Unified Neutral Theory of Biodiversity - Dynamics Under Neutral Hypothesis

Dynamics Under Neutral Hypothesis

All biodiversity patterns previously described are related to time-independent quantities. However, for biodiversity evolution and species preservation, it is crucial to compare the dynamics of ecosystems with models (Leigh, 2007). An easily accessible index of the underlying evolution is the so called species turnover distribution (STD), defined as the probability P(r,t) that the population of any species has varied by a fraction r after a given time t.

A neutral model that can analytically predict both the relative species abundance (RSA) at steady-state and the STD at time t has been presented in . Within this framework the population of any species is represented by a continuous (random) variable x, whose evolution is governed by the following Langevin equation:

$\dot{x}=b-x/\tau+\sqrt{Dx}\xi(t)$

where b is the immigration rate from a large regional community, represents competition for finite resources and D is related to demographic stochasticity; is a Gaussian white noise. The model can also be derived as a continuous approximation of a master equation, where birth and death rates are independent of species, and predicts that at steady-state the RSA is simply a gamma distribution.

From the exact time-dependent solution of the previous equation, one can exactly calculate the STD at time t under stationary conditions:

$P(r,t)=A\frac{\lambda+1}{\lambda}\frac{(e^{t/\tau})^{b/2D}}{1-e^{-t/\tau}}\left(\frac{\sinh(\frac{t}{2\tau})}{\lambda}\right)^{\frac{b}{D}+1}\left(\frac{4\lambda^2}{(\lambda+1)^2e^{t/\tau}-4\lambda}\right)^{\frac{b}{D}+\frac{1}{2}}.$

This formula provides good fits of data collected in the Barro Colorado tropical forest from 1990 to 2000. From the best fit one can estimate ~ 3500 years with a broad uncertainty due to the relative short time interval of the sample. This parameter can be interpreted as the relaxation time of the system, i.e. the time the system needs to recover from a perturbation of species distribution. In the same framework, the estimated mean species lifetime is very close to the fitted temporal scale . This suggests that the neutral assumption could correspond to a scenario in which species originate and become extinct on the same timescales of fluctuations of the whole ecosystem.

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