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Assessing the Impacts of Competition and Dispersal on a Multiple Interactions Type Model

Murtala Bello Aliyu, Mohd Hafiz Mohd and Mohd Salmi Md. Noorani

Pertanika Journal of Science & Technology, Volume 29, Issue 3, July 2021

DOI: https://doi.org/10.47836/pjst.29.3.04

Keywords: Hopf bifurcation, limit cycle, limit point bifurcation, period-doubling bifurcation, stability, transcritical bifurcation

Published on: 31 July 2021

Multiple interactions (e.g., mutualist-resource-competitor-exploiter interactions) type models are known to exhibit oscillatory behaviour as a result of their complexity. This large-amplitude oscillation often de-stabilises multispecies communities and increases the chances of species extinction. What mechanisms help species in a complex ecological system to persist? Some studies show that dispersal can stabilise an ecological community and permit multi-species coexistence. However, previous empirical and theoretical studies often focused on one- or two-species systems, and in real life, we have more than two-species coexisting together in nature. Here, we employ a (four-species) multiple interactions type model to investigate how competition interacts with other biotic factors and dispersal to shape multi-species communities. Our results reveal that dispersal has (de-)stabilising effects on the formation of multi-species communities, and this phenomenon shapes coexistence mechanisms of interacting species. These contrasting effects of dispersal can best be illustrated through its combined influences with the competition. To do this, we employ numerical simulation and bifurcation analysis techniques to track the stable and unstable attractors of the system. Results show the presence of Hopf bifurcations, transcritical bifurcations, period-doubling bifurcations and limit point bifurcations of cycles as we vary the competitive strength in the system. Furthermore, our bifurcation analysis findings show that stable coexistence of multiple species is possible for some threshold values of ecologically-relevant parameters in this complex system. Overall, we discover that the stability and coexistence mechanisms of multiple species depend greatly on the interplay between competition, other biotic components and dispersal in multi-species ecological systems.

  • Abbott, K. C. (2011). A dispersal‐induced paradox: Synchrony and stability in stochastic metapopulations. Ecology Letters, 14(11), 1158-1169. https://doi.org/10.1111/j.1461-0248.2011.01670.x

  • Aliyu, M. B., & Mohd, M. H. (2021). Combined impacts of predation, mutualism and dispersal on the dynamics of a four-species ecological system. Pertanika Journal of Science & Technology, 29(1), 239-244. https://doi.org/10.47836/pjst.29.1.13

  • Allesina, S., & Tang, S. (2012). Stability criteria for complex ecosystems. Nature, 483(7388), 205-208. https://doi.org/10.1038/nature10832

  • Amarasekare, P. (2016). Evolution of dispersal in a multi‐trophic community context. Oikos, 125(4), 514-525. https://doi.org/10.1111/oik.02258

  • Anderson, K. E., & Hayes, S. M. (2018). The effects of dispersal and river spatial structure on asynchrony in consumer–resource metacommunities. Freshwater biology, 63(1), 100-113. https://doi.org/10.1111/fwb.12998

  • Bach, L. A., Thomsen, R., Pertoldi, C., & Loeschcke, V. (2006). Kin competition and the evolution of dispersal in an individual-based model. Ecological Modelling, 192(3-4), 658-666. https://doi.org/10.1016/j.ecolmodel.2005.07.026

  • Baek, H. (2018). Complex dynamics of a discrete-time predator-prey system with ivlev functional response. Mathematical Problems in Engineering, 2018, Article 8635937. https://doi.org/10.1155/2018/8635937

  • Barabás, G., D’Andrea, R., & Stump, S. M. (2018). Chesson’s coexistence theory. Ecological Monographs, 88(3), 277-303. https://doi.org/10.1002/ecm.1302

  • Barabás, G., J. Michalska-Smith, M., & Allesina, S. (2016). The effect of intra-and interspecific competition on coexistence in multispecies communities. The American Naturalist, 188(1), E1-E12. https://doi.org/10.1086/686901

  • Barraquand, F., Louca, S., Abbott, K. C., Cobbold, C. A., Cordoleani, F., DeAngelis, D. L., & Murray, D. L. (2017). Moving forward in circles: Challenges and opportunities in modelling population cycles. Ecology Letters, 20(8), 1074-1092. https://doi.org/10.1111/ele.12789

  • Bashkirtseva, I., Ryashko, L., & Ryazanova, T. (2019). Stochastic variability and transitions to chaos in a hierarchical three-species population model. Chaos, Solitons & Fractals, 119, 276-283. https://doi.org/10.1016/j.chaos.2018.12.035

  • Bassett, A., Krause, A. L., & Van Gorder, R. A. (2017). Continuous dispersal in a model of predator–prey-subsidy population dynamics. Ecological Modelling, 354, 115-122. https://doi.org/10.1016/j.ecolmodel.2017.02.017

  • Becks, L., Ellner, S. P., Jones, L. E., & Hairston Jr, N. G. (2012). The functional genomics of an eco‐evolutionary feedback loop: Linking gene expression, trait evolution, and community dynamics. Ecology Letters, 15(5), 492-501. https://doi.org/10.1111/j.1461- 0248.2012.01763.x

  • Bjørnstad, O. N. (2000). Cycles and synchrony: two historical ‘experiments’ and one experience. Journal of Animal Ecology, 69(5), 869-873. https://doi.org/10.1046/j.1365-2656.2000.00444.x

  • Briggs, C. J., & Hoopes, M. F. (2004). Stabilizing effects in spatial parasitoid–host and predator–prey models: A review. Theoretical Population Biology, 65(3), 299-315. https://doi.org/10.1016/j.tpb.2003.11.001

  • Bullock, J. M., Kenward, R. E., & Hails, R. S. (Eds.). (2002). Dispersal ecology: 42nd symposium of the British ecological society (Vol. 42). Cambridge University Press.

  • Carrara, F., Giometto, A., Seymour, M., Rinaldo, A., & Altermatt, F. (2015). Inferring species interactions in ecological communities: A comparison of methods at different levels of complexity. Methods in Ecology and Evolution, 6(8), 895-906. https://doi.org/10.1111/2041-210X.12363

  • Chaianunporn, T., & Hovestadt, T. (2012). Evolution of dispersal in metacommunities of interacting species. Journal of Evolutionary Biology, 25(12), 2511-2525. https://doi.org/10.1111/j.1420-9101.2012.02620.x

  • Chaianunporn, T., & Hovestadt, T. (2015). Evolutionary responses to climate change in parasitic systems. Global Change Biology, 21(8), 2905-2916. https://doi.org/10.1111/gcb.12944

  • Chesson, P. (2018). Updates on mechanisms of maintenance of species diversity. Journal of Ecology, 106(5), 1773-1794. https://doi.org/10.1111/1365-2745.13035

  • Chow, Y., Jang, S. R. J., & Yeh, N. S. (2018). Dynamics of a population in two patches with dispersal. Journal of Difference Equations and Applications, 24(4), 543-563. https://doi.org/10.1080/10236198.2018.1428962

  • Crooks, K. R., & Sanjayan, M. (Eds.). (2006). Connectivity conservation (Vol. 14). Cambridge University Press.

  • Crowley, P. H. (1981). Dispersal and the stability of predator-prey interactions. The American Naturalist, 118(5), 673-701. https://doi.org/10.1086/283861

  • Dey, S., & Joshi, A. (2006). Stability via asynchrony in Drosophila metapopulations with low migration rates. Science, 312(5772), 434-436. https://doi.org/10.1126/science.1125317

  • Feyrer, F., Hobbs, J., Acuna, S., Mahardja, B., Grimaldo, L., Baerwald, M., Johnson, R. C., & Teh, S. (2015). Metapopulation structure of a semi-anadromous fish in a dynamic environment. Canadian Journal of Fisheries and Aquatic Sciences, 72(5), 709-721. https://doi.org/10.1139/cjfas-2014-0433

  • Fussell, E. F., Krause, A. L., & Van Gorder, R. A. (2019). Hybrid approach to modeling spatial dynamics of systems with generalist predators. Journal of Theoretical Biology, 462, 26-47. https://doi.org/10.1016/j.jtbi.2018.10.054

  • Fussmann, G. F., & Gonzalez, A. (2013). Evolutionary rescue can maintain an oscillating community undergoing environmental change. Interface Focus, 3(6), Article 20130036. https://doi.org/10.1098/rsfs.2013.0036

  • Gandon, S. (1999). Kin competition, the cost of inbreeding and the evolution of dispersal. Journal of Theoretical Biology, 200(4), 345-364. https://doi.org/10.1006/jtbi.1999.0994

  • Gandon, S., & Rousset, F. (1999). Evolution of stepping-stone dispersal rates. Proceedings of the Royal Society of London. Series B: Biological Sciences, 266(1437), 2507-2513. https://doi.org/10.1098/rspb.1999.0953

  • Gause, G. F. (1932). Experimental studies on the struggle for existence: I. Mixed population of two species of yeast. Journal of Experimental Biology, 9(4), 389-402.

  • Gellner, G., & McCann, K. S. (2016). Consistent role of weak and strong interactions in high- and low-diversity trophic food webs. Nature Communications, 7(1), 1-7. https://doi.org/10.1038/ncomms11180

  • Goldwyn, E. E., & Hastings, A. (2008). When can dispersal synchronize populations? Theoretical Population Biology, 73(3), 395-402. https://doi.org/10.1016/j.tpb.2007.11.012

  • Gotelli, N. J. (2008). A primer of ecology, Sunderland. Sinauer Associates.

  • Gouhier, T. C., Guichard, F., & Gonzalez, A. (2010). Synchrony and stability of food webs in metacommunities. The American Naturalist, 175(2), E16-E34. https://doi.org/10.1086/649579

  • Green, D. M. (2009). Coevolution of dispersal in a parasitoid–host system. Population Ecology, 51(2), 253-260. https://doi.org/10.1007/s10144-008-0131-3

  • Grover, J. P., Hudziak, J., & Grover, J. D. (1997). Resource competition (Vol. 19). Springer Science & Business Media

  • Gupta, R. P., & Yadav, D. K. (2020). Complex dynamical behavior of a three species prey–predator system with nonlinear harvesting. International Journal of Bifurcation and Chaos, 30(13), Article 2050195. https://doi.org/10.1142/S0218127420501953

  • Gyllenberg, M., Jiang, J., Niu, L., & Yan, P. (2019). On the dynamics of multi-species Ricker models admitting a carrying simplex. Journal of Difference Equations and Applications, 25(11), 1489-1530. https://doi.org/10.1080/10236198.2019.1663182

  • Hanski, I. (1998). Metapopulation dynamics. Nature, 396(6706), 41-49. https://doi.org/10.1038/23876

  • Hardin, G. (1960). The competitive exclusion principle. Science, 131(3409), 1292-1297.

  • He, X., & Ni, W. M. (2013). The effects of diffusion and spatial variation in Lotka–Volterra competition–diffusion system I: Heterogeneity vs. homogeneity. Journal of Differential Equations, 254(2), 528-546. https://doi.org/10.1016/j.jde.2012.08.032

  • Holyoak, M. (2000). Effects of nutrient enrichment on predator–prey metapopulation dynamics. Journal of Animal Ecology, 69(6), 985-997. https://doi.org/10.1111/j.1365-2656.2000.00453.x

  • Hovestadt, T., Kubisch, A., & Poethke, H. J. (2010). Information processing in models for density-dependent emigration: a comparison. Ecological Modelling, 221(3), 405-410. https://doi.org/10.1016/j.ecolmodel.2009.11.005

  • Hudson, P. J., & Cattadori, I. (1999). The Moran effect: A cause of population synchrony. Trends in Ecology and Evolution, 14(1), 1-2. https://doi.org/10.1016/S0169-5347 (98)

  • Hutchinson, G. E. (1961). The paradox of the plankton. The American Naturalist, 95(882), 137-145. https://doi.org/10.1086/282171

  • Kakishima, S., Morita, S., Yoshida, K., Ishida, A., Hayashi, S., Asami, T., Ito, H., Miller III, D. G., Uehara, T., Mori, S., & Hasegawa, E. (2015). The contribution of seed dispersers to tree species diversity in tropical rainforests. Royal Society Open Science, 2(10), Article 150330. https://doi.org/10.1098/rsos.150330

  • Karakoç, C., Clark, A. T., & Chatzinotas, A. (2020). Diversity and coexistence are influenced by time‐dependent species interactions in a predator–prey system. Ecology Letters, 23(6), 983-993. https://doi.org/10.1111/ele.13500

  • Kendall, B. E., Bjørnstad, O. N., Bascompte, J., Keitt, T. H., & Fagan, W. F. (2000). Dispersal, environmental correlation, and spatial synchrony in population dynamics. The American Naturalist, 155(5), 628-636. https://doi.org/10.1086/303350

  • Kindlmann, P., & Burel, F. (2008). Connectivity measures: A review. Landscape ecology, 23(8), 879-890. https://doi.org/10.1007/s10980-008-9245-4

  • Koch, H., Frickel, J., Valiadi, M., & Becks, L. (2014). Why rapid, adaptive evolution matters for community dynamics. Frontiers in Ecology and Evolution, 2, Article 17. https://doi.org/10.3389/fevo.2014.00017

  • Kokkoris, G. D., Troumbis, A. Y., & Lawton, J. H. (1999). Patterns of species interaction strength in assembled theoretical competition communities. Ecology Letters, 2(2), 70-74. https://doi.org/10.1046/j.1461-0248.1999.22058.x

  • Kondoh, M. (2008). Building trophic modules into a persistent food web. Proceedings of the National Academy of Sciences, 105(43), 16631-16635. https://doi.org/10.1073/pnas.0805870105

  • Kondoh, M., & Mougi, A. (2015). Interaction-type diversity hypothesis and interaction strength: The condition for the positive complexity-stability effect to arise. Population Ecology, 57(1), 21-27. https://doi.org/10.1007/s10144-014-0475-9

  • Kool, J. T., Moilanen, A., & Treml, E. A. (2013). Population connectivity: Recent advances and new perspectives. Landscape Ecology, 28(2), 165-185. https://doi.org/10.1007/s10980-012-9819-z

  • Kouvaris, N., Kugiumtzis, D., & Provata, A. (2011). Species mobility induces synchronization in chaotic population dynamics. Physical Review E, 84(3), Article 036211.

  • Ladeira, D. G., & de Oliveira, M. M. (2019). Chaotic coexistence in a resource–consumer model. Journal of Biological Systems, 27(02), 167-184. https://doi.org/10.1142/S0218339019500086

  • Lampert, A., & Hastings, A. (2016). Stability and distribution of predator–prey systems: Local and regional mechanisms and patterns. Ecology letters, 19(3), 279-288. https://doi.org/10.1111/ele.12565

  • Landi, P., Minoarivelo, H. O., Brännström, Å., Hui, C., & Dieckmann, U. (2018). Complexity and stability of ecological networks: A review of the theory. Population Ecology, 60(4), 319-345. https://doi.org/10.1007/s10144-018-0628-3

  • Lee, A. M., Sæther, B. E., & Engen, S. (2020). Spatial covariation of competing species in a fluctuating environment. Ecology, 101(1), Article e02901. https://doi.org/10.1002/ecy.2901

  • Leibold, M. A., & Chase, J. M. (2017). Metacommunity ecology (Vol. 59). Princeton University Press.

  • Liu, X., & Huang, Q. (2018). The dynamics of a harvested predator–prey system with Holling type IV functional response. Biosystems, 169, 26-39. https://doi.org/10.1016/j.biosystems.2018.05.005

  • Loreau, M. (2010). Linking biodiversity and ecosystems: Towards a unifying ecological theory. Philosophical Transactions of the Royal Society B: Biological Sciences, 365(1537), 49-60. https://doi.org/10.1098/rstb.2009.0155

  • McCann, K., Hastings, A., & Huxel, G. R. (1998). Weak trophic interactions and the balance of nature. Nature, 395(6704), 794-798. https://doi.org/10.1038/27427

  • Mitani, N., & Mougi, A. (2017). Population cycles emerging through multiple interaction types. Royal Society Open Science, 4(9), Article 170536. https://doi.org/10.1098/rsos.170536

  • Mittelbach, G. G., & McGill, B. J. (2019). Community ecology. Oxford University Press.

  • Mohd, M. H. (2018). Numerical bifurcation and stability analyses of partial differential equations with applications to competitive system in ecology. In SEAMS School on Dynamical Systems and Bifurcation Analysis (pp. 117-132). Springer. https://doi.org/10.1007/978-981-32-9832-3_7

  • Mohd, M. H. (2019). Diversity in interaction strength promotes rich dynamical behaviours in a three-species ecological system. Applied Mathematics and Computation, 353, 243-253. https://doi.org/10.1016/j.amc.2019.02.007

  • Mohd, M. H. B. (2016). Modelling the presence-absence of multiple species (Doctoral dissertation). University of Canterbury. http://dx.doi.org/10.26021/1670

  • Mohd, M. H., & Noorani, M. S. M. (2020). Local dispersal, trophic interactions and handling times mediate contrasting effects in prey-predator dynamics. Chaos, Solitons & Fractals, 142, Article 110497. https://doi.org/10.1016/j.chaos.2020.110497

  • Mohd, M. H., Murray, R., Plank, M. J., & Godsoe, W. (2016). Effects of dispersal and stochasticity on the presence–absence of multiple species. Ecological Modelling, 342, 49-59. https://doi.org/10.1016/j.ecolmodel.2016.09.026

  • Mohd, M. H., Murray, R., Plank, M. J., & Godsoe, W. (2017). Effects of biotic interactions and dispersal on the presence-absence of multiple species. Chaos, Solitons & Fractals, 99, 185-194. https://doi.org/10.1016/j.chaos.2017.04.012

  • Mohd, M. H., Murray, R., Plank, M. J., & Godsoe, W. (2018). Effects of different dispersal patterns on the presence-absence of multiple species. Communications in Nonlinear Science and Numerical Simulation, 56, 115-130. https://doi.org/10.1016/j.cnsns.2017.07.029

  • Mondor, E. B., Rosenheim, J. A., & Addicott, J. F. (2005). Predator-induced transgenerational phenotypic plasticity in the cotton aphid. Oecologia, 142(1), 104-108. https://doi.org/10.1007/s00442-004-1710-4

  • Mougi, A. (2012). Unusual predator–prey dynamics under reciprocal phenotypic plasticity. Journal of theoretical biology, 305, 96-102. https://doi.org/10.1016/j.jtbi.2012.04.012

  • Mougi, A. (2016). Stability of an adaptive hybrid community. Scientific reports, 6, Article 28181. https://doi.org/10.1038/srep28181

  • Mougi, A., & Kondoh, M. (2012). Diversity of interaction types and ecological community stability. Science, 337(6092), 349-351. https://doi.org/10.1126/science.1220529

  • Mougi, A., & Kondoh, M. (2014). Adaptation in a hybrid world with multiple interaction types: A new mechanism for species coexistence. Ecological Research, 29(2), 113-119. https://doi.org/10.1007/s11284-013-1111-4

  • Moustafa, M., Mohd, M. H., Ismail, A. I., & Abdullah, F. A. (2020). Dynamical analysis of a fractional-order eco-epidemiological model with disease in prey population. Advances in Difference Equations, 2020(1), Article 48. https://doi.org/10.1186/s13662-020-2522-5

  • Namba, T., Takeuchi, Y., & Banerjee, M. (2018). Stabilizing effect of intra-specific competition on prey-predator dynamics with intraguild predation. Mathematical Modelling of Natural Phenomena, 13(3), Article 29. https://doi.org/10.1051/mmnp/2018033

  • Nath, B., & Das, K. P. (2020). Harvesting in tri-trophic food chain stabilises the chaotic dynamics-conclusion drawn from Hastings and Powell model. International Journal of Dynamical Systems and Differential Equations, 10(2), 95-115. https://doi.org/10.1504/IJDSDE.2020.106025

  • Omaiye, O. J., & Mohd, M. H. (2018). Computational dynamical systems using XPPAUT. In SEAMS School on Dynamical Systems and Bifurcation Analysis (pp. 175-203). Springer. https://doi.org/10.1007/978-981-32-9832-3_10

  • Perrin, N., & Goudet, J. (2001). Inbreeding, kinship, and the evolution of natal dispersal. Dispersal, 123-142.

  • Poethke, H. J., & Hovestadt, T. (2002). Evolution of density–and patch–size–dependent dispersal rates. Proceedings of the Royal Society of London. Series B: Biological Sciences, 269(1491), 637-645. https://doi.org/10.1098/rspb.2001.1936

  • Poethke, H. J., Hovestadt, T., & Mitesser, O. (2003). Local extinction and the evolution of dispersal rates: Causes and correlations. The American Naturalist, 161(4), 631-640. https://doi.org/10.1086/368224

  • Poethke, H. J., Pfenning, B., & Hovestadt, T. (2007). The relative contribution of individual and kin selection to the evolution of density-dependent dispersal rates. Evolutionary Ecology Research, 9(1), 41-50.

  • Poethke, H. J., Weisser, W. W., & Hovestadt, T. (2010). Predator-induced dispersal and the evolution of conditional dispersal in correlated environments. The American Naturalist, 175(5), 577-586. https://doi.org/10.1086/651595

  • Rozhnova, G., Metcalf, C. J. E., & Grenfell, B. T. (2013). Characterizing the dynamics of rubella relative to measles: the role of stochasticity. Journal of The Royal Society Interface, 10(88), Article 20130643. https://doi.org/10.1098/rsif.2013.0643

  • Selvam, A. G. M., & Dhineshbabu, R. (2020). Bifurcation and chaos in a discrete fractional order prey-predator system involving infection in prey. In Mathematical Models of Infectious Diseases and Social Issues (pp. 95-119). IGI Global.

  • Shabunin, A. V., Efimov, A., Tsekouras, G. A., & Provata, A. (2005). Scaling, cluster dynamics and complex oscillations in a multispecies Lattice Lotka–Volterra Model. Physica A: Statistical Mechanics and its Applications, 347, 117-136. https://doi.org/10.1016/j.physa.2004.09.021

  • Shabunin, A., & Provata, A. (2013). Lattice limit cycle dynamics: Influence of long-distance reactive and diffusive mixing. The European Physical Journal Special Topics, 222(10), 2547-2557. https://doi.org/10.1140/epjst/e2013-02036-5

  • Steiner, C. F., Stockwell, R. D., Kalaimani, V., & Aqel, Z. (2013). Population synchrony and stability in environmentally forced metacommunities. Oikos, 122(8), 1195-1206. https://doi.org/10.1111/j.1600-0706.2012.20936.x

  • Travis, J. M. (2001). The color of noise and the evolution of dispersal. Ecological Research, 16(1), 157-163. https://doi.org/10.1046/j.1440-1703.2001.00381.x

  • Tubay, J. M., Ito, H., Uehara, T., Kakishima, S., Morita, S., Togashi, T., Tainaka, K., Niraula, M. P., Casareto, B. E., Suzuki, Y. & Yoshimura, J. (2013). The paradox of enrichment in phytoplankton by induced competitive interactions. Scientific Reports, 3(1), 1-8. https://doi.org/10.1038/srep02835

  • Turchin, P. (2003). Complex population dynamics: A theoretical/empirical synthesis (Vol. 35). Princeton University Press.

  • Upadhyay, R. K., & Roy, P. (2016). Disease spread and its effect on population dynamics in heterogeneous environment. International Journal of Bifurcation and Chaos, 26(01), Article 1650004. https://doi.org/10.1142/S0218127416500048

  • Upadhyay, R. K., Roy, P., & Datta, J. (2015). Complex dynamics of ecological systems under nonlinear harvesting: Hopf bifurcation and Turing instability. Nonlinear Dynamics, 79(4), 2251-2270. https://doi.org/10.1007/s11071-014-1808-0

  • Vasseur, D. A., & Fox, J. W. (2009). Phase-locking and environmental fluctuations generate synchrony in a predator–prey community. Nature, 460(7258), 1007-1010. https://doi.org/10.1038/nature08208

  • Vellend, M. (2020). The theory of ecological communities (MPB-57). Princeton University Press.

  • Verma, T., & Gupta, A. K. (2020). Mean-field dispersal induced synchrony and stability in an epidemic model under patchy environment. Physica A: Statistical Mechanics and its Applications, 541, Article 123300. https://doi.org/10.1016/j.physa.2019.123300

  • Vogwill, T., Fenton, A., & Brockhurst, M. A. (2009). Dispersal and natural enemies interact to drive spatial synchrony and decrease stability in patchy populations. Ecology Letters, 12(11), 1194-1200. https://doi.org/10.1111/j.1461-0248.2009.01374.x

  • Wei, Z., Xia, Y., & Zhang, T. (2020). Stability and bifurcation analysis of an amensalism model with weak Allee effect. Qualitative Theory of Dynamical Systems, 19(1), Article 23. https://doi.org/10.1007/s12346-020-00341-0

  • Williams, P. D., & Hastings, A. (2013). Stochastic dispersal and population persistence in marine organisms. The American Naturalist, 182(2), 271-282.

  • Yaari, G., Ben-Zion, Y., Shnerb, N. M., & Vasseur, D. A. (2012). Consistent scaling of persistence time in metapopulations. Ecology, 93(5), 1214-1227. https://doi.org/10.1890/11-1077.1

  • Zhou, P. (2016). On a Lotka-Volterra competition system: diffusion vs advection. Calculus of Variations and Partial Differential Equations, 55(6), 137. https://doi.org/10.1007/s00526-016-1082-8

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JST-2256-2020

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