## Problems

Problem 4.1: Plot the population size n(t) with respect to time using the recursion equation for logistic growth, n(t + 1) = n(t) + r n(t) (1 - n(t)/K), for the populations of yeast studied by Mable and Otto (2001). Use the parameters estimated from the growth of (a) haploid yeast (Khaploid = 3.7 x 10s, rhaploid = 0.55, n(0) = 3.0 x 10s) and (b) diploid yeast (Kdlploid = 2.3 x 10s, rdlp|0id = 0.55, n(0) = 1.9 x 105). Compare your figures to Figure 3.4. Problem 4.2: Plot the population size n(t) with respect to time using the differential equation for logistic growth, drc/dt = r n(t) (1 - n(t)/K), for the populations of yeast studied by Mable and Otto (2001). Use the parameters estimated from the growth of (a) haploid yeast (Khaplold = 3.7 x 10s, rhaploid = 0.55, n(0) = 3.0 x 105) and (b) diploid yeast (-Kdlploid = 2.3 x 108, rdlplold = 0.55, «(0) = 1.9 X 10s). Compare your figures to Figure 3.4.

Problem 4.3: Plot the population size n(t) with respect to time using the recursion equation for logistic growth, n(t + 1) = n(t) + r n(t) (1 - n(t)/K). Starting with a population of size 100, show that the population grows in a qualitatively similar manner regardless of the carrying capacity (use K = 500, K = 1000, and K = 5000) but with drastically different behavior depending on the growth rate r. That is, show that (a) the population smoothly approaches the carrying capacity without overshooting it when r = 1.0, (b) the population settles upon a two-point cycle when r = 2.1, (c) the population settles upon a four-point cycle when r = 2.5, and (d) the population varies erratically and is chaotic when r = 2.7.

Problem 4.4: Figure 4.20 illustrates the diploid recursion equation (3.13a) as a function of the allele frequency p{t) for a different set of fitnesses than Figure 4.10. (a) Place circles on the diagonal line indicating where the three equilibria are. (b) Starting from an allele frequency of 0.1, use the cobwebbing procedure of Box 4.2 to predict the allele frequency after three generations of selection, (c) Draw a phase-line plot with arrows showing the direction of allele frequency change that is consistent with figure 4.20. (d) For each of the three equilibria, specify whether it is locally stable or unstable.

Problem 4.5: In a plot of the allele frequency in one generation, p(t + 1), versus the allele frequency in the previous generation, p(t), under haploid selection, prove that the slope is never negative. That is, use the recursion equation to calculate and factor the derivative dp(t + l)/dpit) which describes the slope in this plot. By examining the sign of each factor, show that this derivative must always be positive.

Problem 4.6: In a plot of the allele frequency in one generation, p{t +1), versus the allele frequency in the previous generation, p(t), under diploid selection, prove that the slope is never negative. That is, use the recursion equation p(t + 1)

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Figure 4.20: Allele frequency recursion in the diploid model of selection Allele frequency at time t, pit)

to calculate and factor the derivative dp(t + l)/dp(t), which describes the slope in this plot. By examining the sign of each factor, show that this derivative must always be positive.

Problem 4.7: Find the equations for the null clines of the Lotka-Volterra predator-prey model described by equation (3.18):

Specify which equations must be satisfied for the number of prey (/;,) to remain constant and which equations must be satisfied for the number of predators (n2) to remain constant.

Problem 4.8: In the consumer-resource model described by equations (3.17), resources («i) enter the system at a constant rate and are continuously eaten by a consumer species (n2):

For example this model might describe the flow of insect larvae into a pond containing minnow predators. Assume that 1000 new larvae arrive per minute (6 = 1000), that the per capita contact rate is c = 0.01 per minute, that the probability that a minnow will consume a larva once detected is a = 1, that one larva is the energetic equivalent of e = 0.0005 minnows, and that the per capita death rate of minnows is S = 0.001. (a) Draw a vector-field plot for this model, (b) Determine the null clines for this model and add them to the plot, (c) Describe what happens to the number of larvae and minnows over time, and discuss any limitations of the vector-field plot in this case.

Problem 4.9: The logistic equation (3.5a) differs from the recursion equation often considered in mathematical discussions of chaos, which is given by

Show that if we define x(t) = (p. - 1) n(t)/(p. K) and ¡i — 1 + r, we can rearrange the above equation to get (3.5a). This "transformation" explains why the transition to chaos occurs at r* = 2.569944 in this book but at ¡x* = 3.569944 in other treatments.

Problem 4.10: Use a mathematical software package to plot the differential equations for the Lotka-Volterra model of competition between two species given by (3.15):

d nx

Assume that r, = r2 = 0.5, a12 = 0, a21 = 0.5, and iC, = K2 = 1000 and that the initial population size ten for each species (^(O) = 10, n2(0) = 10). Compare the resulting graphs to Figure 4.8. Do the discrete-time and continuous-time graphs differ substantially for this example?

Problem 4.11: After recovering from a cold, we are susceptible to catching another cold, because many different viruses can cause colds. For such diseases, the Susceptible-Infected-Recovered (SIR) model can be simplified to a Susceptible-Infected (SI) model, where individuals that recover from a cold return to the susceptible class. We can modify the differential equations for the SIR model (3.19) to describe the number of individuals susceptible to colds, S, and infected with colds, I:

^ = 9 - dS(t) ~ acS(t) I(t) + PI(t), ~ = a c S(t) I(t) - 81(f) - PI(t).

Let 0 = 4 be the rate at which individuals enter the population, c = 0.01 be the per capita rate of contact between a susceptible and an infected individual, a = 0.1 be the probability of transmission of the disease per contact, p = 0.2 be the rate at which infected individuals recover from a cold, and d = 0.01 and 8 = 0.02 be the death (or emigration) rates of susceptible and infected individuals. Figure 4.21 illustrates a vector-field plot for this model.

Each arrow starts at an (x,y) coordinate equal to some combination of (S, I) and points in the direction of (S + dS/dí, I + dl/dt). To make the arrows easier to see, each arrow was drawn with a constant length of five.

(a) Determine the null-cline along which the number of susceptible individuals, S, remains constant. Write this null cline in terms of what its height must equal (I = •••)• (b) Determine the null cline along which the number of infected individuals, I, remains constant. Write this null cline in terms of what its height must equal (I = ■■■). (c) Add curves for these null clines to the figure, specifying what remains