Logistic growth Scientists often use the logistic growth...

Logistic growth Scientists often use the logistic growth func$\operatorname{tion} P(t)=\frac{P_{0} K}{P_{0}+\left(K-P_{0}\right) e^{-r_{0} t}}$ to model population growth, where $P_{0}$ is the initial population at time $t=0, K$ is the carrying capacity, and $r_{0}$ is the base growth rate. The carrying capacity is a theoretical upper bound on the total population that the surrounding environment can support. The figure shows the sigmoid (S-shaped) curve associated with a typical logistic model. (FIGURE CAN'T COPY) Gone fishing When a reservoir is created by a new dam, 50 fish are introduced into the reservoir, which has an estimated carrying capacity of 8000 fish. A logistic model of the fish population is $P(t)=\frac{400,000}{50+7950 e^{-0.5 t}}$, where $t$ is measured in years. a. Graph $P$ using a graphing utility. Experiment with different windows until you produce an $S$ -shaped curve characteristic of the logistic model. What window works well for this function? b. How long does it take for the population to reach 5000 fish? How long does it take for the population to reach $90 \%$ of the carrying capacity? c. How fast (in fish per year) is the population growing at $t=0 ?$ At $t=5 ?$ d. Graph $P^{\prime}$ and use the graph to estimate the year in which the population is growing fastest.

Logistic growth Scientists often use the logistic growth func$\operatorname{tion} P(t)=\frac{P_{0} K}{P_{0}+\left(K-P_{0}\right) e^{-r_{0} t}}$ to model population growth, where
$P_{0}$ is the initial population at time $t=0, K$ is the carrying capacity, and $r_{0}$ is the base growth rate. The carrying capacity is a theoretical
upper bound on the total population that the surrounding environment can support. The figure shows the sigmoid (S-shaped) curve associated with a typical logistic model.
(FIGURE CAN'T COPY)
Gone fishing When a reservoir is created by a new dam, 50 fish are introduced into the reservoir, which has an estimated carrying capacity of 8000 fish. A logistic model of the fish population is $P(t)=\frac{400,000}{50+7950 e^{-0.5 t}}$, where $t$ is measured in years.
a. Graph $P$ using a graphing utility. Experiment with different windows until you produce an $S$ -shaped curve characteristic of the logistic model. What window works well for this function?
b. How long does it take for the population to reach 5000 fish? How long does it take for the population to reach $90 \%$ of the carrying capacity?
c. How fast (in fish per year) is the population growing at $t=0 ?$ At $t=5 ?$
d. Graph $P^{\prime}$ and use the graph to estimate the year in which the population is growing fastest.

Key Concepts

Logistic Growth Model

The logistic growth model describes how populations grow in environments with limited resources. It is characterized by an initial period of exponential growth that slows as the population approaches a maximum limit, known as the carrying capacity. This model is typically represented by a function that incorporates both exponential growth and a limiting factor to account for environmental constraints.

Carrying Capacity

Carrying capacity is the theoretical maximum population size that an environment can sustain indefinitely given the available resources such as food, habitat, and water. In logistic growth models, it serves as an upper bound that the population asymptotically approaches over time, indicating the environmental limits to growth.

Sigmoidal (S-shaped) Curve

A sigmoidal, or S-shaped, curve is a graphical representation of logistic growth. It displays three distinct phases: a slow start when the population is small, a period of rapid growth during which the population increases quickly, and a leveling off as the population nears the carrying capacity. This S-curve visually represents how growth deceleration happens as environmental limits are reached.

Exponential Functions

Exponential functions are a key component of the logistic model, particularly in the early stages of growth when resources are abundant and population increase is rapid. However, unlike pure exponential growth, logistic growth incorporates a decay in the growth rate as the influence of limiting factors, through the carrying capacity, becomes more significant.

Rate of Change and Derivatives

The derivative of a logistic function represents the rate of change of the population with respect to time. Analyzing this derivative allows one to determine the speed at which the population is growing at any given moment and to identify the instance of maximum growth rate, which is critical for understanding the dynamics of the system.

Inflection Point

In the context of logistic growth, the inflection point is where the growth curve changes from being concave upward to concave downward. This point corresponds to the maximum rate of population increase. Determining the inflection point is important for predicting when the population will experience its fastest growth phase before slowing down as it approaches the carrying capacity.

Transcript

00:01 We're given a question where we're asked to draw a function.

00:04 It's in a logistic curve, and it's this function.

00:08 It's 400, or sorry, 400 ,000 divided by, and then it's 50 plus 7 ,950e raised 2, and then the minus 0 .5x or t.

00:32 Okay, so there's our graph and it asks us i'm actually going to change my axes around just so that i can get this is graph 4 .4.

00:52 I'm sure you've seen something similar but i'm going to put my maximum up to like a really big number here so i can see if i that's too big.

01:03 So let's bring that back down a couple of factors of zero we're still not that's way too big still let's go to 500 then let's make our minimum minus 500 there we go so that gives us a pretty good picture now okay so that's the first part of the question the second part of the question says how long does it take the population to reach 5 ,000 fish so all i'm going to do is put in 5 ,000 and and then if i go up i'm going to actually have to change my axes around again and go maximum like 50 and then in the y go up to at least 5 ,000 so let's go 500 what's rate there so i can calculate that intersection point using this we're going to look at the intersection all right so we're just click here so it takes about 11 years i think t is in years yes so it takes about 11 years 11 .15 years say that's part b that it says how long does it take to reach 90 % of its carrying capacity so we don't we've got to figure out what that carrying capacity is if we sort of scroll up here we can see that our carrying capacity it's going to be 8 ,000 so 90 % of that is 7200 so again we're just going to change this to 70200 and jump to our intersection again we're we're going to change this around to intersection, and jump.

03:14 Sorry, it doesn't seem to want to work, so let's try this again.

03:17 Let's jump to our intersection right here.

03:20 There we go...

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