How Does the Alligator Population Change Over Time in a Swamp?

[saintdick]

This has been posted before but i didn't understand the answer

Suppose the number x(t) (with t in months) of alligators in a swamp satisfies the differential equation

dp/dt = 0.0001x^2 - 0.01x

(a) If there are initially 25 alligators in the swamp, solve this differential equation to determine what happens to the alligator population in the long run.

(b) Repeat for an initial population of 150 alligators."

i am totally clueless about this one?

 

[chiro]

Hey saintdick and welcome to the forums.

From your question it seems that p is actually the same as x(t) [i.e. p = x(t)] from the context of your question.

So basically the model is the same as saying:

dp/dt = 0.0001p^2 - 0.01p.

Using this information, what do you think the next step is?

 

[saintdick]

Ho chiro,

I used separtion of variable to get: dx/x(x-100)=dt/10000

integrated both side, used PFD for left side:

1/100 ln(x-100)-1/100 ln(x)= 1/10000 *t +c

now what? how do i solve this for x?

 

[chiro]

Ho chiro,

I used separtion of variable to get: dx/x(x-100)=dt/10000

integrated both side, used PFD for left side:

1/100 ln(x-100)-1/100 ln(x)= 1/10000 *t +c

now what? how do i solve this for x?

Yes you need to solve for x by using algebra and properties of logs.

Your integration constant will depend on your initial value which is different for each sub-question.

It may help if you post your final answer if you want someone to check it or if you run into trouble.

 

[blue_raver22]

I can provide an explanation for the given differential equation and offer a solution to the problem.

The differential equation given is a model for the population dynamics of alligators in a swamp. It takes into account two factors - the growth rate of the population and the death rate of the population. The growth rate is represented by the term 0.0001x^2, which means that as the population increases, the rate of growth also increases. The death rate is represented by the term -0.01x, which means that as the population increases, the rate of death also increases.

To solve this differential equation, we need to find the function x(t) that satisfies it. This can be done by separating the variables and integrating. The resulting function will give us the population of alligators at any given time t.

(a) If there are initially 25 alligators in the swamp, we can substitute x(0) = 25 into the equation and solve for x(t). This will give us the population of alligators at any time t. In the long run, as t approaches infinity, the population will approach a steady state value, which can be found by setting dp/dt = 0 and solving for x. In this case, the steady state value is x = 1000. This means that in the long run, the alligator population will stabilize at 1000 individuals.

(b) Similarly, if there are initially 150 alligators in the swamp, we can substitute x(0) = 150 into the equation and solve for x(t). In the long run, the population will again approach a steady state value, which is x = 1000. This means that regardless of the initial population, the alligator population will eventually stabilize at 1000 individuals.

In conclusion, the given differential equation predicts that the alligator population in the swamp will reach a stable value of 1000 individuals in the long run, regardless of the initial population. This could be due to factors such as limited resources or competition within the population. Further research and data analysis would be needed to fully understand the dynamics of the alligator population in this swamp.