Its system of ode:
$$\left\{ \begin{array}{l} x' = x(1 -\frac{x}{k}) - \frac{dxy}{1+x} \\ y' = -my + \frac{dxy}{1+x} \end{array} \right.$$
I found 3 equilibriums:
- $(0,0)$ (saddle)
- $(k,0)$ (saddle) and
- $\left(\frac{m}{d-m}, \frac{m(kd - mk - m)}{(d-m)^2 km }\right)$, if $\frac{m}{d-m} < \frac{k-1}{2}$ is stable, is not stable in opposite situation.
How can i show that if $\frac{m}{d-m} < \frac{k-1}{2}$, the periodic solution exists?