The perfect mixing paradox and the logistic equation: Verhulst vs. Lotka

Arditi, Roger; Bersier, Louis‐Félix; Rohr, Rudolf P.

doi:10.1002/ecs2.1599

Cited by 23 publications

(79 citation statements)

References 17 publications

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“…Using the methods explained in the appendices of Arditi et al. () and in the present Appendix S1, it is easy to show that under perfect mixing (

β \to \infty

), model 5 predicts (from A1)

\frac{N_{1}}{γ_{1}} = \frac{N_{2}}{γ_{2}}

and (rewriting A2 to isolate

K_{1} + K_{2}

)

K_{normalT} = (K_{1} + K_{2}) + \frac{r_{1} (1 - \frac{{normalγ}_{1} K_{2}}{{normalγ}_{2} K_{1}}) + r_{2} (1 - \frac{{normalγ}_{2} K_{1}}{{normalγ}_{1} K_{2}})}{r_{1} \frac{γ_{1}}{{normalγ}_{2} K_{1}} + r_{2} \frac{γ_{2}}{{normalγ}_{1} K_{2}}} .

…”

Section: A General Model With Asymmetrical Migrationmentioning

confidence: 92%

“…Incidentally, this is another example that shows the advantage of Verhulst's formalism over Lotka's (see Arditi et al. ).…”

Section: A General Model With Asymmetrical Migrationmentioning

confidence: 93%

“…In Arditi et al. (), we considered the following model for logistic growth in two coupled patches:

\begin{matrix} \frac{d N_{1}}{d t} = r_{1} N_{1} (1 - \frac{N_{1}}{K_{1}}) + β false(N_{2} - N_{1} false), \\ \frac{d N_{2}}{d t} = r_{2} N_{2} (1 - \frac{N_{2}}{K_{2}}) + β false(N_{1} - N_{2} false) . \end{matrix}

…”

mentioning

confidence: 99%

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The perfect mixing paradox and the logistic equation: Verhulst vs. Lotka: Reply

2017

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“…Using the methods explained in the appendices of Arditi et al. () and in the present Appendix S1, it is easy to show that under perfect mixing (

β \to \infty

), model 5 predicts (from A1)

\frac{N_{1}}{γ_{1}} = \frac{N_{2}}{γ_{2}}

and (rewriting A2 to isolate

K_{1} + K_{2}

)

K_{normalT} = (K_{1} + K_{2}) + \frac{r_{1} (1 - \frac{{normalγ}_{1} K_{2}}{{normalγ}_{2} K_{1}}) + r_{2} (1 - \frac{{normalγ}_{2} K_{1}}{{normalγ}_{1} K_{2}})}{r_{1} \frac{γ_{1}}{{normalγ}_{2} K_{1}} + r_{2} \frac{γ_{2}}{{normalγ}_{1} K_{2}}} .

…”

Section: A General Model With Asymmetrical Migrationmentioning

confidence: 92%

“…Incidentally, this is another example that shows the advantage of Verhulst's formalism over Lotka's (see Arditi et al. ).…”

Section: A General Model With Asymmetrical Migrationmentioning

confidence: 93%

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The perfect mixing paradox and the logistic equation: Verhulst vs. Lotka: Reply

2017

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“…The term in parentheses cumulatively acts as ‘friction’ (Arditi et al . ), slowing the rate of increase in species number as diversity increases, in this case modulated by the influence of latitude, global temperature, and their statistical interaction. This equation has a stable equilibrium, which I treat as a statistically estimated ECC, at:

S_{L, T}^{*} = \frac{β_{1}}{({normalβ}_{2} + {normalβ}_{L} L + {normalβ}_{T} T_{i} + {normalβ}_{LT} L T_{i})} .

…”

Section: Methodsmentioning

confidence: 99%

Environmental limits to mammal diversity vary with latitude and global temperature

Brodie

2019

Ecol Lett

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Understanding the mechanisms driving declines in biodiversity with latitude requires assessing if there are ecological limits to the number of species that can coexist, and if these limits vary with latitude, both of which are long‐standing and currently debated questions. Here I show that diversification of North American mammals across the Cenozoic Era slowed as diversity increased. This damping of diversification rates indicates ecological limitation, which occurred even though diversity fluctuated through time and was almost never at an equilibrium ‘saturation’ point. The estimated environmental carrying capacity was correlated with global temperature positively at high latitudes and negatively at low latitudes. Geographical variation in how standing diversity affects diversification rates could help explain the latitudinal biodiversity gradient as well as changes in the strength of the gradient over time.

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“…In Arditi et al. (), they noted that the asymptotic dynamics of system in the case of perfect mixing (i.e., with β → ∞) is different from the asymptotic dynamics of the sum of the two populations in isolation (i.e., with β = 0). In particular, they showed that the equilibrium population size of the system with perfect mixing is different (either larger or smaller) from the sum of equilibrium sizes of the isolated populations.…”

Section: Introductionmentioning

confidence: 99%