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

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

Mengting Fang 1, , Yuanshi Wang 1, , Mingshu Chen 1, and  Donald L. DeAngelis 2,,

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.

Keywords: Spatially distributed population, diffusion, consumer-resource model, uniform persistence, Liapunov stability.
Mathematics Subject Classification: Primary: 34C37, 92D25; Secondary: 37N25.
Citation: Mengting Fang, Yuanshi Wang, Mingshu Chen, Donald L. DeAngelis. Asymptotic population abundance of a two-patch system with asymmetric diffusion. Discrete & 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.  Google Scholar

[2]

R. Arditi and H. Saiah, Empirical evidence of the role of heterogeneity in ratio-dependent consumption, Ecology, 73 (1992), 1544-1551.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[7]

C. Cosner, Variability, vagueness and comparison methods for ecological models, Bull. Math. Biol., 58 (1996), 207-246.  doi: 10.1007/BF02458307.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

R. Arditi and H. Saiah, Empirical evidence of the role of heterogeneity in ratio-dependent consumption, Ecology, 73 (1992), 1544-1551.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[7]

C. Cosner, Variability, vagueness and comparison methods for ecological models, Bull. Math. Biol., 58 (1996), 207-246.  doi: 10.1007/BF02458307.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  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)
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Figure 2.  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)
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Figure 3.  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)
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Figure 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
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