# Essential Mathematical Biology

### by Nick Britton

Here is a list of substantive corrections to the first printing. Any new contributions will be gratefully received and acknowledged here. Please e-mail them to me.

 Page 7, three text lines from the bottom. The last interval should be $(1,\infty)$, i.e. replace '0' by '1'. (Thanks to Stephen Eglen, Edinburgh.) Page 31, example 1.4. (i) The eigenvalues are $\lambda_1$ and $-\lambda_1^{-1}$. (ii) The answer is based on a draft in which I included newborn rabbits in the Leslie matrix, and needs to be modified in the following way. First, the model states explicitly that (after time zero) the number of newborns is equal to the number of adults, so we only need to find the ratio $v_1:v_2$ of juveniles to adults, with a change of notation. Now we are in two dimensions, the eigenvalue equation gives $v_2 = \lambda_1 v_1$, $v_1 + v_2 = \lambda_1 v_2$, so $v_1 = (\lambda_1 - 1)v_2$, and the result follows. (Thanks to Herb Hethcote, Iowa and Steven White, Bath.) Pages 34 and 35. The component $u_{0,n}$ should not appear in the vector $\mathbf{u}_n$ on page 34, '$u_{0,n-i}$' should be '$b_{n-i}$' in (1.10.28), and '$b_n = u_{0,n}$' should be simply '$b_n$' two lines later. (Thanks to Jeffrey Chasnov, Hong Kong University of Science and Technology.) Page 45, third bullet point, should be 'nonlinear difference equation'. (Thanks to Herb Hethcote, Iowa, henceforth HH.) Page 48, line 18. Commensalism and mutualism are not synonymous. Mutualistic (or symbiotic) organisms benefit each other, whereas commensal ones live in close association with but do not have an obvious effect on each other. So replace 'Commensalism' by 'Symbiosis'. Page 51, penultimate line, replace 'predator' by 'parasitoid'. (Thanks to HH.) Page 52, line -9, 'The graph' refers to 'Figure 2.5'. Page 55, bottom, replace 'proportion' by 'ratio'. (Thanks to HH.) Page 57, mid-page equations, replace $a(1-\bar{u})$ by $a(\bar{u}-1)$. (Thanks to HH.) Page 61, first equation of (2.4.13), delete $b$ in numerator of predation term. As a consequence, a few lines later, $v^* = \frac{b-c(1+b)}{b(1-c)^2}$. (Thanks to John Oprea, Cleveland State University, Ohio.) Page 64, exercise 2.6, should be $V_0$ handling prey and $V_1$ searching. (Thanks to HH.) Page 72, $Z^*$ equation, replace $c/a$ by $a/c$; mid-page equation, $c=\frac{d}{A-b/d}$. (Thanks to HH and Selwyn Hollis, Armstrong Atlantic State University.) Page 94, exercise 3.9(b), replace 'die' by 'catch the disease'. (Thanks to HH.) Page 97, penultimate line, replace 'fast' by 'slow'. (Thanks to HH.) Page 98, mid-page, '$u$ may undershoot this value markedly'. Page 100, antepenultimate line, replace '$R_0$' by 'effective $R_0$'. (Thanks to HH.) Page 101, penultimate line, should be 'Successful vaccination ...'. (Thanks to HH.) Page 102, last paragraph before table, note that no vaccine is 100\% successful. (Thanks to HH.) Page 109. A pair of parentheses has been omitted from each equation in exercise 3.19(b): $\gamma_1$ and $\gamma_2$ respectively multiply all terms on the right-hand sides. Page 114, equation (3.9.41), the inequality sign should be reversed. (Thanks to Jeffrey Chasnov, Hong Kong University of Science and Technology.) Page 116, first two bullet points, should be 'effective $R_0$'. Page 120, in the displayed equations. Two of the terms should be squares, $p_{n+1} = p_n^2 + \frac{1}{2} 2 p_n q_n = p_n$, $q_{n+1} = q_n^2 + \frac{1}{2} 2 p_n q_n = q_n$. (Thanks to Markus Owen, Loughborough.) Page 149, the sentence on mean velocity of particles is unclear and should be replaced by the following. 'A concept related to particle flux is the mean velocity $\mathbf{w}$ of the particles, which may be defined by $\mathbf{J}=u\mathbf{w}$, where $u$ is the concentration in particles per unit volume.' (Thanks to Eduardo Sontag, Rutgers University.) Page 151, derivation of equations (5.2.7) and (5.2.8). In both cases, the assumption of constant $v$ should be replaced by the incompressibility condition $\nabla \cdot \mathbf{v} = 0$. Page 157: the derivation of transit time (starting with 'Then the probability...') is incorrect, and should be replaced by the following. 'Let the {\em age} of a particle in $V$ be the time it has spent in $V$, and let $\tau$ be the {\em transit time}, defined to be the average age of a particle leaving $V$, or the average time taken for a particle to cross $V$ from $S_1$ to $S_2$. In an interval of time $\delta t$ each particle in $V$ ages by $\delta t$, while $I\delta t$ particles of average age $\tau$ leave $V$, so that (in steady state) $N \delta t = \tau I \delta t$, $\tau = N/I$.' (Thanks to Eduardo Sontag, Rutgers University.) Page 170, first equation of (5.7.28), $u' = - \frac{1}{c} R_0 u v$. (Thanks to Selwyn Hollis, Armstrong Atlantic State University.) Pages 180-181, each of the four appearances of '$\kappa_d$' should be '$\kappa_e$'; line 5 of page 180, '$k-1$' should be '$k_{-1}$'. (Thanks to Jeffrey Chasnov, Hong Kong University of Science and Technology.) Page 183, exercise 6.4(c), factor $E_0$ missing on RHS. (Thanks to Jeffrey Chasnov, Hong Kong University of Science and Technology.) Page 190, exercise 6.6(b), replace $k_{-1}$ by $k_2$ in $S_2$ equation. Hence, in part (c), $\gamma=K_e^{1/n}K_m^{-1}$. (Thanks to Jens Helge Larsen and others at Norwegian University of Science and Technology, Trondheim.) In part (c), $s_2=K_e^{-1/n}S_2$ and $\gamma = K_e^{-1/n}K_m$. Page 201, three lines after (6.5.46), replace 'millilitre' by 'microlitre' (twice). Page 215, line after (7.4.22), '$a_1$' should be '$-a_1$'. Page 230, lines 3 and 5, replace '$N/K$' by '$n/K$'. Page 243, lines 14 and 15, insert 'specific' before 'growth rate' (twice). Page 244, two lines after (8.4.20), insert 'indefinitely or' after 'increases', 'in either case' after 'so'. Page 254, line 2 from the bottom, replace 'this chapter' by 'chapter 5'. Page 291, the first displayed equation in example C.5 should read $-\nabla^2 F = \lambda F$, i.e. replace 'R' by 'F'. Page 309: the derivation in exercise 3.1(b) is correct but inelegant and unclear, and a better argument is as follows. Let $T$ be the time spent in the infective class. Then $\Prob \left\{ \tau \leq T < \tau + \delta \tau \right\} = p(\tau) - p(\tau + \delta \tau) \approx - \dot{p}(\tau) \delta \tau = \gamma \exp( - \gamma \tau) \delta \tau$, so $\bar{T} = \int_0^{\infty} \tau \gamma \exp(-\gamma\tau) d\tau = 1/\gamma$, as required. (Thanks to HH and Jeffrey Chasnov, Hong Kong University of Science and Technology.) Page 310-311, the answer to exercise 3.8 ($Npw_1$) is missing, and the answers to exercises 3.9 to 3.23 are consequently misnumbered. Exercise 3.9 has no part (c). (Thanks to HH.) Page 311, answer to 3.15(c), labelled 3.14(c), replace '13' by '13-1' and '12' by '12-1', so the corresponding numerical results are 2.75 and 2.87. (Thanks to HH.) Page 326, replace '$\frac{k}{4D}\delta R^2$' by '$\frac{k}{2D}\delta R^2$', and consequently '$\frac{4D}{k}$' by '$\frac{2D}{k}$'.