# Better population models than the logistic equation [closed]

For modelling the population of a single species in a region, the logistic equation is used

$$\frac{dN}{dt} =kN\Big(1-\frac{N}{K}\Big)$$

where $N$ is the number of organisms, $K$ is the carrying capacity, and $k$ is a constant.

However, for most real primary consumers, in the absence of predation, the population usually deviates from the standard sigmoidal curve in two ways

1. The population typically overshoots the carrying capacity before decaying again to the carrying capacity
2. The population usually oscillates around the carrying capacity, rather then remaining constant

Are there any better mathematical models for describing growth mediated by density dependent factors of primary consumers in the absence of predation?

• You should explain what exactly do you mean by better Jun 9 '15 at 18:57
• I agree with @WYSIWYG - this is opinion-based at the moment. Are you asking for models that can accomodate for your points 1 and 2? What other types of models have you looked at? Jun 9 '15 at 19:04
• I don't see how using the world "better" is opinion based. The point of models is to accurately simulate real systems. By "better" I mean more accurate at producing the experimentally observed population versus time graphs, which display the points I mentioned above. Jun 9 '15 at 19:12
• The thing is that many types of models can produce the type of dynamics that you describe. Whether they are suitable for a particular species depends on the biology of this species though, and you haven't included any information about the organism you want to model (prey/predator? generation length? discrete or continuous reproduction? etc etc). And whether something is "better" is rather opinion based, while "What types of models can accomodate for points 1 and 2 in the absence of predation, and describe a population with discrete reproductive events?" can be answered more easily . Jun 9 '15 at 19:23
• I think "better" is pretty well defined in the question, and the edits presumably address the above concerns. Voted to reopen. Jun 10 '15 at 23:25

The behaviour that you describe is totally possible with a discrete time logistic model. In short, consider the recursion equation $n_{t+1} = f(n_t)$. Consider also $\hat n$ as being an equilibrium of your system (typically n=0 or n=k, where k is the carrying capacity). In other words, $\hat n$ satisfies the equation $\hat n = f(\hat n)$. And finally consider the statistics $\lambda$ which is equal to the derivative of $f(n)$ over $n$ evaluated at the equilibrium $\hat n$. If $\lambda < 0$, then you have oscillatory behaviour (and therefore overshoot) at the vicinity of the equilibrium $\hat n$. Using the correct parameters, the logistics equation gives you a lambda smaller than 0 (typically between $-1 > \lambda > 0$), and therefore you can have the oscillatory behaviour you describe.