Population dynamics solution using finite difference method is illustrated in brief here.

fdm

With the definition of the derivate of a smooth function $u$ ata point $x \in \mathbb{R}$

$$\tag{1} u' = \lim_{h \rightarrow 0} \frac{u(x + h) - u(x)}{h}, $$

a differential equation

$$\tag{2} u' = \frac{du}{dx} $$

can be solved numerically

$$\tag{3} u(x + h) = u(x) + h u'(x) + O(h^2) $$

using finite difference method (FDM) with the error of the approximation is proportional to $h^2$ ¹.

num-eqns

Using Eqns (3) and (26b52.5) following equation can be obtained

$$\tag{4} N(t + dt) = N(t) + r N(t) \left( 1 - \frac{N(t)}{K} \right) \Delta t, $$

where $r$ and $K$ stand for not growth rate and carrying capacity, respectively.

result

Result from Eqn (4) is as follows.

Figure 1. Population dynamics for two different set of parameters.

Results in Figure 1 are obrtained using $\Delta t = 1$, $t_{\rm beg} = 0$, and $t_{\rm end} = 120$. For Data 1 other parameters are $N_0 = 1000$, $K = 4500$, and $r = 0.1$, while for Data 2 the parameters are $N_0 = 8000$, $K = 3000$, and $r = 0.05$.

It shows that for Data 1 the growth is positive until it reaching carrying capacity $K$, while for Data 2 the growth is negative until it reaching carrying capacity $K$. Oscillation near the value of carrying capacity is not observed.

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