I'm feeling stuck with this problem, It was taken from the book: Dinamical models in biology - Miklos Farkas: Consider the Lotka-Volterra system, so the equations are and the fixed points are
the next function of Lyapunov is proposed
My question is: How I get $\dot{L}_{(2.2.1)}$ I tried to solve it ( $\nabla L(N,P) \cdot f(N,P)$) but I did not get the calculations, it will be some error of the book or in the end I am
There is no error in the book. The calculations are neither exactly straightforward nor really lengthy.
Let $\hat N,\hat P$ denote the coordinate of equilibrium point (I do not like the bars). Then they satisfy the system $$ \epsilon=\frac{\epsilon}{K}\hat N+\alpha \hat P,\\ -\gamma=\delta \hat P-\beta \hat N.\tag{$\ast$} $$
Now, using $L$ as suggested in the book, $$ \dot L=(N-\hat N)(\epsilon-\frac{\epsilon}{K} N-\alpha P)+\frac{\alpha}{\beta}(P-\hat P)(-\gamma-\delta \hat P+-\beta \hat N). $$ Now replace $\epsilon$ (only first one) and $-\gamma$ with the expressions from $(\ast)$ and simplify. You'll end up with the required expression.