# Mathematics for the Life Sciences

Biology and the life sciences are relatively late in adapting mathematical techniques as compared to e.g. physics. With this book the authors want to bring mathematics early in the curriculum of biology students.

This is a glossy textbook intended for biology students at an undergraduate level. It introduces the mathematics needed for modeling in life sciences. Some elementary notions of mathematics are assumed known, but basically the text is self contained. No formal theorems are formulated and hence no mathematical proofs are included, but there are many examples, and exercises, usually illustrated with matlab code. Matlab is briefly introduced in an appendix, but it is meant to be learned along the way.

There are many alternative books available on the market that deal with "Mathematics for the Life Sciences". Most universities have such a curriculum and professors are using one of these books or have their own lecture notes. Calculus, modeling, probability, and dynamical systems (discrete and/or continuous) constitute a common denominator that shows up in most of them with varying degrees of depth in their theoretical and practical approach. This book can be classified as rather on the "light" side, less theoretical, and with the use of matlab, somewhat oriented towards practice.

A first unit explains descriptive statistics, linear regression, and the visualization (log-scales —needing exp and log functions—; histograms, etc.). Matlab programs are provided and many exercises are listed. The answers to some selected exercises are given in an appendix. Like all other units, it ends with a number of projects to be worked out with guidelines for the students and the teacher.

A second unit is about discrete modelling. That means giving the definition and properties of sequences and their limits, and working with difference equations. First in one variable, later for the multivariate case. It is basically about population growth and similar problems. Solving these problems requires the use of matrices (Leslie matrix) and vectors, and to study steady state solutions and stability also the eigenvalue-eigenvector problems need to be introduced. Hence the mathematical component here is essentially linear algebra.

The next unit introduces elements from probability. This includes some combinatorics, conditional probability, independence, Bayes rules, etc. From the life science side, these newly acquired mathematical skills are applied to population genetics.

The second half of the book (units 4-7) switches from discrete to continuous. This involves some style break. While the first three units were dealing with somehow unrelated topics, the remainder of the book follows more or less the order of a classic calculus course. The topics are built up following a mathematical rationale. First continuity and limits are defined, then derivatives, and integrals, to result in differential equations. Of course examples are taken from biology.

The text is the result of many years of experience and follows the "rule of five" which included the four complementary approaches: verbal, symbolic, numerical, and graphical and the authors have added the fifth: the use of motivating (biological) data that engage the students. The text has been successfully used to teach in small groups as well as in large lecturing auditoriums. Some guidelines resulting from experience by the authors for the selection of topics are provided in the introduction. For example it is obvious that the first part (discrete models and probability) can be used independently from the second half (continuous part) of the book, but some other combinations and ordering of selective units are proposed too.

Although many examples are included and they are all taken from a biological context, the emphasis is still on the mathematics, certainly in the second half of the book where continuous mathematics are treated. The more advanced applications of mathematics in biological sciences are not given, which might have been a stronger and more tempting teaser for what mathematics is capable of. I think the guideline for composing the material was still "What mathematics should be known by students when they start their courses in biology?" Perhaps a more drastic approach could have been: "What are the biological problems we want to solve and what are the mathematics needed for that?". Thus instead of "Here is the mathematics and this is an application", it could have been "Here is a biological problem, and these are the mathematical tools needed to solve it". The topics and the problems that are the subject of mathematical biology are still far off. More spectacular biological phenomena like symmetry, spirals, etc. in morphogenesis or phyllotaxis could be other approaches to illustrate the need for dynamical systems, difference and differential equations etc. But in the end everything reduces to the modeling of biological phenomena which is eventually THE most important target in this context, and at least on this aspect a little insight is given.

**Submitted by Adhemar Bultheel |

**21 / Sep / 2014