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Chapter 3: Problem 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 $$ X_{n+1}=r\left(X_{n}-0.001 X_{n}^{2}\right) $$ 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 \(r\).

Short Answer

Expert verified
As the parameter \(r\) increases from 0 to 3, population patterns transition from stable to chaotic.

Step by step solution

01

Define the Difference Equation

The insect population at time \(n+1\) can be described by the equation \(X_{n+1} = r \left( X_n - 0.001 X_n^2 \right)\). This is a quadratic difference equation where the parameter \(r\) controls the rate of growth of the population.
02

Set Up the Equation in Software

Load either Maple or MATLAB. Define the function for the population difference equation: \(X_{n+1} = r \left( X_n - 0.001 X_n^2 \right)\). Ensure \(X_0\) is initialized to a nonzero value, like 10.
03

Simulate Population Growth

Simulate the equation by iterating it over a range of values for \(r\), starting from \(r = 0\) up to \(r = 3\). For each value of \(r\), run the simulation for a sufficient number of generations to observe stable patterns or growth.
04

Analyze Different Growth Patterns

Evaluate the different types of population growth patterns for the various values of \(r\). You might observe steady state populations, periodic cycles, or chaotic fluctuations as \(r\) increases.
05

Sketch and Label Growth Patterns

For each unique pattern of growth, plot the population over time. Label each plot with its corresponding \(r\) value. You should see distinctive behavior changes at key \(r\) values, such as a transition to chaos.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Difference Equations in Population Dynamics
Difference equations are mathematical expressions used to model the change in population over discrete intervals of time. This approach is particularly useful for populations that breed seasonally, such as many insect species. In our equation, \(X_{n+1} = r \left( X_n - 0.001 X_n^2 \right)\), the change from one generation to the next is determined by the current population size, \(X_n\), and a growth rate parameter, \(r\). Unlike continuous models, which use differential equations to predict changes, difference equations step through time in distinct intervals, capturing the pulsed nature of breeding seasons in insects.

Some key features of difference equations include:
  • Tracking changes in population across generations.
  • Allowing for adjustments based on breeding patterns and resource limitations.
  • Providing insights into population stability or fluctuations.
Insect Population Dynamics: Booms and Busts
Insect populations are often subject to dramatic changes from one generation to the next. A single breeding event followed by the death of all adults is a common lifecycle trait in insects, creating a sharp generational cycle. The model represented by our difference equation captures these cycles and helps understand how populations can explode (boom) or crash (bust) based on environmental factors encapsulated in the growth rate, \(r\).

When the growth rate \(r\) is altered, it affects how the population replenishes and competes for resources. Here鈥檚 what typically happens:
  • Low \(r\): Populations may decline and stabilize at low numbers, failing to sustain themselves.
  • Moderate \(r\): Populations may exhibit stable cycles or periodic booms and busts.
  • High \(r\): Populations may become erratic, leading to unpredictable fluctuations.
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Most popular questions from this chapter

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 $$ B(X)=\beta-(\beta-\alpha) \delta \frac{X}{K}, \quad A(X)=\alpha+(\beta-\alpha)(1-\delta) \frac{X}{K} $$ where \(K\) is the population carrying capacity, \(\beta\) is the intrinsic per- capita birth rate, \(\alpha\) is the intrinsic per-capita death rate and \(\delta\), where \(0 \leq \delta \leq 1\), 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 $$ \frac{d X}{d t}=r X\left(1-\frac{X}{K}\right) $$

Predicting population size. In a population, the initial population is \(x_{0}=100\). Suppose a population can be modelled using the differential equation $$ \frac{d X}{d t}=0.2 X-0.001 X^{2} $$ with an initial population size of \(x_{0}=100\) 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.)

Chemostat. A chemostat is used by microbiologists and ecologists to model aquatic environments, 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 nutrient 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 \(\left(x^{\prime}\right)\) 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 increasing function bounded above. Show that a Michaelis-Menten type function $$ p(S)=\frac{m S}{a+S} $$ with \(m\) and a positive, non-zero constants, satisfies these requirements. What is the maximum 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 \(p(a) .)\)

Quadratic population model. Consider the population model $$ \frac{d N}{d t}=a N-b N^{2}, $$ where \(a\) and \(b\) are positive constants. Here \(b N^{2}\) represents a death term due to overcrowding (i.e., proportional to \(N^{2}\) due 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.

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 $$ \frac{d N}{d t}=a N\left(1-\frac{N}{N_{T}}\right) $$ where \(N_{T}\) is 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. (a) Which terms correspond to the fraction of the population who have not yet adopted the improved pasture technology? (b) According to Banks (1994), \(N_{T}=17,015, a=0.490\) and \(N_{0}=141\). Determine how long it takes for the improved pasture technology to spread to \(80 \%\) of the population.
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