Nicholson–Bailey model

The Nicholson–Bailey model was developed in the 1930s to describe the population dynamics of a coupled host-parasitoid system.^a It is named after Alexander John Nicholson and Victor Albert Bailey. Host-parasite and prey-predator systems can also be represented with the Nicholson-Bailey model. The model is closely related to the Lotka–Volterra model, which describes the dynamics of antagonistic populations (preys and predators) using differential equations.

The model uses (discrete time) difference equations to describe the population growth of host-parasite populations. The model assumes that parasitoids search for hosts at random, and that both parasitoids and hosts are assumed to be distributed in a non-contiguous ("clumped") fashion in the environment. In its original form, the model does not allow for stable coexistence. Subsequent refinements of the model, notably adding density dependence on several terms, allowed this coexistence to happen.

A credible, simple alternative to the Lotka–Volterra predator–prey model and its common prey dependent generalizations (like Nicholson–Bailey) is the ratio-dependent or Arditi–Ginzburg model.^[1] The two are the extremes of the spectrum of predator interference models. According to the authors of the alternative view, the data show that true interactions in nature are so far from the Lotka–Volterra extreme on the interference spectrum that the model can simply be discounted as wrong. They are much closer to the ratio dependent extreme, so if a simple model is needed one can use the Arditi–Ginzburg model as the first approximation.^[2]

Equations

Model derivation

The model is defined in discrete time. It is usually expressed as:^[3]

${\displaystyle {\begin{array}{rcl}H_{t+1}&=&kH_{t}e^{-aP_{t}}\\P_{t+1}&=&cH_{t}\left(1-e^{-aP_{t}}\right)\end{array}}}$

with H the population size of the host, P the population size of the parasitoid, k the reproductive rate of the host, a the searching efficiency of the parasitoid, and c the average number of viable eggs that a parasitoid lays on a single host.

This model can be explained based on probability.^[3] ${\displaystyle e^{-aP_{t}}}$ is the probability that the host will survive ${\displaystyle P_{t}}$ predators; whereas ${\displaystyle 1-e^{-aP_{t}}}$ is that they will not, bearing in mind the parasitoid eventually will hatch into larva and escape.

Model analysis

Simulating this system usually results in oscillations moving away from an unstable equilibrium.

Depending on the parameter values, this system admits on unstable fixed point, at

${\displaystyle {\begin{array}{rcl}H^{\star }&=&{\frac {k\,{\text{ln}}(k)}{(k-1)\,ac}}\\P^{\star }&=&{\frac {{\text{ln}}(k)}{a}}\end{array}}}$

This point has positive abundances of P and H only when k is greater than one.

See also

• Lotka–Volterra inter-specific competition equations
• Population dynamics

Notes

• ^a Parasitoids encompass insects that place their ova inside the eggs or larva of other creatures (generally other insects as well).^[3]

References

1. Arditi, R.; Ginzburg, L. R. (1989). "Coupling in predator-prey dynamics: ratio dependence" (PDF). Journal of Theoretical Biology. 139: 311–326. doi:10.1016/s0022-5193(89)80211-5.
2. Arditi, R.; Ginzburg, L. R. (2012). How Species Interact: Altering the Standard View on Trophic Ecology. Oxford University Press. ISBN 978-0-19-991383-1.
3. Logan, J. David; Wolesensky, Willian R. (2009). Mathematical Methods in Biology. Pure and Applied Mathematics: a Wiley-interscience Series of Texts, Monographs, and Tracts. John Wiley & Sons. p. 214. ISBN 978-0-470-52587-6.

Further reading

• Hopper, J. L. (1987). "Opportunities and Handicaps of Antipodean Scientists: A. J. Nicholson and V. A. Bailey on the Balance of Animal Populations". Historical Records of Australian Science. 7 (2): 179–188. doi:10.1071/hr9880720179.

External links

• Nicholson–Bailey model
• Nicholson-Bailey model with density dependence
• Nicholson-Bailey spatial model