Limited Growth Equation Another differential equation that models limited growth of a population

step 1 This question is a bit of a trickier one. It gives us a differential equation that models a popu

Limited Growth Equation Another differential equation that...

Key Concepts

Comparison of Growth Models

This concept deals with understanding the differences between various population dynamics models, such as the limited growth model and the logistic growth model. While both incorporate the idea of a carrying capacity, the logistic model often includes additional non-linear terms that create a sigmoidal growth curve, reflecting more complex interactions between the population and its environment.

Population Growth Rate Analysis

This concept involves examining the rate at which a population increases or decreases. It often includes identifying conditions for maximum growth. In limited growth models, analyzing the derivative of the population function helps determine when the population grows the fastest, which has implications in resource management and planning.

Initial Conditions in Differential Equations

Initial conditions specify the state of a system at the beginning of the analysis and are essential for determining the particular solution of a differential equation. In population models, initial conditions define the starting population size, which influences the future evolution of the model.

Carrying Capacity

Carrying capacity is a critical concept in population dynamics representing the maximum population size that an environment can sustainably support. In models such as the limited growth equation, the carrying capacity serves as an upper bound toward which the population asymptotically approaches, accounting for limitations in resources or space.

Separable Differential Equations

This concept refers to differential equations that can be rearranged so that all the terms involving one variable and its differential are on one side of the equation and the terms involving the other variable are on the opposite side. Solving these equations typically involves integrating both sides separately, which is a fundamental technique in solving first-order ordinary differential equations used in modeling various phenomena.

Exponential Decay and Growth

This concept involves mathematical functions that describe rates of change proportional to the current state, often resulting in behaviors characterized by exponential increase or decrease. In the context of limited growth models, exponential decay is observed in the transient deviation of the population from the carrying capacity over time.

Transcript

00:01 This question is a bit of a trickier one.

00:02 It gives us a differential equation that models a population using a limited growth equation.

00:08 So there's a couple parts, and we're going to start with part a that wants to show that p equals m minus ae to the negative k t power.

00:17 So what we're going to do is we're going to get d p on one side and d t on the other side.

00:21 So if we multiply both sides by d t and divide both sides by k times m minus p, we get d p over k.

00:30 Times m minus p equals d t and so we're going to integrate and if we integrate we get on the dt side we just get t and then on this side we have one over something that if we integrate we get ln because there's just p to the first power here so if we do that we have to take out a negative 1 over k so we have ln of mk minus pk because if we set that u was pk u was mk minus pk then d u would have been negative k so to be able to integrate that correctly we'd have to take out a negative one over k here and then what we want to do now is multiply both sides by negative k so we get negative k t equals ln of mk minus pk and we'll do e to the both sides we have e to the negative k t equals mk minus pkk and what we want to do is we want to always we have to keep in mind there's a constant here there's possibly a constant here we won't write anything for now but keep that in mind we're going to add pk to both sides subtract e to the negative kt from both sides so we get pkkk minus e to the negative kt if we divide a k from both sides we get p equals m and of course that constant divided by k is when another constant will say a e to the negative k t this is dependent on the initial condition but we've successfully showed that this is what our population will look like or what our equation for it is.

02:08 So then the next part of the question, part b, says, what is the limit of time approaching infinity of p of t? and this is quite simple.

02:18 We know that it's m minus ae to a negative power.

02:21 So if we have infinity, we have something like m minus a over e to infinity.

02:29 This will approach 0, so this just equals m.

02:35 And then part c says for what time t is the population growing the fastest? so what we have to do here is we have to look at dpdt and basically differentiate again because if we have our growth equation and we want to see where is our growth equation growing the fastest, we take the derivative of that, said it equal to zero.

02:55 So if we have dpdd equals k times m minus p and we differentiate, again we get the second derivative of p equals so km of course is a constant so it goes away so we have negative kp left so we get negative k dp d t we plug this back in we get the second derivative equals negative k times k times m minus p and once we do that we see that we have to also replace our p with what we solve for because of course we're trying to look for a t.

03:35 So we'll go to the next page.

03:36 We have the second derivative equals negative k times km.

03:45 K times p, which we solved for was this m minus a, e to the negative kt...

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