**Malthus Model**

: Total population

: Birth-rate per capita

: Death-rate per capita

Solution to D.E.:

where .

**Logistic Equation**

**Logistic Case 1: Increasing population ()**

The second expression can be derived from the first: divide by in both the numerator and denominator.

Logistic Case 2: Decreasing population ()

Logistic Case 3: Constant population ()

Harvesting

Basic Harvesting Model:

: Harvest rate (Amount harvested per unit time)

Maximum harvest rate without causing extinction: .

: Unstable equilibrium population

: Stable equilibrium population

Extinction Time:

Laplace transform of

Tip: Use this equation when the questions contains the words “show from the definition”.

**Inverse transform of **

**Linearity**

## List of common Laplace Transforms

**-shifting**

If , , then

.

Tip: Use this when doing Laplace Transform of a function with an exponential factor . Note that the reverse direction can sometimes be used as well:

**-shifting**

If , then

Tip: Frequently, we use the reverse direction

**Delta function**

: infinitely tall and narrow spike at .

: infinitely tall and narrow spike at .

**Two properties of delta function**

for .

Tip: Use delta function when the keywords “suddenly”, “burst”, etc. appear.

**Unit step function**

For ,

Tip: Use unit step function for questions that require a force to “switch on / switch off” at certain times.