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Chapter 10: Problem 27

Plant-herbivore systems and the qualify of the vegetarion. In problem 17 of Chapter 3 and problem 20 of Chapter 5 we discussed models of plant-herbivore interactions that considered the quality of the vegetation. We now further develop a mathematical framework for dealing with the problem. We shall assume that the vegetation is spatially uniform but that there is a variety in the quality of the plants. By this we mean that some chemical or physical plant trait \(q\) governs the success of herbivores feeding on the vegetation. For example, \(q\) might reflect the succulence, nutritional content, or digestibility of the vegetation, or it may signify the degree of induced chemical substances, which some plants produce in response to herbivory. We shall be primarily interested in the mutual responses of the vegetation and the herbivores to one another. (a) Define \(q(t)=\) quality of the plant at time \(t\), \(h(t)=\) average number of herbivores per plant at time \(t\). Reason that equations describing herbivores interacting with a (single) plant might take the form $$ \frac{d q}{d t}=f(q, h), \quad \frac{d h}{d t}=g(q, h)=h r(q, h) \text {. } $$ What assumptions underly these equations? (b) We wish to define a variable to describe the distribution of plant quality in the vegetation. Consider \(p(q, t)=\) biomass of the vegetation whose quality is \(q\) at time \(t\). Give a more accurate definition by interpreting the following integral: $$ \int_{q}^{p+d p} p(q, t) d q=? $$ (c) Show that the total amount of vegetation and total quality of the plants at time \(t\) is given by $$ \begin{aligned} &P(t)=\int_{0}^{\infty} p(q, t) d q, \\ &Q(t)=\int_{0}^{\infty} q p(q, t) d q . \end{aligned} $$ What would be the average quality \(\bar{Q}(t)\) of the vegetation at time \(t\) ? (d) Suppose that there is no removal (death) or addition of plant material. Write down an equation of conservation for \(p(q, t)\) that describes how the distribution of quality changes as herbivory occurs (Hint: Use an analogy similar to that of Sections \(10.8\) and 10.9.) (e) Suppose that the herbivores are only affected by the average plant quality, \(Q(t)\). What would this mean biologically? What would it imply about the equation for \(d h / d t\) ? (f) Further suppose that the function \(f(q, h)\) is linear in \(q\). (Note: This is probably an unrealistic assumption, but it will be used to illustrate a point.) Assume that $$ f(q, h)=f_{1}(h)+q f_{2}(h) . $$ Interpret the meanings of \(f_{1}\) and \(f_{2}\). *(g) Show that the model thus far can be used to conclude that an equation for the average quality of the vegetation is $$ \frac{d \bar{Q}}{d t}=f_{1}(h)+f_{2}(h) \bar{Q} . $$ [Use the assumptions in parts (d) and (f), the equation you derived in (d), and integration by parts.] (h) Explore what this model would imply about average quality and average number of herbivores per plant if \(f\) and \(r\) are given by $$ \begin{gathered} f(q, h)=K_{1}-K_{2} q h\left(h-h_{0}\right) \\ r(q, h)=K_{3}\left(1-K_{4} h / \bar{Q}\right) \end{gathered} $$

Short Answer

Expert verified
Define plant quality and herbivores, formulate their interactions, interpret distribution instances, derive equations for the average quality, and analyze implications of the given functions.

Step by step solution

01

Define the Variables

Define the quality of the plant, denoted by \( q(t) \), and the average number of herbivores per plant, denoted by \( h(t) \) at time \( t \).
02

Formulate the Interactions

Formulate the equations for the plant quality and herbivore population interactions: \( \frac{d q}{d t} = f(q, h) \) and \( \frac{d h}{d t} = g(q, h) = h r(q, h) \).
03

Assumptions Underlying the Equations

Discuss that the assumptions might include: the change in plant quality depends on both the initial quality and the number of herbivores, and the change in the herbivore population depends on the quality of plants they are consuming.
04

Define Biomass of Vegetation

Consider \( p(q, t) \) to represent the biomass of vegetation with quality \( q \) at time \( t \).
05

Integral Interpretation

Interpret the integral \( \int_{q}^{q + dq} p(q, t) dq = \,? \) which defines the distribution of plant quality in terms of biomass.
06

Calculate Total Vegetation and Quality

Calculate the total amount of vegetation \( P(t) \) and total quality of the plants \( Q(t) \) as \[ P(t) = \int_{0}^{\infty} p(q, t) dq \] and \[ Q(t) = \int_{0}^{\infty} q p(q, t) dq \].
07

Average Quality of Vegetation

Define the average quality \( \bar{Q}(t) \) of the vegetation at time \( t \) as \[ \bar{Q}(t) = \frac{Q(t)}{P(t)} \].
08

Conservation Equation for \( p(q, t) \)

Write down the conservation equation for \( p(q, t) \). If there is no addition or removal of plant material, use the mass conservation principle: \( \frac{\partial p(q, t)}{\partial t} + \frac{\partial (p u)}{\partial q} = 0 \), where \( u \) represents the effect of herbivory on plant quality.
09

Herbivores Affected by Average Quality

If herbivores are only affected by the average quality of the vegetation, i.e., \( Q(t) \), this implies that \( r(q, h) \) should depend only on the average quality \( Q(t) \).
10

Linear Function Assumption

Assume the linear function \( f(q, h) = f_1(h) + q f_2(h) \). The terms \( f_1(h) \) and \( f_2(h) \) represent components of the rate of change in plant quality that are independent of \( q \) and those that scale linearly with \( q \), respectively.
11

Derive the Equation for Average Quality

Using parts (d) and (f), and integrating by parts if necessary, derive the equation \[ \frac{d \bar{Q}}{d t} = f_1(h) + f_2(h) \bar{Q} \].
12

Implications of Given Functions

If \( f(q, h) = K_1 - K_2 q h (h - h_0) \) and \( r(q, h) = K_3 (1 - K_4 h / \bar{Q}) \), analyze what this model implies about the average quality \( \bar{Q}(t) \) and the average number of herbivores per plant.

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Key Concepts

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

mathematical modeling
Mathematical modeling helps us understand complex systems by representing them with mathematical equations. In the case of plant-herbivore interactions, we create equations to describe how the quality of vegetation and the number of herbivores change over time. These models help in predicting the outcomes based on different scenarios.

For example, defining the quality of the plant as \( q(t) \) and the average number of herbivores per plant as \( h(t) \), we can use the following equations:

\[ \frac{d q}{d t} = f(q, h) \quad \text{and} \quad \frac{d h}{d t} = g(q, h) = h r(q, h) \]

These equations help illustrate how the interactions between plants and herbivores affect both their population and quality over time.
vegetation quality
Vegetation quality is a critical factor in understanding plant-herbivore dynamics. It refers to characteristics of plants that affect herbivores' success, such as succulence, nutritional content, digestibility, or presence of chemical substances. These traits are represented in our models by a variable \( q(t) \), which changes over time.

The quality of vegetation influences how herbivores interact with plants and how plants can sustain herbivory. High-quality vegetation might lead to increased herbivore population, which in turn can lead to a decline in plant quality because of excessive feeding. Our models capture this dynamic by using specific functions \( f(q, h) \).
differential equations
Differential equations are mathematical tools used to describe how a quantity changes over time. In our context, we use them to represent how plant quality and herbivore populations evolve. The equations we use are:

\[ \frac{d q}{d t} = f(q, h) \quad \text{and} \quad \frac{d h}{d t} = g(q, h) = h r(q, h) \]

These differential equations require initial conditions and assumptions about how \( q \) and \( h \) interact. For example, the rate of change in plant quality \( \frac{d q}{d t} \) might depend on both the current plant quality and the number of herbivores. Similarly, the rate of change in herbivore population \( \frac{d h}{d t} \) depends on the plant quality and availability.
biomass distribution
Biomass distribution refers to how the total plant material is spread across different quality levels in an ecosystem. It is denoted by \( p(q, t) \), which represents the biomass of vegetation with quality \( q \) at time \( t \). This concept helps us understand the overall status of the vegetation:

Total amount of vegetation \( P(t) \) and total quality of the plants \( Q(t) \) are calculated by:

\[ P(t) = \int_{0}^{\infty} p(q, t) dq \]\[ Q(t) = \int_{0}^{\infty} q p(q, t) dq \]

Knowing \( P(t) \) and \( Q(t) \), we can calculate the average quality \( \bar{Q}(t) \) of the vegetation by:

\[ \bar{Q}(t) = \frac{Q(t)}{P(t)} \].
herbivory impact
Herbivory impact considers how herbivores affect plant quality and overall vegetation. When herbivores feed on plants, they alter the quality of the vegetation. This interaction is modeled as:

\[ \frac{\partial p(q, t)}{\partial t} + \frac{\partial (p u)}{\partial q} = 0 \]

This equation states that any change in the distribution of plant quality is due to herbivory. If herbivores are only affected by the average plant quality \( Q(t) \), it implies that \( r(q, h) \) should depend on \( Q(t) \). For example, if we assume that \( f(q, h) \) is linear in \( q \), we could write:

\[ f(q, h)=f_{1}(h)+q f_{2}(h) \].

This helps in understanding how different factors, such as feeding rate, influence both plants and herbivores. By analyzing these interactions, we can predict how herbivory will impact future plant quality and herbivore populations.

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Most popular questions from this chapter

Many of Skellam's arguments are based directly on random-walk calculations, not derived from the continuous PDEs such as equation (1). In his original article he defines the following: \(a^{2}=\) mean square dispersion per generation, \(R=\) radial distance from point at which population was released, \(\lambda=\) growth rate of the population, \(n=\) number of elapsed generations, \(p=\) proportion of the population lying outside a circle of radius \(R\) after \(n\) generations. He proves that $$ p=\exp \frac{-R^{2}}{n a^{2}} $$ (a) If the population growth is described by $$ N_{n+1}=\lambda N_{n} $$ and initially there is just a single individual, show that the population in the \(n\)th generation is \(N_{n}=\lambda^{n}\). (b) Now consider a region that contains all but a single individual. Show that the radius of this region at the \(n\)th generation is given by $$ R_{s}=\left(n a^{2} \ln N_{*}\right)^{1 / 2} $$ [Hint: Why is it true that \(1 / N_{n}=\exp \left(-R^{2} / n a^{2}\right)\) holds for this radius?] (c) Use parts (a) and (b) to show that $$ R_{n}=n a(\ln \lambda)^{1 / 2} $$ He proves that $$ p=\exp \frac{-R^{2}}{n a^{2}} $$ (a) If the population growth is described by $$ N_{n+1}=\lambda N_{s} $$ and initially there is just a single individual, show that the population in the \(n\)th generation is \(N_{n}=\lambda^{n}\). (b) Now consider a region that contains all but a single individual. Show that the radius of this region at the \(n\)th generation is given by $$ R_{s}=\left(n a^{2} \ln N_{0}\right)^{1 / 2} $$ [Hint: Why is it true that \(1 / N_{n}=\exp \left(-R^{2} / n a^{2}\right)\) holds for this radius?] (c) Use parts (a) and (b) to show that $$ R_{n}=n a(\ln \lambda)^{1 / 2} $$ (d) Show that, save for a proportionality factor, this result agrees with the rate of spread of a population given by equation (4).

Other modeling problems related to tissue cultures (a) Suppose a tissue culture is grown in a chemostat, with constant outflow at some rate \(F\). Give a set of equations to describe the problem. (b) If particles of size \(r_{\max }\) always break apart into \(n\) identical particles of size \(r_{n}\), how would the problem be formulated? *(c) Suppose now that pellets can diminish in size due to shaving off of minute pieces (such as single cells) as a result of friction or turbulence. Assume this takes place at a rate proportional to the pellet radius. How would you model this effect in the following two situations? (1) All minute pieces can then grow into bigger pieces (participate in the overall growth). (2) All fragments die.

Suggest a (set of) partial differential equations to describe the following processes: (a) A predator-prey system in which both species move randomly in a one. dimensional setting. (b) A predator-prey system in which the predator moves towards higher prey densities and the prey moves towards lower predator densities. (c) A pair of reacting and diffusing chemicals such that species 1 activates the formation of both substances and species 2 inhibits the formation of both substances. Design your model so that it has a homogeneous steady state that is stable. (d) A population of cells that secretes a chemical substance. Assume that the cells orient and move chemotactically in gradients of this chemical and also more randomly. Further assume that the chemical diffuses and is gradually broken down at some rate.

Polymorphonuclear (PMN) phagocytes (white blood cells) are generally the first defense mechanism employed in the body in response to bacterial inva- (a) Write a set of equations to describe the motions and interactions of microbes \(b\) and phagocytes \(c\). (b) Additional assumptions made were that \(\begin{aligned} f(b)=& \frac{k_{z} b}{1+b / K_{i}}, \quad d(b, c)=\frac{-k_{d} b c}{K_{b}+b}, \\ & s A(c, b)=h_{0} \frac{A}{V} C_{b}\left(1+\frac{h_{1}}{h_{0}} b\right) \\ \text { where } k_{z}=& \text { bacterial growth rate constant, } \\ k_{a}=\text { phagocytic killing rate constant, } \\ h_{0}=\text { rate of emigration from venules when inflammation is ab- } \\ & \text { sent, } \\ h_{1} &=\text { inflammation- enhanced emigration rate, } \\ / V &=\text { ratio of venule wall-surface area to tissue volume, } \\ &=\text { phagocyte density in the venules. } \end{aligned}\) \(k_{d}=\) phagocytic killing rate constant, \(h_{0}=\) rate of emigration from venules when inflammation is absent, \(h_{1}=\) inflammation-enhanced emigration rate, \(A / V=\) ratio of venule wall-surface area to tissue volume, \(C_{b}=\) phagocyte density in the venules. (1) Explain the meaning of these assumptions. (2) Define \(K_{i}\) and \(K_{b}\) and give dimensions of all parameters above. (c) By dimensional analysis, it is possible to reduce the equations to the following form: $$ \begin{aligned} &\frac{\partial v}{\partial \tau}=\rho \frac{\partial^{2} v}{\partial \xi^{2}}+\frac{\gamma v}{1+v}-\frac{u v}{\kappa+v} \\ &\frac{\partial u}{\partial \tau}=\frac{\partial^{2} u}{\partial \xi^{2}}-\delta \frac{\partial}{\partial \xi}\left(u \frac{\partial v}{\partial \xi}\right)+\alpha(1+\sigma v-u) . \end{aligned} $$ (d) Show that the equations in part (c) have two types of uniform steady-state solutions: (1) \(v=0, \quad u=1\). (2) \(v>0, u=1+\sigma v\). Identify the biological meaning of these steady states.

Segel et al. (1977) calculated the motility coefficients based on equation (6) (where \(r=0\) ) as follows. A capillary tube (cross-sectional area \(=A\) ) is filled with fluid. At time \(t=0\) the open end is placed in a bacterial suspension of concentration \(C_{0}\) and removed at time \(t=T\). The number of bacteria in the tube is counted. Motility \(\mu\) is then computed as follows: $$ \mu=\frac{\pi N^{2}}{4 C \hat{B} A^{2} T} $$ (a) Give justification for or derive their formula. (b) Suppose that (1) the radius of the capillary is approximately \(0.01 \mathrm{~cm}\), (2) the bacterial suspension contains a density \(C_{0}=\left(1 / 7 \times 10^{-7}\right)\) bacteria per milliliter, and (3) the following observations (from Segel et al., 1977 ) are made: $$ \begin{aligned} &\begin{array}{cc} T \text { (min) } & N \text { (bacteria) } \\ \hline 2 & 1800 \\ 5 & 3700 \\ 10 & 4800 \\ 12.5 & 5500 \\ 15 & 6700 \\ \hline \end{array}\\\ &\text { What do you conclude about } \mu ? \end{aligned} $$
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