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Matrix models for size-structured populations: unrealistic fast growth or simply diffusion?

Picard N, Liang J - PLoS ONE (2014)

Bottom Line: For 53 tree species from a tropical rain forest in the Central African Republic, the diffusion resulting from common matrix dimensions was much stronger than would be realistic.Moreover, λ could either increase or decrease when class width increased depending on the species.Overall, even if the class width should be kept small enough to limit diffusion, it had little impact on the estimate of λ for tree species.

View Article: PubMed Central - PubMed

Affiliation: UPR Biens et services des écosystèmes forestiers tropicaux (BSEF), Centre de coopération internationale en recherche agronomique pour le développement (CIRAD), Montpellier, France.

ABSTRACT
Matrix population models are widely used to study population dynamics but have been criticized because their outputs are sensitive to the dimension of the matrix (or, equivalently, to the class width). This sensitivity is concerning for the population growth rate (λ) because this is an intrinsic characteristic of the population that should not depend on the model specification. It has been suggested that the sensitivity of λ to matrix dimension was linked to the existence of fast pathways (i.e. the fraction of individuals that systematically move up a class), whose proportion increases when class width increases. We showed that for matrix population models with growth transition only from class i to class i + 1, λ was independent of the class width when the mortality and the recruitment rates were constant, irrespective of the growth rate. We also showed that if there were indeed fast pathways, there were also in about the same proportion slow pathways (i.e. the fraction of individuals that systematically remained in the same class), and that they jointly act as a diffusion process (where diffusion here is the movement in size of an individual whose size increments are random according to a normal distribution with mean zero). For 53 tree species from a tropical rain forest in the Central African Republic, the diffusion resulting from common matrix dimensions was much stronger than would be realistic. Yet, the sensitivity of λ to matrix dimension for a class width in the range 1-10 cm was small, much smaller than the sampling uncertainty on the value of λ. Moreover, λ could either increase or decrease when class width increased depending on the species. Overall, even if the class width should be kept small enough to limit diffusion, it had little impact on the estimate of λ for tree species.

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Growth of a cohort of 100 Celtis zenkeri trees with a uniform initial diameter distribution.Projection time is  yr. The time step of the matrix model is  yr and the class width is column-wise (A)  cm, (B) 0.9994 cm, and (C) 2.4985 cm. The top panel shows images of the transitions matrices between the initial and final times (i.e. the annual transition matrices raised to the power of ), where the starting class is column-wise, the ending class is row-wise, and the transition rates between classes are shown using heat colors (from white = zero to red = the highest values). The bottom panel shows the predicted dbh distributions: dotted line = initial dbh distribution (uniform across 10–14.997 cm); solid line = final dbh distribution according to the McKendrick continuous model; shaded bins = final dbh distribution according to the Usher matrix model.
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pone-0098254-g001: Growth of a cohort of 100 Celtis zenkeri trees with a uniform initial diameter distribution.Projection time is yr. The time step of the matrix model is yr and the class width is column-wise (A) cm, (B) 0.9994 cm, and (C) 2.4985 cm. The top panel shows images of the transitions matrices between the initial and final times (i.e. the annual transition matrices raised to the power of ), where the starting class is column-wise, the ending class is row-wise, and the transition rates between classes are shown using heat colors (from white = zero to red = the highest values). The bottom panel shows the predicted dbh distributions: dotted line = initial dbh distribution (uniform across 10–14.997 cm); solid line = final dbh distribution according to the McKendrick continuous model; shaded bins = final dbh distribution according to the Usher matrix model.

Mentions: To illustrate fast and slow pathways, we considered the dynamics of an even-aged cohort of 100 trees uniformly distributed between 10 and 15.0 cm at . When cm and yr, the condition was met and the Usher scheme for growth was not dissipative (Figure 0A): there were neither fast pathways nor slow ones in this case. With the exception of the last class, non- transition rates defined one-to-one connections between classes. The only difference between the exact solution of the McKendrick equation and the Usher model followed from the difference between (for the exact solution) and (for the Usher model) for the attenuation of the number of trees. For C. zenkeri, this difference was actually so small that it is not visually perceptible in Figure 1A.


Matrix models for size-structured populations: unrealistic fast growth or simply diffusion?

Picard N, Liang J - PLoS ONE (2014)

Growth of a cohort of 100 Celtis zenkeri trees with a uniform initial diameter distribution.Projection time is  yr. The time step of the matrix model is  yr and the class width is column-wise (A)  cm, (B) 0.9994 cm, and (C) 2.4985 cm. The top panel shows images of the transitions matrices between the initial and final times (i.e. the annual transition matrices raised to the power of ), where the starting class is column-wise, the ending class is row-wise, and the transition rates between classes are shown using heat colors (from white = zero to red = the highest values). The bottom panel shows the predicted dbh distributions: dotted line = initial dbh distribution (uniform across 10–14.997 cm); solid line = final dbh distribution according to the McKendrick continuous model; shaded bins = final dbh distribution according to the Usher matrix model.
© Copyright Policy
Related In: Results  -  Collection

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Show All Figures
getmorefigures.php?uid=PMC4048208&req=5

pone-0098254-g001: Growth of a cohort of 100 Celtis zenkeri trees with a uniform initial diameter distribution.Projection time is yr. The time step of the matrix model is yr and the class width is column-wise (A) cm, (B) 0.9994 cm, and (C) 2.4985 cm. The top panel shows images of the transitions matrices between the initial and final times (i.e. the annual transition matrices raised to the power of ), where the starting class is column-wise, the ending class is row-wise, and the transition rates between classes are shown using heat colors (from white = zero to red = the highest values). The bottom panel shows the predicted dbh distributions: dotted line = initial dbh distribution (uniform across 10–14.997 cm); solid line = final dbh distribution according to the McKendrick continuous model; shaded bins = final dbh distribution according to the Usher matrix model.
Mentions: To illustrate fast and slow pathways, we considered the dynamics of an even-aged cohort of 100 trees uniformly distributed between 10 and 15.0 cm at . When cm and yr, the condition was met and the Usher scheme for growth was not dissipative (Figure 0A): there were neither fast pathways nor slow ones in this case. With the exception of the last class, non- transition rates defined one-to-one connections between classes. The only difference between the exact solution of the McKendrick equation and the Usher model followed from the difference between (for the exact solution) and (for the Usher model) for the attenuation of the number of trees. For C. zenkeri, this difference was actually so small that it is not visually perceptible in Figure 1A.

Bottom Line: For 53 tree species from a tropical rain forest in the Central African Republic, the diffusion resulting from common matrix dimensions was much stronger than would be realistic.Moreover, λ could either increase or decrease when class width increased depending on the species.Overall, even if the class width should be kept small enough to limit diffusion, it had little impact on the estimate of λ for tree species.

View Article: PubMed Central - PubMed

Affiliation: UPR Biens et services des écosystèmes forestiers tropicaux (BSEF), Centre de coopération internationale en recherche agronomique pour le développement (CIRAD), Montpellier, France.

ABSTRACT
Matrix population models are widely used to study population dynamics but have been criticized because their outputs are sensitive to the dimension of the matrix (or, equivalently, to the class width). This sensitivity is concerning for the population growth rate (λ) because this is an intrinsic characteristic of the population that should not depend on the model specification. It has been suggested that the sensitivity of λ to matrix dimension was linked to the existence of fast pathways (i.e. the fraction of individuals that systematically move up a class), whose proportion increases when class width increases. We showed that for matrix population models with growth transition only from class i to class i + 1, λ was independent of the class width when the mortality and the recruitment rates were constant, irrespective of the growth rate. We also showed that if there were indeed fast pathways, there were also in about the same proportion slow pathways (i.e. the fraction of individuals that systematically remained in the same class), and that they jointly act as a diffusion process (where diffusion here is the movement in size of an individual whose size increments are random according to a normal distribution with mean zero). For 53 tree species from a tropical rain forest in the Central African Republic, the diffusion resulting from common matrix dimensions was much stronger than would be realistic. Yet, the sensitivity of λ to matrix dimension for a class width in the range 1-10 cm was small, much smaller than the sampling uncertainty on the value of λ. Moreover, λ could either increase or decrease when class width increased depending on the species. Overall, even if the class width should be kept small enough to limit diffusion, it had little impact on the estimate of λ for tree species.

Show MeSH
Related in: MedlinePlus