Modeling Mosquito Population Growth and Predation

In summary, the conversation discusses a modeling problem involving the population of mosquitoes in a certain area, which increases at a rate proportional to the current population. In the absence of predators, the population doubles each week, starting with an initial population of 200,000. However, predators (such as birds and bats) eat 20,000 mosquitoes per day, which affects the population growth. The conversation also mentions a differential equation for solving the problem and the importance of understanding the problem before attempting to solve it.
  • #1
FrogPad
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Ok, I have this modeling problem that I cannot figure out how to setup.
The problem is given:

The population of mosquitoes in a certain area increases at a rate proportional to the current population, and in the absence of other factors, the population doubles each week. There are 200,000 mosquitoes in the area initially, and predators (birds, bats, so forth) eat 20,000 mosquitoes/day. Determine the population of mosquitoes in the area at any time.


I don't understand what "absense of other factors" means. I'll put down what I have.

Let P(t) = the population of mosquitoes.
P(0)=200,000

dP/dt = 2*P(t) mosquitoes/week - 20,000 mosquitoes/day

(now simply convert week to days)
dP/dt = 2*P(t) mosquitoes/(7*day) - 20,000 mosquitoes/day

So yeah, that's the equation I have to model it. I have NO idea if that is right not, because like I said, I don't really understand what the question is asking.
Now, when I solve this problem I get a different answer then what the book has.
The book gives:
P=201,977.31-1977.31*e^(ln(2))*t, 0<=t<=t_f ~=6.6745 weeks

So, if anybody could point me in a direction of what I am doing wrong with the modeling... or maybe, just maybe my model is fine and I'm solving the DIFFEQ problem wrong...

Anyways... thanks in advance, I appreciate any help.
 
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  • #2
Why do you think the proportionality factor is 2??
Let us consider the exponential growth scenario, that is, with no predators present.
Let time be measured in weeks.
We then have the diff.eq:
[tex]\frac{dp}{dt}=kp[/tex] where we regard k as momentarily an unkown constant.
This equation is easy to solve with an intial value [tex]p(0)=p_{0}\neq{0}[/tex]:
[tex]p(t)=p_{0}e^{kt}[/tex]
Now, we invoke the condition that after 1 week (t=1), the population is doubled ([tex]p(1)=2p_{0}[/tex]):
[tex]2p_{0}=p_{0}e^{k}\to{k}=ln(2)[/tex]

Thus, when we are to formulate the associated problem with predators, we get (with time still measured in weeks):
[tex]\frac{dp}{dt}=ln(2)p-7*a, a=20.000[/tex]
 
  • #3
Ok, so the basic idea for problems like this is you hit condition A with some constant (say C from the integrating factor/solving process) and you hit the next condition with B (say with the constant K).


Yeah, that doubling process makes more sense. I was really just tossing variables and stuff in there, hoping it would work like magic without understanding it. That makes a hell of a lot more sense with what you are saying, and I'll play around with it when I get home tonight. Thanks man, I appreciate the help.
 

1. What is a differential equation?

A differential equation is a mathematical equation that describes the relationship between a function and its derivatives. It is used to model various physical, biological, and economic phenomena, such as population growth, heat transfer, and chemical reactions.

2. What is the purpose of using differential equations in modeling problems?

Differential equations are used in modeling problems because they provide a way to describe the rate of change of a system over time. This allows us to make predictions and understand how a system will behave under different conditions.

3. How do you solve a differential equation?

The method for solving a differential equation depends on its type and complexity. Some common techniques include separation of variables, integrating factors, and using power series. In some cases, numerical methods may be used to approximate a solution.

4. Can differential equations be used to solve real-world problems?

Yes, differential equations are commonly used in a wide range of fields, including physics, engineering, biology, economics, and more. They can be used to model and analyze real-world systems and make predictions about their behavior.

5. What are some practical applications of differential equations?

There are numerous practical applications of differential equations, including calculating the rate of change of a chemical reaction, predicting the spread of diseases in a population, and understanding the behavior of electric circuits. They are also used in fields such as fluid dynamics, control theory, and optimization.

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