Integrodifference equations, Allee effects, and invasions

Wang, Mei-Hui; Kot, Mark; Neubert, Michael G.

doi:10.1007/s002850100116

Cited by 137 publications

(128 citation statements)

References 45 publications

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“…As indicated by equations (14) and figure 2, the subsequent wave should spread at a moderate pace. Both theoretical (Schofield 2002;Wang et al 2002) and empirical (Turelli andHoffmann 1991, 1995) results indicate that these predicted wave speeds may significantly underestimate actual wave speed if the dispersal distribution is highly leptokurtic, with long-distance dispersal more prevalent than expected under a Gaussian dispersal model.…”

Section: Implications For Population Transformationmentioning

confidence: 96%

“…Hence, the direction of wave motion depends on dynamics averaged over all frequencies. In contrast, Fisherian variants essentially always spread, and their wave speed depends only on dynamics at the leading edge, via , the rate of increase f (0) near (Stokes 1976;Rothe 1981 (Mollison 1977;Kot et al 1996), whereas bistable waves are much less sensitive to non-Gaussian dispersal (Wang et al 2002). Stokes (1976) refers to these qualitatively different regimes as "pulled" versus "pushed" waves, respectively.…”

Section: One Dimension Homogeneous Environmentmentioning

confidence: 99%

“…One general result is that whether a variant spreads is independent of the form of the dispersal function, as long as it is symmetrical. Wang et al (2002) have shown that for model (10) with symmetric dispersal, fixation spreads as a traveling wave if and only if 1 (h(p) Ϫ p)dp 1 0.…”

Section: One Dimension Homogeneous Environmentmentioning

confidence: 99%

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Spatial Waves of Advance with Bistable Dynamics: Cytoplasmic and Genetic Analogues of Allee Effects

Barton

Turelli

2011

The American Naturalist

202

400

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JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. abstract: Unlike unconditionally advantageous "Fisherian" variants that tend to spread throughout a species range once introduced anywhere, "bistable" variants, such as chromosome translocations, have two alternative stable frequencies, absence and (near) fixation. Analogous to populations with Allee effects, bistable variants tend to increase locally only once they become sufficiently common, and their spread depends on their rate of increase averaged over all frequencies. Several proposed manipulations of insect populations, such as using Wolbachia or "engineered underdominance" to suppress vector-borne diseases, produce bistable rather than Fisherian dynamics. We synthesize and extend theoretical analyses concerning three features of their spatial behavior: rate of spread, conditions to initiate spread from a localized introduction, and wave stopping caused by variation in population densities or dispersal rates. Unlike Fisherian variants, bistable variants tend to spread spatially only for particular parameter combinations and initial conditions. Wave initiation requires introduction over an extended region, while subsequent spatial spread is slower than for Fisherian waves and can easily be halted by local spatial inhomogeneities. We present several new results, including robust sufficient conditions to initiate (and stop) spread, using a one-parameter cubic approximation applicable to several models. The results have both basic and applied implications. The University of Chicago Press and

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Section: Implications For Population Transformationmentioning

confidence: 96%

Section: One Dimension Homogeneous Environmentmentioning

confidence: 99%

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Spatial Waves of Advance with Bistable Dynamics: Cytoplasmic and Genetic Analogues of Allee Effects

Barton

Turelli

2011

The American Naturalist

202

400

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“…A ''strong'' Allee effect results in negative per capita rate of growth when population density drops below a threshold. A ''weak'' Allee effect, as can be the case with long-lived adults, causes a depressed per capita rate of growth at low population density, but it never becomes negative (27,28). A lack of mating opportunities among sparse or widely spaced individuals can result in an Allee effect and a slowing of the invasion (27)(28)(29).…”

mentioning

confidence: 99%

Pollen limitation causes an Allee effect in a wind-pollinated invasive grass ( Spartina alterniflora )

Davis

Taylor

Lambrinos

et al. 2004

Proc. Natl. Acad. Sci. U.S.A.

187

155

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It is usually assumed that pollen availability does not limit reproduction in wind-pollinated plants. Little evidence either supporting or contradicting this assumption exists, despite the importance of seed production to population persistence and growth. We investigated the role of pollen limitation in an invasive estuarine grass (Spartina alterniflora), with a manipulative pollen supplementation and exclusion experiment in areas of high population density and at the low-density leading edge of the invasion. We also quantified pollen deposition rates on stigmas and pollen traps along a windward to leeward gradient. We found pollen impoverishment at the low-density leading edge of a large invasion, causing an 8-fold reduction in seed set. We found 9-fold more pollen on stigmas of high-density plants than on those of lowdensity plants. Pollen deposition rates on stigmas and traps did not increase downwind of low-density plants but did increase downwind of high-density plants and dropped off precipitously across a gap that lacked pollen donors. The delay of appreciable numbers of seed caused by pollen limitation persists for decades until vegetative growth coalesces plants into continuous meadows, and this Allee effect has slowed the rate of spread of the invasion.

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“…Using various kernels, not necessarily Gaussian, allows one to consider population dynamic consequences of different dispersal behaviour [15][16][17][18][19][20]29]. Application of IDE models to real-world problems has resulted in new insights and helped to resolve several important ecological problems, for example, in invasion biology [3,7,10,25] or habitat fragmentation [5,8].…”

Section: Introductionmentioning

confidence: 99%

The importance of census times in discrete-time growth-dispersal models

Lutscher

Petrovskii

2008

Journal of Biological Dynamics

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Dispersal has been the focus of spatial ecology for a few decades. What should be a proper theoretical framework for understanding and modelling of dispersal processes remains a controversial issue though. Integrodifference equations (IDE) model the spatial dynamics of a population with distinct growth and dispersal stages in their life cycle. Depending on the stage observed, the equations take on different forms, only one of which is usually studied in the literature. Here we reveal that while these different forms are mathematically equivalent, the biological conclusions drawn from the different forms may differ considerably. We provide a summary of similarities and differences and point out the greatest potential caveats when applying IDE.

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Integrodifference equations, Allee effects, and invasions

Cited by 137 publications

References 45 publications

Spatial Waves of Advance with Bistable Dynamics: Cytoplasmic and Genetic Analogues of Allee Effects

Spatial Waves of Advance with Bistable Dynamics: Cytoplasmic and Genetic Analogues of Allee Effects

Pollen limitation causes an Allee effect in a wind-pollinated invasive grass ( Spartina alterniflora )

The importance of census times in discrete-time growth-dispersal models

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