Can somebody explain to me this integrodifferential equation?

[end3r7]

It's a volterra equation with a continuous delay

*I is negative infinity, couldn't figure out how to write it

$$\dot{x} = rx(t)[1 - K^-1\int_{I}^{t} k(t-s)x(s)\,ds]$$

The part in parenthesis is the density dependent factor, but I don't understand how the integral works exactly. I know the function k(t) is a weightfactor which says how much weight should be given to past populations.

So let's see if I get it, feel free to yell at me (AKA reply in CAPS) if I am wrong.

k(t-s)x(s) is same as the function k(t) shifted to the right by s multiplied by a scalar (here x(s) represents a population at time t=s).
Thus the integral is just going to be the summation of all these functions from initial up to current time. This is going to be a function in 't'.

If the max of the kernel occurs at zero then there is almost no delay, right?
Cuz it follows that, say, k(t-s)x(s) will contribute the most to the resulting integral when s=t=now.

On the other hand, if max is at t=T then the major contributor will be when t-s = T, or s = t-T, that is, T generations ago.

If anybody wants to add/correct anything, feel more than free. =)

 

[jvicens]

Hello there! As a fellow scientist, I would like to clarify a few things about this volterra equation with a continuous delay.

First of all, you are correct in understanding that the function k(t-s)x(s) is essentially a weighted contribution of past populations to the current population at time t. The integral is indeed a summation of these contributions over a continuous range of time, from the initial time I to the current time t.

However, I would like to point out that the function k(t) is not just a weight factor, but it represents the interaction strength between individuals in a population. This interaction can be competition, predation, or any other type of interaction that affects the growth rate of the population.

In terms of the delay, you are also correct in your understanding. If the maximum of the kernel occurs at zero, then there is almost no delay in the population growth. On the other hand, if the maximum occurs at t=T, then the major contributor to the integral will be the population T generations ago.

I hope this clarifies things for you. If anyone else has anything to add or correct, please feel free to do so. Let's keep the discussion going!

 

[tacman]

Hi there,

First of all, it's great that you're trying to understand this integrodifferential equation. It can be a bit confusing at first, but with some explanation, I'm sure you'll get the hang of it.

So, let's break down the equation:

\dot{x} = rx(t)[1 - K^-1\int_{I}^{t} k(t-s)x(s)\,ds]

First, let's look at the left side of the equation: \dot{x}. This represents the rate of change of the population x at time t. The dot above the x indicates that we are taking the derivative with respect to time.

Next, we have the term rx(t). This is the intrinsic growth rate of the population x at time t. It is multiplied by the current population size x(t), indicating that the growth of the population is dependent on its current size.

Now, let's look at the term in brackets: [1 - K^-1\int_{I}^{t} k(t-s)x(s)\,ds]. This is the density-dependent factor, as you correctly stated. Let's break it down further:

- K^-1 is the inverse of the carrying capacity of the environment. This is the maximum population size that the environment can sustain.
- The integral represents the sum of all the past population sizes (x(s)) multiplied by the weight factor (k(t-s)) from the initial time (I) up to the current time (t). This is essentially taking into account the effect of past populations on the current population growth.
- The 1 in front of the integral represents the baseline growth rate of the population without any density-dependent factors.

Now, to answer your questions:

- If the maximum of the weight factor (k(t-s)) occurs at zero, then there is almost no delay. This means that the current population growth is mostly affected by the current population size.
- If the maximum of the weight factor (k(t-s)) occurs at t=T, then the major contributor to the integral will be when t-s = T, or s = t-T. This means that T generations ago is the most influential in the current population growth.

I hope this helps clarify the equation for you. Let me know if you have any further questions. Happy learning!