# Mathematics Cheat Sheet for Population Biology

**1 Introduction**

If you fake it long enough, there comes a point where you aren’t faking it any
more. Here are

some tips to help you along the way...

**2 Calculus**

Derivative The definition of a derivative is as follows. For some function f(x),

**2.1 Differentiation Rules **

It is useful to remember the following rules for differentiation . Let f(x) and
g(x) be two functions

**2.1.1 Linearity **

for constants a and b.

**2.1.2 Product rule **

**2.1.3 Chain rule **

**2.1.4 Quotient Rule**

**2.1.5 Some Basic Derivatives**

**2.1.6 Convexity and Concavity**

It is very easy to get confused about the convexity and concavity of a function.
The technical

mathematical definition is actually somewhat at odds with the colloquial usage.
Let f(x) be a

twice differentiable function in an interval I. Then:

If you think about a profit function as a function of
time, a convex function would show

increasing marginal returns, while a concave function would show decreasing
marginal returns.

This leads into an important theorem (particularly for stochastic demography),
known as

Jensen’ s Inequality . For a convex function f(x),

**2.2 Taylor Series**

where denotes the kth
derivative of f evaluated at a, and k! = k(k − 1)(k − 2) . . . (1).

For example, we can approximate e^{r} at a = 0:

Figure 1: Illustration of Jensen’ s Inequality .

Expanding log (1 + x) around a = 0 yields:

**2.3 Jacobian**

For a system of equations , F(x) and G(λ), the Jacobian matrix is

This is very important for the analysis of stability of
interacting models such as those for

epidemics and predator- prey systems . The equilibrium of a system is stable if
and only if the

real parts of all the eigenvalues of J are negative.

**2.4 Integration
**

**Linearity**

**Integration by Parts**

**Some Useful Facts About Integrals**

**2.5 Definite Integrals**

**2.5.1 Expectation**

For a continuous random variable X with probability density function f(x), the
expected value,

or mean, is

where the integral is taken over the set of all possible
outcomes Ω.

For example, the average age of mothers of newborns in a stable population:

Since (from the Euler-Lotka equation) the probability that
a mother will be a years old in a

stable population is

**Some Properties of Expectation **

For two discrete random variables, X and Y ,

**2.5.2 Variance**

For a continuous random variable X with probability density function f(x) and
expected value

μ, the variance is

A useful formula for calculating variances:

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