Mathematical model about the relationship between two animal species

I'd like to relate a mathematical model with two animal species. This is the model:

$$\begin{split} \frac{dN}{dt} & =rN-\frac{rN(N+M)}{k} \\ \frac{dM}{dt} & =r'M-\frac{r'M(N+M)}{k'} \end{split}$$

The model means that there is competition within the population of the first species ($N$) and the population is also affected by the population of another animal species ($M$).

Something similar occurs with the population of species $M$.

My question is:

What kind of animals would fit this model?

I was thinking about white sharks as $N$ and whales as $M$. Clearly the first equation would be correct but the second equation would be saying that the population of whales is affected by sharks, which I think is not true. Is it?

• Welcome to Biology.SE! I'm not sure if you're asking specifically whether populations of whales are affected by sharks (it would also have to apply vice versa) or more generally what kind of relationship has to exist for this model to apply. Can you clarify? (Personally I think Ben's answer covers the 2nd very well, and I also think that would be the more useful question to ask.) – arboviral Jan 11 '18 at 8:30
• @arboviral Thanks 😊. At first I wanted to consider a model related with white sharks, but after reading Ben's answer I definitely should not consider the shark-whale example. Now I just need to find something that can be represented by the model. Do you know of any example so I can consider it? – user441848 Jan 11 '18 at 22:07
• Related question on logistic competition models (that I've answered before), that allows for asymmetrical competition: Stochastic parameters in population growth equations – fileunderwater Jan 19 '18 at 12:52

• the fact that the competition is symmetric (this happens because you are using the sum of the population, $M+N$, and not using a competitive asymmetry parameter, e.g. $M+\alpha_{MN} N$) means that this would be most suited to modeling the competition between species that were very similar to each other (e.g. the same size, resource-gathering ability, etc.)