# [SOLVED]Lotka Volterra

#### dwsmith

##### Well-known member
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

##### Well-known member
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

#### dwsmith

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

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.

#### dwsmith

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

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

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

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

#### dwsmith

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

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?

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​Solved

#### CaptainBlack

##### Well-known member
​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