91Ó°ÊÓ

Any population, \(P\), for which we can ignore immigration, satisfies \(\frac{d P}{d t}=\) Birth rate \(-\) Death rate. For organisms which need a partner for reproduction but rely on a chance encounter for meeting a mate, the birth rate is proportional to the square of the population. Thus, the population of such a type of organism satisfies a differential equation of the form $$ \frac{d P}{d t}=a P^{2}-b P \quad \text { with } a, b>0 $$ This problem investigates the solutions to such an equation. (a) Sketch a graph of \(d P / d t\) against \(P .\) Note when \(d P / d t\) is positive and negative. \(d P / d t<0\) when \(P\) is \begin{tabular}{l} \hline \\ \hline \end{tabular} \(d P / d t>0\) when \(P\) is in (Your answers may involoe a and b. Give your answers as an interval or list of intervals: thus, if \(d P / d t\) is less than zero for \(P\) between 1 and 3 and \(P\) greater than \(4,\) enter \((1,3),(4,\) infinity \() .)\) (b) Use this graph to sketch the shape of solution curves with various initial values: use your answers in part (a), and where \(d P / d t\) is increasing and decreasing to decide what the shape of the curves has to be. Based on your solution curves, why is \(P=b / a\) called the threshold population? If \(P(0)>b / a,\) what happens to \(P\) in the long run? \(P \rightarrow\) If \(P(0)=b / a,\) what happens to \(P\) in the long run? \(P \rightarrow\) If \(P(0)

Step by step solution

01

Understanding the Differential Equation

Given the differential equation \ \( \frac{dP}{dt} = aP^{2} - bP \) with \( a, b > 0 \), identify the behavior of \( \frac{dP}{dt} \) based on the value of \( P \).

02

Setting \( \frac{dP}{dt} \) Equal to Zero

To find where \( \frac{dP}{dt} \) is zero, solve the equation \ \( aP^{2} - bP = 0 \). This gives \ \( P(aP - b) = 0 \). Therefore, \( P = 0 \) or \( P = \frac{b}{a} \).

03

Analyzing the Intervals

The solutions \( P = 0 \) and \( P = \frac{b}{a} \) divide the population into three intervals. Analyze the sign of \( \frac{dP}{dt} \) in each interval: \ 1. If \( P < 0 \), \( \frac{dP}{dt} < 0 \) since both terms in the equation are negative. \ 2. If \( 0 < P < \frac{b}{a} \), \( \frac{dP}{dt} < 0 \) because the death rate term dominates. \ 3. If \( P > \frac{b}{a} \), \( \frac{dP}{dt} > 0 \) because the birth rate term dominates.

04

Sketching the Graph

Plot the graph of \( \frac{dP}{dt} \) against \( P \). The points \( P = 0 \) and \( P = \frac{b}{a} \) are where the curve intersects the \( P \)-axis. The graph will indicate where \( \frac{dP}{dt} \) is positive or negative as described in Step 3.

05

Sketching Solution Curves

Using the graph from Step 4, draw possible solutions \( P(t) \) for different initial values of \( P \). \ * For \( P(0) > \frac{b}{a} \), the population grows indefinitely as \( \frac{dP}{dt} > 0 \) for these values. \ * For \( P(0) < \frac{b}{a} \), the population decreases and approaches zero as \( \frac{dP}{dt} < 0 \). \ * For \( P(0) = \frac{b}{a} \), the population remains constant as \( \frac{dP}{dt} = 0 \) at this point.

06

Threshold Population

Based on the solution curves in Step 5, \( P = \frac{b}{a} \) is called the threshold population because it separates the scenarios where the population grows indefinitely from those where it declines to zero.

07

Long-term Behavior

If \( P(0) > \frac{b}{a} \), in the long run, \( P \rightarrow \infty \). \ If \( P(0) = \frac{b}{a} \), in the long run, \( P \rightarrow \frac{b}{a} \). \ If \( P(0) < \frac{b}{a} \), in the long run, \( P \rightarrow 0 \).

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Key Concepts

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

birth rate

In population dynamics, the **birth rate** is a crucial concept. It refers to the rate at which new individuals are added to a population through reproduction. Assuming organisms need a partner for reproduction, the birth rate is proportional to the square of the population size, which can be modeled mathematically as one part of a differential equation: \ \ \[ \frac{dP}{dt} = aP^2 - bP \ \ \] Here, \(aP^2\) represents the birth rate term. This quadratic relation implies that as the population increases, the number of potential mating pairs increases, leading to a higher birth rate. Therefore, \(dP/dt\) becomes positive when the birth rate term \(aP^2\) outweighs the death rate term \(bP\).

death rate

The **death rate** is another fundamental aspect of population dynamics. It represents the rate at which individuals in the population die. In the differential equation \[ \frac{dP}{dt} = aP^2 - bP \ \ \] the term \(bP\) denotes the death rate. The importance of understanding the death rate lies in knowing how it affects the overall population growth. This linear term means that the number of deaths increases in direct proportion to the current population size. \ \ If the death rate term \(bP\) dominates over the birth rate term \(aP^2\), then the population will decrease over time, leading to a negative \(dP/dt\).

threshold population

The **threshold population** is vital to determining the long-term behavior of a population. This is a critical population size \(P = \frac{b}{a}\) at which birth rates balance death rates, resulting in \(dP/dt = 0\).\ \ \ When the initial population size is exactly at the threshold, \(P = \frac{b}{a}\), the population will remain constant as time progresses. However, the situation changes if the population is not at the threshold: \ \ \ \

• **For \(P < \frac{b}{a}\):** The death rate dominates, and the population will eventually decline to zero.


• **For \(P > \frac{b}{a}\):** The birth rate dominates, and the population will grow indefinitely.

\ Hence, the threshold population effectively acts as a tipping point that determines whether a population will grow or decrease.

solution curves

The shape of **solution curves** \(P(t)\) for a population over time can provide valuable insight into the dynamics at play. These curves are derived from the differential equation \[ \frac{dP}{dt} = aP^2 - bP \ \ \] and show how the population changes with different initial conditions.\ \ \ The solution curves exhibit distinct behaviors based on initial population sizes:

• **If \(P(0) < \frac{b}{a}\):** The population decreases over time and approaches zero, as death rates are higher than birth rates.


• **If \(P(0) = \frac{b}{a}\):** The population remains stable at that threshold level.


• **If \(P(0) > \frac{b}{a}\):** The population increases without bound because birth rates surpass death rates.

\ Understanding these solution curves helps predict population trends and manage natural resources effectively.