e-ISSN 2231-8526
ISSN 0128-7680
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
ISSN 0128-7680
e-ISSN 2231-8526