# developing a differential equation

So I'm a biologist, and I've been in my field for a while. I've recently become interested in developing a model for my system. I don't have a huge background in math so this has been a bit of a challenge, and any help would be appreciated.

I'm developing a set of differential equations to describe the populations of several species based on the environmental conditions, and I'm running into some challenges. The main equation that I've developed describes the population growth of species A. This form of equation has been used a lot in previous research.

$$\frac{dN_{\text{A}}}{dt}={r_{\text{A}}}{N_{\text{A}}}(1+{S_{\text{A}}}-\frac{N_{\text{A}}}{K_{\text{A}}})$$

where

$$N_{\text{A}}$$ is the size of the population for species A

$$r_{\text{A}}$$ is the grown rate of the population for species A

$$S_{\text{A}}$$ is an environmental variable that, in this context, is influencing the growth rate of species A

$$K_{\text{A}}$$ is the carrying capacity for species A

The main difference between this model and previous models is the addition of this $$S_{\text{A}}$$ term. Basically, we think that the value of this environmental variable, $$S_{\text{A}}$$, should be influencing the population growth of the species. $$S_{\text{A}}$$ will vary between -1 and 1. This makes it so when $$S_{\text{A}}$$ is positive the growth rate is increased, while when $$S_{\text{A}}$$ is negative the growth rate is decreased.

The next part is where I'm having trouble though. We have reason to believe that this environmental variable, $$S_{\text{A}}$$, is changing as a result of the density of the population, $$N_{\text{A}}$$. Thus we need a differential equation for $$S_{\text{A}}$$, set up in the form, $$\frac{dS_{\text{A}}}{dt}$$.

I'm having trouble with is setting up a this differential equation. As mentioned, we have reason to believe that $$S_{\text{A}}$$ should be changing through time, dependent on the population size of species A, $$N_{\text{A}}$$. I want the value of $$S_{\text{A}}$$ to be bounded between -1 and 1.

As I mentioned, this is a density dependent equation, such that when $$N_{\text{A}}$$ is rare, or close to 0, $$\frac{dS_{\text{A}}}{dt}$$ should be positive such that $$S_{\text{A}}$$ will be approaching +1. When $$N_{\text{A}}$$ is common, or close to the carrying capacity, $$K_{\text{A}}$$, $$\frac{dS_{\text{A}}}{dt}$$ should be negative such that $$S_{\text{A}}$$ will be approaching -1.

I've gone through a lot of iterations of this, but I've failed to write something that fits these descriptions. Some of the thoughts that I've had are that in writing an equation for $$\frac{dS_{\text{A}}}{dt}$$, I likely need some constant value, c, describing the rate at which $$S_{\text{A}}$$ is changing.

Moreover, I need some way of describing when $$N_{\text{A}}$$ is either rare or common, as that will influence the sign of $$\frac{dS_{\text{A}}}{dt}$$. I've figured I should probably use the carrying capacity described in the first equation, $$K_{\text{A}}$$, as a measure for this. When $$N_{\text{A}} < \frac{K_{\text{A}}}{2}$$, $$\frac{dS_{\text{A}}}{dt}$$ should be positive, while when $$N_{\text{A}} > \frac{K_{\text{A}}}{2}$$, $$\frac{dS_{\text{A}}}{dt}$$ should be negative,

If you have any thoughts on how I might set this up, I'd greatly appreciate the help.

## 1 Answer

Why have you given $$S_A$$ an arbitrary range of $$[-1,1]$$? Also, if $$S_A$$ is changing as a result of the density of the population $$N_A$$ , then surely you'd want to calculate $$\frac{dS_{\text{A}}}{dN_A}$$, or the partial derivative of $$S_A$$ with respect to $$N_A$$ (if $$S_A$$ is a function of more than one variable). Since this would be a more meaningful measure as it shows how $$S_A$$ varies as $$N_A$$ varies.

• I've used the range between -1 and 1 because that's what's realistic in this system. This environmental variable will eventually hit some maximum value. And to be honest, I'm not familiar with how to make a partial derivative with respect to the population as you propose. I originally proposed the derivative as dependent on time because I'm primarily interested in how these variables are changing over time. – ricks.k Sep 27 '18 at 12:55