3.1. Fish farm. In a fish farm, fish are harvested at a constant rate of 2, 100 fish per week. The per-capita death rate for the fish is 0.2 fish per day per fish, and the per-capita birth rate is 0.7 fish per day per fish.
(a) Write a word equation describing the rate of change of the fish population. Hence obtain a differential equation for the number of fish. (Define any symbols you introduce.)
(b) If the fish population at a given time is 240, 000, give an estimate of the number of fish born in one week.
(c) Determine if there are any values for which the fish population is in equilibrium. (That is, look for values of the fish population for which there is no change over time.)
3.2. Modelling the spread of technology. Models for the spread of technology are very similar to the logistic model for population growth. Let N (t) be the number of ranchers who have adopted an improved pasture technology in Uruguay. Then N (t) satisfies the differential equation
dN dt = aN 1 −NN T ,
where NTis the total population of ranchers. It is assumed that the rate of adoption is proportional
to both the number who have adopted the technology and the fraction of the population of ranchers who have not adopted the technology.
3.10 Exercises for Chapter 3 79
(a) Which terms correspond to the fraction of the population who have not yet adopted the improved pasture technology?
(b) According to Banks (1994), NT = 17, 015, a = 0.490 and N0 = 141. Determine how long it
takes for the improved pasture technology to spread to 80% of the population.
Note: This same model can be used to describe the spread of a rumour within an organisation or population.
3.3. Quadratic population model. Consider the population model dN
dt = aN − bN
2
,
where a and b are positive constants. Here bN2 represents a death term due to overcrowding (i.e.,
proportional to N2due to interactions of the population with itself).
(a) Find all the equilibrium points. Are there any conditions on the parameters a and b for the equilibrium population to remain positive?
(b) Determine the stability of each of the equilibrium points.
(c) It is claimed that this model is exactly the same as the logistic growth model. If this claim is true, then express the constants a and b in terms of the intrinsic growth rate r and carrying capacity K. If it is not true, explain why.
3.4. Allee effect model. Consider the population model dN dt = rN (N − b) 1 −NK ,
where r is the intrinsic growth rate, K is the carrying capacity and b is a positive constant where b < K.
(a) It is claimed that this model has the property that for small populations the net reproduc- tion becomes negative but for larger populations behaves like the logistic model in that the reproduction rate is zero at the carrying capacity. Sketch the reproduction function (i.e. the per-capita growth rate) as a function of N and hence verify if the claim is true or false. (b) Find all the equilibrium populations.
(c) Determine the stability of each of the three equilibrium points. (d) What happens if the initial population is below b?
3.5. Density-dependent births. Many animal populations have decreasing per-capita birth rates when the population density increases, as well as increasing per-capita death rates. Suppose the density-dependent per-capita birth rate B(X) and density-dependent death rate A(X) are given by
B(X) = β − (β − α)δXK, A(X) = α + (β − α)(1 − δ)XK,
where K is the population carrying capacity, β is the intrinsic per-capita birth rate, α is the intrinsic per-capita death rate and δ, where 0 ≤ δ ≤ 1, is a parameter describing the extent that density dependence is expressed in births or deaths.
Show that this still gives rise to the standard logistic differential equation dX dt = rX 1 −XK .
3.6. Mouse population model. A population, initially consisting of 1, 000 mice, has a per-capita birth rate of 8 mice per month (per mouse) and a per-capita death rate of 2 mice per month (per mouse). Also, 20 mouse traps are set each week and they are always filled.
First, write a word equation describing the rate of change in the number of mice and then formulate a differential equation for the population with an initial condition.
3.7. Harvesting model. Consider the harvesting model from Section 3.3, dX dt = rX 1 −X K − h.
(a) Show there can be two non-zero equilibrium populations, with the larger value given by Xe=K 2 1 + r 1 − 4h rK ! .
For the parameter values used in Figure 3.7, calculate the value of this equilibrium population. (b) If the harvesting rate h is greater than some critical value hc, the non-zero equilibrium values
do not exist and the population tends to extinction. What is this critical value hc?
(c) If the harvesting rate is h < hc, the population may still become extinct if the initial popu-
lation x0 is below some critical level, perhaps due to an ecological disaster. Show that this
critical initial value is
xc= K 2 1 − r 1 −rK4h ! . (Hint: Consider where X′< 0.)
3.8. Fishing with quotas. In view of the potentially disastrous effects of overfishing causing a population to become extinct, some governments impose quotas that vary depending on estimates of the population at the current time. One harvesting model that takes this into account is
dX dt = rX 1 −KX − hX. (a) Show that the only non-zero equilibrium population is
Xe= K
1 −hr
. (b) At what critical harvesting rate can extinction occur?
Although extinction can occur with this model, as the harvesting parameter h increases towards the critical value the equilibrium population tends to zero. This contrasts with the constant harvesting model in Section 3.3 and Exercise 3.7, where a sudden population crash (from a large population to extinction) can occur as the harvesting rate increases beyond a critical value. 3.9. Predicting population size. In a population, the initial population is x0= 100. Suppose a
population can be modelled using the differential equation dX
dt = 0.2X − 0.001X
2,
with an initial population size of x0 = 100 and a time step of 1 month. Find the predicted
population after 2 months. (Use either an analytical solution or a numerical solution from Maple or MATLAB.)
3.10 Exercises for Chapter 3 81
3.10. Plant biomass. Let the dry weight of some plant (that is, its biomass) at time t be denoted by x(t). And suppose this plant feeds off a fixed amount of some single substrate, or a nutrient medium, for which the amount remaining at time t is denoted by S(t). Assume that the growth rate of the plant is proportional to its dry weight as well as to the amount of nutrient available, and that no material is lost in the conversion of S into x.
(a) Starting with a word equation, model the rate of plant growth dx/dt with an initial plant biomass of x0 and with xf the amount of plant material associated with S = 0. (Use the
fact that S can be written as a function of x and xf: S(t) = xf− x(t).)
(b) Solve the equation using analytical techniques or numerically with Maple or MATLAB. (Sepa- rable techniques together with partial fractions are one way to solve the equation.)
(c) Using this model, why can the plant biomass of xf never be attained? (In spite of this, and
the fact that plants do reach a maximum biomass with finite time, this model does give a reasonable prediction of annual plant growth.)
3.11. Modelling the population of a country. Consider the population of a country. Assume constant per-capita birth and death rates, and that the population follows an exponential growth (or decay) process. Assume there to be significant immigration and emigration of people into and out of the country.
(a) Assuming the overall immigration and emigration rates are constant, formulate a single differential equation to describe the population size over time.
(b) Suppose instead that all immigration and emigration occurs with a neighbouring country, such that the net movement from one country to the other is proportional to the population difference between the two countries and such that people move to the country with the larger population. Formulate a coupled system of equations as a model for this situation. In both (a) and (b), start with appropriate word equations and ensure all variables are defined. Give clear explanations of how the differential equations are obtained from the word equations. 3.12. Newly abandoned dogs. Read the case study (Section 3.6) for a model that incorporates euthanasia as a control for stray dogs.
(a) Let A denote the number of newly abandoned dogs per km2 each year. Assuming this is
a constant annual rate, and is not density dependent or dependent on stray dog numbers, include this in the euthanasia model (3.15).
(b) The first plot in Figure 3.8 illustrates that for ǫ = 0.36 (K = 250, a = 0.34, b = 0.12) euthanasia results in the eradication of stray dogs. Using the model in (a), establish that with the regular abandonment of dogs (h > 0) extinction is no longer a possible outcome, but with control the population approaches a non-zero stable population.
3.13. Scaling. Scaling is a technique commonly applied in mathematical modelling to reduce the number of parameters while retaining the dynamical properties of the original system.
(a) Consider the standard logistic equation, dX dt = rX 1 −XK .
Using the substitution x = X/K, show that this equation can be written as dx
dt = rx (1 − x) .
This latter equation has been scaled. The resulting equation is independent of parameter K but retains the same dynamical properties, and its dynamics can now easily be compared with other logistic models, with different carrying capacities.
(b) Show that model ((3.15)) from the case study on stray dog control (Section 3.6) can be scaled using the transformation n = N/K to give,
dn
dt = rn (1 − n) − ǫn.
3.14. Population density. It is often convenient to measure population abundance (size) as a population density (number of animals per unit area). What difference does it make to the population equations? To find out, let n(t) = N (t)/A, where A is the fixed area where the population resides. Given the population logistic equation
dN dt = rN 1 −NK , what is the differential equation for the density n(t)?
3.15. Sensitivity to initial conditions. Referring to the results generated in Figure 3.13 for two separate initial conditions, x0= 100 and x0 = 101, generate the results with these initial conditions
for r < 2.7, r = 2.5 and r = 1.9. What do you notice about the distance between the two graphs at each time step? (The relevant code is given in Section 3.7.)
3.16. Investigating parameter change. Using Maple or MATLAB, examine the effect of increasing the parameter r on the solution to the equation
Xn+1= Xnea(1−Xn/K), where a = ℓn(r + 1).
Establish (roughly) for what values of r the system undergoes its first two bifurcations. (Code can be adapted from that in Section 3.7.)
3.17. Finding equilibrium solutions. For the following discrete population models find all the equilibrium solutions by setting Nk+1= Nk= N . Determine the stability of each of the equilibrium
populations.
(a) Nk+1= 5Nk (b) Nk+1= 0.8Nk− 0.1Nk2
3.18. Adults die after laying eggs. A difference equation describing female insects with a periodic breeding time is
Xk+1= rXk(1 − Xk/K),
where all the parent insects die after laying their eggs.
(a) Where is this model different from the discrete logistic equation in lectures? (b) Find all the equilibrium solutions, for r > 1, and determine their stability.
3.19. Modelling insect populations. Many insect populations breed only at specific times of the year and all the adults die after breeding. These may be modelled by a difference equation, such as
Xn+1= r(Xn− 0.001Xn2).
Using Maple or MATLAB, investigate what happens as the parameter r (the growth rate) is varied from r = 0 to r = 3. Sketch all the different types of growth patterns observed, labelled with the corresponding value of r.
3.20. Discrete growth with harvesting. Consider the discrete model for linear population growth with a constant positive number h harvested each time period. In this model all adults die after giving birth. The difference equation is
Xk+1= rXk− h,
where r is the per-capita net growth rate (per time step). Find all the equilibrium solutions and determine their stability.
3.10 Exercises for Chapter 3 83
3.21. Ricker model. The Ricker model is sometimes used for fish populations as an alternative to the discrete logistic equation. The difference equation is
Nk+1= Nker(1−Nk/K), r > 0,
where Nkis the population size in generation k, r is the intrinsic growth rate and K is the carrying
capacity, which are positive constants. Find all the equilibrium solutions and determine their stability.
3.22. Stability of 2-cycles. Consider the discrete logistic equation (with K = 1) Xn+1= Xn+ rXn(1 − Xn).
(a) Show that every second term in the sequence X0, X1, X2, . . . satisfies the difference equation
Xn+2= (1 + 2r + r2)Xn− (2r + 3r2+ r3)Xn2
+ (2r2+ 2r3)Xn3− r3Xn4.
(b) For equilibrium solutions, let S = Xn+2 = Xn and obtain a quartic equation (that is, an
equation with the unknown raised to the fourth power, at most). Explain why S = 0 and S = 1 must be solutions of this quartic equation. Hence show that the other two solutions are
S =(2 + r) ± √
r2− 4
2r .
[Note: Comparing with Figure 3.11 (r = 2.2) we see that these two values of the two non- zero equilibrium solutions are the values between which the population oscillates in a 2-cycle. Furthermore, when r increases to where these two equilibrium solutions become unstable, this corresponds to where the 2-cycle changes to a 4-cycle.]
3.23. Linear differential-delay equation. Consider the linear differential-delay equation dX
dt = X(t − 1), X(0) = 1.
Look for an exponential solution of the form X(t) = Cemt, where m is a constant to be determined and C is an arbitrary constant.
3.24. Chemostat. A chemostat is used by microbiologists and ecologists to model aquatic envi- ronments, or waste treatment plants. It consists of a tank filled with a mixture of some medium and nutrients, which microorganisms require to grow and multiply. A fresh nutrient-medium mixture is pumped into the tank at a constant rate F and the tank mixture is pumped from the tank at the same rate. In this way the volume of liquid in the tank remains constant. Let S(t) denote the concentration of the nutrient in the tank at time t, and assume the mixture in the tank is well stirred. Let x(t) denote the concentration of the microorganism in the tank at time t.
(a) Draw a compartmental diagram for the amount of nutrient.
(b) In the absence of the organism, suggest a model for the rate of change of S(t).
(c) If the microorganisms’ per-capita uptake of the nutrient is dependent on the amount of nutri- ent present and is given by p(S), and the per-capita reproduction rate of the microorganism is directly proportional to p(S), extend the model equation above to include the effect of the organism. (The per-capita uptake function measures the rate at which the organism is able to absorb the nutrient when the nutrient’s concentration level is S.)
(d) Now develop an equation describing the rate of change of the concentration of the live organism (x′) in the tank to derive the second equation for the system.
(e) The nutrient uptake function p(S) can be shown experimentally to be a monotonically in- creasing function bounded above. Show that a Michaelis–Menten type function
p(S) = mS a + S,
with m and a positive, non-zero constants, satisfies these requirements. What is the max- imum absorption rate? And why is a called the half-saturation constant? (Hint: The maximum absorption rate is the maximum reached by p(S). For the second part, consider p(a).)
This system of equations is known as the Monod Model for single species growth and was developed by Jaques Monod in 1950.
Chapter 4
Numerical solution of differential equations
This chapter provides a brief overview of numerical procedures on which we rely when em- ploying software as a tool in the solution and analysis of mathematical models. While developments have ensured that in most cases the methods are robust, it is important to understand the trade-offs we must accept when using them, and the possible errors that could accumulate, making interpretation inaccurate. Perhaps the most important part of this chapter is the final discussion.
4.1
Introduction
In finding solutions to differential equations, plotting trajectories and displaying time- dependent graphs, we have come to rely on the performance of computers and software packages. Some packages use analytical solutions where possible, a symbolic approach such as in Maple, but many problems cannot be solved analytically and there is a need to employ numerical schemes to find a solution. In this case the numerical solution, while possibly extremely accurate, is an approximation to the exact analytic solution.
There are some major drawbacks in accepting whatever is produced by a computer, particularly when numerical solvers are used, as errors may accrue for the following reasons: • There are round-off errors and these increase with an increase in the number of cal-
culations performed.
• There are discretisation errors resulting from the estimation of a solution for a discrete set of points; these may decrease with an increase in the number of points used. • There are errors due to the estimating procedure, or method of approximation, used
by the numerical solver.
Many different numerical procedures are available for finding derivatives, integrals, sums, etc. First we provide a very brief outline of a few used in finding solutions to differential equations (the main use of solvers in this book), and we then consider some of the drawbacks in using these numerical approximations.