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Chapter 1: Problem 5

Assume that we are considering the survival of whales and that if the number of whales falls below a minimum survival level \(m\), the species will become extinct. Assume also that the population is limited by the carrying capacity \(M\) of the environment. That is, if the whale population is above \(M\), it will experience a decline because the environment cannot sustain that large a population level. In the following model, \(a_{n}\) represents the whale population after \(n\) years. Discuss the model. $$ a_{n+1}-a_{n}=k\left(M-a_{n}\right)\left(a_{n}-m\right) $$

Short Answer

Expert verified
The model describes whale population dynamics between a stable maximum capacity \(M\) and an unstable minimum survival level \(m\).

Step by step solution

01

Understanding the Equation

The equation provided is a difference equation that models the change in whale population from year to year. It consists of two key factors: the carrying capacity of the environment, represented by \(M\), and the minimum survival level, represented by \(m\). These factors contribute to the overall change in the population as they define the rate at which the population can grow or shrink based on the values of \(a_n\).
02

Identifying Steady States

Steady states occur when the population does not change from year to year, meaning \(a_{n+1} = a_n\). This results in the equation \(0 = k(M - a_n)(a_n - m)\). The solutions to this equation are \(a_n = M\) or \(a_n = m\), indicating that the population will remain stable if it reaches either the carrying capacity \(M\) or the minimum survival level \(m\).
03

Analyzing Stability of Steady States

To determine if these steady states are stable, consider small perturbations around each state. If \(a_n < m\), the population will move further away from \(m\), indicating that \(m\) is an unstable steady state. Conversely, if \(a_n > m\) but less than \(M\), the population will grow towards \(M\) if \(a_n < M\) establish that \(M\) is a stable steady state. If \(a_n > M\), the population will decrease towards \(M\).
04

Implications of Parameters

The parameter \(k\) reflects the rate of population change and influences how quickly the population moves toward stable states \(m\) or \(M\). A larger \(k\) results in faster changes, while a smaller \(k\) indicates slower adjustments. **Carry capacity \(M\)** represents the maximum number of whales that the environment can support, while \(m\) provides a threshold below which the species cannot survive. The relationship between these values and the current population \(a_n\) governs the trends in the population dynamics.

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Key Concepts

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

Carrying Capacity
In population modeling, one significant concept is the carrying capacity, which is represented by the letter "M". Carrying capacity refers to the maximum number of individuals in a population that the environment can support indefinitely. In the case of whales, carrying capacity takes into account the availability of resources like food, space, and favorable living conditions that the environment can continuously provide.

If the whale population exceeds this carrying capacity, resources become stretched thin, leading to increased competition, which can cause the population to decline. Therefore, the carrying capacity serves as a natural limit to population growth. When the population reaches this limit, it is expected to stabilize, maintaining a balance between resource availability and population size.
  • Carrying capacity ensures balance in ecosystems.
  • It represents the maximum sustainable population size.
  • Exceeding this capacity leads to competition and decline.
Difference Equations
Difference equations are mathematical formulas that describe how a certain quantity changes between periods. They are essential in modeling discrete processes, like annual changes in whale population. In our equation: \[ a_{n+1} - a_n = k(M - a_n)(a_n - m) \] the left-hand side represents the change in the whale population from one year to the next.

The term \((M - a_n)\) represents how close the population is to exceeding the carrying capacity, while \((a_n - m)\) indicates how far the current population is from becoming extinct. The parameter \(k\) is a constant that determines the rate of change. It adjusts the sensitivity of the population to these pressures of carrying capacity and survival threshold.
  • Difference equations quantify yearly population changes.
  • They indicate growth, stability, or decline based on current population levels.
  • The constant \(k\) modulates the impact of these differences.
Steady States
Steady states in a difference equation model are population levels where the population does not change year over year. These are critical points in the dynamics of population modeling, where conditions become stable, with no net population growth or decline.

In our model, steady states occur when \(a_{n+1} = a_n\). This results in the equation: \[0 = k(M - a_n)(a_n - m)\] leading to two possible solutions for \(a_n\), which are \(M\) and \(m\). When \(a_n = M\), the population is at the carrying capacity. And when \(a_n = m\), it is at the minimum survival level. Therefore, these points indicate where the population will stabilize under given conditions without any further change.
  • Steady states indicate equilibrium in populations.
  • Population growth or shrinkage halts at these points.
  • These states occur at maximum (\(M\)) and critical survival levels (\(m\)).
Stability Analysis
Stability analysis helps us understand the resilience of steady states when the population undergoes small disturbances. It tells us whether a population will return to its original steady state or diverge away from it when slightly perturbed.

For the model \( a_{n+1} - a_n = k(M - a_n)(a_n - m) \), if \(a_n < m\), the population will tend to decline further and move away from \(m\), indicating that \(m\) is an unstable steady state. On the other hand, if \(a_n\) is greater than \(m\) but less than \(M\), the population shows tendencies to increase towards \(M\), making \(M\) a stable steady state.
When \(a_n > M\), the population will decrease, again moving towards \(M\). Thus, analyzing the stability clarifies which states can sustain deviations, ensuring that the population remains viable.
  • Stability indicates resilience to disturbances.
  • Unstable points lead to population decline.
  • Stable points restore the population after perturbations.

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Most popular questions from this chapter

Your grandparents have an annuity. The value of the annuity increases each month by an automatic deposit of \(1 \%\) interest on the previous month's balance. Your grandparents withdraw 1000 at the beginning of each month for living expenses. Currently, they have 50,000 in the annuity. Model the annuity with a dynamical system. Will the annuity run out of money? When? Hint: What value will \(a_{n}\) have when the annuity is depleted?

Consider the spreading of a rumor through a company of 1000 employees, all working in the same building. We assume that the spreading of a rumor is similar to the spreading of a contagious disease (see Example 3 , Section 1.2) in that the number of people hearing the rumor each day is proportional to the product of the number who have heard the rumor previously and the number who have not heard the rumor. This is given by $$ r_{n+1}=r_{n}+k r_{n}\left(1000-{ }_{n}\right) $$ where \(k\) is a parameter that depends on how fast the rumor spreads and \(n\) is the number of days. Assume \(k=0.001\) and further assume that four people initially have heard the rumor. How soon will all 1000 employees have heard the rumor?

Suppose the spotted owls' primary food source is a single prey: mice. An ecologist wishes to predict the population levels of spotted owls and mice in a wildlife sanctuary. Letting \(M_{n}\) represent the mouse population after \(n\) years and \(O_{n}\) the predator owl population, the ecologist has suggested the model. $$ \begin{aligned} M_{n+1} &=1.2 M_{n}-0.001 O_{n} M_{n} \\ O_{n+1} &=0.7 O_{n}+0.002 O_{n} M_{n} \end{aligned} $$ The ecologist wants to know whether the two species can coexist in the habitat and whether the outcome is sensitive to the starting populations. Find the equilibrium values of the dynamical system for this predator-prey model. a. Compare the signs of the coefficients of the preceding model with the signs of the coefficients of the owls-hawks model in Example 3. Explain the sign of each of the four coefficients \(1.2,-0.001,0.7\), and \(0.002\) in terms of the predator-prey relationship being modeled. b. Test the initial populations in the following table and predict the long- term outcome: $$ \begin{array}{l|c|c} \hline & \text { Owls } & \text { Mice } \\ \hline \text { Case A } & 150 & 200 \\ \text { Case B } & 150 & 300 \\ \text { Case C } & 100 & 200 \\ \text { Case D } & 10 & 20 \\ \hline \end{array} $$ c. Now experiment with different values for the coefficients using the starting values given. Then try different starting values. What is the long- term behavior? Do your experimental results indicate that the model is sensitive to the coefficients? Is it sensitive to the starting values?

Your parents are considering a 30 -year mortgage that charges \(0.5 \%\) interest each month. Formulate a model in terms of a monthly payment \(p\) that allows the mortgage (loan) to be paid off after 360 payments. Your parents can afford a monthly payment of $$\$ 1500$$. Experiment to determine the maximum amount of money they can borrow. Hint \(:\) If \(a_{n}\) represents the amount owed after \(n\) months, what are \(a_{0}\) and \(a_{360}\) ?

For the following problems, find the solution to the difference equation and the equilibrium value if one exists. Discuss the long-term behavior of the solution for various initial values. Classify the equilibrium values as stable or unstable. a. \(a_{n+1}=-a_{n}+2, \quad a_{0}=1\) b. \(a_{n+1}=a_{n}+2, \quad a_{0}=-1\) c. \(a_{n+1}=a_{n}+3.2, \quad a_{0}=1.3\) d. \(a_{n+1}=-3 a_{n}+4, \quad a_{0}=5\)
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