How to model species interactions on population size dynamic?

Question

Here are two classical discrete time models of population size dynamic. The exponential growth model

$$N_{t+1} = r\cdot N_t \qquad\qquad(1)$$

and the logistic growth model

$$N_{t+1} = r\cdot N_t \left(1 - \frac{N_t}{K}\right) \qquad\qquad(2)$$

, where $r$ is the growth rate, $N_t$ is the population size at time $t$ and $K$ is the carrying capacity.

I am wondering how are species interaction generally modelled into this framework. Let's consider three interacting species called "a", "b" and "c" with populations sizes at time $t$, $Na_t$, $Nb_t$, $Nc_t$, respecitively. The effect of these three species on the focal species are $\alpha_a$, $\alpha_b$ and $\alpha_c$.

How do we typically compute, for both a logistic and an exponential discrete time model, the recursive equation of the focal species' population size?

For the exponential model, is it

$$N_{t+1} = r\cdot N_t \cdot (Na_t \alpha_a + Nb_t \alpha_b + Nc_t \alpha_c) \qquad\qquad(3)$$

or

$$N_{t+1} = r\cdot N_t + Na_t \alpha_a + Nb_t \alpha_b + Nc_t \alpha_c \qquad\qquad(4)$$

or something else? And for the logistic model, is it

$$N_{t+1} = r\cdot N_t \left(1 - \frac{N_t}{K}\right) \cdot (Na_t \alpha_a + Nb_t \alpha_b + Nc_t \alpha_c) \qquad\qquad(5)$$

or

$$N_{t+1} = r\cdot N_t \left(1 - \frac{N_t \cdot (Na_t \alpha_a + Nb_t \alpha_b + Nc_t \alpha_c)}{K}\right) \qquad\qquad(6)$$

or something else?

It is possible that we typically define more parameters that just the $\alpha$ values I define above to describe a species interaction even in the simplest models. Ideally, I would love an answer that draws from a source (a textbook in theoretical ecology, a peer-reviewed article or whatever).

Answer 1

You can write these equations in the form of ODEs but it is not really essential.

This is my analysis (just the model formulation and not literature):

The effects of interactor species may or may not be dependent on the population of the species that is being analysed (recipient). Now you have to model based on how you would realistically expect the interactions to have an effect on a species. A simple assumption would be that the interaction improves or reduces fitness (growth rate), additively. So you can model the effects as:

$$N_{t+1} = (r_0 + \beta_a + \beta_b + \beta_c) N_t$$

where $\beta_x$ denotes the effect of species $x$ which in turn you can model as dependent ($N_tN_{x_t}\alpha_x$) or independent ($N_{x_t}\alpha_x$) of the recipient population. Now, you can also impose a condition that the growth rate cannot increase beyond a certain value and so you can model it as a saturating function. A logistic or a Michaelis-Menten like function would work. For e.g:

$$\begin{align} r &= \frac{r'}{K+r'}\\[1em] r' &= r_0 + \beta_a + \beta_b + \beta_c\end{align}$$

You can likewise adapt model using the logistic equation.

Answer 2

What you are looking for can easily found in any undergrad text on ecology and it goes by the name (competitive) Lotka-Volterra Equations. This is a system Ordinary Differential Equations and in my opinion, can easily be converted to recursive or difference equations. Your guess with logistic growth is fairly close to these equations. You can remove the term corresponding to intra-specific competition $\alpha_{ii}$ from those equations and you get the exponential model.

Please let me know if you'd like me to elaborate on my answer.

Lotka-Volterra Equations

Competitive Lotka-Volterra Equations

The book I know (Ecology: From Individuals to Ecosystems) has the 2 species interaction model.