The Lambert W function in ecological and evolutionary models

Lehtonen, Jussi

doi:10.1111/2041-210x.12568

Cited by 54 publications

(40 citation statements)

References 53 publications

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“…When the rate of environmental change is less than this maximum rate of evolution,

k < k_{t i p}

, the steady‐state lag is

\hat{l} = {(V w_{k})}^{1 / 2}

(solid black curve in Fig. C), where

w_{k}

is the solution to

w_{k} \exp false(w_{k} false) = {(k V)}^{2} / {(d σ_{g}^{2} σ_{w})}^{2}

(i.e.,

w_{k} false(x false)

is the Lambert W function, and here

x = {(k V)}^{2} / {(d σ_{g}^{2} σ_{w})}^{2}

; Lehtonen ). If this lag remains biologically valid (real) at the point where the expected long‐run population growth rate becomes zero, there is a critical rate of environmental change,

k_{c}

, that determines persistence.…”

Section: Methods and Resultsmentioning

confidence: 99%

An evolutionary tipping point in a changing environment

Osmond

Klausmeier

2017

Evolution

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Populations can persist in directionally changing environments by evolving. Quantitative genetic theory aims to predict critical rates of environmental change beyond which populations go extinct. Here, we point out that all current predictions effectively assume the same specific fitness function. This function causes selection on the standing genetic variance of quantitative traits to become increasingly strong as mean trait values depart from their optima. Hence, there is no bound on the rate of evolution and persistence is determined by the critical rate of environmental change at which populations cease to grow. We then show that biologically reasonable changes to the underlying fitness function can impose a qualitatively different extinction threshold. In particular, inflection points caused by weakening selection create local extrema in the strength of selection and thus in the rate of evolution. These extrema can produce evolutionary tipping points, where long-run population growth rates drop from positive to negative values without ever crossing zero. Generic early-warning signs of tipping points are found to have little power to detect imminent extinction, and require hard-to-gather data. Furthermore, we show how evolutionary tipping points produce evolutionary hysteresis, creating extinction debts.

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“…When the rate of environmental change is less than this maximum rate of evolution,

k < k_{t i p}

, the steady‐state lag is

\hat{l} = {(V w_{k})}^{1 / 2}

(solid black curve in Fig. C), where

w_{k}

is the solution to

w_{k} \exp false(w_{k} false) = {(k V)}^{2} / {(d σ_{g}^{2} σ_{w})}^{2}

(i.e.,

w_{k} false(x false)

is the Lambert W function, and here

x = {(k V)}^{2} / {(d σ_{g}^{2} σ_{w})}^{2}

; Lehtonen ). If this lag remains biologically valid (real) at the point where the expected long‐run population growth rate becomes zero, there is a critical rate of environmental change,

k_{c}

, that determines persistence.…”

Section: Methods and Resultsmentioning

confidence: 99%

An evolutionary tipping point in a changing environment

Osmond

Klausmeier

2017

Evolution

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“…With the assumption of constant seed rain S , it is possible to find an exact solution for time to canopy closure ( t 1 ):

t_{1} = a_{normalg} - \frac{(][, W_{- 1} ()(, e^{(\frac{- H_{normalg} {normalμ}_{normalc}}{G_{c}} - 1)} (][, \frac{μ_{normalc}^{2} e^{μ_{g} a_{g}}}{G_{c} S c_{1}} - \frac{H_{g} μ_{c}}{G_{normalc}} - 1)) + \frac{H_{g} μ_{c}}{G_{normalc}} + 1)}{μ_{c}}

where W −1 is the nonprincipal branch of Lambert's W function (Corless et al. , Lehtonen ). This closed‐form expression can determine time to canopy closure, using parameters from individual‐level demography, without the need for numerical simulations.…”

Section: Methods and Resultsmentioning

confidence: 99%

“…where W −1 is the nonprincipal branch of Lambert's W function (Corless et al 1996, Lehtonen 2016). This closed-form expression can determine time to canopy closure, using parameters from individual-level demography, without the need for numerical simulations.…”

Section: Figs 4mentioning

confidence: 99%

When does seed limitation matter for scaling up reforestation from patches to landscapes?

Caughlin

Elliott

Lichstein

2016

Ecological Applications

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Restoring forest to hundreds of millions of hectares of degraded land has become a centerpiece of international plans to sequester carbon and conserve biodiversity. Forest landscape restoration will require scaling up ecological knowledge of secondary succession from small-scale field studies to predict forest recovery rates in heterogeneous landscapes. However, ecological field studies reveal widely divergent times to forest recovery, in part due to landscape features that are difficult to replicate in empirical studies. Seed rain can determine reforestation rate and depends on landscape features that are beyond the scale of most field studies. We develop mathematical models to quantify how landscape configuration affects seed rain and forest regrowth in degraded patches. The models show how landscape features can alter the successional trajectories of otherwise identical patches, thus providing insight into why some empirical studies reveal a strong effect of seed rain on secondary succession, while others do not. We show that seed rain will strongly limit reforestation rate when patches are near a threshold for arrested succession, when positive feedbacks between tree canopy cover and seed rain occur during early succession, and when directed dispersal leads to between-patch interactions. In contrast, seed rain has weak effects on reforestation rate over a wide range of conditions, including when landscape-scale seed availability is either very high or very low. Our modeling framework incorporates growth and survival parameters that are commonly estimated in field studies of reforestation. We demonstrate how mathematical models can inform forest landscape restoration by allowing land managers to predict where natural regeneration will be sufficient to restore tree cover. Translating quantitative forecasts into spatially targeted interventions for forest landscape restoration could support target goals of restoring millions of hectares of degraded land and help mitigate global climate change.

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“…Although the number of prey eaten (N e ) appears on both sides of eqn 2, it can be solved using the 'Lambert-W' function (W). The derivation and definition of W is beyond the scope of this manuscript but it is described in detail in Corless et al (1996) and with respect to ecological applications by Lehtonen (2016).…”

Section: T H E a N A T O M Y O F A F U N C T I O N A L R E S P O N S Ementioning

confidence: 99%

“…The derivation and definition of W is beyond the scope of this manuscript but it is described in detail in Corless et al . () and with respect to ecological applications by Lehtonen ().…”

Section: Introductionmentioning

confidence: 99%

frair: an R package for fitting and comparing consumer functional responses

et al. 2017

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Summary1. Consumer-resource interactions (i.e. the functional response) underpin decades of ecological advancements. However, selecting, fitting and comparing functional response models using appropriate methods remains a non-trivial endeavour. 2. The R package FRAIR provides tools for selecting and differentiating various forms of consumer functional response models, a consistent interface for fitting and visualising response curves, and a selection of statistically robust methods for comparing fitted parameters. 3. Using real data from crustacean predator-prey systems, we demonstrate the utility of FRAIR, highlighting best practice and common analytical mistakes.

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The Lambert W function in ecological and evolutionary models

Cited by 54 publications

References 53 publications

An evolutionary tipping point in a changing environment

An evolutionary tipping point in a changing environment

When does seed limitation matter for scaling up reforestation from patches to landscapes?

frair: an R package for fitting and comparing consumer functional responses

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