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Chapter 3: Problem 8

Project: One problem that leads to the oscillations in the Nicholson-Bailey model (equations \(21 \mathrm{a}, \mathrm{b}\) ) is the fact that at low parasitoid densities the host population behaves approximately as follows: $$ N_{i+1}=\lambda N_{i} $$ that is, it grows at an unchecked rate. Notice that this eventually causes the parasitoids to increase in number until their hosts are overwhelmed by the attack. This is what sets up the increasing oscillations in the system. Consider the effect of introducing other types of density dependence into the host population. Two examples include $$ N_{r+1}=\lambda N_{t} \frac{\left(K-N_{0}\right)}{K} e^{-a f_{,}} $$ and, $$ N_{t+1}=\lambda N_{t}^{1-t} e^{-\alpha h}, $$ (see Varley and Gradwell (1963) and Hassell (1978)). Use the literature, computer simulations, and your own analysis to compare the predictions of these models and to comment on their applicability and/or special features. (Note: Stability analysis may be algebraically messy in such models.)

Short Answer

Expert verified
Introduce density dependence into the host growth equations, analyze stability, and compare model predictions using literature review and computer simulations to assess their applicability and special features.

Step by step solution

01

Understand the Nicholson-Bailey Model Equation

The Nicholson-Bailey model describes the dynamics between host and parasitoid populations with the following equations: \[ N_{i+1} = \text{Host population at next time step } (i+1) \] \[ = \text{Growth rate } \lambda \times N_i \text{ (current host population)} \] This equation implies that the host population grows at an unchecked rate under low parasitoid densities. This unchecked growth will lead to an increase in parasitoid numbers, causing oscillations in the system.
02

Introduce Density Dependence in Host Population

To address the unchecked growth, we modify the host population growth equation to include density dependence. Two examples are given: 1. \[ N_{r+1} = \lambda N_t \frac{(K-N_0)}{K} e^{-af} \] 2. \[ N_{t+1} = \lambda N_t^{1-t} e^{-\alpha h} \] These equations introduce factors that limit the host population growth, preventing it from growing without bounds.
03

Literature Review

Consult the literature (e.g., Varley and Gradwell (1963) and Hassell (1978)) to understand the derivation and implications of these modified equations. These sources provide insights into the biological rationale and assumptions behind these density-dependent models.
04

Perform Stability Analysis

Analyze the stability of the modified equations. This process involves determining the equilibrium points and evaluating their stability. Note that the algebra may be complex, but it is crucial to understand the long-term behavior of the system under the modified growth equations.
05

Conduct Computer Simulations

Use computer simulations to model the dynamics of the host and parasitoid populations under the modified equations. By observing the outcomes of these simulations, compare the predictions of each model.
06

Compare and Comment on Models

Compare the outcomes from the analytical study and computer simulations of both models. Comment on their applicability to real-world scenarios and discuss any special features or limitations of each model. Highlight how density dependence affects the oscillations and stability of the population dynamics.

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Key Concepts

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

Host-Parasitoid Dynamics
The Nicholson-Bailey model is a classical ecological framework to describe the interaction between host and parasitoid populations. These interactions are vital in understanding how populations of species that utilize parasitoids (organisms that live on or within a host and eventually kill it) manage to sustain over time.
The core idea is simple. Hosts (often some kind of prey or pests) have a population that can grow quickly, usually unchecked if there are few parasitoids. However, when parasitoids are introduced, they lay eggs in or on the hosts, gradually leading the parasitoid population to increase and the host population to decline.
When host numbers drop significantly, fewer parasitoids are supported. This cyclical pattern creates dynamic fluctuations in the populations of both hosts and parasitoids over time.
Density Dependence
In ecological contexts like the Nicholson-Bailey model, density dependence refers to the effect that population density has on the growth rate and survival of a species. Without density dependence, a host population can grow exponentially, leading to dramatic oscillations when parasitoid interactions are considered.
To control such growth, density-dependent factors are introduced. The two modified equations given in the problem introduce mechanisms for limiting population growth:
  • \(N_{r+1} = \lambda N_t \frac{(K - N_0)}{K} e^{-af} \) implies growth is limited by the population’s carrying capacity \(K\).
  • \(N_{t+1} = \lambda N_t^{1-t} e^{-\textbackslash alpha h}\) assumes an intrinsic limitation in the growth rate itself.
These additions make the model more realistic and suppress the extremes in population oscillations observed in purely exponential growth scenarios.
Population Oscillations
Population oscillations refer to the cyclical rise and fall in the populations of the host and parasitoid over time. In the Nicholson-Bailey model, these oscillations arise due to the feedback loop in the interaction:
  • When the host population is high, parasitoids reproduce more.
  • The increased parasitoid population eventually suppresses the host numbers.
  • As the host population declines, fewer parasitoids are supported, causing their numbers to decline too.
  • With fewer parasitoids, the host population can start growing again, repeating the cycle.
This cycle can lead to sharp peaks and valleys in population sizes, which might stabilize or continue indefinitely depending on various factors including environmental conditions and the exact nature of the density dependence in the model.
Stability Analysis
Stability analysis is a crucial part of understanding ecological models like the Nicholson-Bailey one. It involves examining equilibrium points—where populations don't change from one time step to the next—and assessing whether smallchanges around these points will cause populations to return to equilibrium or move away from it.
This often requires complex algebra but helps predict long-term behaviors of populations governed by the model's equations. For the modified Nicholson-Bailey model:
  • Analyze the fixed points of the system.
  • Check how small disturbances are handled by the system — do populations return to equilibrium (stable) or move away from it (unstable)?
Using such techniques alongside computer simulations, where practicality of hand calculations falls short, provides insights into the long-term viability of populations in ecological scenarios similar to those described by our model.

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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 \(\Delta \tau\) ?

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?

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.

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.

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.
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