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

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Discussion Overview

The discussion revolves around modifying the Lotka-Volterra predator-prey model to account for the effects of insecticide use on both predator and prey populations, specifically insects. Participants explore various formulations of the model that incorporate insecticide-related mortality rates.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • Some participants propose modifications to the original Lotka-Volterra equations by introducing terms for insecticide-induced deaths, represented by variables $\gamma$ and $\rho$, but express uncertainty about whether these should be equal.
  • Another participant suggests that the additional deaths should be proportional to the current populations, indicating that these terms act as modifiers of the growth rates.
  • There is a suggestion to use a single variable $k$ to represent the proportion of deaths for each species, with questions about whether $k$ needs to be defined in terms of the parameters $a$ and $d$ from the original model.
  • Participants discuss the implications of different formulations, including whether the populations would die out under certain conditions based on the values of $k_N$ and $k_P$ compared to $a-bP$ and $cN-d$, respectively.
  • Technical issues regarding LaTeX rendering are mentioned, with participants sharing experiences and suggesting changes to post-editing limits to avoid formatting problems.

Areas of Agreement / Disagreement

Participants express differing views on the appropriate modifications to the model and the implications of those modifications. There is no consensus on whether the insecticide effects should be equal for both predator and prey or how to best represent these effects mathematically.

Contextual Notes

Participants highlight the need for clarity in defining the parameters used in the modified equations, as well as the potential for different interpretations of how insecticide impacts the populations. There are unresolved questions about the mathematical relationships between the new variables and the original model parameters.

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

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

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

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

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?
 
Last edited:
​Solved
 
dwsmith said:
​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
 

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