Biomedical subjects

Alan J McKane

Publications and source records attributed to Alan J McKane.

5 recordsLinked to original sources

PubMed2006

The merits of neutral theory.

Hubbell's neutral theory of biodiversity has challenged the classic niche-based view of ecological community structure. Although there have been many attempts to falsify Hubbell's theory, we argue that falsification should not lead to rejection, because there is more to the theory than neutrality alone. Much of the criticism has focused on the neutrality assumption without full appreciation of other relevant aspects of the theory. Here, we emphasize that neutral theory is also a stochastic theory, a sampling theory and a dispersal-limited theory. These important additional features should be retained in future theoretical developments of community ecology.

Animals↗

PubMed2005

Quantifying stochastic outcomes.

A system consisting of two species in a fluctuating environment, when the interspecies competition for resources is strong, will have a stochastic outcome: only one of the species will survive, but there is no a priori way of knowing which one this will be. It is natural in such a situation to ask what will be the probability of one or another of the species surviving. This probability is calculated as a function of the average growth rates and the strengths of the interaction between the species and of the randomness. This is an example of a class of stochastic problems in which multiple final states are available for occupation. We refer to the choice of final states as state selection, and the probabilities of final states being occupied as state-selection probabilities. The calculation of these probabilities is carried out in the context of a model of the system which consists of two coupled stochastic differential equations. By reformulating these equations in terms of path integrals, the powerful methods based on the use of optimal paths may be utilized to calculate the probability of one outcome or the other. The analytical results obtained by using this technique agree well with numerical simulations when both species have the same growth rate. Although the method adopted rests on the assumption that the strength of the fluctuations, D , is small, remarkably the analytic results are still found to be in good agreement with the numerical results when D is of order 1.

Journal Article↗

PubMed2004

The impact of nonlinear functional responses on the long-term evolution of food web structure.

We investigate the long-term web structure emerging in evolutionary food web models when different types of functional responses are used. We find that large and complex webs with several trophic layers arise only if the population dynamics is such that it allows predators to focus on their best prey species. This can be achieved using modified Lotka-Volterra or Holling/Beddington functional responses with effective couplings that depend on the predator's efficiency at exploiting the prey, or a ratio-dependent functional response with adaptive foraging. In contrast, if standard Lotka-Volterra or Holling/Beddington functional responses are used, long-term evolution generates webs with almost all species being basal, and with additionally many links between these species. Interestingly, in all cases studied, a large proportion of weak links result naturally from the evolution of the food webs.

Animals↗

PubMed2004

Optimal paths and the calculation of state selection probabilities.

The addition of noise to a dynamical system means that initial states near points of instability may no longer decay to a unique stable state. A common example of this behavior occurs in a dynamical system with two degrees of freedom and with two or more stable states. If the initial state of the system is near the separatrices bounding the basins of attraction of these stable states, then the addition of noise to the system means that there is a nonzero probability that the stable state selected is in a different basin of attraction to that of the initial state. We discuss a method of calculating these state-selection probabilities based on a path-integral representation of the stochastic dynamics. The relationship of this approach to a method based on the solution of the backward Fokker-Planck equation is particularly stressed, since this was used in previous studies of problems of this type. However, while the method based on the backward Fokker-Planck equation is a powerful one for systems with one degree of freedom, in systems with more degrees of freedom it is much less useful. Since the standard method of solution in this case involves a series of mappings onto a deterministic dynamics which is simply the classical dynamics associated with the path-integral formulation, we argue that for systems with more than one degree of freedom, the path-integral method is a very natural way of calculating state-selection probabilities. We illustrate this on a simple example taken from population biology, and find that the state-selection probabilities are in excellent agreement with Monte Carlo simulations.

Journal Article↗

PubMed2004

Analytic solution of Hubbell's model of local community dynamics.

Recent theoretical approaches to community structure and dynamics reveal that many large-scale features of community structure (such as species-rank distributions and species-area relations) can be explained by a so-called neutral model. Using this approach, species are taken to be equivalent and trophic relations are not taken into account explicitly. Here we provide a general analytic solution to the local community model of Hubbell's neutral theory of biodiversity by recasting it as an urn model, i.e. a Markovian description of states and their transitions. Both stationary and time-dependent distributions are analysed. The stationary distribution-also called the zero-sum multinomial-is given in closed form. An approximate form for the time-dependence is obtained by using an expansion of the master equation. The temporal evolution of the approximate distribution is shown to be a good representation for the true temporal evolution for a large range of parameter values.

Biodiversity↗