10.1111/ele.12807.  Google Scholar

show all references

References:
[1]

R. ArditiC. Lobry and T. Sari, Is dispersal always beneficial to carrying capacity? New insights from the multi-patch logistic equation, Theor. Popul. Biol., 106 (2015), 45-59.  doi: 10.1016/j.tpb.2015.10.001.  Google Scholar

[2]

R. ArditiC. Lobry and T. Sari, Asymmetric dispersal in the multi-patch logistic equation, Theor. Popul. Biol., 120 (2018), 11-15.  doi: 10.1016/j.tpb.2017.12.006.  Google Scholar

[3]

J. $\hat{A}$str$\ddot{o}$m and T. P$\ddot{a}$rt, Negative and matrix-dependent effects of dispersal corri- dors in an experimental metacommunity, Ecology, 94 (2013), 1939-1970.   Google Scholar

[4]

C. J. Briggs and M. F. Hoopes, Stabilizing effects in spatial parasitoid-host and predator-prey models: A review, Theor. Popul. Biol., 65 (2004), 299-315.  doi: 10.1016/j.tpb.2003.11.001.  Google Scholar

[5]

G. J. ButlerH. I. Freedman and P. Waltman, Uniformly persistent systems, Proc. Amer. Math. Sco., 96 (1986), 425-430.  doi: 10.1090/S0002-9939-1986-0822433-4.  Google Scholar

[6]

L. Fahrig, Effect of habitat fragmentation on the extinction threshold: A synthesis, Ecol. Appl., 12 (2002), 346-353.   Google Scholar

[7]

H. I. Freedman and D. Waltman, Mathematical models of population interactions with dispersal. I. Stability of two habitats with and without a predator, SIAM J Appl Math., 32 (1977), 631-648.  doi: 10.1137/0132052.  Google Scholar

[8] J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge, UK, 1998.  doi: 10.1017/CBO9781139173179.  Google Scholar
[9]

R. D. Holt, Population dynamics in two-patch environments: Some anomalous consequences of an optimal habitat distribution, Theor. Popul. Biol., 28 (1985), 181-208.  doi: 10.1016/0040-5809(85)90027-9.  Google Scholar

[10]

J. N. Holland and D. L. DeAngelis, A consumer-resource approach to the density-dependent population dynamics of mutualism, Ecology, 91 (2010), 1286-1295.   Google Scholar

[11]

V. HutsonY. Lou and K. Mischaikow, Convergence in competition models with small diffusion coefficients, J. Diff. Equa., 211 (2005), 135-161.  doi: 10.1016/j.jde.2004.06.003.  Google Scholar

[12]

V. A. A. Jansen, The dynamics of two diffusively coupled predator-prey populations, Theor. Popul. Biol., 59 (2001), 119-131.  doi: 10.1006/tpbi.2000.1506.  Google Scholar

[13]

J. Jiang, Three- and four-dimensional cooperative systems with every equilibrium stable, J. Math. Anal. Appl., 188 (1994), 92-100.  doi: 10.1006/jmaa.1994.1413.  Google Scholar

[14]

Y. Lou, On the effects of migration and spatial heterogeneity on single and multiple species, J. Diff. Equa., 223 (2006), 400-426.  doi: 10.1016/j.jde.2005.05.010.  Google Scholar

[15]

T. A. Revilla, Numerical responses in resource-based mutualisms: A time scale approach, J. Theor. Biol., 378 (2015), 39-46.  doi: 10.1016/j.jtbi.2015.04.012.  Google Scholar

[16]

S. Rinaldi and M. Scheffer, Geometric analysis of ecological models with slow and fast processes, Ecosystems, 3 (2000), 507-521.  doi: 10.1007/s100210000045.  Google Scholar

[17]

A. Ruiz-Herrera and P. J. Torres, Effects of diffusion on total biomass in simple metacommunities, J. Theor. Biol., 447 (2018), 12-24.  doi: 10.1016/j.jtbi.2018.03.018.  Google Scholar

[18] H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Amer. Math. Soci. Press, New York, USA, 1995.   Google Scholar
[19] H. L. Smith and P. Waltman, The Theory of the Chemostat, New York: Cambridge University Press, 1995.  doi: 10.1017/CBO9780511530043.  Google Scholar
[20]

G. Takimoto and K. Suzuki, Global stability of obligate mutualism in community modules with facultative mutualists, OIKOS, 125 (2015), 535-540.  doi: 10.1111/oik.02741.  Google Scholar

[21]

J. J. Tewksbury et al., Corridors affect plants, animals, and their interactions in fragmented landscapes, Proc. Natl. Acad. Sci. U.S.A., 99 (2002), 12923-12926. Google Scholar

[22]

Y. Wang and D. L. DeAngelis, Comparison of effects of diffusion in heterogeneous and homogeneous with the same total carrying capacity on total realized population size, Theor. Popul. Biol., 125 (2019), 30-37.   Google Scholar

[23]

Y. WangH. Wu and D. L. DeAngelis, Global dynamics of a mutualism-competition model with one resource and multiple consumers, J. Math. Biol., 78 (2019), 683-710.  doi: 10.1007/s00285-018-1288-9.  Google Scholar

[24]

B. ZhangK. AlexM. L. KeenanZ. LuL. R. ArrixW.-M. NiD. L. DeAngelis and J. D. Dyken, Carrying capacity in a heterogeneous environment with habitat connectivity, Ecology Letters, 20 (2017), 1118-1128.  doi: 10.1111/ele.12807.  Google Scholar

Figure 1.  Phase-plane diagram of system (5). Stable and unstable equilibria are identified by solid and open circles, respectively. Vector fields are shown by green arrows. Isoclines of species $ u,v_1 $ are represented by red and blue lines, respectively. Fix $ r = r_1 = c = 1 $. (a) Let $ a_{12} = 2.5, a_{21} = 1.3 $. Equilibrium $ E^+(2.7,2.5) $ is globally asymptotically stable. (b) Let $ a_{12} = 4.5, a_{21} = 0.9 $. There are two positive equilibria $ E^-(1.25,0.14) $ and $ E^+(2.79,1.5) $. (c) Let $ a_{12} = 3.5, a_{21} = 0.9 $. The equilibria $ E^- $ and $ E^+ $ coincide and form a saddle-node point $ E^\pm(1.6,0.45) $. In the cases of (b-c), the separatrices (the black line) of $ E^- $ subdivide the first quadrant into two regions: one is the basin of attraction of $ E_1 $ while the other is that of $ E^+ $. (d) Let $ a_{12} = 2.5, a_{21} = 0.5 $. All positive solutions converge to equilibrium $ E_1(1,0) $
Figure 4.  Comparison of $ T_1(s, 0) $ and $ T_2(s, D) $ when there is diffusion $ D $ as shown in Theorem 4.1(ⅰ). The solid blue line represents $ T_2(s, D) $, while the dash-dot red line represents $ T_1(s, 0) $. Let $ r = 0.2, c = 0.05, a_{12} = 0.1, b = b_1 = 1, s = 1 $, $ a_{21} = 0.5, r_1 = 0.8, r_2 = 0.1 $. (a) When $ s = 1 $, we have $ T_2>T_1 $ for $ D>0 $. (b) When $ s = 10 $, we have $ T_2>T_1 $ as $ 0<D< 0.0135 $ while $ T_2<T_1 $ as $ D> 0.0135 $
Figure 5.  Comparison of $ T_1(s, 0) $ and $ T_2(s,100) $ when there is asymmetry $ s $ in large diffusion as shown in Theorem 4.2(ⅰ). The solid blue line represents $ T_2(s,100) $, while the dash-dot red line represents $ T_1(s, 0) $. Let $ r = 0.2, c = 0.05, a_{12} = 0.1, b = b_1 = 1, D = 100 $, $ a_{21} = 0.5, r_1 = 0.8, r_2 = 0.1 $. Then we have $ T_1(s, 0) = 1.9233 $ and $ T_2(s,100)>T_1(s, 0) $ as $ 0.1935< s< 9.3201 $. Numerical computation shows that the function $ T_2 = T_2(s,100) $ is convex upward with $ T_2(s,100) = 0 $ as $ s\ge 12.1537 $
Figure 2.  Comparison of $ T_1(s, D) $ and $ T_1(s, 0), T_2(s, D) $ and $ T_2(s, 0) $ when there is a small diffusion rate $ D $, as shown in Theorem 4.1(ⅰ). The solid red and blue lines represent $ T_1(s, D) $ and $ T_2(s, D) $, while the dash-dot red and blue lines represent $ T_1(s, 0) $ and $ T_2(s, 0) $, respectively. Let $ r = 0.2, c = 0.05, a_{12} = 0.1, b = b_1 = 1, D = 0.1, s = 1 $, $ a_{21} = 0.5, r_1 = 0.8, r_2 = 0.1 $. Then we have $ T_1(1, 0.1)<T_1(1,0) $ but $ T_2(1,0.1)>T_2(1,0) $
Figure 3.  Comparison of $ T_1(s, D) $ and $ T_1(s, 0), T_2(s, D) $ and $ T_2(s, 0) $ when there is a large diffusion rate $ D $, as shown in Theorem 4.2(ⅰ). The solid red and blue lines represent $ T_1(s, D) $ and $ T_2(s, D) $, while the dash-dot red and blue lines represent $ T_1(s, 0) $ and $ T_2(s, 0) $, respectively. Let $ r = 0.2, c = 0.05, a_{12} = 0.1, b = b_1 = 1, D = 100, s = 0.1 $, $ a_{21} = 0.5, r_1 = 0.8, r_2 = 0.1 $. Then we have $ T_1(0.1,100)<T_1(0.1,0) $ but $ T_2(0.1,100)>T_2(0.1,0) $
Figure 6.  The surface of $ T_2 = T_2(s,D) $ when both of $ s $ and $ D $ varies. Let $ r = 0.2, c = 0.05, a_{12} = 0.1, b = 4, b_1 = 1 $, $ a_{21} = 0.5, r_1 = 0.8, r_2 = 0.1, 0<s<6, 0<D<6 $. Numerical computation shows that for fixed $ s $, the surface decreases monotonically, which is consistent with Fig. 4. For fixed $ D $, the surface is convex upward, which is consistent with Fig. 5
[1]

Ting Liu, Guo-Bao Zhang. Global stability of traveling waves for a spatially discrete diffusion system with time delay. Electronic Research Archive, , () : -. doi: 10.3934/era.2021003

[2]

Shao-Xia Qiao, Li-Jun Du. Propagation dynamics of nonlocal dispersal equations with inhomogeneous bistable nonlinearity. Electronic Research Archive, , () : -. doi: 10.3934/era.2020116

[3]

Kimie Nakashima. Indefinite nonlinear diffusion problem in population genetics. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3837-3855. doi: 10.3934/dcds.2020169

[4]

Ebraheem O. Alzahrani, Muhammad Altaf Khan. Androgen driven evolutionary population dynamics in prostate cancer growth. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020426

[5]

Attila Dénes, Gergely Röst. Single species population dynamics in seasonal environment with short reproduction period. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020288

[6]

Mengting Fang, Yuanshi Wang, Mingshu Chen, Donald L. DeAngelis. Asymptotic population abundance of a two-patch system with asymmetric diffusion. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3411-3425. doi: 10.3934/dcds.2020031

[7]

Divine Wanduku. Finite- and multi-dimensional state representations and some fundamental asymptotic properties of a family of nonlinear multi-population models for HIV/AIDS with ART treatment and distributed delays. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021005

[8]

Weiwei Liu, Jinliang Wang, Yuming Chen. Threshold dynamics of a delayed nonlocal reaction-diffusion cholera model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020316

[9]

Hui Zhao, Zhengrong Liu, Yiren Chen. Global dynamics of a chemotaxis model with signal-dependent diffusion and sensitivity. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021011

[10]

Xueli Bai, Fang Li. Global dynamics of competition models with nonsymmetric nonlocal dispersals when one diffusion rate is small. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3075-3092. doi: 10.3934/dcds.2020035

[11]

Raffaele Folino, Ramón G. Plaza, Marta Strani. Long time dynamics of solutions to $ p $-Laplacian diffusion problems with bistable reaction terms. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020403

[12]

Shipra Singh, Aviv Gibali, Xiaolong Qin. Cooperation in traffic network problems via evolutionary split variational inequalities. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020170

[13]

Yuxin Zhang. The spatially heterogeneous diffusive rabies model and its shadow system. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020357

[14]

Kaixuan Zhu, Ji Li, Yongqin Xie, Mingji Zhang. Dynamics of non-autonomous fractional reaction-diffusion equations on $ \mathbb{R}^{N} $ driven by multiplicative noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020376

[15]

Yong-Jung Kim, Hyowon Seo, Changwook Yoon. Asymmetric dispersal and evolutional selection in two-patch system. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3571-3593. doi: 10.3934/dcds.2020043

[16]

Laurent Di Menza, Virginie Joanne-Fabre. An age group model for the study of a population of trees. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020464

[17]

Fathalla A. Rihan, Hebatallah J. Alsakaji. Stochastic delay differential equations of three-species prey-predator system with cooperation among prey species. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020468

[18]

Touria Karite, Ali Boutoulout. Global and regional constrained controllability for distributed parabolic linear systems: RHUM approach. Numerical Algebra, Control & Optimization, 2020  doi: 10.3934/naco.2020055

[19]

Feifei Cheng, Ji Li. Geometric singular perturbation analysis of Degasperis-Procesi equation with distributed delay. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 967-985. doi: 10.3934/dcds.2020305

[20]

Dominique Chapelle, Philippe Moireau, Patrick Le Tallec. Robust filtering for joint state-parameter estimation in distributed mechanical systems. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 65-84. doi: 10.3934/dcds.2009.23.65

2019 Impact Factor: 1.27

Article outline

Figures and Tables

[Back to Top]