Skip to main content

Advertisement

SpringerLink
Log in

Discrete-time growth-dispersal models with shifting species ranges

  • Original Paper
  • Published:
Theoretical Ecology Aims and scope Submit manuscript

Abstract

Many species are responding to global climate change by shifting their ranges poleward in latitude or upward in elevation. We analyze an integrodifference equation that combines growth, dispersal, and a constant-speed, climate-induced range shift and find that a shifting population can die out, even if the width of its range remains constant. We show how to determine the critical range-shift speed (for extinction) and study the effects of the growth rate and of the shape and scale of the dispersal kernel on persistence.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  • Allee WC (1938) The social life of animals. Norton, New York

    Google Scholar 

  • Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J, Croz JD, Greenbaum A, Hammarling S, McKenney A, Sorensen D (1999) LAPACK user’s guide. Society for Industrial and Applied Mathematics, Philadelphia

    Google Scholar 

  • Bakkenes M, Alkemade JRM, Ihle F, Leemans R, Latour JB (2002) Assessing effects of forecasted climate change on the diversity and distribution of European higher plants for 2050. Glob Chang Biol 8:390–407

    Article  Google Scholar 

  • Berestycki H, Diekmann O, Nagelkerke CJ, Zegeling PA (2009) Can a species keep pace with a shifting climate? Bull Math Biol 71:399–429

    Article  CAS  PubMed  Google Scholar 

  • Beverton RJH, Holt SJ (1957) On the dynamics of exploited fish populations. Her Majesty’s Stationery Office, London

    Google Scholar 

  • Box EO (1981) Macroclimate and Plant Forms: an introduction to predictive modeling in phytogeography. Dr. W. Junk, The Hague

    Google Scholar 

  • Brent RP (1973) Algorithms for minimization without derivatives. Prentice-Hall, Inc., Englewood Cliffs

    Google Scholar 

  • Cantrell RS, Cosner C (2003) Spatial ecology via reaction-diffusion equations. J. Wiley, Chichester

    Google Scholar 

  • Clark JS (1998) Why trees migrate so fast: confronting theory with dispersal biology and the paleorecord. Am Nat 152:204–224

    Article  CAS  PubMed  Google Scholar 

  • Crone EE, Schultz CB (2003) Movement behavior and minimum patch size for butterfly population persistence. In: Boggs CL, Watt WB, Ehrlich PR (eds) Butterflies: ecology and evolution taking flight. University of Chicago Press, Chicago, pp 561–576

    Google Scholar 

  • Crone EE, Schultz CB (2008) Old models explain new observations of butterfly movement at patch edges. Ecology 89:2061–2067

    Article  PubMed  Google Scholar 

  • Delves LM, Walsh J (1974) Numerical solution of integral equations. Clarendon Press, Oxford

    Google Scholar 

  • Dwyer G, Morris WF (2006) Resource-dependent dispersal and the speed of biological invasions. Am Nat 167:165–176

    Article  PubMed  Google Scholar 

  • Fagan WF, Lewis M, Neubert MG, Aumann C, Apple JL, Bishop JG (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  PubMed  Google Scholar 

  • Fisher RA (1937) The wave of advance of advantageous genes. Ann Eugen 7:355–369

    Google Scholar 

  • Galassi M, Davies J, Theiler J, Gough B, Jungman G, Aiken P, Booth M, Rossi F (2009) GNU scientific library: reference manual. Network Theory Ltd., Bristol

    Google Scholar 

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

    Google Scholar 

  • Guisan A, Thuiller W (2005) Predicting species distribution: offering more than simple habitat models. Ecol Lett 8:993–1009

    Article  Google Scholar 

  • Hart DR, Gardner RH (1997) A spatial model for the spread of invading organisms subject to competition. J Math Biol 35:935–948

    Article  Google Scholar 

  • Hastings A, Higgins K (1994) Persistence of transients in spatially structured ecological models. Science 263:1133–1136

    Article  CAS  PubMed  Google Scholar 

  • Hastings A, Cuddington K, Davies KF, Dugaw CJ, Elmendorf S, Freestone A, Harrison S, Holland M, Lambrinos J, Malvadkar U, Melbourne BA, Moore K, Taylor C, Thomson D (2005) The spatial spread of invasions: new developments in theory and evidence. Ecol Lett 8:91–101

    Article  Google Scholar 

  • Horiguchi T, Fukui Y (1996) A variation of the Jentzsch theorem for a symmetric integral kernel and its application. Interdiscip Inf Sci 2:139–144

    Google Scholar 

  • Hughes L (2000) Biological consequences of global warming: is the signal already apparent? Trends Ecol Evol 15:56–61

    Article  PubMed  Google Scholar 

  • Hutson V, Pym JS (1980) Applications of functional analysis and operator theory. Academic Press, London

    Google Scholar 

  • IPCC (2007) Climate change 2007: synthesis report. Contribution of working groups I, II and III to the fourth assessment report of the intergovernmental panel on climate change. Core Writing Team and Pachauri, R. K. and Reisinger, A., IPCC, Geneva

  • Jeffree CE, Jeffree EP (1996) Redistribution of the potential geographical ranges of mistletoe and Colorado beetle in Europe in response to the temperature component of climate change. Funct Ecol 10:562–577

    Article  Google Scholar 

  • Jentzsch R (1912) Über integralgleichungen mit positivem kern. J Reine Angew Math 141:235–244

    Google Scholar 

  • Kadmon R, Farber O, Danin A (2003) A systematic analysis of factors affecting the performance of climatic envelope models. Ecol Appl 13:853–867

    Article  Google Scholar 

  • Kang Y, Armbruster D, Kuang Y (2008) Dynamics of a plant–herbivore model. J Biol Dyn 2:89–101

    Article  Google Scholar 

  • Kareiva P (1990) Population dynamics in spatially complex environments: theory and data. Philos Trans: Biol Sci 330:175–190

    Article  Google Scholar 

  • Karlin S (1964) The existence of eigenvalues for integral operators. Trans Am Math Soc 113:1–17

    Article  Google Scholar 

  • Keeling M (1999) Spatial models of interacting populations. In: McGlade JM (ed) Advanced ecological theory: principles and applications. Blackwell Science, Malden, pp 64–99

    Chapter  Google Scholar 

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

    Article  CAS  PubMed  Google Scholar 

  • Kot M, Schaffer WM (1986) Discrete-time growth-dispersal models. Math Biosci 80:109–136

    Article  Google Scholar 

  • Kot M, Lewis MA, Van Den Driessche P (1996) Dispersal data and the spread of invading organisms. Ecology 77:2027–2042

    Article  Google Scholar 

  • Kot M, Medlock J, Reluga T, Walton DB (2004) Stochasticity, invasions, and branching random walks. Theor Popul Biol 66:175–184

    Article  PubMed  Google Scholar 

  • Krzemiński S (1977) Comment on ‘A simple proof of the Perron–Frobenius theorem for positive symmetric matrices’. J Phys A Math Gen 10:1437–1438

    Article  Google Scholar 

  • Latore J, Gould P, Mortimer AM (1998) Spatial dynamics and critical patch size of annual plant populations. J Theor Biol 190:277–285

    Article  PubMed  Google Scholar 

  • Latore J, Gould P, Mortimer AM (1999) Effects of habitat heterogeneity and dispersal strategies on population persistence in annual plants. Ecol Model 123:127–139

    Article  Google Scholar 

  • Letcher TM (2009) Climate change: observed impacts on planet Earth. Elsevier, Amsterdam

    Google Scholar 

  • Lewis MA (1997) Variability, patchiness, and jump dispersal in the spread of an invading population. In: Tilman D, Kareiva P (eds) Spatial ecology: the role of space in population dynamics and interspecific interactions. Princeton University Press, Princeton, pp 46–69

    Google Scholar 

  • Lewis MA, Neubert MG, Caswell H, Clark JS, Shea K (2006) A guide to calculating discrete-time invasion rates from data. In: Cadotte MW, McMahon SM, Fukami T (eds) Conceptual ecology and invasions biology: reciprocal approaches to nature. Springer, Dordrecht, pp 169–192

    Chapter  Google Scholar 

  • Lockwood DR, Hastings A, Botsford LW (2002) The effects of dispersal patterns on marine reserves: does the tail wag the dog? Theor Popul Biol 61:297–309

    Article  PubMed  Google Scholar 

  • Lovejoy TE, Hannah L (2005) Climate change and biodiversity. Yale University Press, New Haven

    Google Scholar 

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

    Article  Google Scholar 

  • Lutscher F (2008) Density-dependent dispersal in integrodifference equations. J Math Biol 56:499–524

    Article  PubMed  Google Scholar 

  • May RM (1973) On relationships among various types of population models. Am Nat 107:46–57

    Article  Google Scholar 

  • May RM, Hassell MP, Anderson RM, Tonkyn DW (1981) Density dependence in host–parasitoid models. J Anim Ecol 50:855–865

    Article  Google Scholar 

  • Maynard Smith J (1968) Mathematical ideas in biology. Cambridge University Press, London

    Book  Google Scholar 

  • McCarty JP (2001) Ecological consequences of recent climate change. Conserv Biol 15:320–331

    Article  Google Scholar 

  • Mitikka V, Heikkinen RK, Luoto M, Araújo MB, Saarinen K, Pöyry J, Fronzek S (2008) Predicting range expansion of the map butterfly in Northern Europe using bioclimatic models. Biodivers Conserv 17:623–641

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Neubert MG, Parker IM (2004) Projecting rates of spread for invasive species. Risk Anal 24:817–831

    Article  PubMed  Google Scholar 

  • Neubert MG, Kot M, Lewis MA (1995) Dispersal and pattern formation in a discrete-time predator-prey model. Theor Popul Biol 48:7–43

    Article  Google Scholar 

  • Neubert MG, Kot M, Lewis MA (2000) Invasion speeds in fluctuating environments. Proc R Soc Lond B 267:1603–1610

    Article  CAS  Google Scholar 

  • Nicholson AJ (1933) The balance of animal populations. J Anim Ecol 2:132–178

    Article  Google Scholar 

  • Okubo A (1980) Diffusion and ecological problems: mathematical models. Springer, Berlin

    Google Scholar 

  • Osborne JL, Loxdale HD, Woiwod IP (2002) Monitoring insect dispersal: methods and approaches. In: Bullock JM, Kenward RE, Hails RS (eds) Dispersal ecology: the 42nd symposium of the British ecological society held at the University of Reading 2–5 April 2001. Blackwell Science, Malden, pp 24–49

    Google Scholar 

  • Parmesan C (2006) Ecological and evolutionary responses to recent climate change. Annu Rev Ecol Evol Systemat 37:637–669

    Article  Google Scholar 

  • Parmesan C, Ryrholm N, Stefanescu C, Hill JK, Thomas CD, Descimon H, Huntley B, Kaila L, Kullberg J, Tammaru T, Tennent WJ, Thomas JA, Warren M (1999) Poleward shifts in geographical ranges of butterfly species associated with regional warming. Nature 399:579–583

    Article  CAS  Google Scholar 

  • Pearson RG, Dawson TP (2003) Predicting the impacts of climate change on the distribution of species: are bioclimate envelope models useful? Glob Ecol Biogeogr 12:361–371

    Article  Google Scholar 

  • Pipkin AC (1991) A course on integral equations. Springer-Verlag, New York

    Google Scholar 

  • Potapov AB, Lewis MA (2004) Climate and competition: the effect of moving range boundaries on habitat invasibility. Bull Math Biol 66:975–1008

    Article  CAS  PubMed  Google Scholar 

  • Press WH, Teukolsky SA, Vetterling WT, Flannery BP (1992) Numerical recipes in C: the art of scientific computing. Cambridge University Press, Cambridge

    Google Scholar 

  • Ricker WE (1954) Stock and recruitment. J Fish Res Board Can 11:559–623

    Google Scholar 

  • Schultz CB (1998) Dispersal behavior and its implications for reserve design in a rare oregon butterfly. Conserv Biol 12:284–292

    Article  Google Scholar 

  • Schultz CB, Crone EE (1998) Burning prairie to restore butterfly habitat: a modeling approach to management tradeoffs for the Fender’s blue. Restor Ecol 6:244–252

    Article  Google Scholar 

  • Schultz CB, Crone EE (2001) Edge-mediated dispersal behavior in a prairie butterfly. Ecology 82:1879–1892

    Article  Google Scholar 

  • Schultz CB, Hammond PC (2003) Using population viability analysis to develop recovery criteria for endangered insects: case study of the Fender’s blue butterfly. Conserv Biol 17:1372–1385

    Article  Google Scholar 

  • Schultz CB, Hammond PC, Wilson MV (2003) Biology of the Fender’s blue butterfly (Icaricia icarioides fenderi Macy), an endangered species of western Oregon native prairies. Nat Areas J 23:61–71

    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 the range expansion of species: effects of long-distance dispersal. In: Bullock J, Kenward R, Hails R (eds) Dispersal ecology. Blackwell Science, Oxford, pp 350–373

    Google Scholar 

  • Skellam JG (1951) Random dispersal in theoretical populations. Biometrika 38:196–218

    CAS  PubMed  Google Scholar 

  • Taylor CM, Hastings A (2005) Allee effects in biological invasions. Ecol Lett 8:895–908

    Article  Google Scholar 

  • Turchin P (1998) Quantitative analysis of movement. Sinauer Associates, Inc., Sunderland

    Google Scholar 

  • US Fish and Wildlife Service (2000) Endangered and threatened wildlife and plants; endangered status for Erigeron decumbens var. decumbens (Willamette daisy) and Fender’s blue butterfly (Icaricia icarioides fenderi) and threatened status for Lupinus sulphureus ssp. kincaidii (Kincaid’s lupine). Fed Regist 65:3875–3890

    Google Scholar 

  • Van Kirk RW, Lewis MA (1997) Integrodifference models for persistence in fragmented habitats. Bull Math Biol 59:107–137

    Article  Google Scholar 

  • Van Kirk RW, Lewis MA (1999) Edge permeability and population persistence in isolated habitat patches. Nat Resour Model 12:37–64

    Article  Google Scholar 

  • Walther GR, Post E, Convey P, Menzel A, Parmesan C, Beebee TJC, Fromentin JM, Hoegh-Guldberg O, Bairlein F (2002) Ecological responses to recent climate change. Nature 416:389–395

    Article  CAS  PubMed  Google Scholar 

  • Wang MH, Kot M, Neubert MG (2002) Integrodifference equations, Allee effects, and invasions. J Math Biol 44:150–168

    Article  PubMed  Google Scholar 

  • Weinberger HF (1978) Asymptotic behavior of a model in population genetics. Lect Notes Math 648:47–96

    Article  Google Scholar 

  • Weinberger HF (1982) Long-time behavior of a class of biological models. SIAM J Math Anal 13:353–396

    Article  Google Scholar 

  • Westerling AL, Hidalgo HG, Cayan DR, Swetnam TW (2006) Warming and earlier spring increase western U.S. forest wildfire activity. Science 313:940–943

    Article  CAS  PubMed  Google Scholar 

  • Wilson MV, Erhart T, Hammond PC, Kaye TN, Kuykendall K, Liston A, Robinson Jr AF, Schultz CB, Severns PM (2003) Biology of Kincaid’s lupine(Lupinus sulphureus ssp. kincaidii [Smith] Phillips), a threatened species of western Oregon native prairies, USA. Nat Areas J 23:72–83

    Google Scholar 

Download references

Acknowledgements

MK acknowledges Mark A. Lewis and Michael G. Neubert for stimulating and helpful discussions that helped inspire and further this research. We are grateful to the reviewers for their helpful comments.

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, University of Washington, Seattle, USA

    Ying Zhou & Mark Kot

Authors
  1. Ying Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mark Kot
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ying Zhou.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zhou, Y., Kot, M. Discrete-time growth-dispersal models with shifting species ranges. Theor Ecol 4, 13–25 (2011). https://doi.org/10.1007/s12080-010-0071-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12080-010-0071-3

Keywords

Search

Navigation