# American Institute of Mathematical Sciences

February  2022, 27(2): 977-1000. doi: 10.3934/dcdsb.2021077

## Dynamics of consumer-resource systems with consumer's dispersal between patches

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

Received  August 2020 Revised  December 2020 Published  February 2022 Early access  March 2021

Fund Project: The second author is supported by NSF grant of China (12071495, 11571382)

This paper considers consumer-resource systems with Holling II functional response. In the system, the consumer can move between a source and a sink patch. By applying dynamical systems theory, we give a rigorous analysis on persistence of the system. Then we show local/global stability of equilibria and prove Hopf bifurcation by the Kuznetsov Theorem. It is shown that dispersal in the system could lead to results reversing those without dispersal. Varying a dispersal rate can change species' interaction outcomes from coexistence in periodic oscillation, to persistence at a steady state, to extinction of the predator, and even to extinction of both species. By explicit expressions of stable equilibria, we prove that dispersal can make the consumer reach overall abundance larger than if non-dispersing, and there exists an optimal dispersal rate that maximizes the abundance. Asymmetry in dispersal can also lead to those results. It is proven that the overall abundance is a ridge-like function (surface) of dispersal rates, which extends both previous theory and experimental observation. These results are biologically important in protecting endangered species.

Citation: 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
##### 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. [2] R. Arditi, C. Lobry and T. Sari, Asymmetric dispersal in the multi-patch logistic equation, Theor. Popul. Biol., 120 (2018), 11-15. [3] J. Astr$\ddot{o}$m and T. P$\ddot{a}$rt, Negative and matrix-dependent effects of dispersal corridors in an experimental metacommunity., Ecology, 94 2013), 1939-1970. [4] G. Butler, H. I. Freedman and P. Waltman, Uniformly persistent systems, Proc. Amer. Math. Soc., 96 (1986), 425-430.  doi: 10.1090/S0002-9939-1986-0822433-4. [5] L. Fahrig, Effect of habitat fragmentation on the extinction threshold: A synthesis, Ecol. Appl., 12 (2002), 346-353. [6] W. Feng, B. Rock and J. Hinson, On a new model of two-patch predator-prey system with migration of both species, J. Appl. Anal. Comput., 1 (2011), 193-203.  doi: 10.11948/2011013. [7] D. Franco and A. Ruiz-Herrera, To connect or not to connect isolated patches, J. Theor. Biol., 370 (2015), 72-80.  doi: 10.1016/j.jtbi.2015.01.029. [8] 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. [9] E. E. Goldwyn and A. Hastings, When can dispersal synchronize populations?, Theor. Popul. Biol., 73 (2008), 395-402. [10] J. K. Hale, Ordinary Differential Equations, Wiley-Interscience, 1969. [11] J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge, UK, 1998.  doi: 10.1017/CBO9781139173179. [12] R. D. Holt, Population dynamics in two-patch environments: Some anomalous consequences of an optimal habitat distribution, Theoret. Population Biol., 28 (1985), 181-208.  doi: 10.1016/0040-5809(85)90027-9. [13] Y. Huang and O. Diekmann, Predator migration in response to prey density: What are the consequences?, J. Math. Biol., 43 (2001), 561-581.  doi: 10.1007/s002850100107. [14] R. Huang, Y. Wang and H. Wu, Population abundance in predator-prey systems with predator's dispersal between two patches, Theor. Popu. Biol., 135 (2020), 1-8. [15] 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. [16] V. A. Jansen, The dynamics of two diffusively coupled predator-prey populations, Theo. Popu. Biol., 59 (2001), 119-131. [17] Y. Kang, S. K. Sasmal and K. Messan, A two-patch prey-predator model with predator dispersal driven by the predation strength, Math. Biosci. Eng., 14 (2017), 843-880.  doi: 10.3934/mbe.2017046. [18] Y. A. Kuznetsov, Elements of Applied Bifurcation Theory in: Applied Mathematical Sciences, Vol. 112, third ed., Springer-Verlag, New York, 2004. doi: 10.1007/978-1-4757-3978-7. [19] Y. Liu, The Dynamical Behavior of a Two Patch Predator-Prey Model, Honor Thesis, from The College of William and Mary, 2010. [20] 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. [21] 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. [22] H. L. Smith and P. Waltman, The Theory of the Chemostat, Cambridge University Press, New York, 1995. [23] Y. Wang, Pollination-mutualisms in a two-patch system with dispersal, J. Theor. Biol., 476 (2019), 51-61.  doi: 10.1016/j.jtbi.2019.06.004. [24] Y. Wang, Asymptotic state of a two-patch system with infinite diffusion, Bull. Math. Biol., 81 (2019), 1665-1686.  doi: 10.1007/s11538-019-00582-4. [25] 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. [26] Y. Wang, H. Wu, Y. He, Z. Wang and K. Hu, Population abundance of two-patch competitive systems with asymmetric dispersal, J. Math. Biol., 81 (2020), 315-341.  doi: 10.1007/s00285-020-01511-z. [27] H. Wu, Y. Wang, Y. Li and D. L. DeAngelis, Dispersal asymmetry in a two-patch system with source-sink populations, Theor. Popul. Biol., 131 (2020), 54-65. [28] 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, Ecol. Lett., 20 (2017), 1118-1128. [29] B. Zhang, D. L. DeAngelis, W. M. Ni, Y. Wang, L. Zhai, A. Kula, S. Xu and J. D. Van Dyken, Effect of stressors on the carrying capacity of spatially-distributed metapopulations, The American Naturalis, 196 (2020), 46-60. [30] B. Zhang, X. Liu, D. L. DeAngelis, W.-M. Ni and G. 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.

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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. [2] R. Arditi, C. Lobry and T. Sari, Asymmetric dispersal in the multi-patch logistic equation, Theor. Popul. Biol., 120 (2018), 11-15. [3] J. Astr$\ddot{o}$m and T. P$\ddot{a}$rt, Negative and matrix-dependent effects of dispersal corridors in an experimental metacommunity., Ecology, 94 2013), 1939-1970. [4] G. Butler, H. I. Freedman and P. Waltman, Uniformly persistent systems, Proc. Amer. Math. Soc., 96 (1986), 425-430.  doi: 10.1090/S0002-9939-1986-0822433-4. [5] L. Fahrig, Effect of habitat fragmentation on the extinction threshold: A synthesis, Ecol. Appl., 12 (2002), 346-353. [6] W. Feng, B. Rock and J. Hinson, On a new model of two-patch predator-prey system with migration of both species, J. Appl. Anal. Comput., 1 (2011), 193-203.  doi: 10.11948/2011013. [7] D. Franco and A. Ruiz-Herrera, To connect or not to connect isolated patches, J. Theor. Biol., 370 (2015), 72-80.  doi: 10.1016/j.jtbi.2015.01.029. [8] 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. [9] E. E. Goldwyn and A. Hastings, When can dispersal synchronize populations?, Theor. Popul. Biol., 73 (2008), 395-402. [10] J. K. Hale, Ordinary Differential Equations, Wiley-Interscience, 1969. [11] J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge, UK, 1998.  doi: 10.1017/CBO9781139173179. [12] R. D. Holt, Population dynamics in two-patch environments: Some anomalous consequences of an optimal habitat distribution, Theoret. Population Biol., 28 (1985), 181-208.  doi: 10.1016/0040-5809(85)90027-9. [13] Y. Huang and O. Diekmann, Predator migration in response to prey density: What are the consequences?, J. Math. Biol., 43 (2001), 561-581.  doi: 10.1007/s002850100107. [14] R. Huang, Y. Wang and H. Wu, Population abundance in predator-prey systems with predator's dispersal between two patches, Theor. Popu. Biol., 135 (2020), 1-8. [15] 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. [16] V. A. Jansen, The dynamics of two diffusively coupled predator-prey populations, Theo. Popu. Biol., 59 (2001), 119-131. [17] Y. Kang, S. K. Sasmal and K. Messan, A two-patch prey-predator model with predator dispersal driven by the predation strength, Math. Biosci. Eng., 14 (2017), 843-880.  doi: 10.3934/mbe.2017046. [18] Y. A. Kuznetsov, Elements of Applied Bifurcation Theory in: Applied Mathematical Sciences, Vol. 112, third ed., Springer-Verlag, New York, 2004. doi: 10.1007/978-1-4757-3978-7. [19] Y. Liu, The Dynamical Behavior of a Two Patch Predator-Prey Model, Honor Thesis, from The College of William and Mary, 2010. [20] 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. [21] 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. [22] H. L. Smith and P. Waltman, The Theory of the Chemostat, Cambridge University Press, New York, 1995. [23] Y. Wang, Pollination-mutualisms in a two-patch system with dispersal, J. Theor. Biol., 476 (2019), 51-61.  doi: 10.1016/j.jtbi.2019.06.004. [24] Y. Wang, Asymptotic state of a two-patch system with infinite diffusion, Bull. Math. Biol., 81 (2019), 1665-1686.  doi: 10.1007/s11538-019-00582-4. [25] 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. [26] Y. Wang, H. Wu, Y. He, Z. Wang and K. Hu, Population abundance of two-patch competitive systems with asymmetric dispersal, J. Math. Biol., 81 (2020), 315-341.  doi: 10.1007/s00285-020-01511-z. [27] H. Wu, Y. Wang, Y. Li and D. L. DeAngelis, Dispersal asymmetry in a two-patch system with source-sink populations, Theor. Popul. Biol., 131 (2020), 54-65. [28] 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, Ecol. Lett., 20 (2017), 1118-1128. [29] B. Zhang, D. L. DeAngelis, W. M. Ni, Y. Wang, L. Zhai, A. Kula, S. Xu and J. D. Van Dyken, Effect of stressors on the carrying capacity of spatially-distributed metapopulations, The American Naturalis, 196 (2020), 46-60. [30] B. Zhang, X. Liu, D. L. DeAngelis, W.-M. Ni and G. 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.
Effect of dispersal $D_1$ on dynamics of system (1). Fix $r = 1, r_1 = 0.2, r_2 = 0.1, a_{12} = 0.9, a_{21} = 0.5, b = 1, K = 4, D_2 = 0.1$. (a-b) Let $D_1 = 0.03$ and $D_1 = 0.15$, respectively. The resource and consumer coexist in periodic oscillations, while the amplitude decreases with the increase of $D_1$. (c) Let $D_1 = 0.28$. The resource and consumer coexist at a steady state $P^*( 2.1250, 1.6272, 2.2787)$. (d) Let $D_1 = 0.9$. The consumer goes to extinction even though the resource species persists
Dynamics of system (1). Let $r = 1, r_1 = 0.2, r_2 = 0.1, a_{12} = 0.9, a_{21} = 0.8, b = 1, K = 4, D_1 = 0.3, D_2 = 0.1$. Numerical simulations display that all positive solutions (except $P^*$) of systems (1) converge to the unique limit cycle and exhibit periodic oscillations
Comparison of $T_1(D_1, D_2)$ and $T_0$. The red and black lines represent $T_1$ and $T_0$, respectively. Fix $r = 1, r_1 = 0.2, r_2 = 0.1, a_{12} = 2, a_{21} = 0.8, b = 1, K = 2$. (a) Fix $D_2 = 0.1$ but let $D_1$ vary. $T_1$ approaches its maximum $T_{1max} = 1.5089$ at $\bar{D_1} = 0.4276$, and the curve is hump-shaped. We have $T_1>T_0$ if $D_1<0.6305$; $T_1<T_0$ if $D_1>0.6305$ as shown in Proposition 4(ii). (b) Fix $D_1 = 0.86$ but let $D_2$ vary. $T_1$ approaches its maximum $T_{1max} = 1.5089$ at $\bar{D_2} = 0.3023$. The curve is hump-shaped if $D_2<0.6014$ but is convex if $D_2>0.6014$. We have $T_1>T_0$ if $D_2>\hat{D_2} = 0.1728$; $T_1<T_0$ if $D_2<0.1728$
The surface of $T_1(D_1, D_2)$ when both $D_1$ and $D_2$ vary. Fix $r = 1, r_1 = 0.2, r_2 = 0.1, a_{12} = 2, a_{21} = 0.8, b = 1, K = 2$. Then $T_1$ approaches its maximum $T_{1max} = 1.5089$ at a line $D_1 = 0.213+2.137D_2$, as shown in proposition 5(iii). This figure provides an intuition of the surface of $T_1 = T_1(D_1, D_2)$, which is a combination of Figs. 3a-b, i.e., when $D_2$ is fixed, the surface becomes Fig. 3a; when $D_1$ is fixed, the surface becomes Fig. 3b
Dynamics of system (1). Fix $r = 1, r_1 = 0.2, r_2 = 0.1, a_{12} = 0.9, b = 1, K = 4, D_1 = 0.28, D_2 = 0.1$. (a) Let $a_{21} = 0.4$. The consumer goes to extinction even though the resource species persists. (b) Let $a_{21} = 0.5$. The resource and consumer coexist at a steady state $P^*(2.1250, 1.6272, 2.2787)$. (c-d) Let $a_{21} = 0.65$ and $a_{21} = 0.8$, respectively. The resource and consumer coexist in periodic oscillation, while the amplitude increases with the increase of $a_{21}$
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