# American Institute of Mathematical Sciences

June  2020, 40(6): 3411-3425. doi: 10.3934/dcds.2020031

## Asymptotic population abundance of a two-patch system with asymmetric diffusion

 1 School of Mathematics, Sun Yat-sen University, Guangzhou 510275, China 2 U.S. Geological Survey, Wetland and Aquatic Research Center, Gainesville, FL 32653, USA

* Corresponding author: Donald DeAngelis

Received  January 2019 Revised  June 2019 Published  October 2019

Fund Project: The second author is supported by NSF grant of China (11571382)

This paper considers a two-patch system with asymmetric diffusion rates, in which exploitable resources are included. By using dynamical system theory, we exclude periodic solution in the one-patch subsystem and demonstrate its global dynamics. Then we exhibit uniform persistence of the two-patch system and demonstrate uniqueness of the positive equilibrium, which is shown to be asymptotically stable when the diffusion rates are sufficiently large. By a thorough analysis on the asymptotic population abundance, we demonstrate necessary and sufficient conditions under which the asymmetric diffusion rates can lead to the result that total equilibrium population abundance in heterogeneous environments is larger than that in heterogeneous/homogeneous environments with no diffusion, which is not intuitive. Our result extends previous work to the situation of asymmetric diffusion and provides new insights. Numerical simulations confirm and extend our results.

Citation: Mengting Fang, Yuanshi Wang, Mingshu Chen, Donald L. DeAngelis. Asymptotic population abundance of a two-patch system with asymmetric diffusion. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3411-3425. doi: 10.3934/dcds.2020031
##### References:
 [1] R. Arditi, N. Perrin and H. Saiah, Functional responses and heterogeneities: An experimental test with cladocerans, Oikos, 60 (1991), 69-75.  doi: 10.2307/3544994. [2] R. Arditi and H. Saiah, Empirical evidence of the role of heterogeneity in ratio-dependent consumption, Ecology, 73 (1992), 1544-1551. [3] 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. [4] 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. [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] B. J. Cardinale, M. A. Palmer, C. M. Swan, S. Brooks and N. Leroy Poff, The influence of substrate heterogeneity on biofilm metabolism in a stream ecosystem, Ecology, 83 (2002), 412-422. [7] C. Cosner, Variability, vagueness and comparison methods for ecological models, Bull. Math. Biol., 58 (1996), 207-246.  doi: 10.1007/BF02458307. [8] D. L. DeAngelis, W. Ni and B. Zhang, Dispersal and heterogeneity: Single species, J. Math. Biol., 72 (2016), 239-254.  doi: 10.1007/s00285-015-0879-y. [9] D. L. DeAngelis, W. Ni and B. Zhang, Effects of diffusion on total biomass in heterogeneous continuous and discrete-patch systems, Theoretical Ecology, 9 (2016), 443-453. [10] J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge, 1998.  doi: 10.1017/CBO9781139173179. [11] V. Hutson, Y. Lou and K. Mischaikow, Convergence in competition models with small diffusion coefficients, J. Differential Equations, 211 (2005), 135-161.  doi: 10.1016/j.jde.2004.06.003. [12] Y. Lou, On the effects of migration and spatial heterogeneity on single and multiple species, J. Differential Equations, 223 (2006), 400-426.  doi: 10.1016/j.jde.2005.05.010. [13] J. C. Poggiale and P. Auger, Fast Oscillating Migrations in a predator-prey model, Math. Models Methods Appl. Sci., 6 (1996), 217-226.  doi: 10.1142/S0218202596000559. [14] J. C. Poggiale, From behavioral to population level: Growth and competition, Aggregation and emergence in population dynamics,, Math. Comput. Modelling, 27 (1998), 41-49.  doi: 10.1016/S0895-7177(98)00004-1. [15] J. C. Poggiale, P. Auger, D. Nerini, C. Mante and F. Gilbert, Global production increased by spatial heterogeneity in a population dynamics model, Acta, Biotheor, 53 (2005), 359-370. [16] A. Ruiz-Herrera and P. J. Torres, Effects of diffusion on total biomass in simple metacommunities, J. Theoret. Biol., 447 (2018), 12-24.  doi: 10.1016/j.jtbi.2018.03.018. [17] 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. [18] 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. [19] B. Zhang, X. Liu, D. L. DeAngelis, W. -M. Ni and G. Wang, Effects of dispersal on total biomass in a patchy, heterogeneous system: Analysis and experiment, Math. Biosci., 264 (2015), 54-62.  doi: 10.1016/j.mbs.2015.03.005.

show all references

##### References:
 [1] R. Arditi, N. Perrin and H. Saiah, Functional responses and heterogeneities: An experimental test with cladocerans, Oikos, 60 (1991), 69-75.  doi: 10.2307/3544994. [2] R. Arditi and H. Saiah, Empirical evidence of the role of heterogeneity in ratio-dependent consumption, Ecology, 73 (1992), 1544-1551. [3] 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. [4] 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. [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] B. J. Cardinale, M. A. Palmer, C. M. Swan, S. Brooks and N. Leroy Poff, The influence of substrate heterogeneity on biofilm metabolism in a stream ecosystem, Ecology, 83 (2002), 412-422. [7] C. Cosner, Variability, vagueness and comparison methods for ecological models, Bull. Math. Biol., 58 (1996), 207-246.  doi: 10.1007/BF02458307. [8] D. L. DeAngelis, W. Ni and B. Zhang, Dispersal and heterogeneity: Single species, J. Math. Biol., 72 (2016), 239-254.  doi: 10.1007/s00285-015-0879-y. [9] D. L. DeAngelis, W. Ni and B. Zhang, Effects of diffusion on total biomass in heterogeneous continuous and discrete-patch systems, Theoretical Ecology, 9 (2016), 443-453. [10] J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge, 1998.  doi: 10.1017/CBO9781139173179. [11] V. Hutson, Y. Lou and K. Mischaikow, Convergence in competition models with small diffusion coefficients, J. Differential Equations, 211 (2005), 135-161.  doi: 10.1016/j.jde.2004.06.003. [12] Y. Lou, On the effects of migration and spatial heterogeneity on single and multiple species, J. Differential Equations, 223 (2006), 400-426.  doi: 10.1016/j.jde.2005.05.010. [13] J. C. Poggiale and P. Auger, Fast Oscillating Migrations in a predator-prey model, Math. Models Methods Appl. Sci., 6 (1996), 217-226.  doi: 10.1142/S0218202596000559. [14] J. C. Poggiale, From behavioral to population level: Growth and competition, Aggregation and emergence in population dynamics,, Math. Comput. Modelling, 27 (1998), 41-49.  doi: 10.1016/S0895-7177(98)00004-1. [15] J. C. Poggiale, P. Auger, D. Nerini, C. Mante and F. Gilbert, Global production increased by spatial heterogeneity in a population dynamics model, Acta, Biotheor, 53 (2005), 359-370. [16] A. Ruiz-Herrera and P. J. Torres, Effects of diffusion on total biomass in simple metacommunities, J. Theoret. Biol., 447 (2018), 12-24.  doi: 10.1016/j.jtbi.2018.03.018. [17] 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. [18] 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. [19] B. Zhang, X. Liu, D. L. DeAngelis, W. -M. Ni and G. Wang, Effects of dispersal on total biomass in a patchy, heterogeneous system: Analysis and experiment, Math. Biosci., 264 (2015), 54-62.  doi: 10.1016/j.mbs.2015.03.005.
Phase-plane diagram of system (6). Stable equilibrium is displayed by solid circle. Vector fields are shown by gray arrows. Isoclines of nutrient $u_1$ and consumer $v_1$ are represented by red and blue lines, respectively. According to parameter values in experiments by Zhang et al. (2017), let $N_{01} = 0.02, r = 0.1, k_1 = 0.1, \gamma = 0.01, g_1 = 0.0001$. Then $u_{01} = 0.2 < r^2/g_1$. Numerical simulations show that all positive solutions of (6) converge to equilibrium $E_1^+$, which is consistent with Theorem 2.1(ⅱ)
Numerical simulations for comparison of $T_1$ and $T_0$ when $s$ varies, Let $r = 0.1, u_{01} = 0.06, u_{02} = 0.0002, g_1 = 0.001, g_2 = 0.0005, D = 100$. When $s = 0.1$, we obtain $T_1 = 11.9531 >8.5095 = T_0$ by numerical computations on (4)
Numerical simulations for comparison of $T_1$ and $T_2$ when $s$ varies, Let $r = 0.1, u_{01} = 0.06, u_{02} = 0.0002, g_1 = 0.001, g_2 = 0.0005, D = 100$. Then $u_{mean} = 0.0301$. When $s = 0.1$, we obtain $T_1 = 13.4364> 13.2452 = T_2$ by numerical computations on (4)
Numerical simulations for comparison of $T_1$ and $T_2$ when $s$ varies, Let $r = 0.1, u_{01} = 0.06, u_{02} = 0.0002, g_1 = 0.001, g_2 = 0.0005, D = 100$. When the initial values are $(1.4, 1.4, 1.4, 1.4)$, $(3.4, 3.4, 3.4, 3.4)$, $(4, 4, 4, 4)$, $(7, 7, 7, 7)$ and $(8, 8, 8, 8)$, numerical computations on (4) show that all solutions converge to the same equilibrium $(0.1549, 4.4737, 0.0002, 8.9457)$, while the component $v_1(t)$ is displayed in this figure
 [1] Wonlyul Ko, Inkyung Ahn, Shengqiang Liu. Asymptotical behaviors of a general diffusive consumer-resource model with maturation delay. Discrete and Continuous Dynamical Systems - B, 2015, 20 (6) : 1715-1733. doi: 10.3934/dcdsb.2015.20.1715 [2] Robert Stephen Cantrell, Chris Cosner, Shigui Ruan. Intraspecific interference and consumer-resource dynamics. Discrete and Continuous Dynamical Systems - B, 2004, 4 (3) : 527-546. doi: 10.3934/dcdsb.2004.4.527 [3] Kun Hu, Yuanshi Wang. Dynamics of consumer-resource systems with consumer's dispersal between patches. Discrete and Continuous Dynamical Systems - B, 2022, 27 (2) : 977-1000. doi: 10.3934/dcdsb.2021077 [4] Zhihua Liu, Pierre Magal, Shigui Ruan. Oscillations in age-structured models of consumer-resource mutualisms. Discrete and Continuous Dynamical Systems - B, 2016, 21 (2) : 537-555. doi: 10.3934/dcdsb.2016.21.537 [5] 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 [6] Keng Deng, Yixiang Wu. Extinction and uniform strong persistence of a size-structured population model. Discrete and Continuous Dynamical Systems - B, 2017, 22 (3) : 831-840. doi: 10.3934/dcdsb.2017041 [7] Antoine Perasso. Global stability and uniform persistence for an infection load-structured SI model with exponential growth velocity. Communications on Pure and Applied Analysis, 2019, 18 (1) : 15-32. doi: 10.3934/cpaa.2019002 [8] Edoardo Beretta, Dimitri Breda. Discrete or distributed delay? Effects on stability of population growth. Mathematical Biosciences & Engineering, 2016, 13 (1) : 19-41. doi: 10.3934/mbe.2016.13.19 [9] Hal L. Smith, Horst R. Thieme. Persistence and global stability for a class of discrete time structured population models. Discrete and Continuous Dynamical Systems, 2013, 33 (10) : 4627-4646. doi: 10.3934/dcds.2013.33.4627 [10] Kaifa Wang, Aijun Fan. Uniform persistence and periodic solution of chemostat-type model with antibiotic. Discrete and Continuous Dynamical Systems - B, 2004, 4 (3) : 789-795. doi: 10.3934/dcdsb.2004.4.789 [11] Zhihua Liu, Hui Tang, Pierre Magal. Hopf bifurcation for a spatially and age structured population dynamics model. Discrete and Continuous Dynamical Systems - B, 2015, 20 (6) : 1735-1757. doi: 10.3934/dcdsb.2015.20.1735 [12] Abed Boulouz. A spatially and size-structured population model with unbounded birth process. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022038 [13] Guangrui Li, Ming Mei, Yau Shu Wong. Nonlinear stability of traveling wavefronts in an age-structured reaction-diffusion population model. Mathematical Biosciences & Engineering, 2008, 5 (1) : 85-100. doi: 10.3934/mbe.2008.5.85 [14] Ting Liu, Guo-Bao Zhang. Global stability of traveling waves for a spatially discrete diffusion system with time delay. Electronic Research Archive, 2021, 29 (4) : 2599-2618. doi: 10.3934/era.2021003 [15] Bin Fang, Xue-Zhi Li, Maia Martcheva, Li-Ming Cai. Global stability for a heroin model with two distributed delays. Discrete and Continuous Dynamical Systems - B, 2014, 19 (3) : 715-733. doi: 10.3934/dcdsb.2014.19.715 [16] Yukihiko Nakata, Yoichi Enatsu, Yoshiaki Muroya. On the global stability of an SIRS epidemic model with distributed delays. Conference Publications, 2011, 2011 (Special) : 1119-1128. doi: 10.3934/proc.2011.2011.1119 [17] Miljana Jovanović, Vuk Vujović. Stability of stochastic heroin model with two distributed delays. Discrete and Continuous Dynamical Systems - B, 2020, 25 (7) : 2407-2432. doi: 10.3934/dcdsb.2020016 [18] Seung-Yeal Ha, Mitsuru Yamazaki. $L^p$-stability estimates for the spatially inhomogeneous discrete velocity Boltzmann model. Discrete and Continuous Dynamical Systems - B, 2009, 11 (2) : 353-364. doi: 10.3934/dcdsb.2009.11.353 [19] Keng Deng. On a nonlocal reaction-diffusion population model. Discrete and Continuous Dynamical Systems - B, 2008, 9 (1) : 65-73. doi: 10.3934/dcdsb.2008.9.65 [20] Henri Berestycki, Jean-Michel Roquejoffre, Luca Rossi. The periodic patch model for population dynamics with fractional diffusion. Discrete and Continuous Dynamical Systems - S, 2011, 4 (1) : 1-13. doi: 10.3934/dcdss.2011.4.1

2021 Impact Factor: 1.588