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# 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

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 er 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

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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