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Dynamics of predator-prey systems with prey’s dispersal between patches. (English) Zbl 1489.34054

Summary: This paper analyzes dispersal in predator-prey systems, where the prey can move between a source and a sink patch and the predation interaction is described by the Holling II functional response. By applying dynamical systems theory, we present a complete study on persistence of the system, and show local/global stability of equilibria. Then we prove Hopf bifurcation in the system by computing Lyapunov coefficient, and show that dispersal could lead to results reversing those if non-dispersing. By explicit expressions of stable equilibria, we show that dispersal could make the prey approach total abundance larger than if non-dispersing, even larger than its carrying capacity when the predator persists. Asymmetry in dispersal could also lead to the results. Total abundance of the prey is shown to be a distorted function (surface) of dispersal rates, which extends both previous theory and experimental observations. It is proven that there exists an optimal dispersal that drives the predator into extinction and makes the prey reach the maximal abundance. These results are biologically important in preserving endangered species.

MSC:

34C12 Monotone systems involving ordinary differential equations
34C28 Complex behavior and chaotic systems of ordinary differential equations
37G20 Hyperbolic singular points with homoclinic trajectories in dynamical systems
37N25 Dynamical systems in biology
92D25 Population dynamics (general)
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References:

[1] I. Hanski, Metapopulation ecology. Oxford University Press, 1999.
[2] Zhang, B.; DeAngelis, DL; Ni, WM; Wang, Y.; Zhai, L.; Kula, A.; Xu, S.; Van Dyken, JD, Effect of stressors on the carrying capacity of spatially-distributed metapopulations, The American Naturalist, 196, 46-60 (2020) · doi:10.1086/709293
[3] Aström, J.; Pärt, T., Negative and matrix-dependent effects of dispersal corridors in an experimental metacommunity, Ecology, 94, 1939-1970 (2013) · doi:10.1890/11-1795.1
[4] Holt, RD, Population dynamics in two-patch environments: Some anomalous consequences of an optimal habitat distribution, Theor. Popul. Biol., 28, 181-207 (1985) · Zbl 0584.92022 · doi:10.1016/0040-5809(85)90027-9
[5] Wu, H.; Wang, Y.; Li, Y.; DeAngelis, DL, Dispersal asymmetry in a two-patch system with source-sink populations, Theor. Popul. Biol., 131, 54-65 (2020) · Zbl 1516.92099 · doi:10.1016/j.tpb.2019.11.004
[6] Zhang, B.; Liu, X.; DeAngelis, DL; Ni, W-M; Wang, GG, Effects of dispersal on total biomass in a patchy, heterogeneous system: analysis and experiment, Math. Biosci., 264, 54-62 (2015) · Zbl 1371.92083 · doi:10.1016/j.mbs.2015.03.005
[7] Zhang, B.; Alex, K.; Keenan, ML; Lu, Z.; Arrix, LR; Ni, W-M; DeAngelis, DL; Van Dyken, JD, Carrying capacity in a heterogeneous environment with habitat connectivity, Ecology Letters, 20, 1118-1128 (2017) · doi:10.1111/ele.12807
[8] H.I. Freedman, 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. · Zbl 0362.92006
[9] Arditi, R.; Lobry, C.; Sari, T., Is dispersal always beneficial to carrying capacity? New insights from the multi-patch logistic equation, Theor. Popul. Biol., 106, 45-59 (2015) · Zbl 1343.92389 · doi:10.1016/j.tpb.2015.10.001
[10] Arditi, R.; Lobry, C.; Sari, T., Asymmetric dispersal in the multi-patch logistic equation, Theoretical Population Biology, 120, 11-15 (2018) · Zbl 1397.92555 · doi:10.1016/j.tpb.2017.12.006
[11] Goldwyn, EE; Hastings, A., When can dispersal synchronize populations?, Theor. Popul. Biol., 73, 395-402 (2008) · Zbl 1210.92051 · doi:10.1016/j.tpb.2007.11.012
[12] E.C. Haskell and J. Bell, Pattern formation in a predator-mediated coexistence model with prey-texis . Disc. Cont. Dyna. Syst.-B, 25(2020): 2895-2921. · Zbl 1509.35035
[13] Huang, Y.; Diekmann, O., Predator migration in response to prey density: What are the consequences?, Journal of Mathematical Biology, 43, 561-581 (2001) · Zbl 0995.92044 · doi:10.1007/s002850100107
[14] Hutson, V.; Lou, Y.; Mischaikow, K., Convergence in competition models with small diffusion coefficients, J. Differential Equations, 211, 135-161 (2005) · Zbl 1074.35054 · doi:10.1016/j.jde.2004.06.003
[15] Lou, Y., On the effects of migration and spatial heterogeneity on single and multiple species, J. Differential Equations, 223, 400-426 (2006) · Zbl 1097.35079 · doi:10.1016/j.jde.2005.05.010
[16] Ruiz-Herrera, A., Metapopulation dynamics and total biomass: Understanding the effects of diffusion in complex networks, Theor. Popul. Biol., 121, 1-11 (2018) · Zbl 1397.92600 · doi:10.1016/j.tpb.2018.03.002
[17] Ruiz-Herrera, A.; Torres, PJ, Effects of diffusion on total biomass in simple metacommunities, Journal of Theoretical Biology, 447, 12-24 (2018) · Zbl 1397.92601 · doi:10.1016/j.jtbi.2018.03.018
[18] Wang, X.; Zanette, L.; Zou, X., Modelling the fear effect in predator-prey interactions, Journal of Mathematical Biology, 73, 1179-1204 (2016) · Zbl 1358.34058 · doi:10.1007/s00285-016-0989-1
[19] Wang, Y., Asymptotic state of a two-patch system with infinite diffusion, Bulletin of Mathematical Biology, 81, 1665-1686 (2019) · Zbl 1415.92215 · doi:10.1007/s11538-019-00582-4
[20] Wang, Y.; Wu, H.; He, Y.; Wang, Z.; Hu, K., Population abundance of two-patch competitive systems with asymmetric dispersal, Journal of Mathematical Biology, 81, 315-341 (2020) · Zbl 1448.34102 · doi:10.1007/s00285-020-01511-z
[21] Tiwari, V.; Tripathi, JP; Mishra, S.; Upadhyay, RK, Modeling the fear effect and stability of non-equilibrium patterns in mutually interfering predator-prey systems, Appl. Math. Comp., 371 (2020) · Zbl 1433.92042 · doi:10.1016/j.amc.2019.124948
[22] Huang, R.; Wang, Y.; Wu, H., Population abundance in predator-prey systems with predator’s dispersal between two patches, Theor. Popu. Biol., 135, 1-8 (2020) · Zbl 1516.92084 · doi:10.1016/j.tpb.2020.06.002
[23] Hofbauer, J.; Sigmund, K., Evolutionary Games and Population Dynamics (1998), Cambridge, UK: Cambridge University Press, Cambridge, UK · Zbl 0914.90287 · doi:10.1017/CBO9781139173179
[24] J.K. Hale, Ordinary differential equations, Wiley-Interscience, 1969. · Zbl 0186.40901
[25] Smith, HL; Waltman, P., The Theory of the Chemostat (1995), New York: Cambridge University Press, New York · Zbl 0860.92031 · doi:10.1017/CBO9780511530043
[26] Butler, G.; Freedman, HI; Waltman, P., Uniformly persistent systems, Proceedings of the American Mathematical Society, 83, 425-430 (1986) · Zbl 0603.34043 · doi:10.1090/S0002-9939-1986-0822433-4
[27] Li, MY; Shuai, Z., Global-stability problem for coupled systems of differential equations on networks, J. Differential Equations, 248, 1-20 (2010) · Zbl 1190.34063 · doi:10.1016/j.jde.2009.09.003
[28] Y.A. Kuznetsov, Elements of Applied Bifurcation Theory in: Applied Mathematical Sciences, Vol. 12, third ed., Springer-Verlag, New York, 2004. · Zbl 1082.37002
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