$$\frac{x_{t+1}}{x_t}=1-b\left(\frac{x_t}{y_t}-a\right)$$ $$y_{t+1}+x_t=hy_t+k$$
Here, all parameters $(b,a,h,k)$ are strictly positive. And $h>1$.
What are the conditions on parameters such that nontrivial optimal points exists.
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What I did is that
the optimal (steady state) is found when $x_{t+1}=x_t=x^*$ and $y_{t+1}=y_t=y^*$ are satisfied.
And in this way, I found that $$(x^*,y^*)=\left(\frac{ak}{a-(h-1)}, \frac{k}{a-(h-1)}\right)$$ are the optimal points which exists when $a>h-1$.
I cannot find any condition on parameters expect for this.
Please share your ideas with me and show me a way to find conditions.
The equilibria of you dynamical system can be found by setting $x_{t+1}=x_{t} = x^*$ and $y_{t+1}=y_{t} = y^*$.
The equilibrium is
$$(x^*,y^*)=\left(\frac{ak}{a-(h-1)}, \frac{k}{a-(h-1)}\right).$$
To be an equilibrium, then $(x^*,y^*) \in \mathbb{R}^2$. It is straightforward to notice that for $a = h-1$, the denominators of both $x^*$ and $y^*$ is $0$. Recall that dividing any number (in this case, $ak$ and $k$) by $0$ is not a well-posed mathematical operation. Hence, for $a=h-1$, the equilibrium $(x^*,y^*)$ is not defined (or equivalently it does not exist, or equivalently the equilibrium corresponds to "infinity").