MHB Modifying the Lotka-Volterra Predator-Prey Model for Insecticide Use

[Dustinsfl]

The Lotka Volterra predator prey is:

$$
\frac{dN}{dt} = N(a-bP)\quad\text{and}\quad\frac{dP}{dt} = P(cN-d)
$$

How can this model be modified to demonstrate farmers who continual spray insecticides that kill both predator and prey (the predators and prey are insects)?

$$
\frac{dN}{dt} = N(a-bP) - \gamma\quad\text{and}\quad\frac{dP}{dt} = P(cN-d)-\rho
$$

Where $\gamma$ and $\rho$ are the insecticide deaths. Could $\gamma=\rho$? Or would it affect them different?

I just made up gamma and rho as the variables for the deaths. I am not sure if they would kill equally the predator and prey or the same.

 

[CaptainBlack]

The Lotka Volterra predator prey is:

$$
\frac{dN}{dt} = N(a-bP)\quad\text{and}\quad\frac{dP}{dt} = P(cN-d)
$$

How can this model be modified to demonstrate farmers who continual spray insecticides that kill both predator and prey (the predators and prey are insects)?

$$
\frac{dN}{dt} = N(a-bP) - \gamma\quad\text{and}\quad\frac{dP}{dt} = P(cN-d)-\rho
$$

Where $\gamma$ and $\rho$ are the insecticide deaths. Could $\gamma=\rho$? Or would it affect them different?

I just made up gamma and rho as the variables for the deaths. I am not sure if they would kill equally the predator and prey or the same.

Since \(N\) and \(P\) are actual populations the additional deaths should be proportional to the current populations. So in effect they are modifiers of \(a\) and \(d\)

(or population densities if you will)

CB

 

[Dustinsfl]

$$
\frac{dN}{dt} = N(a-bP) - ka\quad\text{and}\quad\frac{dP}{dt} = P(cN-d)-kd
$$

Just simply that then?

Also, I know why sometimes the latex won't show. I know you said it was a delimiter not appearing but I know why it occurs now. It occurs when I right click my Latex, show source, and copy the Latex so I don't have to re-type it. With that being the issue, can we not have an edit post limit? By going into edit and copying the Latex, this won't occur.

 

[Dustinsfl]

$$
\frac{dN}{dt} = N(a-bP) - ka\quad\text{and}\quad\frac{dP}{dt} = P(cN-d)-kd
$$

Just simply that then?

Also, I know why sometimes the latex won't show. I know you said it was a delimiter not appearing but I know why it occurs now. It occurs when I right click my Latex, show source, and copy the Latex so I don't have to re-type it. With that being the issue, can we not have an edit post limit? By going into edit and copying the Latex, this won't occur.

Can I use the same k as the the proportion that die for each species?

Does k has to written as an expression involving a for the first and d for the second equation?

If so, I am not sure about how to come up with it.

 

[CaptainBlack]

$$
\frac{dN}{dt} = N(a-bP) - ka\quad\text{and}\quad\frac{dP}{dt} = P(cN-d)-kd
$$

Just simply that then?

Also, I know why sometimes the latex won't show. I know you said it was a delimiter not appearing but I know why it occurs now. It occurs when I right click my Latex, show source, and copy the Latex so I don't have to re-type it. With that being the issue, can we not have an edit post limit? By going into edit and copying the Latex, this won't occur.

$$
\frac{dN}{dt} = N(a-bP) - k_N N\quad\text{and}\quad\frac{dP}{dt} = P(cN-d)-k_P P
$$

CB

 

[Dustinsfl]

$$
\frac{dN}{dt} = N(a-bP) - k_N N\quad\text{and}\quad\frac{dP}{dt} = P(cN-d)-k_P P
$$

CB

That is it?

I don't have to define k in terms of a and d?

So if $k_N> a-bP$ then the population of N would die out and similar if $k_P<cN-d$ the P population would die out correct?

 

[Dustinsfl]

​Solved

 

[CaptainBlack]

​Solved

Check, but you should be able to mark a thread as solved from the thread tools menu at the top of the thread page.

CB