Two Interacting Populations

  • Frithjof Lutscher
Part of the Interdisciplinary Applied Mathematics book series (IAM, volume 49)


Most biological populations do not exist in isolation but interact with other species in many ways that may increase or decrease their reproductive ability, affect their survival, or alter their dispersal behavior. Species interactions can lead to phenomena such as sustained population oscillations or competitive exclusion. In this chapter, we present some of the spatial aspects of population interaction in the context of IDEs. We begin with a brief background on nonspatial models before we move to study critical patch-sizes for predator and prey systems. Some of the most surprising and beautiful results in this section relate to dispersal-induced pattern formation in these systems. Spatial invasion dynamics of predator and prey show rich and complex behavior. We then present the phenomenon of anomalous spreading speeds in mutualism systems. Finally, we consider several aspects of persistence and invasion of competing species.


  1. Adler, F. (1993). Migration alone can produce persistence of host–parasitoid models. The American Naturalist, 141, 642–650.CrossRefGoogle Scholar
  2. Allen, L. (2006). An introduction to mathematical biology. New York: Pearson.Google Scholar
  3. Allen, E., Allen, L., & Gilliam, X. (1996). Dispersal and competition models for plants. Journal of Mathematical Biology, 34, 455–481.MathSciNetzbMATHCrossRefGoogle Scholar
  4. Allen, J., Brewster, C., & Slone, D. (2001). Spatially explicit ecological models: A spatial convolution approach. Chaos, Solitons & Fractals, 12, 333–347.zbMATHCrossRefGoogle Scholar
  5. Alonso, D., Bartumeus, F., & Catalan, J. (2002). Mutual interference between predators can give rise to Turing spatial patterns. Ecology, 83(1), 28–34.CrossRefGoogle Scholar
  6. Assaneo, F., Coutinho, R.M., Lin, Y., Mantilla, C., & Lutscher, F. (2013). Dynamics and coexistence in a system with intraguild mutualism. Ecological Complexity, 14, 64–74.CrossRefGoogle Scholar
  7. 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.CrossRefGoogle Scholar
  8. Beddington, J. R., Free, C. A., & Lawton, J. H. (1975). Dynamic complexity in predator–prey models framed in difference equations. Nature, 255(5503), 58.CrossRefGoogle Scholar
  9. Boucher, D. (1982). The ecology of mutualism. Annual Review of Ecology and Systematics, 13, 315–347.CrossRefGoogle Scholar
  10. Bramburger, J., & Lutscher, F. (2019) Analysis of integrodifference equations with a separable dispersal kernel. Acta Applicandae Mathematicae, 161(1), 127–151.MathSciNetzbMATHCrossRefGoogle Scholar
  11. Carrillo, C., Cherednichenko, K., Britton, N., & Mogie, M. (2009). Dynamic coexistence of sexual and asexual invasion fronts in a system of integro-difference equations. Bulletin of Mathematical Biology, 71, 1612–1625.MathSciNetzbMATHCrossRefGoogle Scholar
  12. Castillo-Chavez, C., Li, B., & Wang, H. (2013). Some recent developments on linear determinacy. Mathematical Biosciences and Engineering, 10, 1419–1436.MathSciNetzbMATHCrossRefGoogle Scholar
  13. Cobbold, C., Lewis, M., Lutscher, F., & Roland, J. (2005). How parasitism affects critical patch size in a host–parasitoid system: Application to forest tent caterpillar. Theoretical Population Biology, 67(2), 109–125.zbMATHCrossRefGoogle Scholar
  14. Dwyer, G., & Morris, W. (2006). Resource-dependent dispersal and the speed of biological invasions. The American Naturalist, 167(2), 165–176.CrossRefGoogle Scholar
  15. Edelstein-Keshet, L. (2005). Mathematical models in biology. Philadelphia: SIAM.zbMATHCrossRefGoogle Scholar
  16. Elliott, E., & Cornell, S. (2012). Dispersal polymorphism and the speed of biological invasions. PLoS ONE, 7(7), e40496.CrossRefGoogle Scholar
  17. Fagan, W., Lewis, M., Neubert, M., & van den Driessche, P. (2002). Invasion theory and biological control. Ecology Letters, 5, 148–157.CrossRefGoogle Scholar
  18. Fagan, W., Lewis, M., Neubert, M., Aumann, C., Apple, J., & Bishop, J. (2005). When can herbivores slow or reverse the spread of an invading plant? A test case from mount St. Helens. The American Naturalist, 166, 669–685.CrossRefGoogle Scholar
  19. Fasani, S., & Rinaldi, S. (2011). Factors promoting or inhibiting Turing instability in spatially extended prey–predator systems. Ecological Modelling, 222, 3449–3452.CrossRefGoogle Scholar
  20. Fort, J. (2012). Synthesis between demic and cultural diffusion in the neolithic transition in Europe. Proceedings of the National Academy of Sciences of the United States of America, 109(46), 18669–18673.CrossRefGoogle Scholar
  21. Fort, J., Pérez-Losada, J., Suñol, J., Escoda, L., & Massaneda, J. (2008). Integro-difference equations for interacting species and the neolithic transition. New Journal of Physics, 10, 043045.CrossRefGoogle Scholar
  22. Gouhier, T., Guichard, F., & Menge, B. (2010). Ecological processes can synchronize marine population dynamics over continental scales. Proceedings of the National Academy of Sciences of the United States of America, 107(18), 8281–8286.CrossRefGoogle Scholar
  23. Hart, D., & Gardner, R. (1997). A spatial model for the spread of invading organisms subject to competition. Journal of Mathematical Biology, 35, 935–948.MathSciNetzbMATHCrossRefGoogle Scholar
  24. Hassell, M. P. (1978). The dynamics of arthropod predator–prey systems. Princeton: Princeton University Press.zbMATHGoogle Scholar
  25. Holzer, M. (2014). Anomalous spreading in a system of coupled Fisher–KPP equations. Physica D, 270, 1–10.MathSciNetzbMATHCrossRefGoogle Scholar
  26. Hughes, J., Cobbold, C., Haynes, K., & Dwyer, G. (2015). Effects of forest spatial structure on insect outbreaks: Insights from a host–parasitoid model. The American Naturalist, 185(5), E130–E152.CrossRefGoogle Scholar
  27. Kanary, L., Musgrave, J., Locke, A., Tyson, R., & Lutscher, F. (2014). Modelling the dynamics of invasion and control of competing green crab genotypes. Theoretical Ecology, 7(4), 391–404.CrossRefGoogle Scholar
  28. Keener, J. (2000). Principles of applied mathematics. Boulder: Westview.zbMATHGoogle Scholar
  29. Kot, M. (1989). Diffusion-driven period-doubling bifurcations. BioSystems, 22, 279–287.CrossRefGoogle Scholar
  30. Kot, M. (1992). Discrete-time travelling waves: Ecological examples. Journal of Mathematical Biology, 30, 413–436.MathSciNetzbMATHCrossRefGoogle Scholar
  31. Kot, M. (2001). Elements of mathematical ecology. Cambridge: Cambridge University Press.zbMATHCrossRefGoogle Scholar
  32. Kot, M., & Phillips, A. (2015). Bounds for the critical speed of climate-driven moving-habitat models. Mathematical Biosciences, 262, 65–72.MathSciNetzbMATHCrossRefGoogle Scholar
  33. Kot, M., & Schaffer, W. (1986). Discrete-time growth-dispersal models. Mathematical Biosciences, 80, 109–136.MathSciNetzbMATHCrossRefGoogle Scholar
  34. 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.CrossRefGoogle Scholar
  35. Lewis, M., Li, B., & Weinberger, H. (2002). Spreading speed and linear determinacy for two-species competition models. Journal of Mathematical Biology, 45, 219–233.MathSciNetzbMATHCrossRefGoogle Scholar
  36. Lewis, M., Petrovskii, S., & Potts, J. (2016). The mathematics behind biological invasions. Berlin: Springer.zbMATHCrossRefGoogle Scholar
  37. Li, B. (2009). Some remarks on traveling wave solutions in competition models. Discrete and Continuous Dynamical Systems - Series B, 12, 389–399.MathSciNetzbMATHCrossRefGoogle Scholar
  38. Li, K., Huang, J., Li, X., & He, Y. (2016b). Asymptotic behavior and uniqueness of traveling wave fronts in a competitive recursion system. Zeitschrift für Angewandte Mathematik und Physik, 67(6), 144.MathSciNetzbMATHCrossRefGoogle Scholar
  39. Li, K., & Li, X. (2012a). Asymptotic behavior and uniqueness of traveling wave solutions in Ricker competition system. Journal of Mathematical Analysis and Applications, 389, 486–497.MathSciNetzbMATHCrossRefGoogle Scholar
  40. Li, K., & Li, X. (2012b). Travelling wave solutions in integro-difference competition system. IMA Journal of Applied Mathematics, 78(3), 633–650.MathSciNetzbMATHCrossRefGoogle Scholar
  41. Li, B., Weinberger, H., & Lewis, M. (2005). Spreading speeds as slowest wave speeds for cooperative systems. Mathematical Biosciences, 196, 82–98.MathSciNetzbMATHCrossRefGoogle Scholar
  42. Lin, G. (2015). Traveling wave solutions for integro-difference systems. Journal of Differential Equations, 258, 2908–2940.MathSciNetzbMATHCrossRefGoogle Scholar
  43. Lin, H.-T. (1995). On a system of integrodifference equations modelling the propagation of genes. SIAM Journal on Mathematical Analysis, 26(1), 35–76.MathSciNetzbMATHCrossRefGoogle Scholar
  44. Lin, G., & Li, W.-T. (2010). Spreading speeds and traveling wavefronts for second order integrodifference equations. Journal of Mathematical Analysis and Applications, 361(2), 520–532.MathSciNetzbMATHCrossRefGoogle Scholar
  45. Lin, G., Li, W.-T., & Ruan, S. (2011). Spreading speeds and traveling waves in competitive recursion systems. Journal of Mathematical Biology, 62(2), 165–201.MathSciNetzbMATHCrossRefGoogle Scholar
  46. Lui, R. (1989a). Biological growth and spread modeled by systems of recursions. I Mathematical theory. Mathematical Biosciences, 93, 269–295.MathSciNetzbMATHCrossRefGoogle Scholar
  47. Lutscher, F. (2008). Density-dependent dispersal in integrodifference equations. Journal of Mathematical Biology, 56(4), 499–524.MathSciNetzbMATHCrossRefGoogle Scholar
  48. Lutscher, F., & Iljon, T. (2013). Competition, facilitation and the Allee effect. Oikos, 122(4), 621–631.CrossRefGoogle Scholar
  49. May, M. (1973). Stability and complexity in model ecosystems. Princeton: Princeton University Press.Google Scholar
  50. May, R. M., Hassell, M. P., Anderson, R. M., & Tonkyn, D. W. (1981). Density dependence in host–parasitoid models. Journal of Animal Ecology, 50(3), 855–865.MathSciNetCrossRefGoogle Scholar
  51. Murray, J. D. (2001). Mathematical biology I: An introduction. Berlin: Springer.Google Scholar
  52. Murray, J. D. (2002). Mathematical biology II: Spatial models and biomedical applications. Berlin: Springer.CrossRefGoogle Scholar
  53. Neubert, M., Caswell, H., & Murray, J. (2002). Transient dynamics and pattern formation: Reactivity is necessary for Turing instabilities. Mathematical Biosciences, 175, 1–11.MathSciNetzbMATHCrossRefGoogle Scholar
  54. Neubert, M., & Kot, M. (1992). The subcritical collapse of predator populations in discrete time predator–prey models. Mathematical Biosciences, 110, 45–66.MathSciNetzbMATHCrossRefGoogle Scholar
  55. Neubert, M., Kot, M., & Lewis, M. A. (1995). Dispersal and pattern formation in a discrete-time predator–prey model. Theoretical Population Biology, 48(1), 7–43.zbMATHCrossRefGoogle Scholar
  56. Nicholson, A. (1954). An outline of the dynamics of animal populations. Australian Journal of Zoology, 2, 9–65.CrossRefGoogle Scholar
  57. Nicholson, A., & Bailey, V. (1935). The balance of animal populations. Part I. Proceedings of the Zoological Society of London, 105(3), 551–598.CrossRefGoogle Scholar
  58. Okubo, A., & Levin, S. A. (2001). Diffusion and ecological problems: Modern perspectives. New York: Springer.zbMATHCrossRefGoogle Scholar
  59. Okubo, A., Maini, P., Williamson, M., & Murray, J. (1989). On the spatial spread of the grey squirrel in Britain. Proceedings of the Royal Society B, 238, 113–125.Google Scholar
  60. Owen, M., & Lewis, M. (2001). How predation can slow, stop, or reverse a prey invasion. Bulletin of Mathematical Biology, 63, 655–684.MathSciNetzbMATHCrossRefGoogle Scholar
  61. Pan, S., & Lin, G. (2011). Propagation of second order integrodifference equations with local monotonicity. Nonlinear Analysis: Real World Applications, 12, 535–544.MathSciNetzbMATHCrossRefGoogle Scholar
  62. Pan, S., & Lin, G. (2014). Coinvasion-coexistence travelling wave solutions of an integro-difference competition system. Journal of Difference Equations and Applications, 20(4), 511–525.MathSciNetzbMATHCrossRefGoogle Scholar
  63. Pan, S., & Yang, P. (2014). Traveling wave solutions in a Lotka-Volterra type competition recursion. Advances in Difference Equations, 2014(1), 173.MathSciNetzbMATHCrossRefGoogle Scholar
  64. Ramanantoanina, A., Ouhinou, A., & Hui, C. (2014). Spatial assortment of mixed propagules explains the acceleration of range expansion. PLoS ONE, 9(8), e103409.CrossRefGoogle Scholar
  65. Ramanantoanina, A., Ouhinou, A., & Hui, C. (2015). Correction: Spatial assortment of mixed propagules explains the acceleration of range expansion. PLoS ONE, 10(8), e0136479.CrossRefGoogle Scholar
  66. Rietkerk, M., & van de Koppel, J. (2008). Regular pattern formation in real ecosystems. Trends in Ecology & Evolution, 23(3), 169–175.CrossRefGoogle Scholar
  67. Sherratt, J., Eagan, B., & Lewis, M. (1997). Oscillations and chaos behind predator–prey invasion: Mathematical artifact or ecological reality? Philosophical Transactions of the Royal Society of London B, 352, 21–38.CrossRefGoogle Scholar
  68. Tilman, D. (1982). Resource competition and community structure. Princeton: Princeton University Press.Google Scholar
  69. Turing, A. (1952). The chemical basis of morphogenesis. Philosophical Transactions of the Royal Society B, 237, 5–72.MathSciNetzbMATHGoogle Scholar
  70. Wang, H., & Castillo-Chavez, C. (2012). Spreading speeds and traveling waves for non-cooperative integro-difference systems. Discrete and Continuous Dynamical Systems - Series B, 17(6), 2243–2266.MathSciNetzbMATHCrossRefGoogle Scholar
  71. Wei, H., & Lutscher, F. (2013). From individual movement rules to population level patterns: The case of central-place foragers. In M. Lewis, P. Maini, & S. Petrovskii (Eds.), Dispersal, individual movement and spatial ecology. Lecture notes in mathematics (vol. 2071). Berlin: Springer.Google Scholar
  72. Weinberger, H. (1982). Long-time behavior of a class of biological models. SIAM Journal on Mathematical Analysis, 13, 353–396.MathSciNetzbMATHCrossRefGoogle Scholar
  73. Weinberger, H., Lewis, M., & Li, B. (2002). Analysis of linear determinacy for spread in cooperative models. Journal of Mathematical Biology, 45, 183–218.MathSciNetzbMATHCrossRefGoogle Scholar
  74. Weinberger, H., Lewis, M., & Li, B. (2007). Anomalous spreading speeds of cooperative recursion systems. Journal of Mathematical Biology, 55(2), 207–222.MathSciNetzbMATHCrossRefGoogle Scholar
  75. White, S., & White, K. (2005). Relating coupled map lattices to integro-difference equations: Dispersal-driven instabilities in coupled map lattices. Journal of Theoretical Biology, 235, 463–475.MathSciNetCrossRefGoogle Scholar
  76. Wright, R., & Hastings, A. (2007). Spontaneous patchiness in a host–parasitoid integrodifference model. Bulletin of Mathematical Biology, 69, 2693–2709.MathSciNetzbMATHCrossRefGoogle Scholar
  77. Zhang, Y., & Zhao, X.-Q. (2012). Bistable travelling waves in competitive recursion systems. Journal of Differential Equations, 252, 2630–2647.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Frithjof Lutscher
    • 1
  1. 1.Mathematics and StatisticsUniversity of OttawaOttawaCanada

Personalised recommendations