Endangered species The population of an endangered species...

Key Concepts

Population Dynamics

Population dynamics is the study of how and why the population size of a species changes over time. It often involves models that incorporate factors such as birth rates, death rates, and other influences, which can be represented mathematically using differential equations.

Extinction Time Calculation

Extinction time refers to the point in time when the population reaches zero. By solving the integrated population model for the time variable when the population equals zero, one can determine when extinction occurs, and variations in initial conditions further illustrate how sensitive these models are to starting population sizes.

Initial Value Problems

An initial value problem involves a differential equation along with a given initial condition, which makes the solution unique. In population dynamics, the starting population size is used as an initial condition to determine the constant of integration, ensuring a specific solution to the model.

Differential Equations

A differential equation connects a function with its derivative and is a fundamental tool for modeling real-world systems that change over time. In population dynamics, it can describe how a population size changes as a function of time based on various growth factors or limitations.

Integration

Integration is the reverse process of differentiation and is used to determine the original function from its rate of change. In solving differential equations for population models, integration helps in reconstructing the population function from the given rate of change.

Transcript

00:01 Okay, here we have a population of endangered species.

00:05 It starts at 300, that population of 300, and it decays according to our change, our rate function, 30 minus 20t.

00:16 To figure this out, what i can do first is, what we can do is figure out what pft is, because we're trying to figure out what the population is after five years for our first problem.

00:28 So to figure out what p of t is p of t, we're going to find an antiter derivative.

00:33 So anti -terributive of 30 is 30t.

00:38 For this, it's 202 squared over 2 plus c.

00:46 So if we want to figure out what c is, we know it's 300.

00:50 So 300 equals 30 times 0 minus 20, and 0 squared over 2.

00:59 So plus c so c has to be equal to 300 because these are cancelled because they're both equals to zero if we get that we get 30 t minus 10 t squared because 20 divided by 2 is 10 plus 300 or if you want to do it in a more normal way it's going to look like this so moving on it asks us to figure out what the population is after five years plugin 5 here we get negative 10 times 5 squared plus 30 times 5 plus 300 and we get negative 250 plus 150 plus 150 plus 300 which adds up to equal 200 individuals moving on it asks us when do or not this species extinct so we get zero equals negative 10 t squared plus 30 times t plus 300.

02:13 This is a tricky one because it doesn't factor easily.

02:16 So we're going to go ahead and use the quadratic formula.

02:20 So x equals negative b, so we have negative 30 plus or minus the square root of b squared, which is going to be 900 minus 4a .c.

02:34 So 4 times a, which is negative 10, is negative 40 times 300 is negative 12 ,000.

02:43 And because we're subtracting, we can go ahead and make this a plus.

02:47 So plus 12 ,000, all divided by negative 20.

02:54 Because 2a, so 2 times negative 10.

02:59 If we calculate that out, we get 900 plus 12 ,000, which is 12 ,900.

03:07 Get the square root of that.

03:09 It's approximately 113 .6.

03:27 Now, we want this to be positive, of course.

03:30 Because we're looking for a positive number, because years don't go backwards like that.

03:34 So because the negative 30, and this is also negative 20, the only way to go positive is for this top part to be negative.

03:45 So we have to do negative 30 minus 113.

03:50 And if we do that and then divide by 20, we get approximately 7 .18 years.

04:03 So that is when a population like this would go extinct.

04:15 Let me just rewrite that box.

04:17 Those are really bad bucks.

04:19 So if we keep going, our last problem is asking us, how does our extinction time change depending on the population? so instead of having 300, we're going to have 100...

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