Problem 4

Question

When the initial parasitoid density is $P_{0}=0$, the NicholsonBailey model reduces to $$ N_{t+1}=b N_{t} $$ as shown in the previous problem. For which values of $b$ is the host density increasing if $N_{0}>0 ?$ For which values of $b$ is it decreasing? (Assume that $b>0$.)

Step-by-Step Solution

Verified

Answer

The host density increases when $ b > 1 $ and decreases when $ 0 < b < 1 $.

1Step 1: Understanding the Model

The model given is a reduction of the Nicholson-Bailey model where initially there are no parasitoids, meaning the parasitism rate is zero. Therefore, the equation simplifies to describe the host population only: \[ N_{t+1} = b N_t \] where $ N_t $ is the host population at time $ t $, and $ b $ is a growth factor.

2Step 2: Identifying Increasing Population

The host population $ N_{t+1} $ is increasing for all time periods when the growth factor $ b > 1 $. This is because the multiplication by $ b > 1 $ results in making $ N_{t+1} $ greater than $ N_t $. Therefore, for the host density to increase, we need:\[ b > 1 \]

3Step 3: Identifying Decreasing Population

The host population $ N_{t+1} $ decreases when the growth factor $ b < 1 $. This results in $ N_{t+1} $ being less than $ N_t $, as each successive multiplication reduces the population. Thus, for the host density to decrease, we look for:\[ 0 < b < 1 \]

Key Concepts

Understanding Host Population DynamicsThe Role of the Growth FactorPopulation Growth Rate Effects

Understanding Host Population Dynamics

The concept of host population dynamics is a crucial part of understanding ecological models like the Nicholson-Bailey model. Here, "host" refers to a particular species that is being studied, while "population dynamics" describes how this population changes over time. In the simplified version of the Nicholson-Bailey model, where parasitoids are absent initially (i.e., the parasitoid density is zero), the focus is entirely on the changes within the host population. This is reflected in the equation:
\[ N_{t+1} = b N_t \]
Here, $N_t$ represents the population size at time $t$, and $N_{t+1}$ indicates the population size at the next time interval. By examining this, we can observe how specific factors, such as the growth factor, influence the population's growth. Understanding these dynamics helps ecologists predict future changes in population size and understand the equilibrium state of the host population.

The Role of the Growth Factor

A key element in this model is the growth factor, denoted here by $b$. It essentially determines how the host population changes from one time period to the next. The role of the growth factor involves multiplying the current population size to predict the next period's population size.

If $b > 1$, the population grows—that is, the population at the next time step is larger than the current population.
If $b < 1$, the population decreases, meaning the population size shrinks over time.

To further clarify, think of $b$ as a "multiplier." If $b$ is greater than 1, each host produces more than one equivalent host in the subsequent time period, implying population growth. Conversely, if $b$ is less than 1, each host produces less than one equivalent host, leading to population decline. Understanding the growth factor helps ecologists manage and predict changes in host populations effectively.

Population Growth Rate Effects

Understanding how the population growth rate affects the host population is fundamental when applying ecological models. The "population growth rate" in this context is directly linked to the growth factor $b$, determining the direction and magnitude of population change.
When the initial host population $N_0$ is greater than zero, and $b > 1$, this indicates a positive population growth rate, leading to an increase in population size over successive periods. Over time, this can result in significant population expansion if the conditions remain constant.

For $b > 1$: Expect exponential growth in the population. This means the population size could double or more, as the growth compounding grows quickly.
For $0 < b < 1$: The population is in decline. This negative growth rate eventually leads the population size towards zero if sustained over numerous periods.

The insights gained from studying population growth rates help predict long-term ecological changes and can inform conservation efforts as well as the management of pest species. Recognizing these effects allows for better planning and action concerning ecological balance.

Problem 4

Other exercises in this chapter

Problem 3

In Problems 1-14, use the properties of limits to calculate the following limits: $$ \lim _{(x, y) \rightarrow(2,-1)}\left(x^{2} y^{3}-3 x y\right) $$

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Problem 4

Let $f(x, y)=\ln \left(x y-x^{2}\right)$ with $x(t)=t^{2}$ and $y(t)=t$. Find the derivative of $w=f(x, y)$ with respect to $t$ when $t=5$.

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Problem 4

Describe in words the set of all points in $\mathbf{R}^{3}$ that satisfy the following expressions: (a) $x=0$ (b) $y=0$ (c) $z=0$ (d) $z \geq 0$ (e) \

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Problem 4

The tangent plane at the indicated point $\left(x_{0}, y_{0}, z_{0}\right)$ exists. Find its equation. $$ f(x, y)=\sin x+\cos y ;(0,0,1) $$

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