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Chapter 3: Problem 15

Consider the following model for leaf-eating herbivores whose population size (number of individuals) is \(h_{n}\) on a tree whose leaf mass is \(0_{n}\) * $$ \begin{aligned} &v_{n+1}=f v_{n}\left(e^{-\alpha \hat{\omega}}\right), \\ &h_{a+1}=r h_{n}\left(\delta-\frac{h_{2}}{v_{n}}\right), \quad \text { where } v_{n} \neq 0 \end{aligned} $$ and where \(f, a, r, \delta\) are positive constants. (a) Find the steady state(s) \(\bar{v}\) and \(\bar{h}\) of this system. What happens if \(f=1 ?\) What restrictions on the parameters should be met for a biologically reasonable steady state? (b) Show that by rescaling the equations, it is possible to reduce the number of parameters. To do this, define $$ V_{n}=v_{n} / \bar{v}, \quad H_{n}=h_{n} / \bar{h} \text {. } $$ Show that the system of equations can then be converted to the following form: $$ \begin{aligned} &V_{n+1}=V_{n} \exp k\left(1-H_{n}\right) \\ &H_{n+1}=b H_{n}\left(1+\frac{1}{b}-\frac{H_{n}}{V_{n}}\right) \end{aligned} $$ What is the connection between \(b, k\), and the previous four parameters \(f\), \(a, r\), and \(\delta\) ? (c) Show that the equations in part (b) now have steady-state solutions $$ \bar{Q}=\bar{H}=1 \text {. } $$ (d) Determine whether the functions in the equations of the model fall into the general category described in Section 3.5. (e) Determine when the steady state will be stable.

Short Answer

Expert verified
Find steady states by solving equilibrium conditions. Rescale equations with new variables, then analyze stability using the Jacobian matrix.

Step by step solution

01

Identify the steady states

Set the system at steady state by letting the population size and leaf mass not change over time. Hence, set \(v_{n+1} = v_n = \bar{v}\) and \(h_{n+1} = h_n = \bar{h}\). Solve for \(\bar{v}\) and \(\bar{h}\) by setting the equations equal to these steady state values.
02

Solve for \(\bar{v}\)

Using \(v_{n+1} = fv_{n}\left(e^{-\alpha \hat{\omega}}\right)\), since \(v_{n+1} = \bar{v}\) and \(v_n = \bar{v}\), we get \[ \bar{v} = f \bar{v} e^{-\alpha \hat{\omega}} \]. Solve for \(\bar{v}\).
03

Solve for \(\bar{h}\)

Next, use the equation \(h_{n+1} = r h_n \left(\delta - \frac{h_n}{v_n}\right)\). Since at steady state \(h_{n+1} = \bar{h}\) and \(h_n = \bar{h}\), divide by \(v_n = \bar{v}\) to get \[ \bar{h} = r \bar{h} \left(\delta - \frac{\bar{h}}{\bar{v}}\right) \] and solve for \(\bar{h}\).
04

Considering \(f=1\)

If \(f=1\), we get \[ \bar{v} = \bar{v} e^{-\alpha \hat{\omega}} \]. Which simplifies to \[ e^{-\alpha \hat{\omega}} = 1 \Rightarrow \alpha \hat{\omega} = 0 \Rightarrow \hat{\omega} = 0 \]. This result and the conditions on parameters must be examined for biological feasibility.
05

Restrictions for a biologically reasonable steady state

For a biologically reasonable steady state, \(\bar{h}\) and \(\bar{v}\) should be positive values. Thus, parameters must be set so that this condition holds, namely \(\delta\) must be greater than zero, among others.
06

Rescale the equations

Define \(V_n = \frac{v_n}{\bar{v}}\) and \(H_n = \frac{h_n}{\bar{h}}\). Substitute these into the original equations: \[ V_{n+1} = V_n \exp k (1 - H_n) \]\[ H_{n+1} = b H_n \left(1 + \frac{1}{b} - \frac{H_n}{V_n}\right) \]where \(b\) and \(k\) are mappings of the original parameters.
07

Connection between parameters

Determine the values of \(b\) and \(k\) from the original parameters \(f, a, r, \delta\) by comparison.
08

Steady-state solutions in the rescaled equations

Show that the steady-state solutions \(\bar{V} = 1\) and \(\bar{H} = 1\) satisfy the scaled equations.
09

General category check

Determine if the functions in the equations fall into the specified general category described in Section 3.5 by comparing properties.
10

Stability conditions

Analyze the stability of the steady states by finding eigenvalues of the Jacobian matrix at the steady state.

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Key Concepts

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

steady state
In mathematical modeling, the steady state of a system refers to a situation where the variables (such as population sizes) do not change over time. That means the future state is the same as the current state. For the given herbivore dynamics model, the steady-state conditions are described by the equations \(v_{n+1} = v_n = \bar{v}\) and \(h_{n+1} = h_n = \bar{h}\).

To find the steady state solutions, we substitute \(\bar{v}\) and \(\bar{h}\) into the equations and simplify. This gives us \(\bar{v} = f \bar{v} e^{-\frac{\bar{h}}{\bar{v}}}\) for \(\bar{v}\) and \(\bar{u} = r \bar{h} (\bar{v} - \bar{h}/\bar{v})\) for \(\bar{h}\).

If one of the parameters \(f\) is equal to 1, it simplifies the task to finding when \(e^{-\frac{\bar{h}}{\bar{v}}} = 1\). This introduces additional constraints on the variables ensuring the results are biologically feasible.
population dynamics
Population dynamics explore how and why the number of individuals in a population changes over time. In this model, we consider the dynamics of herbivores (\(h_n\)) and the leaf mass (\(v_n\)) on a tree.

The model consists of two equations that describe the changes:
  • \(v_{n+1}=f v_{n}\big(\text{exp}^{-\frac{\bar{h}}{\bar{v}}}\big)\) represents how leaf mass affects the next time step.
  • \(h_{a+1}=r h_{n}\big(1-\frac{h_n}{v_n}\big)\) shows how the herbivore population increases or decreases based on the available leaf mass.
Understanding these dynamics is crucial for predicting whether the populations will grow, stabilize, or crash under different conditions.

Considering both \(r\) (herbivore reproduction rate) and \(f\) (leaf mass growth rate) is essential for realistic predictions.
rescaling equations
Rescaling is a technique used to simplify equations by reducing the number of parameters. This involves normalizing variables, making models easier to analyze.

For the herbivore dynamics model, we define new variables \(V_n = \frac{v_n}{\bar{v}}\) and \(H_n = \frac{h_n}{\bar{h}}\). Substituting these into the original equations gives us:
  • \(V_{n+1}=V_{n} \text{exp } (1-H_{n})\).
  • \(H_{n+1}=b H_{n}\big(1+\frac{1}{b}-\frac{H_{n}}{V_{n}}\big)\).
This process results in fewer parameters, making it easier to analyze the system behavior and stability. Parameters \(b\) and \(k\) now map from the original parameters, simplifying the complex system.

This rescaling approach retains the model's essence while facilitating a clearer analysis.
stability analysis
Understanding the stability of a system involves determining if small disturbances will die out or amplify over time. For the herbivore dynamics model, this is checked using stability analysis.

In mathematical terms, we do this by examining the eigenvalues of the Jacobian matrix at the steady state. If all eigenvalues' real parts are negative, the system is stable at that steady state.

For the rescaled equations \(V_{n+1} = V_n \text{exp} (1 - H_n)\) and \(H_{n+1}= b H_n\big(1 +\frac{1}{b}-\frac{H_n}{V_n}\big)\), finding steady-state values \(\bar{V}= 1\) and \(\bar{H}= 1\) and analyzing their stability involves calculating the Jacobian:

    \(\begin{pmatrix}\frac{\text{d}f(V,H)}{\text{d}V}&\frac{\text{d}f(V,H)}{\text{d}H}\frac{\text{d}g(V,H)}{\text{d}V}&\frac{\text{d}g(V,H)}{\text{d}H}ewlineewline\right)\text{ at} (1,1)\right)\).
      Analyzing eigenvalues here yields stability conditions.
biological feasibility
For the herbivore dynamics model to be biologically meaningful, steady state values must be realistic. This means ensuring positive population sizes and non-negative growth rates.
  • Positive \(\bar{v}\) and \(\bar{h}\) are required for meaningful leaf mass and herbivore populations.
  • Parameters \(r\) and \(f\) should represent realistic growth rates for herbivores and leaves respectively.
  • Any parameter configurations must prevent non-physical results like negative leaf mass or population.
For example, setting \(f=1\) might lead to results where \(e^{-\frac{\bar{h}}{\bar{v}}}=1\) implying that conditions like \(\hat{\tau}=0\) must be examined closely to ensure they align with ecological constraints.

Ensuring biological feasibility thus sets the boundary conditions for realistic and applicable models.

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Most popular questions from this chapter

(Note to the instructor: Problem 17 gives the student good practice at formulating a model and gradually increasing its complexity. Do not expect the later stages of this problem to be as amenable to further direct analysis as the more elementary model. More advanced students may wish to implement their models in computer simulations.) As indicated near the end of Section 3.5, an important influence on herbivores is the quality of the host vegetation, not just its abundance. If the plants have induced chemical defenses or other protective responses, the population of attackers may suffer from increased mortality, decreased fecundity, and lower growth rates. (a) Define a new variable \(q_{n}\) as the average quality of the vegetation in generation \(n\). Suggest what general equations might then be used to model the \(v_{n}, q_{\infty}, h_{n}\) system. (b) Instead of treating the vegetation as a uniform collection of identical plants, consider making a distinction between plants that are nutritious (or chemically undefended) and those that are not. For example, with \(h_{n}\) as before, let \(u_{n}=\) number of undefended plants, \(v_{n}=\) number of defended plants. Now suppose that plants make the transition from \(u\) to \(v\) after attack by herbivores and back from \(v\) to \(u\) at some constant rate that represents the loss of chemical defenses. Formulate a model for \(v_{n}, u_{\infty}\), and \(h_{n}\). (c) As a final step, consider vegetation in which plants can range in quality \(q\) from 0 (very hostile to herbivores) to 1 (very nutritious to herbivores). Assume that the change in quality of a given plant takes place in very small steps every \(\Delta \tau\) days at a rate that depends on herbivory in time interval \(n\) and on previous vegetation quality. That is, $$ q_{n+1}=q_{n}+F\left(q_{n}, h_{n}\right) \Delta \tau_{n} $$ where \(F\) can be positive or negative. Let \(v_{n}(q)\) be the percentage of the vegetation whose quality is \(q\) at the nth step of the process, and assume that the total biomass $$ V_{n}=\int_{0}^{1} v_{*}(q) d q, $$ does not change over the time scale of the problem. Can you formulate an equation that describes how the distribution of vegetation quality \(v_{n}(q)\) changes as each plant undergoes the above defense response? Figure for problem \(I 7(c)\). Hint: Consider subdividing the interval \((0,1)\) into \(n\) quality classes, each of range \(\Delta q\). How many plants leave or enter a given quality class during time \(\Delta \tau\) ?

In this problem we consider a host population that has refuges from parasitoid attack. (a) Explain the assertion that \(E K / N_{i}\) is the fraction of the population that can retreat to a refuge. (b) Explain why the fraction of hosts not parasitized is $$ f\left(N_{1}, P_{1}\right)=\frac{E K}{N_{1}}+\left(1-\frac{E K}{N_{t}}\right) e^{-\mathrm{a} h} $$ (c) Write an equation for \(N_{i+1}\). (d) Explain why at time \(t+1\) the parasitoid density is given by $$ P_{t+1}=\left(N_{t}-E K\right)\left(1-e^{-a A}\right) $$ *(e) Longer Project: Consult the literature and use computer simulations and/ or other analysis to explore the results and predictions of this model.

In negative assortative matings, like individuals do not mate with each other. Different types of models may be obtained, depending on assumptions made about the permissible matings. In the questions that follow it is assumed that homozygous females mate only with homozygous males of opposite type and that heterozygous females \((A a)\) mate with \(A A\) and \(a a\) males depending on their relative prevalence. The permitted matings are then as follows: \begin{tabular}{rrr} \hline Females & Males \\ \hline AA & \(\times\) & \(a a\) \\ \(a a\) & \(\times\) & \(A A\) \\ \(A a\) & \(\times\) & \(A A\) \\ \(A a\) & \(\times\) & \(a a\) \\ \hline \end{tabular} Assuming that a single male can fertilize any number of females results in a mating table that depends largely on the female frequencies, as shown in the mating table. It is assumed that \(u+v+w=1\). (a) Explain the entries in the table. (b) Derive an offspring table by accounting for all possible products of the matings shown in the above mating table. (c) Show that the fractions of \(A A, A a\), and aa offspring denoted by \(u_{n+1}\), \(c_{n+1}\), and \(w_{m+1}\) satisfy the equations $$ \begin{aligned} &u_{n+1}=\frac{1}{2} v_{n} \frac{u_{n}}{u_{n}+w_{n}} \\ &v_{n+1}=u_{n}+w_{n}+\frac{1}{2} v_{n} \\ &w_{n+1}=\frac{1}{2} v_{n} \frac{w_{n}}{u_{n}+w_{n}} \end{aligned} $$ (d) Show that \(u+v+w=1\) in the \((n+1)\) st generation. Use this fact to eliminate \(w\) from the equations. (e) Show that the equations you obtain have a steady state with \(\bar{v}=\frac{2}{3}\). Is there a unique value of \((\bar{u}, \vec{w})\) for this steady state? Is this steady state stable? (f) Show that the ratio \(u / w\) does not change from one generation to the next.

The model for density-dependent growth due to Varley, Gradwell, and Hassell (1973) given by equation (3), has the undesirable feature that, for low population levels, the fraction of survivors \(f=N_{s} / N_{t}=\frac{1}{\alpha}\left(N_{t}\right)^{-b}\) is greater than 1 . (a) Explain why this is faulty. (b) Find the critical population level, \(N_{c}\), below which this happens. \(\left(N_{e}\right.\) should be expressed in terms of \(\alpha\) and \(b\).) (c) Suggest how the model might be modified to alleviate this problem.

Now consider the following modification of the random mating assumption: Suppose that individuals mate only with those of like genotype (e.g., \(A a\) with \(A a, A A\) with \(A A\), and so forth). This is called positive assortative mating. How would you set up this problem, and what conclusions do you reach?
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