Nicholson–Bailey model

The Nicholson–Bailey model was developed in the 1930s to describe the

The model is defined in discrete time. It is usually expressed as [1]^[2]

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

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.

Analysis of the Nicholson–Bailey model

When ${\displaystyle 0<k<1}$, ${\displaystyle ({\bar {H}},{\bar {P}})=(0,0)}$ is the unique non-negative fixed point and all non-negative solutions converge to ${\displaystyle (0,0)}$. When ${\displaystyle k=1}$, all non-negative solutions lie on level curves of the function ${\displaystyle z=H+P-{\text{ln}}(P)}$ and converge to a fixed point on the ${\displaystyle P}$-axis.^[4] When ${\displaystyle k>1}$, this system admits one unstable positive fixed point, at

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

It has been proven^[5] that all positive solutions whose initial conditions are not equal to ${\displaystyle ({\bar {H}},{\bar {P}})}$ are unbounded and exhibit oscillations with infinitely increasing amplitude.

Variations

Density dependence can be added to the model, by assuming that the growth rate of the host decreases at high abundances. The equation for the parasitoid is unchanged, and the equation for the host is modified:

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

The host rate of increase k is replaced by r, which becomes negative when the host population density reaches K.

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