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Considering the discrete time logistic growth model,

$$N_{t+1} = \lambda N_t\left(1-\frac{N_t}{K}\right)$$

where $\lambda = 1+ b -d >0$ is the net reproductive rate and $K>0$ is a parameter that affects how the population grows when the population is large.

I have determined the two equilibrium points of this system which are,

$$N^*=0$$

$$N^* = K\left(1-\frac{1}{\lambda}\right)$$

found the conditions for each of the equilibria that I found above to be unstable.

Here is my explanation below,

$$f'(x) = \lambda -\frac{2x\lambda}{K}$$

Using $x=0$

$$f'(0) = \lambda$$

So it is unstable when $\lambda>1$

Using $x = K\left(1-\frac{1}{\lambda}\right)$ it is unstable when $\lambda >3, 0 < \lambda <1$

I believe I have done everything correct to this point.

My question is how do I give the values of $N_0>0$, $K>0$ and $\lambda>0$ such that $N_1<0$, and how do I calculate $N_1$? Thanks!

user123
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1 Answers1

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Yes. All of your assumptions are correct. I checked it. But according to your problem if $N_0, K$ and $\lambda$ are positive, then from the first iteration $$N_1=\lambda N_0\left(1-\frac{N_0}{K}\right),$$ you always get $N_1$ as non negative since $N_0<K$ (the population is at most its carrying capacity).

But what is your actual motivation according to the problem? I think it may be positiveness of population in the model.

SAHEB PAL
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