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Spatial assortment of mixed propagules explains the acceleration of range expansion.

Ramanantoanina A, Ouhinou A, Hui C - PLoS ONE (2014)

Bottom Line: Range expansion of spreading organisms has been found to follow three types: (i) linear expansion with a constant rate of spread; (ii) bi-phase expansion with a faster linear expansion following a slower linear expansion; and (iii) accelerating expansion with a continuously increasing rate of spread.We found that individuals with different dispersal abilities were spatially sorted with the stronger dispersers situated at the expanding range front, causing the velocity of expansion to accelerate.Aside from better managing of invasive species, the derived formula could conceivably also be applied to conservation management of relocated, endangered or extra-limital species.

View Article: PubMed Central - PubMed

Affiliation: Centre for Invasion Biology, Department of Mathematical Sciences, Stellenbosch University, Matieland, South Africa; Mathematical and Physical Biosciences, African Institute for Mathematical Sciences, Muizenberg, South Africa.

ABSTRACT
Range expansion of spreading organisms has been found to follow three types: (i) linear expansion with a constant rate of spread; (ii) bi-phase expansion with a faster linear expansion following a slower linear expansion; and (iii) accelerating expansion with a continuously increasing rate of spread. To date, no overarching formula exists that can be applied to all three types of range expansion. We investigated how propagule pressure, i.e., the initial number of individuals and their composition in terms of dispersal ability, affects the spread of a population. A system of integrodifference equations was then used to model the spatiotemporal dynamics of the population. We studied the dynamics of dispersal ability as well as the instantaneous and asymptotic rate of spread. We found that individuals with different dispersal abilities were spatially sorted with the stronger dispersers situated at the expanding range front, causing the velocity of expansion to accelerate. The instantaneous rate of spread was found to be fully determined by the growth and dispersal abilities of the population at the advancing edge of the invasion. We derived a formula for the asymptotic rate of spread under different scenarios of propagule pressure. The results suggest that data collected from the core of the invasion may underestimate the spreading rate of the population. Aside from better managing of invasive species, the derived formula could conceivably also be applied to conservation management of relocated, endangered or extra-limital species.

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Asymptotic rate of spread of a population with two dispersal abilities.Theoretical (black lines) and computed asymptotic rate of spread when two dispersal abilities are present in the population. Initial propagules are  (blue triangle) and  (magenta triangle). Other parameter values are  (1)  (2)  and (3) . A: Using Gaussian dispersal kernels. B: Using Laplace dispersal kernels.
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pone-0103409-g002: Asymptotic rate of spread of a population with two dispersal abilities.Theoretical (black lines) and computed asymptotic rate of spread when two dispersal abilities are present in the population. Initial propagules are (blue triangle) and (magenta triangle). Other parameter values are (1) (2) and (3) . A: Using Gaussian dispersal kernels. B: Using Laplace dispersal kernels.

Mentions: The asymptotic rate of spread for type II bi-phase expansion (i.e. the rate of spread at the second phase of expansion) was heuristically derived as follows. We note that the solution of the model (Eq.(1)) becomes a travelling wave of the which spreads at the following rate:(3)where is the moment generating function of ki, and the interval I is for Gaussian kernels and is for Laplace kernels [7], [56]. Recall that we are interested in the spread of the total population . Because and for Gaussian and Laplace kernels, we have as x approaches infinity, meaning that the spreading rate of the total population (Eq.(2)) is given by . An approximation has been derived by Lutscher [57]:(4)where is the kurtosis of the dispersal kernel . In particular, for Gaussian kernels we have:(5)For Laplace dispersal kernels we have:(6)The derived rate of spread c fits well with the asymptotic rate of spread obtained from the numerical simulations with a Gaussian dispersal kernel and also sets a close upper bound for the rate of spread with a Laplace dispersal kernel (Fig. 2).


Spatial assortment of mixed propagules explains the acceleration of range expansion.

Ramanantoanina A, Ouhinou A, Hui C - PLoS ONE (2014)

Asymptotic rate of spread of a population with two dispersal abilities.Theoretical (black lines) and computed asymptotic rate of spread when two dispersal abilities are present in the population. Initial propagules are  (blue triangle) and  (magenta triangle). Other parameter values are  (1)  (2)  and (3) . A: Using Gaussian dispersal kernels. B: Using Laplace dispersal kernels.
© Copyright Policy
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4126666&req=5

pone-0103409-g002: Asymptotic rate of spread of a population with two dispersal abilities.Theoretical (black lines) and computed asymptotic rate of spread when two dispersal abilities are present in the population. Initial propagules are (blue triangle) and (magenta triangle). Other parameter values are (1) (2) and (3) . A: Using Gaussian dispersal kernels. B: Using Laplace dispersal kernels.
Mentions: The asymptotic rate of spread for type II bi-phase expansion (i.e. the rate of spread at the second phase of expansion) was heuristically derived as follows. We note that the solution of the model (Eq.(1)) becomes a travelling wave of the which spreads at the following rate:(3)where is the moment generating function of ki, and the interval I is for Gaussian kernels and is for Laplace kernels [7], [56]. Recall that we are interested in the spread of the total population . Because and for Gaussian and Laplace kernels, we have as x approaches infinity, meaning that the spreading rate of the total population (Eq.(2)) is given by . An approximation has been derived by Lutscher [57]:(4)where is the kurtosis of the dispersal kernel . In particular, for Gaussian kernels we have:(5)For Laplace dispersal kernels we have:(6)The derived rate of spread c fits well with the asymptotic rate of spread obtained from the numerical simulations with a Gaussian dispersal kernel and also sets a close upper bound for the rate of spread with a Laplace dispersal kernel (Fig. 2).

Bottom Line: Range expansion of spreading organisms has been found to follow three types: (i) linear expansion with a constant rate of spread; (ii) bi-phase expansion with a faster linear expansion following a slower linear expansion; and (iii) accelerating expansion with a continuously increasing rate of spread.We found that individuals with different dispersal abilities were spatially sorted with the stronger dispersers situated at the expanding range front, causing the velocity of expansion to accelerate.Aside from better managing of invasive species, the derived formula could conceivably also be applied to conservation management of relocated, endangered or extra-limital species.

View Article: PubMed Central - PubMed

Affiliation: Centre for Invasion Biology, Department of Mathematical Sciences, Stellenbosch University, Matieland, South Africa; Mathematical and Physical Biosciences, African Institute for Mathematical Sciences, Muizenberg, South Africa.

ABSTRACT
Range expansion of spreading organisms has been found to follow three types: (i) linear expansion with a constant rate of spread; (ii) bi-phase expansion with a faster linear expansion following a slower linear expansion; and (iii) accelerating expansion with a continuously increasing rate of spread. To date, no overarching formula exists that can be applied to all three types of range expansion. We investigated how propagule pressure, i.e., the initial number of individuals and their composition in terms of dispersal ability, affects the spread of a population. A system of integrodifference equations was then used to model the spatiotemporal dynamics of the population. We studied the dynamics of dispersal ability as well as the instantaneous and asymptotic rate of spread. We found that individuals with different dispersal abilities were spatially sorted with the stronger dispersers situated at the expanding range front, causing the velocity of expansion to accelerate. The instantaneous rate of spread was found to be fully determined by the growth and dispersal abilities of the population at the advancing edge of the invasion. We derived a formula for the asymptotic rate of spread under different scenarios of propagule pressure. The results suggest that data collected from the core of the invasion may underestimate the spreading rate of the population. Aside from better managing of invasive species, the derived formula could conceivably also be applied to conservation management of relocated, endangered or extra-limital species.

Show MeSH
Related in: MedlinePlus