### 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 - Part 1

### Per Capita Rates

It is important to relate the basic population parameters (such as births or deaths) to the size of the whole population. This allows us to make a better decision if a population is at risk.

We define the per capita birth rate (or nativity rate) as the number of births per individual per unit time interval: $b=\frac {B}{N}$ .

Similarly we define the per capita death rate (or mortality rate) as the number of deaths per individual per unit time interval:  $d=\frac {D}{N}$

### The First Model

Recall the population equation from before: $$N_{t+1}=N_t+B-D$$ Because per capita birth and death rates do not change with the size (or density) of the population, we can rewrite our model in terms of per capita rates: $$N_{t+1}=N_t+bN_t-dN_t=N_t+(b-d)N_t$$ This model is said to be density-independent.

We call the term $r=b-d$, the geometric rate of increase. Note that $r=\frac{\Delta N_t}{N_t}$ , so r can be interpreted as the per capita rate of change of population size.

The equation for our model becomes: \begin{align*} N_{t+1}&=N_t+rN_t \\ &=(1+r)N_t \\ &=\lambda N_t \end{align*} where $\lambda=1+r$ is defined as the finite rate of increase. Note that $\lambda=\frac{N_{t+1}}{N_t}$ , so $\lambda$ can be interpreted as the ratio between the population size at one time to another time.

How do you think we can solve this new equation? Go here for more information.

Do you think this model is valid in reality? What problems do you think might occur? Think about environmental resources and density-independence. An investigation of these problems can be found here.

Question:

If 20 sea otters from a total population of 850 are fatally affected by disease, what is the mortality as a per capita rate?

Given the population is initially 850, and increases to 1000 after one year, what is the value of $\lambda$?

Use this to find the per capita birth rate, and find the population size in ten years.