Skip to main content

Advertisement

SpringerLink
Log in

Saddle-Point Approximations, Integrodifference Equations, and Invasions

  • Original Article
  • Published:
Bulletin of Mathematical Biology Aims and scope Submit manuscript

Abstract

Invasion, the growth in numbers and spatial spread of a population over time, is a fundamental process in ecology. Governments and businesses expend vast sums to prevent and control invasions of pests and pestilences and to promote invasions of endangered species and biological control agents. Many mathematical models of biological invasions use nonlinear integrodifference equations to describe the growth and dispersal processes and to predict the speed of invasion fronts. Linear models have received less attention, perhaps because they are difficult to simulate for large times.

In this paper, we use the saddle-point method, alias the method of steepest descent, to derive asymptotic approximations for the solutions of linear integrodifference equations. We work through five examples, for Gaussian, Laplace, and uniform dispersal kernels in one dimension and for asymmetric Gaussian and radially symmetric Laplace kernels in two dimensions. Our approximations are extremely close to the exact solutions, even for intermediate times. We also employ an empirical saddle-point approximation to predict densities using dispersal data. We use our approximations to examine the effects of censored dispersal data on estimates of invasion speed and population density.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Andersen, M., 1991. Properties of some density-dependent integrodifference equation population models. Math. Biosci. 104, 135–157.

    Article  MATH  MathSciNet  Google Scholar 

  • Britton, N.F., 1986. Reaction-Diffusion Equations and Their Applications to Biology. Academic, London.

    MATH  Google Scholar 

  • Brown, J.K.M., Hovmoller, M.S., 2002. Aerial dispersal of pathogens on the global and continental scales and its impact on plant disease. Science 297, 537–541.

    Article  Google Scholar 

  • Bullock, J.M., Clarke, R.T., 2000. Long distance seed dispersal by wind: measuring and modelling the tail of the curve. Oecologia 124, 506–521.

    Article  Google Scholar 

  • Butler, R.W., 2007. Saddlepoint Approximations with Applications. Cambridge University Press, Cambridge.

    MATH  Google Scholar 

  • Cain, M.L., Milligan, B.G., Strand, A.E., 2000. Long-distance seed dispersal in plant populations. Am. J. Bot. 87, 1217–1227.

    Article  Google Scholar 

  • Caswell, H., Lensink, R., Neubert, M.G., 2003. Demography and dispersal: life table response experiments for invasion speed. Ecology 84, 1968–1978.

    Article  Google Scholar 

  • Clark, J.S., 1998. Why trees migrate so fast: Confronting theory with dispersal biology and the paleorecord. Am. Nat. 152, 204–224.

    Article  Google Scholar 

  • Clark, J.S., Horvath, L., Lewis, M., 2001. On the estimation of spread rate for a biological population. Stat. Probab. Lett. 51, 225–234.

    Article  MATH  MathSciNet  Google Scholar 

  • Cobbold, C.A., Lewis, M.A., Lutscher, F., Roland, J., 2005. How parasitism affects critical patch-size in a host–parasitoid model: application to the forest tent caterpillar. Theor. Popul. Biol. 67, 109–125.

    Article  MATH  Google Scholar 

  • Cohen, A., 1991. A Padé approximant to the inverse Langevin function. Rheol. Acta 30, 270–273.

    Article  Google Scholar 

  • Daniels, H.E., 1954. Saddlepoint approximations in statistics. Ann. Math. Stat. 25, 631–650.

    Article  MATH  MathSciNet  Google Scholar 

  • Domb, C., Offenbacher, E.L., 1978. Random walks and diffusion. Am. J. Phys. 46, 49–56.

    Article  Google Scholar 

  • Fagan, W.F., Lewis, M., Neubert, M.G., Aumann, C., Apple, J.L., Bishop, J.G., 2005. When can herbivores slow or reverse the spread of an invading plant? A test case from Mount St. Helens. Am. Nat. 166, 669–685.

    Article  Google Scholar 

  • Feller, W., 1971. An Introduction to Probability Theory and its Applications, vol. II. Wiley, New York.

    MATH  Google Scholar 

  • Feuerverger, A., 1989. On the empirical saddlepoint approximation. Biometrika 76, 457–464.

    Article  MATH  MathSciNet  Google Scholar 

  • Fisher, R.A., 1937. The wave of advance of advantageous genes. Ann. Eugen. 7, 355–369.

    Google Scholar 

  • Fort, J., 2007. Fronts from complex two-dimensional dispersal kernels: Theory and application to Reid’s paradox. J. Appl. Phys. 101, 094701.

    Article  Google Scholar 

  • Giffin, W.C., 1975. Transform Techniques for Probability Modeling. Academic, New York.

    MATH  Google Scholar 

  • Good, I.J., 1957. Saddle-point methods for the multinomial distribution. Ann. Math. Stat. 28, 861–881.

    Article  MathSciNet  Google Scholar 

  • Goutis, C., Casella, G., 1999. Explaining the saddlepoint approximation. Am. Stat. 53, 216–224.

    Article  MathSciNet  Google Scholar 

  • Hart, D.R., Gardner, R.H., 1997. A spatial model for the spread of invading organisms subject to competition. J. Math. Biol. 35, 935–948.

    Article  MATH  MathSciNet  Google Scholar 

  • Higgins, S.I., Richardson, D.M., 1999. Predicting plant migration rates in a changing world: the role of long-distance dispersal. Am. Nat. 153, 464–475.

    Article  Google Scholar 

  • Hughes, B.D., 1995. Random Walks and Random Environments. Volume 1: Random Walks. Oxford University Press, Oxford.

    Google Scholar 

  • Jacquemyn, H., Brys, R., Neubert, M.G., 2005. Fire increases invasive spread of Molinia caerulea mainly through changes in demographic parameters. Ecol. Appl. 15, 2097–2108.

    Article  Google Scholar 

  • Jakeman, E., Pusey, P.N., 1976. A model for non-Rayleigh sea echo. IEEE Trans. Antennas Propag. 24, 806–814.

    Article  Google Scholar 

  • Kot, M., 1992. Discrete-time travelling waves: ecological examples. J. Math. Biol. 30, 413–436.

    Article  MATH  MathSciNet  Google Scholar 

  • Kot, M., Schaffer, W.M., 1986. Discrete-time growth–dispersal models. Math. Biosci. 80, 109–136.

    Article  MATH  MathSciNet  Google Scholar 

  • Kot, M., Lewis, M.A., van den Driessche, P., 1996. Dispersal data and the spread of invading organisms. Ecology 77, 2027–2042.

    Article  Google Scholar 

  • Kotz, S., Kozubowski, T.J., Podgorski, K., 2001. The Laplace Distribution and Generalizations: A Revisit with Applications to Communications, Economics, Engineering, and Finance. Birkhäuser, Boston.

    MATH  Google Scholar 

  • Krkosek, M., Lauzon-Guay, J.S., Lewis, M.A., 2007. Relating dispersal and range expansion of California sea otters. Theor. Popul. Biol. 71, 401–407.

    Article  MATH  Google Scholar 

  • Lewis, M.A., 1997. Variability, patchiness, and jump dispersal in the spread of an invading population. In D. Tilman, P. Kareiva (Eds.) Spatial Ecology: The Role of Space in Population Dynamics and Interspecific Interactions, pp. 46–69. Princeton University Press, Princeton.

    Google Scholar 

  • Lewis, M.A., Neubert, M.G., Caswell, H., Clark, J., Shea, K., 2006. A guide to calculating discrete-time invasion rates from data. In M.W. Cadotte, S.M. McMahon, T. Fukami (Eds.) Conceptual Ecology and Invasions Biology: Reciprocal Approaches to Nature, pp. 169–192. Springer, Dordrecht.

    Chapter  Google Scholar 

  • Lobatschewsky, N., 1842. Probabilité des résultats moyens tirés d’observations répetées. J. Reine Angew. Math. 24, 164–170.

    MATH  Google Scholar 

  • Lui, R., 1983. Existence and stability of travelling wave solutions of a nonlinear integral operator. J. Math. Biol. 16, 199–220.

    Article  MATH  MathSciNet  Google Scholar 

  • Lusk, E.J., Wright, H., 1982. Deriving the probability density for sums of uniform random variables. Am. Stat. 36, 128–130.

    Article  Google Scholar 

  • Lutscher, F., 2007. A short note on short dispersal events. Bull. Math. Biol. 69, 1615–1630.

    Article  MATH  MathSciNet  Google Scholar 

  • McKay, A.T., 1932. A Bessel function distribution. Biometrika 24, 39–44.

    Google Scholar 

  • Mistro, D.C., Rodrigues, L.A.D., Ferreira, W.C., 2005. The Africanized honey bee dispersal: a mathematical zoom. Bull. Math. Biol. 67, 281–312.

    Article  MathSciNet  Google Scholar 

  • Mollison, D., 1991. Dependence of epidemic and population velocities on basic parameters. Math. Biosci. 107, 255–287.

    Article  MATH  Google Scholar 

  • Murray, J.D., 1974. Asymptotic Analysis. Oxford University Press, Oxford.

    MATH  Google Scholar 

  • Nathan, R., Perry, G., Cronin, J.T., Strand, A.E., Cain, M.L., 2003. Methods for estimating long-distance dispersal. Oikos 103, 261–273.

    Article  Google Scholar 

  • Neubert, M.G., Caswell, H., 2000. Demography and dispersal: calculation and sensitivity analysis of invasion speed for structured populations. Ecology 81, 1613–1628.

    Article  Google Scholar 

  • Neubert, M.G., Parker, I.M., 2004. Projecting rates of spread for invasive species. Risk Anal. 24, 817–831.

    Article  Google Scholar 

  • Neubert, M.G., Kot, M., Lewis, M.A., 1995. Dispersal and pattern formation in a discrete-time predator–prey model. Theor. Popul. Biol. 48, 7–43.

    Article  MATH  Google Scholar 

  • Okubo, A., 1980. Diffusion and Ecological Problems: Mathematical Models. Springer, Berlin.

    MATH  Google Scholar 

  • Paolella, M.S., 2007. Intermediate Probability: A Computational Approach. Wiley, Chichester.

    MATH  Google Scholar 

  • Petrova, S.S., Solov’ev, A.D., 1997. The origin of the method of steepest descent. Hist. Math. 24, 361–375.

    Article  MATH  MathSciNet  Google Scholar 

  • Petrovskii, S.V., Li, B.-L., 2006. Exactly Solvable Models of Biological Invasions. Chapman & Hall/CRC, Boca Raton.

    Google Scholar 

  • Powell, J.A., Slapnicar, I., van der Werf, W., 2005. Epidemic spread of a lesion-forming plant pathogen—analysis of a mechanistic model with infinite age structure. Linear Algebra Appl. 398, 117–140.

    Article  MATH  MathSciNet  Google Scholar 

  • Radcliffe, J., Rass, L., 1997. Discrete time spatial models arising in genetics, evolutionary game theory, and branching processes. Math. Biosci. 140, 101–129.

    Article  MATH  MathSciNet  Google Scholar 

  • Reid, N., 1988. Saddlepoint methods and statistical inference. Stat. Sci. 3, 213–227.

    Article  MATH  Google Scholar 

  • Renshaw, E., 2000. Applying the saddlepoint approximation to bivariate stochastic processes. Math. Biosci. 168, 57–75.

    Article  MATH  MathSciNet  Google Scholar 

  • Rényi, A., 1970. Probability Theory. North-Holland, Amsterdam.

    Google Scholar 

  • Shaw, M.W., 1995. Simulation of population expansion and spatial pattern when individual dispersal distributions do not decline exponentially with distance. Proc. R. Soc. Lond. B 259, 243–248.

    Article  Google Scholar 

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

    Google Scholar 

  • Shigesada, N., Kawasaki, K., 2002. Invasion and range expansion of species: effects of long-distance dispersal. In J.M. Bullock, R.E. Kenward, R.S. Hails (Eds.) Dispersal Ecology, pp. 350–373. Blackwell, Malden.

    Google Scholar 

  • Skarpaas, O., Shea, K., 2007. Dispersal patterns, dispersal mechanisms, and invasion wave speeds for invasive thistles. Am. Nat. 170, 421–430.

    Article  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  • Tufto, J., Ringsby, T.H., Dhondt, A.A., Adriaensen, F., Matthysen, E., 2005. A parametric model for estimation of dispersal patterns applied to five passerine spatially structured populations. Am. Nat. 165, E13–E26.

    Article  Google Scholar 

  • Weinberger, H.F., 1978. Asymptotic behavior of a model in population genetics. In J. Chadam (Ed.) Nonlinear Partial Differential Equations and Applications, pp. 47–98. Springer, New York.

    Chapter  Google Scholar 

  • Weinberger, H.F., 1982. Long-time behavior of a class of biological models. SIAM J. Math. Anal. 13, 353–396.

    Article  MATH  MathSciNet  Google Scholar 

  • Willson, M.F., 1992. The ecology of seed dispersal. In M. Fenner (Ed.) Seeds: The Ecology of Regeneration in Plant Communities, pp. 61–85. CAB International, Wallingford.

    Google Scholar 

  • Zayed, A.I., 1996. Handbook of Function and Generalized Function Transformations. CRC Press, Boca Raton.

    MATH  Google Scholar 

  • Zhang, S., Jin, J., 1996. Computation of Special Functions. Wiley, New York.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, University of Washington, Box 352420, Seattle, WA, 98195-2420, USA

    Mark Kot

  2. Biology Department, MS 34, Woods Hole Oceanographic Institution, Woods Hole, MA, 02543-1049, USA

    Michael G. Neubert

Authors
  1. Mark Kot
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Michael G. Neubert
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mark Kot.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kot, M., Neubert, M.G. Saddle-Point Approximations, Integrodifference Equations, and Invasions. Bull. Math. Biol. 70, 1790–1826 (2008). https://doi.org/10.1007/s11538-008-9325-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11538-008-9325-2

Keywords

Search

Navigation