  Let $z=x+iy \in \mathbb{H}:=\{z= x+ i y\in\mathbb{C}: y>0\}$ and $ \theta
(s;z)=\sum_{(m,n)\in\mathbb{Z}^2 } e^{-s \frac{\pi }{y }|mz+n|^2}$ be the theta
function associated with the lattice $\Lambda ={\mathbb Z}\oplus z{\mathbb Z}$.
In this paper we consider the following pair of minimization problems $$ \min_{
\mathbb{H} } \theta (2;\frac{z+1}{2})+\rho\theta (1;z),\;\;\rho\in[0,\infty),$$
$$ \min_{ \mathbb{H} } \theta (1; \frac{z+1}{2})+\rho\theta (2;
z),\;\;\rho\in[0,\infty),$$
  where the parameter $\rho\in[0,\infty)$ represents the competition of two
intertwining lattices. We find that as $\rho$ varies the optimal lattices admit
a novel pattern: they move from rectangular (the ratio of long and short side
changes from $\sqrt3$ to 1), square, rhombus (the angle changes from $\pi/2$ to
$\pi/3$) to hexagonal; furthermore, there exists a closed interval of $\rho$
such that the optimal lattices is always square lattice. This is in sharp
contrast to optimal lattice shapes for single theta function ($\rho=\infty$
case), for which the hexagonal lattice prevails. As a consequence, we give a
partial answer to optimal lattice arrangements of vortices in competing systems
of Bose-Einstein condensates as conjectured (and numerically and experimentally
verified) by Mueller-Ho \cite{Mue2002}.
