mathematics

Nicholson–Bailey model

Nicholson–Bailey model is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Nicholson–Bailey model rather than just read about it. In short: 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.

Key takeaways

Nicholson–Bailey model belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Nicholson–Bailey model to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Nicholson–Bailey model from memory before moving on to harder problems.

Reference excerpt

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.

Equations

Derivation The model is defined in discrete time. It is usually expressed as

H t + 1 = k H t e − a P t P t + 1 = c H t ( 1 − e − a P t ) {\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. e − a P t {\displaystyle e^{-aP_{t}}} is the probability that the host will survive P t {\displaystyle P_{t}} predators; whereas 1 − e − a P t {\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 0 < k < 1 {\displaystyle 0<k<1} , ( H ¯ , P ¯ ) = ( 0 , 0 ) {\displaystyle ({\bar {H}},{\bar {P}})=(0,0)} is the unique non-negative fixed point and all non-negative solutions converge to ( 0 , 0 ) {\displaystyle (0,0)} . When k = 1 {\displaystyle k=1} , all non-negative solutions lie on level curves of the function z = H + P − ln ( P ) {\displaystyle z=H+P-{\text{ln}}(P)} and converge to a fixed point on the P {\displaystyle P} -axis. When k > 1 {\displaystyle k>1} , this system admits one unstable positive fixed point, at

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Nicholson–Bailey model

Start with the simplest possible case. Write down what Nicholson–Bailey model claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Nicholson–Bailey model before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Nicholson–Bailey model ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Nicholson–Bailey model

In research: Nicholson–Bailey model appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Nicholson–Bailey model in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Nicholson–Bailey model is common in secondary-school and first-year university syllabi. It links to neighbouring topics Mathematical and theoretical biology, Population models, Predation, so understanding it makes those chapters shorter.
In everyday life: Look for Nicholson–Bailey model outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

Ask Teacher Smith questions about this articleOpens your AI tutor with a question about “Nicholson–Bailey model” →

Affiliate

Preply — study more efficiently by working with a personal tutor. 50% off.

How to study Nicholson–Bailey model in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Nicholson–Bailey model means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Nicholson–Bailey model out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Nicholson–Bailey model in simple terms?

Why does Nicholson–Bailey model matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Nicholson–Bailey model?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Nicholson–Bailey model.