Chapter 8

Let \(N(t)\) denote the size of a population at time \(t .\) Assume that the population evolves according to the logistic equation. Assume also that the intrinsic growth rate is 5 and that the carrying capacity is 30 . (a) Find a differential equation that describes the growth of this population. (b) Without solving the differential equation in (a), sketch solution curves of \(N(t)\) as a function of \(t\) when (i) \(N(0)=10\), (ii) \(N(0)=20\), and (iii) \(N(0)=40\).

Logistic growth is described by the differential equation $$\frac{d N}{d t}=r N\left(1-\frac{N}{K}\right)$$ The solution of this differential equation with initial condition \(N(0)=N_{0}\) is given by $$N(t)=\frac{K}{1+\left(\frac{K}{N_{0}}-1\right) e^{-r t}}$$ (a) Show that $$r=\frac{1}{t} \ln \left(\frac{K-N_{0}}{N_{0}}\right)+\frac{1}{t} \ln \left(\frac{N(t)}{K-N(t)}\right)$$ by solving \((8.47)\) for \(r\). (b) Equation \((8.48)\) can be used to estimate \(r\). Suppose we follow a population that grows according to the logistic equation and find that \(N(0)=10, N(5)=22, N(100)=30\), and \(N(200)=30\). Estimate \(r\).

Selection at a Single Locus We consider one locus with two alleles, \(A_{1}\) and \(A_{2}\), in a randomly mating diploid population. That is, each individual in the population is either of type \(A_{1} A_{1}, A_{1} A_{2}\), or \(A_{2} A_{2}\). We denote by \(p(t)\) the frequency of the \(A_{1}\) allele and by \(q(t)\) the frequency of the \(A_{2}\) allele in the population at time \(t\). Note that \(p(t)+q(t)=1\). We denote the fitness of the \(A_{i} A_{j}\) type by \(w_{i j}\) and assume that \(w_{11}=1, w_{12}=1-s / 2\), and \(w_{22}=1-s\), where \(s\) is a nonnegative constant less than or equal to \(1 .\) That is, the fitness of the heterozygote \(A_{1} A_{2}\) is halfway between the fitness of the two homozygotes, and the type \(A_{1} A_{1}\) is the fittest. If \(s\) is small, we can show that, approximately, $$\frac{d p}{d t}=\frac{1}{2} s p(1-p) \quad \text { with } p(0)=p_{0}$$ (a) Use separation of variables and partial fractions to find the solution of \((8.49)\). (b) Suppose \(p_{0}=0.1\) and \(s=0.01\); how long will take until \(p(t)=0.5 ?\) (c) Find \(\lim _{t \rightarrow \infty} p(t)\). Explain in words what this limit means.

Solve each differential equation with the given initial condition. \(\frac{d y}{d x}=2 \frac{y}{x}\), with \(y_{0}=1\) if \(x_{0}=1\)

Solve each differential equation with the given initial condition. \(\frac{d y}{d x}=\frac{x+1}{y}\), with \(y_{0}=2\) if \(x_{0}=0\)

Solve each differential equation with the given initial condition. \(\frac{d y}{d x}=\frac{y}{x+1}\), with \(y_{0}=1\) if \(x_{0}=0\)

Solve each differential equation with the given initial condition. \(\frac{d y}{d x}=(y+1) e^{-x}\), with \(y_{0}=2\) if \(x_{0}=0\)

Solve each differential equation with the given initial condition. \(\frac{d y}{d x}=x^{2} y^{2}\), with \(y_{0}=1\) if \(x_{0}=1\)

Solve each differential equation with the given initial condition. \(\frac{d y}{d x}=\frac{y+1}{x-1}\), with \(y_{0}=5\) if \(x_{0}=2\)

Solve each differential equation with the given initial condition. \(\frac{d u}{d t}=\frac{\sin t}{u^{2}+1}\), with \(u_{0}=3\) if \(t_{0}=0\)

Solve each differential equation with the given initial condition. \(\frac{d r}{d t}=r e^{-t}\), with \(r_{0}=1\) if \(t_{0}=0\)

Solve each differential equation with the given initial condition. \(\frac{d x}{d y}=\frac{1}{2} \frac{x}{y}\), with \(x_{0}=2\) if \(y_{0}=3\)

In a case study by Taylor et al. (1980) in which the maximal rate of oxygen consumption (in \(\mathrm{ml} \mathrm{s}^{-1}\) ) for nine species of wild African mammals was plotted against body mass (in \(\mathrm{kg}\) ) on a log-log plot, it was found that the data points fall on a straight line with slope approximately equal to \(0.8 .\) Find a differential equation that relates maximal oxygen consumption to body mass.

Consider the following differential equation, which is important in population genetics: $$a(x) g(x)-\frac{1}{2} \frac{d}{d x}[b(x) g(x)]=0$$ Here, \(b(x)>0\). (a) Define \(y=b(x) g(x)\), and show that \(y\) satisfies $$\frac{a(x)}{b(x)} y-\frac{1}{2} \frac{d y}{d x}=0.$$ (b) Separate variables in \((8.50)\), and show that if \(y>0\), then $$y=C \exp \left[2 \int \frac{a(x)}{b(x)} d x\right].$$

When phosphorus content in Daphnia was plotted against phosphorus content of its algal food on a log-log plot, a straight line with slope \(1 / 7.7\) resulted. (See Sterner and Elser, \(2002 ;\) data from DeMott et al., \(1998 .\) Find a differential equation that relates the phosphorus content of Daphnia to the phosphorus content of its algal food.

This problem addresses Malthus's concerns. Assume that a population size grows exponentially according to $$N(t)=1000 e^{t}.$$ and the food supply grows linearly according to $$F(t)=3 t$$ (a) Write a differential equation for each of \(N(t)\) and \(F(t)\). (b) What assumptions do you need to make to be able to compare whether and, if so, when food supply will be insufficient? Does exponential growth eventually overtake linear growth? Explain. (c) Do a Web search to determine whether food supply has grown linearly, as claimed by Malthus. 57\. At the beginning of this section, we modified the exponentialgrowth equation to include oscillations in the per capita growth rate. Solve the differential equation we obtained, namely, $$\frac{d N}{d t}=2(1+\sin (2 \pi t)) N(t)$$ with \(N(0)=5\).

At the beginning of this section, we modified the exponentialgrowth equation to include oscillations in the per capita growth rate. Solve the differential equation we obtained, namely, $$ \frac{d N}{d t}=2(1+\sin (2 \pi t)) N(t) $$ with \(N(0)=5\)