ODE | Population models

1. Formula
   Observation

   The board shows $P(t)$ on the left side under the title.

2. Audio
   Observation

   The speaker says, "So let's start with a population, let's call it P of t."

$P(t)$

Population as a function of time.

Time-dependent population quantity; the video does not state an explicit domain or range.

1. Formula
   Observation

   The board shows β next to "birth rate" and the parenthetical definition "births per unit of population per unit of time".

2. Audio
   Observation

   The speaker says the birth rate is beta and defines it as the number of births per unit of population per unit of time.

β

Birth rate, defined as births per unit of population per unit of time.

Rate per unit population per unit time; examples of units are mentioned in speech but no fixed numerical domain is given.

1. Formula
   Observation

   The board shows δ next to "death rate" and the parenthetical note "same but for deaths".

2. Audio
   Observation

   The speaker says the death rate is delta and defines it as the number of deaths per unit of population per unit of time.

δ

Death rate, defined as deaths per unit of population per unit of time.

Rate per unit population per unit time; the video gives verbal examples of possible units but no explicit numerical domain.

1. Formula
   Observation

   The constant k appears in all three displayed differential equations at the top right.

2. Audio
   Observation

   The speaker reads dP/dt = kP when introducing exponential growth and later refers to dP/dt = kP^2.

1. The video does not explicitly define k, its sign, or whether it is constant in time.

k

Proportionality constant appearing in the displayed population-model differential equations.

Unspecified in the clip; used multiplicatively in the model equations.

1. Formula
   Observation

   The symbol M appears only inside the middle equation dP/dt = kP(M-P).

2. Audio
   Observation

   The speaker names the middle equation the logistic equation but does not verbally define M in this clip.

1. The clip does not explain what M represents, although it is part of the displayed logistic equation.

M

Parameter appearing in the logistic equation dP/dt = kP(M-P).

Unspecified in the clip.

1. Formula
   Observation

   The derivative notation dP/dt appears in all three top-right equations.

2. Audio
   Observation

   The speaker says, "In other words, dP/dt equals kP," while describing growth proportional to population size.

dP/dt

Instantaneous rate of change of population with respect to time.

Derivative of $P(t)$ with respect to t.

1. Formula
   Observation

   $P(t)$ population

$P(t)$

Population at time t

function of time

1. Formula
   Observation

   β birth rate (births per unit of population per unit of time)

2. Audio
   Observation

   that would be beta if I was to choose the units accordingly

$\beta$

Birth rate, defined as births per unit of population per unit of time

may depend on P and/or t

1. Formula
   Observation

   δ death rate (same but for deaths)

2. Audio
   Observation

   Beta and delta may depend on the population size or the time.

$\delta$

Death rate, defined as deaths per unit of population per unit of time

may depend on P and/or t

1. Formula
   Observation

   $ΔP = βPΔt - δPΔt$

2. Audio
   Observation

   We'll call that change delta P.

$\Delta P$

Change in population over a small time interval

finite difference

1. Formula
   Observation

   How much does P change in some small time Δt?

2. Audio
   Observation

   take the limit as delta t goes to zero

$\Delta t$

Small interval of time

positive real number approaching zero

1. Formula
   Observation

   dP/dt = (β - δ)P

2. Audio
   Observation

   And that's dP dt equals beta minus delta P.

$\frac{dP}{dt}$

Instantaneous rate of change of population with respect to time

derivative

1. Audio
   Observation

   The speaker introduces the topic by saying that modeling population is an interesting application of differential equations.

2. Diagram
   Observation

   The board title reads "Population models".

The clip frames population models as differential-equation applications. The visible title and spoken introduction establish that the lesson is about writing and comparing ODE models for population change rather than solving a specific initial-value problem.

1. The discussion is about mathematical models of population dynamics.

2. No specific biological species or dataset is fixed in this opening statement.

1. Audio
   Observation

   The speaker says the exponential growth model applies when population grows at a rate proportional to the size of the population itself.

2. Formula
   Observation

   The first displayed equation is dP/dt = kP.

1. The clip states the proportionality rule but does not solve the equation or discuss the sign of k.

The exponential growth model is presented as the case where the instantaneous population growth rate is proportional to the current population size. In symbols, the model is dP/dt = kP, with $P(t)$ denoting population and k the proportionality constant shown on the board.

1. Population growth rate is proportional to population size.

2. $P(t)$ is treated as a differentiable function of time.

1. Population modeling as an application of differential equations

1. Audio
   Observation

   The speaker says the second written model is called the logistic equation and calls it a better model for population in applications than exponential growth.

2. Formula
   Observation

   The second displayed equation is dP/dt = kP(M-P).

1. The claim that it is "better" is qualitative and not justified within this clip.

2. M is not defined verbally in the clip.

The second displayed population model is named the logistic equation. Its formula on the board is dP/dt = kP(M-P). The speaker contrasts it with exponential growth and says it is a better applied model, but the clip does not yet derive why.

1. The model is one of several population ODEs listed on the board.

2. The parameter M appears in the formula but is not explained in this excerpt.

1. Population modeling as an application of differential equations

1. Audio
   Observation

   The speaker says he does not know a proper name for the third model and suggests calling it a blow-up model because population blows up in finite time.

2. Formula
   Observation

   The third displayed equation is dP/dt = kP^2.

1. The finite-time blow-up behavior is asserted verbally but not proved in this clip.

The third displayed equation is dP/dt = kP^2. The speaker says he does not know a standard name and proposes "blow-up model" because, in his description, the population becomes unbounded in finite time under this model.

1. This is presented as another population ODE example.

2. The clip does not specify constraints on k or initial data needed for the blow-up claim.

1. Population modeling as an application of differential equations

1. Audio
   Observation

   The speaker says that instead of deriving each model individually, he will discuss a general population model from which these equations can be derived, among others.

1. The actual derivation is not completed within this 120-second clip.

The lecture announces a methodological shift: rather than treating each displayed ODE separately, the speaker plans to introduce a general population model that can yield the exponential, logistic, and quadratic examples as special cases. This sets up the later definitions of birth and death rates.

1. The general model is intended to encompass multiple specific population equations.

2. Only the plan is stated in this clip, not the full derivation.

1. Exponential growth model
2. Logistic equation
3. Blow-up model dP/dt = kP^2

1. Formula
   Observation

   The board writes β birth rate (births per unit of population per unit of time).

2. Audio
   Observation

   The speaker defines beta as the number of births per unit of population per unit of time and gives examples of possible population and time units.

Birth rate β is defined as births per unit of population per unit of time. The speaker emphasizes that the "unit of population" and "unit of time" are flexible modeling choices, giving examples such as one individual, ten thousand individuals, twenty thousand bacteria, and seconds, years, or decades.

1. β measures births normalized by population size and time.

2. The unit of population and the unit of time must be chosen consistently in a model.

1. General population model as a common source

1. Formula
   Observation

   The board writes δ death rate (same but for deaths).

2. Audio
   Observation

   The speaker says delta is the death rate and defines it as the number of deaths per unit of population per unit of time.

Death rate δ is defined analogously to birth rate, but for deaths: deaths per unit of population per unit of time. The board compresses this as "same but for deaths," and the speaker states the full verbal definition.

1. δ measures deaths normalized by population size and time.

2. It is defined in parallel with β.

1. Birth rate β

1. Formula
   Observation

   $P(t)$ population; β birth rate; δ death rate

2. Audio
   Observation

   Beta and delta may depend on the population size or the time.

The population model uses $P(t)$ to denote the population at time $t$. The parameters $\beta$ and $\delta$ represent the birth rate and death rate, respectively, defined as births or deaths per unit of population per unit of time. Crucially, $\beta$ and $\delta$ are not restricted to being constants; they may depend on the population size $P$ and/or time $t$.

1. $\beta$ and $\delta$ may depend on $P$ and/or $t$

1. Formula
   Observation

   dP/dt = (β - δ)P

2. Audio
   Observation

   the result is a differential equation which is the what I call the master population model

The master population model is the differential equation $\frac{dP}{dt} = (\beta - \delta)P$. It describes the instantaneous rate of change of the population as the product of the current population and the net growth rate ($\beta - \delta$). Other specific population models are derived from this general form by making assumptions about $\beta$ and $\delta$.

1. $\beta$ and $\delta$ may depend on $P$ and/or $t$

1. Definition of Population Model Variables

1. Formula
   Observation

   The board shows $ΔP = βPΔt - δPΔt$ ⇒ dP/dt = (β - δ)P and labels the boxed result "master" population model.

2. Audio
   Observation

   The speaker explains that the model tracks how much P changes in a small time Δt from births minus deaths.

The segment presents a general population ODE built from per-capita birth and death rates. Over a small time interval, population increase from births is βPΔt and decrease from deaths is δPΔt, giving the differential form dP/dt = (β - δ)P.

1. P is the population.

2. β is the birth rate per unit population per unit time.

3. δ is the death rate per unit population per unit time.

4. β and δ may depend on P and/or t.

1. $P(t)$
2. β
3. δ
4. Δt
5. ΔP

1. Formula
   Observation

   The board lists dP/dt = kP with the note "β, δ constant".

2. Audio
   Observation

   The speaker says, "In the exponential model, exponential growth, beta and delta are constants."

The exponential model is identified as the case where both birth and death rates are constant, producing a linear growth law in P.

1. β and δ are constant.

1. Master population model without harvesting
2. β
3. δ
4. k

1. Formula
   Observation

   The board lists dP/dt = kP(M-P) with the note "δ constant" and "β decreases linearly with P".

2. Audio
   Observation

   The speaker says, "For the logistic equation, delta is constant and beta decreases linearly with the population."

The logistic model is presented as a population law in which the death rate remains constant while the birth rate declines linearly with population size.

1. δ is constant.

2. β decreases linearly with P.

1. Master population model without harvesting
2. β
3. δ
4. k
5. M

1. Audio
   Observation

   The speaker says that if a population grows at a rate proportional to the size of the population itself, then the result is exponential growth.

2. Formula
   Observation

   The corresponding displayed equation is dP/dt = kP.

1. The clip does not prove the implication; it states it as a model identification.

If the population growth rate is proportional to the population size, then the governing model is dP/dt = kP, which the speaker identifies as exponential growth.

1. Population is represented by $P(t)$.

2. Growth rate is proportional to current population size.

For a population whose rate of change is proportional to its size.

1. Audio
   Observation

   The speaker says the logistic equation is actually a better model for population in applications than the exponential growth model.

1. This is a qualitative assertion made in the clip without supporting derivation or data in this excerpt.

The speaker claims that the logistic equation is a better model for population in applications than the exponential growth model.

1. Comparison is between the displayed exponential model and the displayed logistic model.

In applications discussed by the speaker.

1. Audio
   Observation

   The speaker says you could call dP/dt = kP^2 a blow-up model because population blows up in finite time in this particular model.

2. Formula
   Observation

   The displayed equation is dP/dt = kP^2.

1. The finite-time blow-up statement is not proved in this clip and no conditions on k or initial population are stated.

For the model dP/dt = kP^2, the speaker asserts that population blows up in finite time.

1. The population follows dP/dt = kP^2.

In the particular model just written.

1. Audio
   Observation

   The speaker says he will discuss a general population model from which each of these equations can be derived, among others.

1. The derivation itself is outside this clip.

The speaker states that a general population model will be introduced from which the displayed equations can be derived, among others.

1. The previously displayed equations are treated as special cases of a broader modeling framework.

For the listed population models and possibly others.

1. Audio
   Observation

   The speaker says he looked up CDC data on the United States and found that in 2009 there were 13.5 births per 1,000 individuals.

1. The clip cuts off before any further interpretation or comparison with death rate data.

Using CDC data for the United States, the speaker reports 13.5 births per 1,000 individuals in 2009 as an example of what a birth-rate quantity might look like.

1. The example is drawn from CDC data for the United States.

2. The year specified is 2009.

For the cited U.S. dataset in 2009.

1. Audio
   Observation

   The speaker explains that a unit of population could be one individual, ten thousand individuals, or twenty thousand bacteria, and a unit of time could be a second, a year, or a decade.

2. Formula
   Observation

   The board definitions for β and δ both use "per unit of population per unit of time".

1. Expression

   $$$$
   Explanation

   Start from the displayed definitions of β and δ as rates per unit population per unit time.

   Justification

   These definitions are written directly on the board.

   Shown in the video
2. Expression

   $$$$
   Explanation

   Recognize that "unit of population" is a modeling choice rather than a fixed biological constant.

   Justification

   The speaker gives examples ranging from one individual to thousands of individuals or bacteria.

   Shown in the video
3. Expression

   $$$$
   Explanation

   Recognize that "unit of time" is likewise flexible, with examples including seconds, years, and decades.

   Justification

   The speaker explicitly lists these time-unit examples.

   Shown in the video
4. Expression

   $$$$
   Explanation

   Conclude that β and δ are normalized rates whose numerical values depend on the chosen population and time units.

   Justification

   This follows from combining the board definitions with the spoken unit examples.

   Derived from the video

The quantities β and δ are not absolute counts; they are normalized rates whose interpretation depends on the selected unit of population and unit of time.

1. Formula
   Observation

   $ΔP = βPΔt - δPΔt$ ⇒ dP/dt = (β - δ)P

2. Audio
   Observation

   we divide both sides by delta t and take the limit as delta t goes to zero. And the result is a differential equation

1. Expression

   $$\Delta P = \beta P \Delta t - \delta P \Delta t$$
   Explanation

   The change in population $\Delta P$ over a small time interval $\Delta t$ is the number of births minus the number of deaths.

   Justification

   By definition, births per unit time is $\beta P$ and deaths per unit time is $\delta P$. Multiplying by $\Delta t$ gives the total births and deaths in that interval.

   Shown in the video
2. Expression

   $$\frac{\Delta P}{\Delta t} = \beta P - \delta P$$
   Explanation

   Divide both sides of the finite difference equation by $\Delta t$.

   Justification

   Algebraic manipulation to find the average rate of change.

   Derived from the video
3. Expression

   $$\lim_{\Delta t \to 0} \frac{\Delta P}{\Delta t} = \lim_{\Delta t \to 0} (\beta P - \delta P)$$
   Explanation

   Take the limit as $\Delta t$ approaches zero.

   Justification

   Definition of the derivative $\frac{dP}{dt}$.

   Shown in the video
4. Expression

   $$\frac{dP}{dt} = (\beta - \delta)P$$
   Explanation

   The resulting differential equation is the master population model.

   Justification

   Evaluating the limits yields the derivative on the left and the net growth rate times population on the right.

   Shown in the video

The master population model is derived as $\frac{dP}{dt} = (\beta - \delta)P$.

1. Formula
   Observation

   The board writes $ΔP = βPΔt - δPΔt$ and then the implication arrow to dP/dt = (β - δ)P.

2. Audio
   Observation

   The speaker frames the question as how much P changes in a small time Δt and identifies the two contributions as born and died.

1. Expression

   $$\Delta P = \beta P \Delta t - \delta P \Delta t$$
   Explanation

   Over a small time interval, births add βPΔt to the population and deaths remove δPΔt.

   Justification

   Definition of per-unit-population birth and death rates shown on the board.

   Shown in the video
2. Expression

   $$\frac{\Delta P}{\Delta t} = (\beta - \delta)P$$
   Explanation

   Dividing by Δt converts the finite change into a rate of change per unit time.

   Justification

   Algebraic rearrangement of the displayed finite-difference equation.

   Derived from the video
3. Expression

   $$\frac{dP}{dt} = (\beta - \delta)P$$
   Explanation

   Passing to the infinitesimal limit gives the differential equation called the "master" population model.

   Justification

   Standard transition from a small-time balance to an ODE, as indicated by the implication arrow on the board.

   Shown in the video

The basic population ODE is dP/dt = (β - δ)P, with β and δ allowed to depend on P and/or t.

1. Formula
   Observation

   The board appends -hΔt under the label "harvested" and rewrites the boxed equation as dP/dt = (β - δ)P - h.

2. Audio
   Observation

   The speaker explains that catching 100 fish every year is not part of birth or death rate, so it is added as a harvesting rate.

1. Expression

   $$\Delta P = \beta P \Delta t - \delta P \Delta t - h \Delta t$$
   Explanation

   A constant external removal contributes an additional loss term hΔt over the same small time interval.

   Justification

   The speaker introduces harvesting as a separate mechanism independent of births and deaths.

   Shown in the video
2. Expression

   $$\frac{dP}{dt} = (\beta - \delta)P - h$$
   Explanation

   Converting the finite balance to a rate equation yields the master model with an extra constant subtraction term.

   Justification

   Same limiting passage used for the unharvested model, now including the harvested term.

   Shown in the video

Adding constant harvesting modifies the master population model to dP/dt = (β - δ)P - h.

1. Formula
   Observation

   $ΔP = βPΔt - δPΔt - hΔt$ => dP/dt = (β - δ)P - h

1. Expression

   $$\Delta P = \beta P \Delta t - \delta P \Delta t - h \Delta t$$
   Explanation

   The change in population ΔP over a small time interval Δt is calculated as the number of births (βPΔt) minus the number of deaths (δPΔt) minus the number harvested (hΔt).

   Justification

   Definition of rates and accounting for population changes.

   Shown in the video
2. Expression

   $$\frac{dP}{dt} = (\beta - \delta)P - h$$
   Explanation

   Dividing by Δt and taking the limit as Δt approaches 0 yields the instantaneous rate of change, resulting in the master population model ODE.

   Justification

   Definition of the derivative.

   Shown in the video

The master population model is dP/dt = (β - δ)P - h.

1. Audio
   Observation

   The speaker gives a concrete numerical example from CDC data: in 2009 there were 13.5 births per 1,000 individuals in the United States.

1. The clip ends before the example is used in a calculation or connected to a formula for dP/dt.

Give a real-world sense of how large a birth rate can be by citing U.S. data.

1. Source: CDC data for the United States.

2. Year: 2009.

3. Reported quantity: 13.5 births per 1,000 individuals.

Illustrate the meaning of a birth-rate quantity with a concrete numerical example.

1. Expression

   $$$$
   Explanation

   Choose a public health dataset as a reference point for realistic rates.

   Justification

   The speaker says he looked up CDC data on the United States.

   Shown in the video
2. Expression

   $$13.5 \text{ births per } 1000 \text{ individuals}$$
   Explanation

   State the reported birth figure for the chosen year.

   Justification

   This value is spoken directly in the clip.

   Shown in the video

13.5 births per 1,000 individuals in the United States in 2009.

No verification is performed within the clip; the example is only stated.

1. Audio
   Observation

   The speaker says, "Suppose we have a fish pond... every year I catch 100 fish out of that pond," then asks how to incorporate that into the model.

2. Formula
   Observation

   The board updates the model to include -hΔt and the final ODE dP/dt = (β - δ)P - h.

1. The numerical value 100 fish/year is spoken but not written explicitly as $h = 100$ on the board in this clip.

A fish population in a pond follows the general population model, but an external agent removes 100 fish every year regardless of population size. How should this be incorporated?

1. Population is governed by the master model dP/dt = (β - δ)P before modification.

2. A fixed number of fish is caught each year.

3. The catch does not depend on the current population, assuming enough fish are present.

Modify the population ODE to account for constant harvesting.

1. Expression

   $$$$
   Explanation

   Recognize that the yearly catch is not a birth or death process because it is independent of population size.

   Justification

   Stated directly by the speaker.

   Shown in the video
2. Expression

   $$\Delta P = \beta P \Delta t - \delta P \Delta t - h \Delta t$$
   Explanation

   Introduce a harvesting term hΔt as an additional loss over the small time interval.

   Justification

   The board adds -hΔt labeled "harvested".

   Shown in the video
3. Expression

   $$\frac{dP}{dt} = (\beta - \delta)P - h$$
   Explanation

   Rewrite the finite-change balance as a differential equation with constant removal.

   Justification

   Directly shown in the updated boxed formula.

   Shown in the video

The harvested model is dP/dt = (β - δ)P - h, where h represents the constant removal rate.

The speaker checks the modeling logic by noting that the term is separate from β and δ because the catch is independent of population size.

1. Diagram
   Observation

   The blackboard shows the title "Population models" at upper left, three differential equations across the upper right, and definitions of $P(t)$, β, and δ down the left side.

2. Animation
   Observation

   A white pointer dot moves among the equations and the definitions as the speaker discusses them.

1. Title text "Population models".

2. Three equations: dP/dt = kP, dP/dt = kP(M-P), dP/dt = kP^2.

3. Definitions for $P(t)$, β, and δ.

4. White pointer dot.

1. The pointer moves from the first equation to the second and third equations during the model overview.

2. The pointer later shifts to the left-side definitions when the speaker introduces $P(t)$, β, and δ.

1. The board content itself remains unchanged throughout the clip.

2. The three displayed equations stay visible simultaneously.

The visual organization supports a comparison among several population ODEs and then transitions to the basic quantities needed for a more general model.

1. Animation
   Observation

   Handwritten text and equations appear sequentially on a black background.

1. Title 'Population models'

2. Variable definitions $P(t)$, β, δ

3. Red annotation 'might depend on P and/or t!'

4. Question 'How much does P change in some small time Δt?'

5. Finite difference equation $ΔP = βPΔt - δPΔt$

6. Differential equation dP/dt = (β - δ)P boxed in yellow

1. Definitions are written first.

2. Red arrows and text are added to emphasize parameter dependence.

3. The derivation question and finite difference equation are written.

4. The final differential equation is derived and boxed.

1. The background remains black.

2. The text is handwritten in various colors (white, cyan, red, yellow).

The visual progression mirrors the logical steps of defining variables, stating assumptions, setting up a discrete change equation, and taking the continuous limit to derive the governing ODE.

1. Diagram
   Observation

   The blackboard is organized with definitions on the left, three example ODEs across the top right, and the finite-change derivation plus boxed master model along the bottom.

1. Title "Population models"

2. Definitions of $P(t)$, β, and δ

3. Three displayed ODE examples

4. Bottom derivation $ΔP = βPΔt - δPΔt$

5. Boxed master model

1. At about 68 seconds, a new term -hΔt is written into the bottom equation.

2. At about 70 seconds, the boxed master model is updated to dP/dt = (β - δ)P - h.

1. The left-side definitions and the three top-right example models remain visible throughout the clip.

2. The overall structure stays as a whiteboard lecture with handwritten formulas.

The visual layout separates general definitions, special-case models, and the derivation of the master equation, then shows how harvesting is appended as an extra term.

1. Diagram
   Observation

   Static whiteboard showing definitions, derivation, and three specific ODE models.

1. Title 'Population models'

2. Variable definitions $P(t)$, β, δ

3. Derivation of ΔP and dP/dt

4. Three specific ODEs: dP/dt = kP, dP/dt = kP(M-P), dP/dt = kP^2

5. Annotations for parameters β, δ, k, M

1. All text and formulas remain static throughout the 1-second clip.

The visual presents a comprehensive summary of basic population dynamics, moving from fundamental definitions to a general differential equation, and then listing three specific cases (exponential, logistic, and quadratic growth) with their parameter behaviors.

1. Audio
   Observation

   The speaker explicitly says a unit of population could be anything from a single individual to ten thousand individuals to twenty thousand bacteria.

One might think that "per unit of population" always means per single organism.

The clip stresses that the population unit is a modeling choice and can be one individual, many individuals, or a batch such as twenty thousand bacteria.

1. Audio
   Observation

   The speaker says a unit of time could be a second, a year, or a decade.

One might assume rate definitions implicitly use a single universal time unit.

The speaker shows that the time unit is also flexible and must be specified in the model.

1. Formula
   Observation

   might depend on P and/or t!

2. Audio
   Observation

   There's nothing that restricts this model to have beta and delta be constant.

Assuming that birth rate ($\beta$) and death rate ($\delta$) must be constants in a population model.

The video explicitly states and highlights in red that $\beta$ and $\delta$ may depend on the population size $P$ and/or time $t$, providing examples like food restrictions affecting birth rates as population increases.

1. Audio
   Observation

   The speaker explicitly says, "I can't incorporate it through a birth rate or a death rate because I catch 100 fish every year no matter what the population is..."

One might try to model a fixed annual catch by changing β or δ.

Because the catch is independent of population size in this example, it must be represented as a separate harvesting term -h rather than folded into the per-capita birth or death rates.

1. Audio
   Observation

   The speaker says he will discuss a general population model from which each of these equations can be derived.

1. The actual derivation is not shown in this clip.

The announced general population model is presented as a broader framework that includes the exponential growth equation as one derivable case.

1. Audio
   Observation

   The speaker says the general model can derive each of the displayed equations, among others.

1. The derivation is outside this clip.

The logistic equation is one of the specific models said to follow from the more general population framework.

1. Audio
   Observation

   The speaker says the general model can derive each of the displayed equations, among others.

1. The derivation is outside this clip.

The quadratic dP/dt = kP^2 model is also presented as a special case obtainable from the general population model.

1. Formula
   Observation

   The board defines β as births per unit population per unit time and δ as the same but for deaths.

2. Audio
   Observation

   The speaker defines δ in the same fashion as β, replacing births with deaths.

Birth rate and death rate are parallel normalized rates; they differ in whether the counted events are births or deaths.

1. Audio
   Observation

   The speaker compares the logistic equation with exponential growth and says the logistic model is better for applications.

1. The comparison is qualitative in this clip.

The logistic equation is contrasted with the exponential growth model as a more realistic applied population model.

1. Audio
   Observation

   the model by from which the other population models are derived.

2. Formula
   Observation

   Top right shows dP/dt = kP, dP/dt = kP(M-P), dP/dt = kP^2 with conditions on β, δ.

The master population model $\frac{dP}{dt} = (\beta - \delta)P$ is a general framework. Specific models (like exponential, logistic, etc., shown in the top right corner) are obtained by making specific assumptions about how $\beta$ and $\delta$ depend on $P$ and $t$.

1. Formula
   Observation

   The board places dP/dt = kP next to the note "β, δ constant" under the general population-model heading.

2. Audio
   Observation

   The speaker identifies this as the exponential model.

The exponential model is the special case of the master population model in which β and δ are constant.

1. Formula
   Observation

   The board lists dP/dt = kP(M-P) with the note that δ is constant and β decreases linearly with P.

2. Audio
   Observation

   The speaker calls this the logistic equation.

The logistic model arises from the master framework by letting the birth rate decline linearly with population while keeping the death rate constant.

1. Formula
   Observation

   The board lists dP/dt = kP^2 with the notes $δ = 0$ and β proportional to P.

2. Audio
   Observation

   The speaker names this the blow-up model.

The blow-up model is obtained from the master framework by setting deaths to zero and making the birth rate proportional to population.

1. Formula
   Observation

   The boxed equation changes from dP/dt = (β - δ)P to dP/dt = (β - δ)P - h after adding the harvested term.

2. Audio
   Observation

   The speaker says this is the master population model with a bonus harvesting term.

The harvested equation extends the master population model by allowing an additional constant external removal term.

1. Formula
   Observation

   dP/dt = kP

The equation dP/dt = kP is a special case of the master model where $δ = 0$, $h = 0$, and $β = k$ (constant).

1. Formula
   Observation

   dP/dt = kP(M-P)

The logistic model dP/dt = kP(M-P) is a special case where the death rate δ decreases linearly with P (specifically, δ = kP, and β = kM, $h = 0$).

1. Audio
   Observation

   The speaker introduces the exponential growth model and writes dP/dt = kP.

2. Formula
   Observation

   The first equation is visible on the board.

1. Exponential growth model

1. Audio
   Observation

   The speaker names the second equation the logistic equation.

2. Formula
   Observation

   The board shows dP/dt = kP(M-P).

1. Logistic equation

1. Audio
   Observation

   The speaker says dP/dt = kP^2 could be called a blow-up model because population blows up in finite time.

1. Blow-up model dP/dt = kP^2

1. Formula
   Observation

   The board defines β as births per unit of population per unit of time.

2. Audio
   Observation

   The speaker repeats and expands this definition with unit examples.

1. Birth rate β

1. Formula
   Observation

   The board defines δ as death rate, same but for deaths.

2. Audio
   Observation

   The speaker states it is deaths per unit of population per unit of time.

1. Death rate δ

1. Audio
   Observation

   The speaker gives examples of possible population units and time units.

1. Birth rate β
2. Death rate δ

1. Audio
   Observation

   The speaker says he will discuss a general population model from which the displayed equations can be derived.

1. General population model as a common source

1. Audio
   Observation

   The speaker cites CDC data showing 13.5 births per 1,000 individuals in 2009.

1. CDC birth-rate example

1. Audio
   Observation

   how does the dependent variable change when the independent variable changes... take the limit as delta t goes to zero

1. Master Population Model Equation
2. Derivation of the Master Population Model

1. Formula
   Observation

   might depend on P and/or t!

1. Definition of Population Model Variables
2. Misconception about Constant Birth and Death Rates

1. Formula
   Observation

   The boxed equation is labeled "master" population model.

1. Master population model without harvesting

1. Audio
   Observation

   The speaker explains why a fixed annual catch is not part of birth or death rate.

1. Harvesting as an external constant removal term
2. Do not absorb constant harvesting into birth or death rates