Problem 51

Question

Denote the size of a population at time $t$ by $N(t)$, and assume that $N(0)=50$ and $|d N / d t| \leq 20$ for all $t \in[0,5] .$ What can you say about $N(5) ?$ [Hint: Remember also that it is impossible for the number of organisms to become negative].

Step-by-Step Solution

Verified

Answer

The population $N(5)$ is between 0 and 150.

1Step 1: Understand the Constraints and Initial Condition

We are given that $N(0) = 50$ as the initial size of the population. Additionally, the absolute rate of change of the population, $\left| \frac{dN}{dt} \right|$, is bounded by 20 for the time interval $[0, 5]$. This implies that the population can increase or decrease by at most 20 units per unit of time.

2Step 2: Calculate the Maximum Possible Increase

Since the population can increase by at most 20 organisms per unit time, over 5 units of time, the population can increase by a maximum of $20 \times 5 = 100$ organisms. Thus, the maximum possible value for $N(5)$ is $50 + 100 = 150$.

3Step 3: Calculate the Minimum Possible Decrease

The absolute rate of change indicates that the population can also decrease by 20 organisms per unit time. Over 5 units, the maximum possible decrease is $20 \times 5 = 100$ organisms. However, negative population sizes are impossible according to the hint, so the smallest value $N(5)$ can take is 0.

4Step 4: Conclusion on Population Size

Combining the results from the previous steps, the value of $N(5)$ must be bounded by the non-negative constraint and the maximum increase. Thus, the possible range of $N(5)$ is from 0 to 150.

Key Concepts

Population DynamicsDifferential EquationsRate of Change

Population Dynamics

Population dynamics is a fascinating field in biology that studies how populations of organisms change over time. It helps us understand crucial behaviors like growth, decline, and patterns of migration. In mathematical terms, we use functions to represent the size of a population at any given time. For instance, in the exercise above, the population size is denoted by $N(t)$, where $t$ represents time.
This function allows us to model the changes in population sizes due to various factors, like birth rates, death rates, and migration.

Birth rates and immigration contribute to population growth.
Death rates and emigration lead to population decline.

Understanding these dynamics is essential for managing wildlife, conserving species, and analyzing the potential impacts of humans and climate change on ecosystems. The knowledge gained from studying these patterns can help in making informed decisions in conservation planning and even urban development.

Differential Equations

Differential equations are mathematical tools used to describe the relationship between a function and its derivatives. They are especially useful in modeling situations where change is constant, like in our population dynamics problem.
In the context of the exercise, the differential equation $\frac{dN}{dt}$ represents the rate of change of the population over time. The derivative $\frac{dN}{dt}$ captures how fast the population size $N(t)$ is increasing or decreasing at any moment.

When $\frac{dN}{dt} > 0$, the population increases.
When $\frac{dN}{dt} < 0$, the population decreases.

In this particular problem, we also consider the constraint $|\frac{dN}{dt}| \leq 20$, which tells us that in any given timeframe, the population can change by at most 20 units per unit time. This boundary is crucial as it limits how quickly the population can grow or shrink, giving us a clearer picture of what might happen over the given time span from time $t=0$ to $t=5$.

Rate of Change

Rate of change is a core concept when dealing with differential equations in population dynamics. It measures how a quantity, such as population size, changes over time. In simpler terms, it tells us how fast something is happening, like how quickly a population grows or shrinks.
In the exercise, the rate of change is given by $\frac{dN}{dt}$, which indicates how the population is changing over time. Knowing the rate of change helps us predict future population sizes—vital for ecological planning and managing resources.
For example, if a wildlife reserve knows the rate at which a bird population is increasing, it can plan if additional habitat space or food resources are necessary.
Furthermore, understanding rate of change is important for:

Predicting trends in population size.
Identifying factors influencing population changes.
Planning interventions to conserve or control population levels effectively.

Through these insights, conservationists and researchers can work towards maintaining balanced and sustainable ecosystems.

Problem 51

Other exercises in this chapter

Problem 50

Find the limits in Problems 1-60; not all limits require use of l'Hôpital's rule. $$ \lim _{x \rightarrow \infty}(1+x)^{1 / x} $$

View solution

Problem 51

Find the general solution of the differential equation. $$ \frac{d y}{d t}=t(1-t), t \geq 0 $$

View solution

Problem 51

Find the limits in Problems 1-60; not all limits require use of l'Hôpital's rule. $$ \lim _{x \rightarrow 0} x e^{x} $$

View solution

Problem 52

Find the general solution of the differential equation. $$ \frac{d y}{d t}=t^{2}\left(1+t^{2}\right), t \geq 0 $$

View solution