Solving Notation Question for Logistic Equation in Population Dynamics

[PhDP]

Let's say I have a simple difference equation (the logistic equation from population dynamics);

$$\frac{x_{t+1}}{x_t} = \alpha(t) - \beta x_t$$

Now, let's say $$\alpha(t)$$ depends on the time of the year, and t = 3 months, I could built a vector for the value of $$\alpha(t)$$ for the year;

$$\alpha = \left[\begin{array}{c} 1.0\\1.2\\1.5\\1.1 \end{array}\right]$$

Now, obviously, at time t=5, it would have to get back to $$\alpha(1)$$. I was thinking about using something like;

$$\frac{x_{t+1}}{x_t} = \alpha([(t-1) \mbox{ mod}(4)]+1) - \beta x_t$$

But I wondered if there was a better notation for this.

 

[jvicens]

Hello,

Thank you for sharing your thoughts on using the logistic equation for population dynamics. Your approach of using a vector for \alpha(t) that depends on the time of year is a great way to incorporate seasonal variations into the model.

I believe the notation you suggested, \frac{x_{t+1}}{x_t} = \alpha([(t-1) \mbox{ mod}(4)]+1) - \beta x_t, is a valid way to express the idea of cycling through the vector of \alpha(t) values. Another way to represent this could be \frac{x_{t+1}}{x_t} = \alpha(t \mbox{ mod}(4)) - \beta x_t, where t \mbox{ mod}(4) would give the remainder when t is divided by 4, ensuring that the index for \alpha(t) stays within the range of the vector.

In general, there may be different ways to represent the same idea in mathematical notation, so it ultimately depends on personal preference and clarity of communication. As long as the notation is clear and consistent, either approach should work well.

 

[tacman]

In population dynamics, the logistic equation is commonly used to model the growth of a population over time. In this equation, the population at time t+1 is dependent on the population at time t and two parameters, alpha and beta. In this scenario, alpha is a variable that changes with time, specifically with the time of year. To incorporate this into the equation, a vector can be created with the values of alpha for each month of the year. This allows for a more accurate representation of the changing values of alpha over time.

To solve for the population at a specific time, t=5, the equation can be modified to incorporate the vector of alpha values. One way to do this is to use the modulus operator to cycle through the values in the vector. This would result in the equation you have proposed, where the value of alpha at time t is determined by the remainder of (t-1) divided by 4, plus 1 to account for the first value in the vector being at t=1. This notation works well and is commonly used in this type of scenario.

However, if you are looking for a more concise notation, you could also use the subscript notation for the vector of alpha values. This would look like \alpha_{(t-1) \mbox{ mod}(4)+1}. This notation essentially means that the value of alpha at time t is determined by the value at the index of (t-1) mod 4 + 1 in the vector. This notation may be easier to understand and use, especially if you are working with more complex equations or multiple variables.

In conclusion, both the notation you have proposed and the subscript notation for the vector of alpha values are valid ways to incorporate time-dependent variables into the logistic equation for population dynamics. The choice ultimately depends on personal preference and the complexity of the equation.