### Ball Bearings

If a is the radius of the axle, b the radius of each ball-bearing, and c the radius of the hub, why does the number of ball bearings n determine the ratio c/a? Find a formula for c/a in terms of n.

### Overarch 2

Bricks are 20cm long and 10cm high. How high could an arch be built without mortar on a flat horizontal surface, to overhang by 1 metre? How big an overhang is it possible to make like this?

### Cushion Ball

The shortest path between any two points on a snooker table is the straight line between them but what if the ball must bounce off one wall, or 2 walls, or 3 walls?

# Population Dynamics

##### Stage: 5 Challenge Level:

A hive of bees, a colony of ants and a parliament of owls.

These are just a few examples of animal groups, or populations. A population is dynamic; this means it is constantly changing in size and demographics. New animals are born, old animals die and other factors such as drought, fire and lack of predators, all cause a change in the population.

The population growth is the change in the number of individuals in a population, per unit time. For example, if a population has ten births and five deaths per year, then the population growth is five individuals per year.

In the following pages, we aim to represent populations and changes in populations using mathematics. This involves using differential equations and even probability.

Links to pages on differential equations:

 Part 1 A First Model Part 2 Exponential and Geometric Models Part 3 The Logistic Equation Part 4 The Logistic Map Part 5 The Lotka-Volterra Equations Part 6 Modified L-V Equations

Links to pages on probability:

### Beginning the Model

We are able to describe population growth by making some generalizations and using simple differential equations:

The size, $N_t$, of a population depends upon:

• The initial number of individuals, $N_0$
• The number of births, B
• The number of deaths, D
• The number of immigrants, I
• The number of migrants, E

This gives us the equation: $$N_t=N_0+B-D+I-E$$

When a population is closed, there is no immigration or emigration. This often occurs on remote islands, such as the Galapagos Islands. Our equation then becomes $N_t=N_0+B-D$ , or equivalently $$N_{t+1}=N_t+B-D$$

Clearly the population will increase if $B> D$, and will decrease if $B< D$.

A population is in equilibrium if on average the population size remains constant over a long period of time. Mathematically, this means:   $N_t=N_{t+\Delta t}$

Question:

We can rewrite the equation $N_{t+1}=N_t+B-D$ , as: $$N_{t+1}-N_t=\Delta N_t=B-D$$ Intuitively, why does this make sense? Think of an example of a population to explain why.