In this paper, the authors consider the following problem for a single population with a total of \( n\geq 2 \) patches: \[ \frac{du_i(t)}{dt}=\sum\limits_{j=1}^n[d_{ij}u_j(t)-d_{ji}u_i(t)]+\mu u_i(t)[r_i-u_i(t-\tau)],\quad t>0,\tag{1} \] where \(u_i\) denotes the population density in a patch \(i\), \(1\leq i\leq n\); \(r_i\geq 0\) is a constant, and at least one \(r_j>0\), \(1\leq j\leq n\); \(d_{ij}\geq 0\) denotes the constant dispersal rate of individuals of patch \(j\) to patch \(i\) and \(d_{ii}=0;\) \(\tau\geq 0\) is the time delay and is assumed to be constant, and \(\mu\) is a constant.
The authors study special cases of system (1) when \(n=2\), and for each case determine the existence of a positive equilibrium. They also use the Hopf bifurcation theorem to obtain the existence of periodic orbits. Furthermore, they use the center manifold theorem and the normal form method to obtain results about the stability and bifurcation direction of periodic orbits. In addition, they illustrate their results with numerical examples.