Skip to main content
Log in

Integrodifference models for persistence in fragmented habitats

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

Abstract

Integrodifference models of growth and dispersal are analyzed on finite domains to investigate the effects of emigration, local growth dynamics and habitat heterogeneity on population persistence. We derive the bifurcation structure for a range of population dynamics and present an approximation that allows straighforward calculation of the equilibrium populations in terms of local growth dynamics and dispersal success rates. We show how population persistence in a heterogeneous environment depends on the scale of the heterogeneity relative to the organism's characteristic dispersal distance. When organisms tend to disperse only a short distance, population persistence is dominated by local conditions in high quality patches, but when dispersal distance is relatively large, poor quality habitat exerts a greater influence.

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

Access this article

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

  • Amarasekare, P. 1994. Spatial population structure in the banner-tailed kangaroo rat,Dipodomys spectabilis.Oecologia 100, 166–176.

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Anderson, P. K. 1989. Dispersal in rodents: a resident fitness hypothesis. Special Publications No. 9, American Society of Mammalogists, Provo, UT.

    Google Scholar 

  • Andow, D. A., P. M. Kareiva, S. A. Levin and A. Okubo. 1990. Spread of invading organisms.Landscape Ecology 4, 177–188.

    Article  Google Scholar 

  • Augspurger, C. K. and K. P. Hogan. 1983. Wind dispersal of fruits with variable seed numbers in a tropical tree (Lonchocarpus pentaphyllus: Leguminosae).Am. J. Botany 70, 1031–1037.

    Article  Google Scholar 

  • Barash, D. P. 1989.Marmots: Social Behavior and Ecology. Stanford, CA: Stanford University Press.

    Google Scholar 

  • Bennett, A. F., K. Henein and G. Merriam. 1994. Corridor use and the elements of corridor quality: chipmunks and fencerows in a farmland mosaic.Biological Conservation 68, 155–165.

    Article  Google Scholar 

  • Berger, E. M. 1983. Population genetics of marine gastropods and bivalves. InMollusca, W. D. Russell-Hunter (Ed), Vol. 6, pp. 563–596. Orlando, FL: Academic Press.

    Google Scholar 

  • Beverton R. J. H. and S. J. Holt, 1957. On the dynamics of exploited fish populations. Fisheries Investigations, Series 2, No. 19, Ministry of Agriculture, Fisheries and Food, London.

    Google Scholar 

  • Bownan, T. J. and R. J. Robel. 1977. Brood break-up, dispersal, and mortality of juvenile prairie chickens.J. Wildlife Management 41, 27–34.

    Google Scholar 

  • Broadbent, S. R. and D. G. Kendall. 1953. The random walk ofTrichostrongylus retortaeformis.Biometrics 9, 460–466.

    Article  Google Scholar 

  • Brown, J. H. 1984. On the relationship between abundance and distribution of species.Am. Naturalist 124, 225–279.

    Article  Google Scholar 

  • Brown, J. H. 1995.Macroecology. Chicago: University of Chicago Press.

    Google Scholar 

  • Carl, E. A. 1971. Population control in Arctic ground squirrels.Ecology 52, 395–413.

    Article  Google Scholar 

  • Cockburn, A., M. P. Scott and D. J. Scotts. 1985. Inbreeding avoidance and male-biased natal dispersal inAnteclinus spp. (Marsupiala: Dasyuridae).Animal Behaviour 33, 908–915.

    Article  Google Scholar 

  • Cushing, D. H. 1981.Fisheries Biology. Madison, WI: University of Wisconsin Press.

    Google Scholar 

  • Davis G. J. and R. W. Howe. 1992. Juvenile dispersal, limited breeding sites and the dynamics of metapopulations.Theor. Pop. Biol. 41, 184–207.

    Article  MATH  Google Scholar 

  • Dennis, B. 1989. Allee effects: population growth, critical density, and the chance of extinction.Natural Resource Modeling 3, 481–538.

    MATH  MathSciNet  Google Scholar 

  • Doak, D. 1987. Spotted owls and old growth logging in the Pacific Northwest.Conservation Biol. 3, 389–396.

    Article  Google Scholar 

  • Doak, D. F., P. C. Marino and P. M. Kareiva. 1992. Spatial scale mediates the influence of habitat fragmentation.Theor. Pop. Biol. 41, 315–336.

    Article  Google Scholar 

  • Dobson F. S. 1979. An experimental study of dispersal in the California ground squirrel.Ecology 60 1103–1109.

    Article  Google Scholar 

  • Doedel E., X. Wang and T. Fairgrieve. 1994. Software for continuation and bifurcation problems in ordinary differential equations. Technical report, California Institute of Technology, Pasadena, CA.

    Google Scholar 

  • Dunning J. B., B. J. Danielson and H. R. Pulliam. 1992. Ecological processes that affect populations in complex landscapes.Oikos 65, 169–175.

    Google Scholar 

  • Eastham M. S. P. 1973.The Spectral Theory of Periodic Differential Equations. Edinburgh: Scottish Academic Press

    MATH  Google Scholar 

  • Fleming T. H. 1979.Ecology of Small Mammals. New York: Wiley.

    Google Scholar 

  • Flowerdew J. R. 1987.Mammals Their Reproductive Biology and Population Ecology. London: Edward Arnold Ltd.

    Google Scholar 

  • Forrest S. C., D. E. Biggins, L. Richardson, T. W. Clark, T. M. CampbellIII, K. A. Fagerstone and E. T. Thorne. 1988. Population attributes for the black-footed ferret (Mustela nigripes) at Meeteetse, Wyoming.J. Mammalogy 69, 261–273.

    Article  Google Scholar 

  • Forsman E. D., E. C. Meslow and H. M. Wright. 1984. Distribution and biology of the spotted owl in Oregon.Wildlife Monographs 87, 1–64.

    Google Scholar 

  • Freedman H. I., J. B. Shukla and Y. Takeuchi. 1989 Population diffusion in a two-patch environment.Math. Biosci..95, 111–123.

    Article  MATH  MathSciNet  Google Scholar 

  • Gaines M. S. and L. R. Mc ClenaghanJr. 1980. Dispersal in small mammals.Ann. Rev. Ecology and Systematics 11, 163–196.

    Article  Google Scholar 

  • Greenwood P. J. 1980. Mating systems, Philopatry and dispersal in birds and mammals.Animal Behaviour 28, 1140–1162.

    Article  Google Scholar 

  • Greenwood P. J. and P. H. Harvey. 1982. The natal and breeding dispersal of birds.Ann. Rev. Ecology and Systematics 13, 1–21.

    Article  Google Scholar 

  • Gurney W. S. C. and R. M. Nisbet. 1975. The regulation of inhomogeneous populations.J. Theor. Biol. 52, 441–457.

    Article  Google Scholar 

  • Hanski I. and D. Zhang. 1993. Migration, metapopulation dynamics and fugitive co-existence.J. Theor. Biol. 163, 491–514.

    Article  Google Scholar 

  • Hansson L. 1991. Dispersal and connectivity in metapopulations.Biol. J. Linnean Soc. 42, 89–103.

    Google Scholar 

  • Harcourt D. G. 1971. Population dynamics ofLeptinitarsa decemlineata (Say) in eastern Ontario. III. Major population processes.Canad. Entomologist 103, 1049–1061.

    Article  Google Scholar 

  • Hardin D. P., P. Takac and G. F. Webb. 1988a. Asymptotic properties of a continuous-space discrete-time population model in a random environment.J. Math. Biol. 26 361–374.

    MATH  MathSciNet  Google Scholar 

  • Hardin D. P., P. Takac and G. F. Webb. 1988b. A comparison of dispersal strategies for survival of spatially heterogeneous populations.SIAM J. Appl. Math. 48, 1396–1423.

    Article  MATH  MathSciNet  Google Scholar 

  • Hardin D. P., P. Takac and G. F. Webb. 1990. Dispersion population models discrete in time and continuous in space.J. Math. Biol. 28, 1–20.

    Article  MATH  MathSciNet  Google Scholar 

  • Harris L. 1984.The Fragmented Forest: Island Biogeography Theory and the Preservation of Biotic Diversity. Chicago: University of Chicago Press.

    Google Scholar 

  • Harris L. D. 1988. Edge effects and conservation of biotic diversity.Conservation Biol. 2, 330–332.

    Article  Google Scholar 

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

    Google Scholar 

  • Henderson M. T., G. Merriam and J. Wegner. 1985. Patchy environments and species survival: chipmunks in an agricultural mosaic.Biol. Conservation 31, 95–105.

    Article  Google Scholar 

  • Hengeveld R. 1990.Dynamic Biogeography. Cambridge: Cambridge University Press.

    Google Scholar 

  • Hengeveld R. and J. Haeck. 1982. The distribution of abundance. I. Measurements.J. Biogeography 9, 303–316.

    Article  Google Scholar 

  • Hobbs R. C., L. W. Botsford and A. Thomas. 1992. Influence of hydrographic conditions and wind forcing on the distribution and abundance of Dungeness crabCancer magister farvae.Canad. J. Fisheries Aquatic Sci. 49, 1379–1388

    Article  Google Scholar 

  • Holmes E. E., M. A. Lewis, J. E. Banks and R. R. Veit. 1994. Partial differential equations in ecology spatial interactions and population dynamics.Ecology 75, 18–29.

    Article  Google Scholar 

  • Hoogland J. L. 1982. Prairie dogs avoid extreme inbreeding.Science 215, 1639–1641.

    Google Scholar 

  • Howard W. E. 1960. Innate and environmental dispersal of individual vertebrates.Am. Midland Naturalist 63, 152–161.

    Article  Google Scholar 

  • Howe H. F. and J. Smallwood. 1982. Ecology of seed dispersal.Ann. Rev. Ecology and Systematics 13, 201–228.

    Article  Google Scholar 

  • Howe R. W., G. J. Davis and V. Mosca. 1991. The demographic significance of “sink” populations.Biol. Conservation 57, 239–255.

    Article  Google Scholar 

  • Janzen, D. H. 1983. No park is an island: increase in interference from outside as park size decreases.Oikos 41, 402–410.

    Google Scholar 

  • Janzen D. H. 1986. The eternal external threat. InConservation Biology: The Science of Scarcity and Diversity, M. E. Soule (Ed), pp. 286–303. Sunderland, MA: Sinauer Associates.

    Google Scholar 

  • Jenkins D., A. Watson and G. R. Miller. 1964. Predation and red grouse populations.J. Appl. Ecol. 1, 183–195.

    Article  Google Scholar 

  • Johnson M. and M. S. Gaines. 1990. Evolution of dispersal: theoretical models and empirical tests using birds and mammals.Ann. Rev. Ecology and Systematics 21, 449–480.

    Article  Google Scholar 

  • Kadmon R. and A. Shmida. 1990. Spatiotemporal demographic processes in plant populations: an approach and a case study.Am. Naturalist 135, 382–397.

    Article  Google Scholar 

  • Keppie D. M. 1979. Dispersal, overwinter mortality and recruitment of spruce grouse.J. Wildlife Management 43, 717–727.

    Google Scholar 

  • Kot M. 1989. Diffusion-driven period-doubling bifurcations.BioSystems 22, 279–287.

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

    Article  MATH  MathSciNet  Google Scholar 

  • Kot, M., M. A. Lewis and P. van den Driessche. 1996. Dispersal data and the spread of invading organisms.Ecology, to appear.

  • Krasnoselskii M. A. 1964.Positive Solutions of Operator Equations. Groningen, The Netherlands: Noordhoff.

    Google Scholar 

  • Krasnoselskii M. A. and P. P. Zabreiko. 1984.Geometrical Methods of Nonlinear Analysis. Berlin: Springer-Verlag.

    Google Scholar 

  • Krebs C. J. 1992. The role of dispersal in cyclic rodent populations. InAnimal Dispersal, N. C. Stenseth and W. Z. LidickerJr. (Eds), pp. 160–175. London: Chapman and Hall.

    Google Scholar 

  • Krebs C. J., B. L. Keller and R. H. Tamarin. 1969.Microtus population biology: demographic changes in fluctuating populations ofM. ochrogaster andM. pennsylvanicus in southern Indiana.Ecology 50, 587–607.

    Article  Google Scholar 

  • Krein M. G. and M. A. Rutman. 1950. Linear operators which leave a cone in Banach space invariant.Am. Math. Soc. Transl. 26, 3–128.

    MathSciNet  Google Scholar 

  • Lamberson R. H., R. McKelvey, B. R. Noon and C. Voss. 1992. A dynamic analysis of northern spotted owl viability in a fragmented forest landscape.Conservation Biol. 6, 505–512.

    Article  Google Scholar 

  • Lefkovitch L. P. and L. Fahrig. 1985. Spatial characteristics of habitat patches and population survival.Ecological Modelling 30, 297–308.

    Article  Google Scholar 

  • Li T-Y. and J. A. Yorke. 1975. Period three implies chaos.Am. Math. Monthly 82, 985–992.

    Article  MATH  MathSciNet  Google Scholar 

  • Li T-Y., M. Misiurewicz, G. Pianigiani and J. A. Yorke. 1982. Odd chaos.Phys. Lett. 87A, 271–273.

    MathSciNet  Google Scholar 

  • Lidicker W. Z.Jr. 1962. Emigration as a possible mechanism permitting the regulation of population density below the carrying capacity.Am. Naturalist 96, 29–34.

    Article  Google Scholar 

  • Lubina J. A. and S. A. Levin. 1988. The spread of a reinvading species: range expansion in the California sea otter.Am. Naturalist 131, 526–543.

    Article  Google Scholar 

  • Ludwig D. D. G. Aronson and H. F. Weinberger. 1979. Spatial patterning of the spruce budworm.J. Math. Biol. 8, 217–258.

    MATH  MathSciNet  Google Scholar 

  • Lui R. 1982a. A nonlinear integral operator arising from a model in population genetics. I. Monotone initial data.SIAM J. Math. Anal. 13, 913–937

    Article  MATH  MathSciNet  Google Scholar 

  • Lui R. 1982b. A nonlinear integral operator arising from a model in population genetics. II. Initial data with compact support.SIAM J. Math. Anal. 13, 938–953.

    Article  MATH  MathSciNet  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 

  • Lui R. 1985. A nonlinear integral operator arising from a model in population genetics. III. Heterozygote inferior case.SIAM J. Math. Anal. 16, 1180–1206.

    Article  MATH  MathSciNet  Google Scholar 

  • Lui R. 1986. A nonlinear integral operator arising from a model in population genetics. IV. Clines.SIAM J. Math. Anal. 17, 152–168.

    Article  MATH  MathSciNet  Google Scholar 

  • Lui R. 1989a. Biological growth and spread modeled by systems of recursions. I. Mathematical theory.Math. Biosci. 93, 269–295.

    Article  MATH  MathSciNet  Google Scholar 

  • Lui R. 1989b. Biological growth and spread modeled by systems of recursions. II. Biological theory.Math. Biosci. 93, 297–312.

    Article  MATH  MathSciNet  Google Scholar 

  • Magnus W. and S. Winkler. 1979.Hill's Equation. New York: Dover.

    Google Scholar 

  • Maurer B. A. 1994.Geographical Population Analysis: Tools for the Analysis of Biodiversity. Oxford: Blackwell Scientific Publications.

    Google Scholar 

  • May R. M. 1974. Biological populations with nonoverlapping generations: stable points, stable cycles and chaos.Science 186, 645–647.

    Google Scholar 

  • May R. M. 1975. Biological populations obeying difference equations: stable points, stable cycles and chaos.J. Theor. Biol. 51, 511–524.

    Article  Google Scholar 

  • May R. M. 1976. Simple mathematical models with very complicated dynamics.Nature 261, 459–467.

    Article  Google Scholar 

  • Metzgar L. H. 1967. An experimental comparison of screech owl predations on resident and transient white-footed mice (Peromyscus leucopus).J. Mammalogy 48, 387–391.

    Article  Google Scholar 

  • Moore J. and R. Ali. 1984. Are dispersal and inbreeding avoidance related?.Animal Behaviour 32, 94–112.

    Article  Google Scholar 

  • Murray J. D. 1989.Mathematical Biology. Berlin: Springer-Verlag

    MATH  Google Scholar 

  • Murray J. D., E. A. Stanley and D. L. Brown. 1986. On the spread of rabies among foxes.Proc. Roy. Soc. London Ser. B 229 111–150.

    Article  Google Scholar 

  • Namba T. 1980. Density-dependent dispersal and spatial distribution of a population.J. Theor. Biol. 86, 351–363.

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

  • Newmark W. D. 1991. Tropical forest fragmentation and the local extinction of understory birds in the eastern Usambara mountains, Tanzania.Conservation Biol. 5, 67–78.

    Article  Google Scholar 

  • Nicholson A. J. 1954. An outline of the dynamics of animal populations.Austral. J. Zoology 2, 9–65.

    Article  Google Scholar 

  • Peitgen H.-O. and P. H. Richter. 1986The Beauty of Fractals: Images of Complex Dynamical Systems. Berlin: Springer.

    MATH  Google Scholar 

  • Pulliam H. R. 1988. Sources, sinks, and population regulation.Am. Naturalist 132, 652–661.

    Article  Google Scholar 

  • Pulliam H. R. and B. J. Danielson. 1991. Sources, sinks and habitat selection: a landscape perspective on population dynamics.Am. Naturalist 137, S50-S66.

    Article  Google Scholar 

  • Ribble D. O. 1992. Dispersal in a monogamous rodent,Peromyscus califormicus.Ecology 73, 859–866.

    Article  Google Scholar 

  • Richter C. J. 1970. Aerial dispersal in relation to habitat in eight wolf spider specie (Pardosa, Araneae, Lycosidae).Oecologia 5, 200–214.

    Article  Google Scholar 

  • Ricker W. E. 1954. Stock and recruitment.J. Fisheries Research Board of Canada 11, 559–623.

    Google Scholar 

  • Roughgarden J. and Y. Iwasa. 1986. Dynamics of a metapopulation with space-limited subpopulations.Theor. Pop. Biol. 29, 235–261.

    Article  MATH  MathSciNet  Google Scholar 

  • Saunders D. A., R. J. Hobbs and C. R. Margules. 1991. Biological consequencs of ecosystem fragmentation: a review.Conservation Biol. 5, 18–32

    Article  Google Scholar 

  • Shields W. M. 1987. Dispersal and mating systems: investigating their causal connections. InMammalian Dispersal Patterns B. D. Chepko-Sade and Z. T. Halpin (Eds), pp. 3–24. Chicago: University of Chicago Press.

    Google Scholar 

  • Shigesada N. and K. Kawasaki. 1986. Traveling periodic waves in heterogeneous environments.Theor. Pop. Biol. 30, 143–160.

    Article  MATH  MathSciNet  Google Scholar 

  • Sievert P. R. and L. B. Keith. 1985. Survival of snowshoe hares at a geographic range boundary.J. Wildlife Management 49, 854–866

    Google Scholar 

  • Simberloff D. and L. G. Abele. 1982. Refuge design and island biogeographic theory: effects of fragmentation.Am. Naturalist 120, 41–50.

    Article  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  • Slatkin M. 1975. Gene flow and selection in a two-locus system.Genetics 81, 787–802.

    Google Scholar 

  • Slatkin M. 1978. Spatial patterns in the distribution of polygenic characters.J. Theor. Biol. 70, 213–228.

    Article  Google Scholar 

  • Small R. J., J. C. Holzwart and D. H. Rusch 1993. Are ruffed grouse more vulnerable to mortality during dispersal?Ecology 74, 2020–2026.

    Article  Google Scholar 

  • Smith A. T. 1974. The distribution and dispersal of pikes: consequences of insular population structure.Ecology 55, 1112–1119.

    Article  Google Scholar 

  • Smith A. T. 1980. Temporal changes in insular populations of the pike (Ochotona princeps).Ecology 61, 8–13.

    Article  Google Scholar 

  • Stefan P. 1977. A theorem of Sarkovskii on the existence of periodic orbits of continuous endomorphisms of the real line.Commun. Math. Phys. 54, 237–248.

    Article  MATH  MathSciNet  Google Scholar 

  • Van Kirk, R. W. 1995. Integrodigerence models of biological growth and dispersal. Ph.D. thesis, University of Utah.

  • Veit R. R. and M. A. Lewis. 1996. Dispersal, population growth and the Allee effect: dynamics of the house finch invasion of eastern North America.Ecology 148, 255–274.

    Google Scholar 

  • Waser P. 1985. Does competition drive dispersal?Ecology 66, 1170–1175.

    Article  Google Scholar 

  • Weinberger H. F. 1978. Asymptotic behavior of a model in population genetics. InNonlinear Partial Differential Equations and Applications. Lecture Notes in Mathematics, J. M. Chadam (Ed), Vol. 648, pp. 47–96. Berlin: Springer-Verlag.

    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 

  • Wiens J. A. 1976. Population responses to patchy environments.Ann. Rev. Ecology and Systematics 7, 81–120.

    Article  Google Scholar 

  • Wilcox B. A. and D. D. Murphy. 1985. Conservation strategy: the effects of fragmentation on extinction.Am. Naturalist 125, 879–887.

    Article  Google Scholar 

  • Williams E. J. 1961. The distribution of larvae of randomly moving insects.Austral. J. Biol. Sci. 14 598–604.

    Google Scholar 

  • Wolff J. O., K. I. Lundy and R. Baccus. 1988. Dispersal, inbreeding avoidance and reproductive success in white-footed mice.Animal Behaviour 36, 456–465.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Van Kirk, R.W., Lewis, M.A. Integrodifference models for persistence in fragmented habitats. Bltn Mathcal Biology 59, 107–137 (1997). https://doi.org/10.1007/BF02459473

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02459473

Keywords

Navigation