Why solutions of Lotka-Volterra model are cycles

Question

I need to analyze Lotka-Volterra model, I have everything except the fact that curves around $(\frac{a}{b},\frac{d}{c})$ are closed, I found one proff in internet, but i don't understand it. $$\frac{dN}{dt}=N(a-b\cdot P)$$ $$\frac{dP}{dt}=P(c\cdot N-d)$$ I will be very glad for as simple as possible explanation.

Answer 1

Compute $$ \frac{\dot N}{N}(cN−d) - \frac{\dot P}{P}(a−bP) =(a−bP)(cN−d)-(cN−d)(a−bP)=0 $$ As this expression is of the form $f'(N)\dot N+g'(P)\dot P=0$, one can directly integrate to obtain $f(N)+g(P)=C$, $$ cN-d\ln|N|-a\ln|P|+bP=C. $$

As this is a first integral of the differential equation, all solution curves follow the level sets of this function. As the level sets are compact, the solution curves are closed.