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Reaction–Diffusion Models: Single Species

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The Mathematics Behind Biological Invasions

Part of the book series: Interdisciplinary Applied Mathematics ((IAM,volume 44))

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Abstract

We revisit the baseline model of biological invasion consisting of a single partial differential equation of reaction–diffusion type. In spite of being one of the oldest models of biological invasion, it remains a valid and useful tool for understanding the spatiotemporal population dynamics of invasive species. We first apply this model to alien species establishment and show how to decide whether an initial population distribution results in extinction or survival. We then use the model to reveal the properties characterizing invasive species spread.

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References

  1. Andow, D.A., Kareiva, P.M., Levin, S.A., Okubo, A.: Spread of invading organisms. Landsc. Ecol. 4(2–3), 177–188 (1990). doi:10.1007/ bf00132860

    Article  Google Scholar 

  2. Aronson, D.G., Weinberger, H.F.: Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation. In: Goldstein, J.A. (ed.) Partial Differential Equations and Related Topics, pp. 5–49. Springer, New York (1975)

    Chapter  Google Scholar 

  3. Aronson, D.G., Weinberger, H.F.: Multidimensional nonlinear diffusion arising in population genetics. Adv. Math. 30, 33–76 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  4. Brauer, F., Soudack, A.C.: Response of predator–prey systems to nutrient enrichment and proportional harvesting. Int. J. Control 27(1), 65–86 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  5. Britton, N.F.: Reaction–Diffusion Equations and Their Applications to Biology. Academic, London (1986)

    MATH  Google Scholar 

  6. Crank, J.: The Mathematics of Diffusion. Oxford University Press, Oxford (1975)

    MATH  Google Scholar 

  7. Dennis, B.: Allee effects: population growth, critical density, and the chance of extinction. Nat. Resour. Model. 3(4), 481–538 (1989)

    MathSciNet  MATH  Google Scholar 

  8. Elton, C.S.: The Ecology of Invasions by Animals and Plants. Methuen, London (1958)

    Book  Google Scholar 

  9. Fife, P.: Mathematical Aspects of Reacting and Diffusing Systems. Springer, New York (1979)

    Book  MATH  Google Scholar 

  10. Fisher, R.A.: The wave of advance of advantageous genes. Ann. Eugen. 7, 355–369 (1937)

    Article  MATH  Google Scholar 

  11. Grindrod, P.: The Theory and Applications of Reaction–Diffusion Equations: Patterns and Waves. Oxford University Press, Oxford (1996)

    MATH  Google Scholar 

  12. Haberman, R.: Applied Partial Differential Equations: With Fourier Series and Boundary Value Problems, 5th edn. Pearson, Boston (2012)

    Google Scholar 

  13. Hadeler, K.P., Lewis, M.A.: Spatial dynamics of the diffusive logistic equation with a sedentary compartment. Can. Appl. Math. Q. 10(4), 473–499 (2002)

    MathSciNet  MATH  Google Scholar 

  14. Hadeler, K.P., Rothe, F.: Travelling fronts in nonlinear diffusion equations. J. Math. Biol. 2, 251–263 (1975). doi:10.1007/bf00277154

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

  16. Kanel, I.Y.: Stabilization of the solutions of the equations of combustion theory with finite initial functions. Matematicheskii Sbornik 65(107)(3), 398–413 (1964)

    Google Scholar 

  17. Kay, A.L., Sherratt, J.A., McLeod, J.B.: Comparison theorems and variable speed waves for a scalar reaction–diffusion equation. Proc. R. Soc. Edinb. A 131, 1133–1161 (2001). doi:10.1017/s030821050000130x

    Article  MathSciNet  MATH  Google Scholar 

  18. Keener, J.P.: A geometrical theory for spiral waves in excitable media. SIAM J. Appl. Math. 46, 1039–1056 (1986). doi:10.1137/0146062

    Article  MathSciNet  MATH  Google Scholar 

  19. Kierstead, H., Slobodkin, L.B.: The size of water masses containing plankton blooms. J. Mar. Res. 12 (1953)

    Google Scholar 

  20. Kolmogorov, A.N., Petrovskii, I.G., Piskunov, N.S.: A study of the diffusion equation with increase in the quantity of matter, and its application to a biological problem. Bull. Moscow Univ. Math. Ser. A 1, 1–25 (1937)

    Google Scholar 

  21. Kolmogorov, A.N., Petrovskii, I.G., Piskunov, N.S.: A study of the diffusion equation with increase in the amount of substance, and its application to a biological problem. In: Tikhomirov, V.M. (ed.) Selected Works of A. N. Kolmogorov, pp. 242–270. Kluwer Academic, Dordrecht (1991)

    Google Scholar 

  22. Lande, R.: Risks of population extinction from demographic and environmental stochasticity and random catastrophes. Am. Nat. 142(6), 911–927 (1993). doi:10.1086/285580

    Article  MathSciNet  Google Scholar 

  23. Larson, D.A.: Transient bounds and time-asymptotic behavior of solutions to nonlinear equations of Fisher type. J. Appl. Math. 34(1), 93–103 (1978). doi:10.1137/0134008

    MathSciNet  MATH  Google Scholar 

  24. Legović, T.: A recent increase in jellyfish populations: a predator–prey model and its implications. Ecol. Model. 38, 243–256 (1987). doi:10. 1016/0304-3800(87)90099-8

    Google Scholar 

  25. Lewis, M.A., Kareiva, P.: Allee dynamics and the spread of invading organisms. Theor. Popul. Biol. 43(2), 141–158 (1993). doi:10.1006/tpbi. 1993.1007

    Article  MATH  Google Scholar 

  26. Lewis, M.A., Schmitz, G.: Biological invasion of an organism with separate mobile and stationary states: modeling and analysis. Forma 11, 1–25 (1996)

    MathSciNet  MATH  Google Scholar 

  27. Lubina, J.A., Levin, S.A.: The spread of a reinvading species: range expansion in the California sea otter. Am. Nat. 131(4), 526–543 (1988). doi:10.1086/284804

    Article  Google Scholar 

  28. Ludwig, D., Aronson, D.G., Weinberger, H.F.: Spatial patterning of the spruce budworm. J. Math. Biol. 8, 217–258 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  29. Malchow, H., Petrovskii, S.V., Venturino, E.: Spatiotemporal Patterns in Ecology and Epidemiology: Theory, Models, and Simulation. Mathematical and Computational Biology Series. Chapman & Hall/CRC Press, Boca Raton (2008)

    MATH  Google Scholar 

  30. McKean, H.P.: Application of Brownian motion to the equation of Kolmogorov-Petrovskii-Piskunov. Commun. Pure Appl. Math. 28, 323–331 (1975). doi:10.1002/cpa.3160280302

    Article  MathSciNet  MATH  Google Scholar 

  31. Mollison, D.: Modelling biological invasions: chance, explanation, prediction. Philos. Trans. R. Soc. Lond. B 314, 675–693 (1986)

    Article  Google Scholar 

  32. Mollison, D.: Dependence of epidemic and population velocities on basic parameters. Math. Biosci. 107(2), 255–287 (1991). doi:10.1016/ 0025-5564(91)90009-8

    Article  MATH  Google Scholar 

  33. Murray, J.D.: Mathematical Biology. Springer, Berlin (1989)

    Book  MATH  Google Scholar 

  34. Neubert, M.G., Parker, I.M.: Projecting rates of spread for invasive species. Risk Anal. 24(4), 817–831 (2004). doi:10.1111/j.0272-4332. 2004.00481.x

    Article  Google Scholar 

  35. Petrovskii, S., Shigesada, N.: Some exact solutions of a generalized Fisher equation related to the problem of biological invasion. Math. Biosci. 172, 73–94 (2001). doi:10.1016/s0025-5564(01)00068-2

    Article  MathSciNet  MATH  Google Scholar 

  36. Petrovskii, S., Li, B.L., Malchow, H.: Transition to spatiotemporal chaos can resolve the paradox of enrichment. Ecol. Complex. 1, 37–47 (2004). doi:10.1016/j.ecocom.2003.10.001

    Article  Google Scholar 

  37. Petrovskii, S.V.: Critical nucleation parameters in an active bistable medium. Tech. Phys. 39, 747–749 (1994)

    Google Scholar 

  38. Petrovskii, S.V., Li, B.L.: Exactly Solvable Models of Biological Invasion. Chapman & Hall/CRC, Boca Raton (2006)

    MATH  Google Scholar 

  39. Protter, M.H., Weinberger, H.F.: Maximum Principles in Differential Equations. Springer, New York (1984)

    Book  MATH  Google Scholar 

  40. Renshaw, E.: Modelling Biological Populations in Space and Time. Cambridge University Press, Cambridge (1991)

    Book  MATH  Google Scholar 

  41. Rothe, F.: Convergence to travelling fronts in semilinear parabolic equations. Proc. R. Soc. Edinb. A 80, 213–234 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  42. Rothe, F.: Convergence to pushed fronts. Rocky Mt. J. Math. 11(4), 617–633 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  43. Sherratt, J.A., Marchant, B.P.: Algebraic decay and variable speeds in wavefront solutions of a scalar reaction–diffusion equation. IMA J. Appl. Math. 56, 289–302 (1996). doi:10.1093/imamat/56.2.289

    Article  MathSciNet  MATH  Google Scholar 

  44. Shigesada, N., Kawasaki, K.: Biological Invasions: Theory and Practice. Oxford University Press, Oxford (1997)

    Google Scholar 

  45. Shigesada, N., Kawasaki, K., Teramoto, E.: Traveling periodic waves in heterogeneous environments. Theor. Popul. Biol. 30(1), 143–160 (1986). doi:10.1016/0040-5809(86)90029-8

    Article  MathSciNet  MATH  Google Scholar 

  46. Skellam, J.G.: Random dispersal in theoretical populations. Biometrika 38(1–2), 196–218 (1951). doi:10.2307/2332328

    Article  MathSciNet  MATH  Google Scholar 

  47. Tikhonov, A.N., Samarskii, A.A.: Equations of Mathematical Physics. Dover, New York (1990)

    MATH  Google Scholar 

  48. Volpert, A.I., Hudjaev, S.I.: Analysis in Classes of Discontinuous Functions and Equations of Mathematical Physics. Martinus Nijhoff, Dordrecht (1985)

    Google Scholar 

  49. Volpert, V., Petrovskii, S.: Reaction–diffusion waves in biology. Phys. Life Rev. 6, 267–310 (2009). doi:10.1016/j.plrev.2009.10.002

    Article  Google Scholar 

  50. Volpert, A.I., Volpert, V.A., Volpert, V.A.: Traveling Wave Solutions of Parabolic Systems. American Mathematical Society, Providence (1994)

    MATH  Google Scholar 

  51. Wang, Q.R., Zhao, X.Q.: Spreading speed and traveling waves for the diffusive logistic equation with a sedentary compartment. Dyn. Continuous Discrete Impulsive Syst. A 13(2), 231–246 (2006)

    MathSciNet  MATH  Google Scholar 

  52. Wilson, W.G.: Resolving discrepancies between deterministic population models and individual-based simulations. Am. Nat. 151, 116–134 (1998). doi:10.1086/286106

    Article  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics & Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada

    Mark A. Lewis

  2. Department of Mathematics, University of Leicester, Leicester, UK

    Sergei V. Petrovskii

  3. School of Mathematics & Statistics, University of Sheffield, Sheffield, UK

    Jonathan R. Potts

Authors
  1. Mark A. Lewis
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  2. Sergei V. Petrovskii
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  3. Jonathan R. Potts
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Lewis, M.A., Petrovskii, S.V., Potts, J.R. (2016). Reaction–Diffusion Models: Single Species. In: The Mathematics Behind Biological Invasions. Interdisciplinary Applied Mathematics, vol 44. Springer, Cham. https://doi.org/10.1007/978-3-319-32043-4_3

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