American Institute of Mathematical Sciences

November  2019, 24(11): 6261-6278. doi: 10.3934/dcdsb.2019138

Dynamics of a chemostat system with two patches

 a. School of Mathematics, Sun Yat-sen University, Guangzhou 510275, China b. Department of Biostatistics, The University of Texas MD Anderson Cancer Center, Houston, USA, 77030

Received  July 2018 Revised  January 2019 Published  November 2019 Early access  July 2019

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

This paper studies a diffusion model with two patches, which is derived from experiments and includes exploitable resources. Our aim is to provide theoretical proof for experimental observations and extend previous theory to consumer-resource systems with external resource inputs. First, we exhibit nonnegativeness and boundedness of solutions of the model. For one-patch subsystems, we demonstrate the global dynamics by excluding periodic solutions. For the two-patch system, we exhibit uniform persistence of the system and asymptotic stability of the positive equilibria, while the equilibria converge to a unique positive point as the diffusion tends to infinity. Then we demonstrate that homogeneously distributed resources support higher total population abundance than heterogeneously distributed resources with diffusion, which coincides with empirical observation but refutes previous theory. Meanwhile, we exhibit new conditions under which populations diffusing in heterogeneous environments can reach higher total size than if non-diffusing. A new finding of our study is that these results hold even with source-sink populations, and varying the diffusion rate can result in survival/extinction of the species. Our results are consistent with experimental observations and provide new insights.

Citation: Shikun Wang. Dynamics of a chemostat system with two patches. Discrete and Continuous Dynamical Systems - B, 2019, 24 (11) : 6261-6278. doi: 10.3934/dcdsb.2019138
References:
 [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] 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. [3] C. Cosner, Variability, vagueness and comparison methods for ecological models, Bull. Math. Biol., 58 (1996), 207-246.  doi: 10.1007/BF02458307. [4] 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. [5] 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. [6] 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. [7] J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge, UK, 1998.  doi: 10.1017/CBO9781139173179. [8] 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. [9] 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. [10] 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. [11] H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence, Graduate Studies in Mathematics, 118. American Mathematical Society, Providence, RI, 2011. [12] 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. [13] L. B. Crowder, S. J. Lyman, W. F. Figueira and J. Priddy, Source-sink population dynamics and the problem of siting marine reserves, Bulletin of Marine Science-Miami, 66 (2000), 799-820. [14] A. R. Watkinson and W. J. Sutherland, Sources, sinks and pseudo-sinks, J. Anim. Ecol., 64 (1995), 126-130. [15] 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. [16] 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, athematical Biosciences, 264 (2015), 54-62.  doi: 10.1016/j.mbs.2015.03.005.

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References:
 [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] 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. [3] C. Cosner, Variability, vagueness and comparison methods for ecological models, Bull. Math. Biol., 58 (1996), 207-246.  doi: 10.1007/BF02458307. [4] 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. [5] 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. [6] 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. [7] J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge, UK, 1998.  doi: 10.1017/CBO9781139173179. [8] 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. [9] 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. [10] 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. [11] H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence, Graduate Studies in Mathematics, 118. American Mathematical Society, Providence, RI, 2011. [12] 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. [13] L. B. Crowder, S. J. Lyman, W. F. Figueira and J. Priddy, Source-sink population dynamics and the problem of siting marine reserves, Bulletin of Marine Science-Miami, 66 (2000), 799-820. [14] A. R. Watkinson and W. J. Sutherland, Sources, sinks and pseudo-sinks, J. Anim. Ecol., 64 (1995), 126-130. [15] 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. [16] 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, athematical Biosciences, 264 (2015), 54-62.  doi: 10.1016/j.mbs.2015.03.005.
Phase-plane diagram of subsystem (3). Stable and unstable equilibria are identified by solid and open circles, respectively. Vector fields are shown by gray arrows. Isoclines of the nutrient and consumer are represented by red and blue lines, respectively. Let $N_{01} = 0.02, r_1 = k = 0.1, \gamma = m_1 = 0.01$. All positive solutions of (3) converge to equilibrium $E^+(0.111, 8.85)$
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