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Revisiting discrepancies between stochastic agent-based and deterministic models

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Abstract

Predicting which species will present (or absent) across geographical regions, and where, remains one of the important issues in ecology. From a methodical viewpoint, one of the concerns in examining species presence–absence across an environmental gradient is about the robustness of model-based predictions, which are given by distinct modelling frameworks used. Generally, different complexities and ecological factors are incorporated into such models, e.g. abiotic environments, spatial dispersal process, and stochasticity. Motivated by these ecological issues, we revisit a single-species logistic growth problem by employing stochastic agent-based model (ABM) and deterministic system and extend these frameworks to incorporate the effects of spatially changing environments. We observe that our ABM, which is formulated using random walk theory and birth–death process, demonstrates important qualitative behaviours that are consistent with the underlying theories of stochastic process. The results of ABM with large population sizes also agree with those of the deterministic equation. However, some discrepancies are observed when the population size is small. The ABM densities seem to underestimate the deterministic solutions, which illustrate the effects of stochasticity on small populations with some individuals may go extinct simply by chance. To quantify the underestimation of ABM as opposed to deterministic predictions, we employ certain probabilistic techniques: while the means of quasistationary probabilities distribution appear to give a counterintuitive prediction particularly near the edge of species ranges, the expected values given by state probabilities distribution are in agreement with the ABM densities observed for small population sizes across spatial locations. These salient observations depict emergent behaviours of stochastic ABM, which can contribute to additional insights on the dynamics of ecological species. It also shows how such small-scale interactions coupled with local dispersal and spatial phenomena occurring at a microscopic level can affect macroscopic-level dynamics. As such, comparing and contrasting the dynamics of different models can help in understanding the generality of ecological results and may offer important insights into the robustness of model-based predictions of species presence–absence.

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References

  • Acevedo, M., Urban, D., & Shugart, H. (1996). Models of forest dynamics based on roles of tree species. Ecological Modelling, 87, 267–284.

    Article  Google Scholar 

  • Adamson, M., & Morozov, A. Y. (2012). Revising the role of species mobility in maintaining biodiversity in communities with cyclic competition. Bulletin of Mathematical Biology, 74, 2004–2031.

    Article  PubMed  CAS  Google Scholar 

  • Allen, L. J. (2003). An introduction to stochastic processes with applications to biology. New Jersey: Pearson Education Upper Saddle River.

    Google Scholar 

  • Allen, L. J., & Allen, E. J. (2003). A comparison of three different stochastic population models with regard to persistence time. Theoretical Population Biology, 64, 439–449.

    Article  PubMed  Google Scholar 

  • Allen, L. J. S., & van den Driessche, P. (2013). Relations between deterministic and stochastic thresholds for disease extinction in continuous- and discrete-time infectious disease models. Mathematical Biosciences, 243, 99–108.

    Article  PubMed  CAS  Google Scholar 

  • Allen, L. J. S., & Lahodny, G. E. (2012). Extinction thresholds in deterministic and stochastic epidemic models. Journal of Biological Dynamics, 6, 590–611.

    Article  PubMed  Google Scholar 

  • Bahn, V., Oconnor, J., & Krohn, W. R. B. (2006). Effect of dispersal at range edges on the structure of species ranges. Oikos, 115, 89–96.

    Article  Google Scholar 

  • Bastille-Rousseau, G., Murray, D. L., Schaefer, J. A., Lewis, M. A., Mahoney, S. P., & Potts, J. R. (2018). Spatial scales of habitat selection decisions: Implications for telemetry-based movement modelling. Ecography, 41, 437–443.

    Article  Google Scholar 

  • Bennett, A. F., & Saunders, D. A. (2010). Habitat fragmentation and landscape change. Conservation Biology for All, 93, 1544–1550.

    Google Scholar 

  • Bergman, C. M., Schaefer, J. A., & Luttich, S. (2000). Caribou movement as a correlated random walk. Oecologia, 123, 364–374.

    Article  PubMed  CAS  Google Scholar 

  • Birand, A., Vose, A., & Gavrilets, S. (2012). Patterns of species ranges, speciation, and extinction. The American Naturalist, 179, 1–21.

    Article  PubMed  Google Scholar 

  • Bobbink, R., Beltman, B., Verhoeven, J., & Whigham, D. (2007). Wetlands: Functioning, biodiversity conservation, and restoration. Berlin: Springer.

    Google Scholar 

  • Byers, J. A. (1999). Effects of attraction radius and flight paths on catch of scolytid beetles dispersing outward through rings of pheromone traps. Journal of Chemical Ecology, 25, 985–1005.

    Article  CAS  Google Scholar 

  • Byers, J. A. (2001). Correlated random walk equations of animal dispersal resolved by simulation. Ecology, 82, 1680–1690.

    Article  Google Scholar 

  • Cadotte, M. W. (2006). Metacommunity influences on community richness at multiple spatial scales: A microcosm experiment. Ecology, 87, 1008–1016.

    Article  PubMed  Google Scholar 

  • Cadotte, M. W., & Fukami, T. (2005). Dispersal, spatial scale, and species diversity in a hierarchically structured experimental landscape. Ecology Letters, 8, 548–557.

    Article  PubMed  Google Scholar 

  • Cain, M., Milligan, B., & Strand, A. (2000). Long-distance seed dispersal in plant populations. American Journal of Botany, 87, 1217–1227.

    Article  PubMed  CAS  Google Scholar 

  • Cantrell, R. S., & Cosner, C. (1998). On the effects of spatial heterogeneity on the persistence of interacting species. Journal of Mathematical Biology, 37, 103–145.

    Article  Google Scholar 

  • Cantrell, R., & Cosner, C. (2003). Spatial ecology via reaction-diffusion equations. London: Wiley.

    Google Scholar 

  • Case, T. J., Holt, R. D., McPeek, M. A., & Keitt, T. H. (2005). The community context of species’ borders: Ecological and evolutionary perspectives. Oikos, 108, 28–46.

    Article  Google Scholar 

  • Caughley, G. (1994). Directions in conservation biology. Journal of Animal Ecology, 2, 215–244.

    Article  Google Scholar 

  • Charles, K. J. (2009). Ecology: The experimental analysis of distribution and abundance (6th ed.). San Francisco: Person International Edition.

    Google Scholar 

  • Codling, E. A., Plank, M. J., & Benhamou, S. (2008). Random walk models in biology. Journal of The Royal Society Interface, 5, 813–834.

    Article  PubMed  PubMed Central  Google Scholar 

  • Comins, H., & Noble, I. (1985). Dispersal, variability, and transient niches: Species coexistence in a uniformly variable environment. The American Naturalist, 126, 706–723.

    Article  Google Scholar 

  • Davidson, J. (1938). On the ecology of the growth of the sheep population in south Australia. Transactions of the Royal Society of South Australia, 62, 141–148.

    Google Scholar 

  • DeAngelis, D. L., & Matsinos, Y. (1995). Individual-based population models: Linking behavioral and physiological information at the individual level to population dynamics. Ecologia Austral, 6, 23–31.

    Google Scholar 

  • DeAngelis, D. L., & Mooij, W. M. (2005). Individual-based modeling of ecological and evolutionary processes. Annual Review of Ecology Evolution and Systematics, 36, 147–168.

    Article  Google Scholar 

  • DeAngelis, D., Zhang, B., Ni, W. M., & Wang, Y. (2020). Carrying capacity of a population diffusing in a heterogeneous environment. Mathematics, 8, 49.

    Article  Google Scholar 

  • Dieckmann, U., Law, R., & Metz, J. A. (2000). The geometry of ecological interactions: Simplifying spatial complexity. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  • Dockery, J., Hutson, V., Mischaikow, K., & Pernarowski, M. (1998). The evolution of slow dispersal rates: A reaction diffusion model. Journal of Mathematical Biology, 37, 61–83.

    Article  Google Scholar 

  • Durrett, R., & Levin, S. (1994). The importance of being discrete (and spatial). Theoretical Population Biology, 46, 363–394.

    Article  Google Scholar 

  • Fagan, W. F., Lewis, M. A., Auger-Méthé, M., Avgar, T., Benhamou, S., Breed, G., et al. (2013). Spatial memory and animal movement. Ecology Letters, 16, 1316–1329.

    Article  PubMed  Google Scholar 

  • Faugeras, B., & Maury, O. (2007). Modeling fish population movements: From an individual-based representation to an advection-diffusion equation. Journal of Theoretical Biology, 247, 837–848.

    Article  PubMed  Google Scholar 

  • Fisher, R. A. (1937). The wave of advance of advantageous genes. Annals of Eugenics, 7, 355–369.

    Article  Google Scholar 

  • Gail, M. H., & Boone, C. W. (1970). The locomotion of mouse fibroblasts in tissue culture. Biophysical Journal, 10, 980.

    Article  PubMed  PubMed Central  CAS  Google Scholar 

  • Gaston, K. J. (2003). The structure and dynamics of geographic ranges. Oxford: Oxford University Press.

    Google Scholar 

  • Gaston, K. J. (2009). Geographic range limits of species. Proceedings of the Royal Society of London B Biological Sciences, 276, 1391–1393.

    Article  CAS  Google Scholar 

  • Gaston, K., & He, F. (2002). The distribution of species range size: A stochastic process. Proceedings of the Royal Society B Biological sciences, 269, 1079–1086.

    Article  PubMed Central  Google Scholar 

  • Godsoe, W., Murray, R., & Plank, M. (2015). Information on biotic interactions improves transferability of distribution models. The American Naturalist, 185, 281–290.

    Article  PubMed  Google Scholar 

  • Godsoe, W., Murray, R., & Plank, M. J. (2015). The effect of competition on species’ distributions depends on coexistence, rather than scale alone. Ecography, 38, 1071–1079.

    Article  Google Scholar 

  • Griffin, J. N., Jenkins, S. R., Gamfeldt, L., Jones, D., Hawkins, S. J., & Thompson, R. C. (2009). Spatial heterogeneity increases the importance of species richness for an ecosystem process. Oikos, 118, 1335–1342.

    Article  Google Scholar 

  • Guisan, A., Tingley, R., Baumgartner, J. B., Naujokaitis-Lewis, I., Sutcliffe, P. R., Tulloch, A. I., et al. (2013). Predicting species distributions for conservation decisions. Ecology Letters, 16, 1424–1435.

    Article  PubMed  PubMed Central  Google Scholar 

  • Hamilton, W. D. (1967). Extraordinary sex ratios. Science, 156, 477–488.

    Article  PubMed  CAS  Google Scholar 

  • Hastings, A., & Gross, L. J. (2012). Encyclopedia of theoretical ecology. New York: University of California Press.

    Google Scholar 

  • Holmes, E. E., Lewis, M. A., Banks, J., & Veit, R. (1994). Partial differential equations in ecology: Spatial interactions and population dynamics. Ecology, 75, 17–29.

    Article  Google Scholar 

  • Holt, R. D., & Barfield, M. (2009). Trophic interactions and range limits: The diverse roles of predation. Proceedings of the Royal Society B Biological Sciences, 276, 1435–1442.

    Article  PubMed Central  Google Scholar 

  • Holt, R. D., Keitt, T. H., Lewis, M. A., Maurer, B. A., & Taper, M. L. (2005). Theoretical models of species-borders: Single species approaches. Oikos, 108, 18–27.

    Article  Google Scholar 

  • Kandler, A., & Unger, R. (2010). Population Dispersal Via Diffusion-reaction equations. Technical University, Fak. für Mathematik.

  • Kearney, M., & Porter, W. (2009). Mechanistic niche modelling: Combining physiological and spatial data to predict species-ranges. Ecology Letters, 12, 334–350.

    Article  PubMed  Google Scholar 

  • Krebs, C. J. (1972). The experimental analysis of distribution and abundance. Ecology New York Harper and Row, 2, 1–14.

    Google Scholar 

  • Kudryashov, N. A., & Zakharchenko, A. S. (2014). A note on solutions of the generalized fisher equation. Applied Mathematics Letters, 32, 53–56.

    Article  Google Scholar 

  • Law, R., Murrell, D. J., & Dieckmann, U. (2003). Population growth in space and time: Spatial logistic equations. Ecology, 84, 252–262.

    Article  Google Scholar 

  • Levin, S., Carpenter, S., Godfray, H., Kinzig, A., Loreau, M., Losos, J., et al. (2009). The Princeton guide to ecology. Princeton University Press: Princeton.

    Book  Google Scholar 

  • Levin, S. A., Cohen, D., & Hastings, A. (1984). Dispersal strategies in patchy environments. Theoretical Population Biology, 26, 165–191.

    Article  Google Scholar 

  • Levin, L. A., Sibuet, M., Gooday, A. J., Smith, C. R., & Vanreusel, A. (2010). The roles of habitat heterogeneity in generating and maintaining biodiversity on continental margins: An introduction. Berlin: Springer.

    Google Scholar 

  • Lewis, M. A., Petrovskii, S. V., & Potts, J. R. (2016). The mathematics behind biological invasions. Berlin: Springer.

    Book  Google Scholar 

  • Lotka, A. J. (1925). Elements of physical biology. New York: Williams and Wilkins.

    Google Scholar 

  • MacLean, W. P., & Holt, R. D. (1979). Distributional patterns in st. croix sphaerodactylus lizards: The taxon cycle in action. Biotropica, 11, 189–195.

    Article  Google Scholar 

  • Mittelbach, G. (2012). Community Ecology. London: Incorporated Sinauer Associates.

    Google Scholar 

  • Mohd, M. H. (2018). How can modelling tools inform environmental and conservation policies? International Journal of Engineering Technology, 7, 555.

    Google Scholar 

  • 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.

    Article  Google Scholar 

  • Mohd, M. H. (2022). Effects of dimensionality of space on the presence/absence of multiple species. Environmental Modeling Assessment, 27, 327–342.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Méndez, V., Campos, D., & Bartumeus, F. (2013). Stochastic foundations in movement ecology: Anomalous diffusion, front propagation and random searches. Berlin: Springer.

    Google Scholar 

  • Nathan, R., & Muller-Landau, H. (2000). Spatial patterns of seed dispersal, their determinants and consequences for recruitment. Trends in Ecology Evolution, 15, 278–285.

    Article  PubMed  CAS  Google Scholar 

  • Nisbet, R., & Gurney, W. (1982). Modelling fluctuating populations. New York: Wiley.

    Google Scholar 

  • Nowak, M. A., & May, R. M. (1992). Evolutionary games and spatial chaos. Nature, 359, 826–829.

    Article  Google Scholar 

  • Okubo, A., & Levin, S. A. (2013). Diffusion and ecological problems: Modern perspectives. Berlin: Springer.

    Google Scholar 

  • Pacala, S. W., & Silander, J., Jr. (1985). Neighborhood models of plant population dynamics. I. Single-species models of annuals. The American Naturalist, 125, 385–411.

    Article  Google Scholar 

  • Pearson, R. G., & Dawson, T. P. (2003). Predicting the impacts of climate change on the distribution of species: Are bioclimate envelope models useful? Global Ecology and Biogeography, 12, 361–371.

    Article  Google Scholar 

  • Potts, J. R., & Lewis, M. A. (2016). How memory of direct animal interactions can lead to territorial pattern formation. Journal of the Royal Society Interface, 13, 20160059.

    Article  PubMed  PubMed Central  Google Scholar 

  • Renshaw, E. (1993). Modelling biological populations in space and time. Cambridge: Cambridge University Press.

    Google Scholar 

  • Roughgarden, J. (1979). Theory of population genetics and evolutionary ecology: An introduction. London: Macmillan Publishing.

    Google Scholar 

  • Sexton, J. P., McIntyre, P. J., Angert, A. L., & Rice, K. J. (2009). Evolution and ecology of species range limits. Annual Review of Ecology Evolution and Systematics, 40, 415–436.

    Article  Google Scholar 

  • Shigesada, N., & Kawasaki, K. (1997). Biological invasions: Theory and practice. Oxford: Oxford University Press.

    Google Scholar 

  • Shmida, A., & Wilson, M. V. (1985). Biological determinants of species diversity. Journal of Biogeography, 2, 1–20.

    Article  Google Scholar 

  • Simonis, J. L. (2012). Demographic stochasticity reduces the synchronizing effect of dispersal in predator-prey metapopulations. Ecology, 93, 1517–1524.

    Article  PubMed  Google Scholar 

  • Skellam, J. (1951). Random dispersal in theoretical populations. Biometrika, 2, 196–218.

    Article  Google Scholar 

  • Soberón, J. (2007). Grinnellian and eltonian niches and geographic distributions of species. Ecology Letters, 10, 1115–1123.

    Article  PubMed  Google Scholar 

  • Verhulst, P. F. (1845). Recherches mathématiques sur la loi d’accroissement de la population. Nouveaux Memoires de lAcademie Royale des Sciences et Belles Lettres de Bruxelles, 18, 14–54.

    Google Scholar 

  • Volterra, V. (1926). Fluctuations in the abundance of a species considered mathematically. Nature, 118, 558–560.

    Article  Google Scholar 

  • Van Nes, E. H., & Scheffer, M. (2007). Slow recovery from perturbations as a generic indicator of a nearby catastrophic shift. The American Naturalist, 169, 738–747.

    Article  PubMed  Google Scholar 

  • Vries Gd (2006) A course in mathematical biology: Quantitative modeling with mathematical and computational methods, Society for Industrial and Applied Mathematics.

  • Weisberg, M., & Reisman, K. (2008). The robust Volterra principle. Philosophy of Science, 75, 106–131.

    Article  Google Scholar 

  • Weiss, G. H. (1983). Random walks and their applications: Widely used as mathematical models, random walks play an important role in several areas of physics, chemistry, and biology. American Scientist, 71, 65–71.

    CAS  Google Scholar 

  • Wilson, W. G. (1998). Resolving discrepancies between deterministic population models and individual-based simulations. The American Naturalist, 151, 116–134.

    Article  PubMed  CAS  Google Scholar 

  • Wilson, W., McCauley, E., & De Roos, A. (1995). Effect of dimensionality on Lotka-Volterra predator-prey dynamics: Individual based simulation results. Bulletin of Mathematical Biology, 57, 507–526.

    Article  Google Scholar 

  • Wisz, M. S., Pottier, J., Kissling, W. D., Pellissier, L., Lenoir, J., Damgaard, C. F., et al. (2013). The role of biotic interactions in shaping distributions and realised assemblages of species: Implications for species distribution modelling. Biological Reviews, 88, 15–30.

    Article  PubMed  Google Scholar 

  • Yang, Z., Liu, X., Zhou, M., Ai, D., Wang, G., Wang, Y., et al. (2015). The effect of environmental heterogeneity on species richness depends on community position along the environmental gradient. Scientific Reports, 5, 1–7.

    Google Scholar 

  • Zhang, B., DeAngelis, D. L., & Ni, W. M. (2021). Carrying capacity of spatially distributed metapopulations. Trends in Ecology Evolution, 36, 164–173.

    Article  PubMed  Google Scholar 

  • Zhang, B., Kula, A., Mack, K. M., Zhai, L., Ryce, A. L., Ni, W. M., et al. (2017). Carrying capacity in a heterogeneous environment with habitat connectivity. Ecology Letters, 20, 1118–1128.

    Article  PubMed  Google Scholar 

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Acknowledgements

The author acknowledges the support from the Fundamental Research Grant Scheme with Project Code: FRGS/1/2022/STG06/USM/02/1 by the Ministry of Higher Education, Malaysia (MOHE).

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    Mohd Hafiz Mohd

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Mohd, M.H. Revisiting discrepancies between stochastic agent-based and deterministic models. COMMUNITY ECOLOGY 23, 453–468 (2022). https://doi.org/10.1007/s42974-022-00118-2

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