Article Contents
Article Contents

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

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.

Mathematics Subject Classification: 34C12, 37N25, 34C28, 37G20.

 Citation:

• 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] R. Arditi, C. 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. [2] R. Arditi, C. 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. [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. [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. [5] G. J. Butler, H. I. Freedman and P. Waltman, Uniformly persistent systems, Proc. Amer. Math. Sco., 96 (1986), 425-430.  doi: 10.1090/S0002-9939-1986-0822433-4. [6] L. Fahrig, Effect of habitat fragmentation on the extinction threshold: A synthesis, Ecol. Appl., 12 (2002), 346-353. [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. [8] J. Hofbauer and  K. Sigmund,  Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge, UK, 1998.  doi: 10.1017/CBO9781139173179. [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. [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. [11] V. Hutson, Y. 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. [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. [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. [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. [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. [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. [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. [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. [19] H. L. Smith and  P. Waltman,  The Theory of the Chemostat, New York: Cambridge University Press, 1995.  doi: 10.1017/CBO9780511530043. [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. [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. [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. [23] Y. Wang, H. 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. [24] B. Zhang, K. Alex, M. L. Keenan, Z. Lu, L. R. Arrix, W.-M. Ni, D. 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.

Figures(6)