Given the following discrete dynamic model:
$P_0=360$ and $R_0=80$ and $t[0,200]$
$P_t=1,18*P_{t-1}-0,002*R_{t-1}*P_{t-1}$and $R_t=0,86*R_{t-1}+0,0004*P_{t-1}*R_{t-1}$
One can rewrite the difference equations as: $P_t=P_{t-1}+0,18*P_{t-1}-0,002*R_{t-1}*P_{t-1}$ and $R_t=R_{t-1}-0,14*R_{t-1}+0,0004*P_{t-1}*R_{t-1}$
Reasoning that in a stable situation $P_t=P_{t-1}$ and $R_t=R_{t-1}$ calculate $P_0$ and $R_0$ such that a stable situation is established.
Suppose $\left(P,R\right)$ with $P,R\neq0$ is a fixed point of the system. That is, if $P_t=P$ and $R_t=R$, we get $P_{t+1}=P$ and $R_{t+1}=R$ from the equations above.
Then, $$ P=1.18P-0.002RP $$ and $$ R=0.86R+0.0004PR. $$
Dividing these equations through by $P$ and $R$ respectively yields $$ 1=1.18-0.002R $$ and $$ 1=0.86+0.0004P. $$
Solving these equations yields $$ R=\frac{1.18-1}{0.002}=90 $$ and $$ P=\frac{1-0.86}{0.0004}=350. $$