Skip to main content
SpringerLink
Log in

Further Topics and Related Models

  • Chapter
  • First Online:
Integrodifference Equations in Spatial Ecology

Part of the book series: Interdisciplinary Applied Mathematics ((IAM,volume 49))

Abstract

Even though IDEs are a relatively recent modeling framework in spatial ecology, their theory and applications contain many more aspects than can fit in a single book. In this final chapter, we mention further topics in the study of IDEs, some related to applications, some to the mathematical theory. We also briefly indicate a number of closely related model formulations and techniques. Some of these models are related in terms of the questions studied, others in terms of the mathematical structure of the equation.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 16.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 79.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  • Abbott, K., & Dwyer, G. (2008). Using mechanistic models to understand synchrony in forest insect populations: The North American gypsy moth as a case study. The American Naturalist, 172, 613–624.

    Article  Google Scholar 

  • Aydogmus, O., Kang, Y., Kavgaci, M., & Bereketoglu, H. (2017). Dynamical effects of nonlocal interactions in discrete-time growth-dispersal models with logistic-type nonlinearities. Ecological Complexity, 31, 88–95.

    Article  Google Scholar 

  • Bainov, D., & Simeonov, P. (1993). Impulsive differential equations: Periodic solutions and applications. Boca Raton: CRC Press.

    MATH  Google Scholar 

  • Baskett, M., Weitz, J., & Levin, S. (2007). The evolution of dispersal in reserve networks. The American Naturalist, 170, 59–78.

    Article  Google Scholar 

  • Best, A., Johst, K., Münkemüller, T., & Travis, J. (2007). Which species will successfully track climate change? The influence of intraspecific competition and density dependent dispersal on range shifting dynamics. Oikos, 116, 1531–1539.

    Article  Google Scholar 

  • Borer, E., Hosseini, P., Seabloom, E., & Dobson, A. (2007). Pathogen-induced reversal of native dominance in a grassland community. Proceedings of the National Academy of Sciences, 104(13), 5473–5478.

    Article  Google Scholar 

  • Brewster, C., & Allen, L. (1997). Spatiotemporal model for studying insect dynamics in large-scale cropping systems. Environmental Entomology, 26(3), 473–482.

    Article  Google Scholar 

  • Britton, N. (1989). Aggregation and the competitive exclusion principle. Journal of Theoretical Biology, 136, 57–66.

    Article  MathSciNet  Google Scholar 

  • Britton, N. (2009). Evolution in a host–parasite system. In Biomat 2008: International Symposium on Mathematical and Computational Biology (pp. 157–169). Singapore: World Scientific.

    Chapter  Google Scholar 

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

    MATH  Google Scholar 

  • Carrillo, C., & Fife, P. (2005). Spatial effects in discrete generation population models. Journal of Mathematical Biology, 50, 161–188.

    Article  MathSciNet  MATH  Google Scholar 

  • Cosner, C. (2014). Reaction-diffusion-advection models for the effects and evolution of dispersal. Discrete and Continuous Dynamical Systems - Series B, 35(5), 1701–1745.

    MathSciNet  MATH  Google Scholar 

  • Cuddington, K., & Hastings, A. (2004). Invasive engineers. Ecological Modelling, 178, 335–347.

    Article  Google Scholar 

  • de Camino-Beck, T., & Lewis, M. (2009). Invasion with stage-structured coupled map lattices: Application to the spread of scentless chamomile. Ecological Modelling, 220(23), 3394–3403.

    Article  Google Scholar 

  • Deng, P., de Vargas Roditi, L., van Ditmarsch, D., & Xavier, J. (2014). The ecological basis of morphogenesis: Branching patterns in swarming colonies of bacteria. New Journal of Physics, 16, 015006.

    Article  Google Scholar 

  • Derrida, B., & Spohn, H. (1988). Polymers on disordered trees, spin glasses, and traveling waves. Journal of Statistical Physics, 51(5–6), 817–840.

    Article  MathSciNet  MATH  Google Scholar 

  • Dewar, M., Scerri, K., & Kadirkamanathan, V. (2009). Data-driven spatio-temporal modeling using the integro-difference equation. IEEE Transactions of Signal Processing, 57(1), 83–91.

    Article  MathSciNet  MATH  Google Scholar 

  • Diekmann, O. (1978). Thresholds and travelling waves for the geographical spread of infection. Journal of Mathematical Biology, 6, 109–130.

    Article  MathSciNet  MATH  Google Scholar 

  • Doebeli, M., & Ruxton, G. (1998). Stabilization through spatial pattern formation in metapopulations with long–range dispersal. Proceedings of the Royal Society of London B, 265(1403), 1325–1332.

    Article  Google Scholar 

  • Dwyer, G., & Morris, W. (2006). Resource-dependent dispersal and the speed of biological invasions. The American Naturalist, 167(2), 165–176.

    Article  Google Scholar 

  • Easterling, M., Ellner, S., & Dixon, P. (2000). Size-specific sensitivity: Applying a new structured population model. Ecology, 81, 694–708.

    Article  Google Scholar 

  • Elderd, B., Rehill, B., Haynes, K., & Dwyer, G. (2013). Induced plant defenses, host–pathogen interactions, and forest insect outbreaks. Ecological Applications, 110(37), 14978–14983.

    Google Scholar 

  • Ellner, S., Childs, D., & Rees, M. (2016). Data-driven modelling of structured populations. Berlin: Springer.

    Book  MATH  Google Scholar 

  • Gaff, H., Joshi, H., & Lenhart, S. (2007). Optimal harvesting during an invasion of a sublethal plant pathogen. Environment and Development Economics, 12, 673–686.

    Article  Google Scholar 

  • Geritz, S., & Kisdi, É. (2004). On the mechanistic underpinning of discrete-time population models with complex dynamics. Journal of Theoretical Biology, 228, 261–269.

    Article  MathSciNet  Google Scholar 

  • Geritz, S., Kisdi, É., Meszéna, G., & Metz, J. (1998). Evolutionarily singular strategies and the adaptive growth and branching of the evolutionary tree. Evolutionary Ecology, 12(1), 35–57.

    Article  Google Scholar 

  • Gilbert, M., Grégoire, J.-C., Freise, J., & Heitland, W. (2004). Long-distance dispersal and human population density allow the prediction of invasive patterns in the horse chestnut leafminer cameraria ohridella. Journal of Animal Ecology, 73, 459–468.

    Article  Google Scholar 

  • Goodsman, D., Cooke, B., Coltman, D., & Lewis, M. (2014). The genetic signature of rapid range expansions: How dispersal, growth and invasion speed impact heterozygosity and allele surfing. Theoretical Population Biology, 98, 1–10.

    Article  MATH  Google Scholar 

  • Haefner, J., & Dugaw, C. (2000). Individual-based models solved using fast Fourier transforms. Ecological Modelling, 125, 159–172.

    Article  Google Scholar 

  • Hall, R., Hastings, A., & Ayres, D. (2006). Explaining the explosion: Modelling hybrid invasions. Proceedings of the Royal Society B, 273, 1385–1389.

    Google Scholar 

  • Hardin, D., Takáč, P., & Webb, G. (1990). Dispersion population models discrete in time and continuous in space. Journal of Mathematical Biology, 28, 1–20.

    Article  MathSciNet  MATH  Google Scholar 

  • Hastings, A. (1983). Can spatial variation alone lead to selection for dispersal? Theoretical Population Biology, 24, 244–251.

    Article  MathSciNet  MATH  Google Scholar 

  • Hooten, M., & Wikle, C. (2008). A hierarchical Bayesian non-linear spatio-temporal model for the spread of invasive species with application to Eurasian Collard-Dove. Environmental and Ecological Statistics, 15, 59–70.

    Article  MathSciNet  Google Scholar 

  • Hutson, V., Martinez, S., Mischaikow, K., & Vickers, G. (2003). The evolution of dispersal. Journal of Mathematical Biology, 46, 483–517.

    Article  MathSciNet  MATH  Google Scholar 

  • Jane White, K. A., Lenhart, S., & Martinez, M. V. (2015). Optimal control of integrodifference equations in pest-pathogen systems. Discrete & Continuous Dynamical Systems - Series B, 20(6), 1759–1783.

    Article  MathSciNet  MATH  Google Scholar 

  • Jongejans, E., Shea, K., Skarpaas, O., Kelly, D., & Ellner, S. (2011). Importance of individual and environmental variation for invasive species spread: A spatial integral projection model. Ecology, 92(1), 86–97.

    Article  Google Scholar 

  • Joshi, H., Lenhart, S., & Gaff, H. (2006). Optimal harvesting in an integro-difference population model. Optimal Control Applications & Methods, 27(2), 61–75.

    Article  MathSciNet  Google Scholar 

  • Joshi, H., Lenhart, S., Lou, H., & Gaff, H. (2007). Harvesting control in an integrodifference population model with concave growth term. Nonlinear Analysis: Hybrid Systems, 1, 417–429.

    MathSciNet  MATH  Google Scholar 

  • Kelly, Jr., M., Xing, Y., & Lenhart, S. (2016). Optimal fish harvesting for a population modeled by a nonlinear parabolic partial differential equation. Natural Resource Modeling, 29(1), 36–70.

    Article  MathSciNet  Google Scholar 

  • Kura, K., Khamis, D., El Mouden, C., & Bonsall, M. (2019). Optimal control for disease vector management in SIT models: An integrodifference equation approach. Journal of Mathematical Biology, 78, 1821–1839.

    Article  MathSciNet  MATH  Google Scholar 

  • Lakshmikantham, V., & Simeonov, P. (1989). Theory of impulsive differential equations (vol. 6). Singapore: World scientific.

    Google Scholar 

  • Legaspi, Jr., B., Allen, J., Brewster, C., Morales-Ramos, J., & King, E. (1998). Areawide management of the cotton boll weevil: Use of a spatio-temporal model in augmentative biological control. Ecological Modelling, 110, 151–164.

    Article  Google Scholar 

  • Lenhart, S., & Workman, J. (2007). Optimal control applied to biological models. London: CRC Press.

    MATH  Google Scholar 

  • Lewis, M., & Li, B. (2012). Spreading speed, traveling waves and the minimal patch-size in impulsive reaction-diffusion models. Bulletin of Mathematical Biology, 74(10), 2383–2402.

    Article  MathSciNet  MATH  Google Scholar 

  • Lewis, M., Marculis, N., & Shen, Z. (2018). Integrodifference equations in the presence of climate change: Persistence criterion, travelling waves and inside dynamics. Journal of Mathematical Biology, 77, 1649–1687.

    Article  MathSciNet  MATH  Google Scholar 

  • Lewis, M., Nelson, W., & Xu, C. (2010). A structured threshold model for mountain pine beetle outbreak. Bulletin of Mathematical Biology, 72, 565–589.

    Article  MathSciNet  MATH  Google Scholar 

  • Lewis, M., Neubert, M., Caswell, H., Clark, J., & Shea, K. (2006). A guide to calculating discrete-time invasion rates from data. In M. Cadotte, S. McMahon, & T. Fukami (Eds.), Conceptual ecology and invasions biology: Reciprocal approaches to nature (pp. 169–192). Berlin: Springer.

    Chapter  Google Scholar 

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

    Book  MATH  Google Scholar 

  • Lui, R. (1982a). A nonlinear integral operator arising from a model in population genetics, I. Monotone initial data. SIAM Journal on Mathematical Analysis, 13(6), 913–937.

    Article  MathSciNet  MATH  Google Scholar 

  • Lui, R. (1982b). A nonlinear integral operator arising from a model in population genetics, II. Initial data with compact support. SIAM Journal on Mathematical Analysis, 13(6), 938–953.

    Article  MathSciNet  MATH  Google Scholar 

  • Lui, R. (1985). A nonlinear integral operator arising from a model in population genetics, III. Heterozygote inferior case. SIAM Journal on Mathematical Analysis, 16(6), 1180–1206.

    Article  MathSciNet  MATH  Google Scholar 

  • Lui, R. (1986). A nonlinear integral operator arising from a model in population genetics, IV. clines. SIAM Journal on Mathematical Analysis, 17(1), 152–168.

    Article  MathSciNet  MATH  Google Scholar 

  • Lutscher, F. (2008). Density-dependent dispersal in integrodifference equations. Journal of Mathematical Biology, 56(4), 499–524.

    Article  MathSciNet  MATH  Google Scholar 

  • Marculis, N., Lewis, M., & Lui, R. (2017). Neutral genetic patterns for expanding populations with nonoverlapping generations. Bulletin of Mathematical Biology, 79, 828–852.

    Article  MathSciNet  MATH  Google Scholar 

  • Medlock, J., & Kot, M. (2003). Spreading diseases: Integro-differential equations new and old. Mathematical Biosciences, 184, 201–222.

    Article  MathSciNet  MATH  Google Scholar 

  • Merchant, S. (2001). Analysis of an integrodifference model for biological invasions with a quasi-local interaction. Master’s Thesis, University of British Columbia.

    Google Scholar 

  • Ming, P., & Albrecht, J. (2004). Integrated framework for the simulation of biological invasions in a heterogeneous landscape. Transactions in GIS, 8(3), 309–334.

    Article  Google Scholar 

  • Muthukrishnan, R., West, N. M., Davis, A., Jordan, N., & Forester, J. (2015). Evaluating the role of landscape in the spread of invasive species: The case of the biomass crop miscanthus × giganteus. Ecological Modelling, 317, 6–15.

    Article  Google Scholar 

  • Pachepsky, E., Nisbet, R. M., & Murdoch, W. W. (2008). Between discrete and continuous: Consumer-resource dynamics with synchronized reproduction. Ecology, 89, 280–288.

    Article  Google Scholar 

  • Phillips, B., Brown, G., Travis, J., & Shine, R. (2008). Reid’s paradox revisited: The evolution of dispersal kernels during range expansion. The American Naturalist, 172, S34–S48.

    Article  Google Scholar 

  • Roff, D. (1974). Spatial heterogeneity and the persistence of populations. Oecologia, 15, 245–258.

    Article  Google Scholar 

  • Scerri, K., Dewar, M., Aram, P., Freestone, D., Kadirkamanathan, V., & Grayden, D. (2011). Balanced reduction of an IDE-based spatio-temporal model. In Computational Tools 2011.

    Google Scholar 

  • Scerri, K., Dewar, M., & Kadirkamanathan, V. (2009). Estimation and model selection for an IDE-based spatio-temporal model. IEEE Transactions on Signal Processing, 57(2), 482–492.

    Article  MathSciNet  MATH  Google Scholar 

  • Sigrist, F., Künsch, H., & Stahel, W. (2012). A dynamic nonstationary spatio-temporal model for short term prediction of precipitation. The Annals of Applied Statistics, 6(4), 1452–1477.

    Article  MathSciNet  MATH  Google Scholar 

  • Skellam, J. G. (1951). Random dispersal in theoretical populations. Biometrika, 38, 196–218.

    Article  MathSciNet  MATH  Google Scholar 

  • Slatkin, M. (1973). Gene flow and selection in a cline. Genetics, 75, 733–756.

    MathSciNet  Google Scholar 

  • Thieme, H. (1979). Density-dependent regulation of spatially distributed populations and their asymptotic speed of spread. Journal of Mathematical Biology, 8, 173–187.

    Article  MathSciNet  MATH  Google Scholar 

  • Thieme, H., & Zhao, X.-Q. (2003). Asymptotic spreads of speed and traveling waves for integral equations and delayed reaction-diffusion models. Journal of Differential Equations, 195, 430–470.

    Article  MathSciNet  MATH  Google Scholar 

  • Travis, J., Harris, C., Park, K., & Bullock, J. (2011). Improving prediction and management of range expansions by combining analytical and individual-based modelling approaches. Methods in Ecology and Evolution, 2, 477–488.

    Article  Google Scholar 

  • Vasilyeva, O., Lutscher, F., & Lewis, M. (2016). Analysis of spread and persistence for stream insects with winged adult stages. Journal of Mathematical Biology, 72(4), 851–875.

    Article  MathSciNet  MATH  Google Scholar 

  • Veit, R. R., & Lewis, M. A. (1996). Dispersal, population growth, and the Allee effect: Dynamics of the house finch invasion in eastern North America. The American Naturalist, 148(2), 255–274.

    Article  Google Scholar 

  • Wang, J., Cui-Ping Cheng, C.-P., & Shuibo Huang, S. (2016). Evolution of dispersal in a spatially periodic integrodifference model. Nonlinear Analysis: Real World Applications, 32, 10–34.

    Article  MathSciNet  MATH  Google Scholar 

  • Wang, X., & Lutscher, F. (2019). Turing patterns in a predator–prey model with seasonality. Journal of Mathematical Biology, 78(3), 711–737.

    Article  MathSciNet  MATH  Google Scholar 

  • Webb, C. (2011). Exact asymptotics of the freezing transition of a logarithmically correlated energy model. Journal of Statistical Physics, 145(6) 1595–1619.

    Article  MathSciNet  MATH  Google Scholar 

  • Weinberger, H. (1982). Long-time behavior of a class of biological models. SIAM Journal on Mathematical Analysis, 13, 353–396.

    Article  MathSciNet  MATH  Google Scholar 

  • Wikle, C. (2001). A kernel-based spectral approach for spatio-temporal dynamic models. In Proceedings of the 1st Spanish Workshop on Spatio-Temporal Modelling of Environmental Processes (METMA) (pp. 167–180)

    Google Scholar 

  • Wikle, C. (2002). A kernel-based spectral model for non-Gaussian spatio-temporal processes. Statistical Modelling, 2(4), 299–314.

    Article  MathSciNet  MATH  Google Scholar 

  • Wikle, C. (2003). Hierarchical Bayesian models for predicting the spread of ecological processes. Ecology, 84(6), 1382–1394.

    Article  Google Scholar 

  • Wikle, C., & Holan, S. (2011). Polynomial nonlinear spatio-temporal integro-difference equation models. Journal of Time Series Analysis, 32(4), 339–350.

    Article  MathSciNet  MATH  Google Scholar 

  • Wikle, C., & Hooten, M. (2006). Hierarchical Bayesian spatio-temporal models for population spread. In J. Clark, & A. Gelfand (Eds.), Hierarchical modelling for the environmental sciences. Oxford: Oxford University Press.

    Google Scholar 

  • Williams, J., Snyder, R., & Levine, J. (2016). The influence of evolution on population spread through patchy landscapes. The American Naturalist, 188(1), 15–26.

    Article  Google Scholar 

  • Xu, K., Wikle, C., & Fox, N. I. (2005). A kernel-based spectral approach for spatio-temporal dynamical model for nowcasting weather radar reflectivities. Journal of the American Statistical Association, 100, 1133–1144.

    Article  MathSciNet  MATH  Google Scholar 

  • Zammit-Mangion, A., Dewar, M., Kadirkamanathan, V., & Sanguinetti, G. (2012). Point process modelling of the Afghan war diary. Proceedings of the National Academy of Sciences, 109, 12414–12419.

    Google Scholar 

  • Zhao, X.-Q. (2009). Spatial dynamics of some evolution systems in biology. In Y. Du, H. Ishii, & W.-Y. Lin (Eds.), Recent progress on reaction-diffusion systems and viscosity solutions (pp. 332–363). Singapore: World Scientific.

    Chapter  Google Scholar 

  • Zhong, P. (2011). Optimal Theory Applied in Integrodifference Equation Models and in a Cholera Differential Equation Model. Ph.D. Thesis, University of Tennessee.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematics and Statistics, University of Ottawa, Ottawa, ON, Canada

    Frithjof Lutscher

Authors
  1. Frithjof Lutscher
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Lutscher, F. (2019). Further Topics and Related Models. In: Integrodifference Equations in Spatial Ecology. Interdisciplinary Applied Mathematics, vol 49. Springer, Cham. https://doi.org/10.1007/978-3-030-29294-2_17

Download citation

Publish with us

Policies and ethics

Search

Navigation