Figure 1

Numerical simulation of equations (3a)-(3d). The solution trajectory approaches oscillatory solution \((0,\tilde {z}(t))\) as time passes. Here, all parameters are chosen to satisfy the conditions in Theorem 4.1, i.e., \({c_{1}}=0.15\), \({c_{2}}=0.8\), \({c_{3}}=0.35\), \({c_{4}}=0.9\), \({d_{1}}=0.9\), \({d_{2}}=0.5\), \({d_{3}}=0.1\), \({k_{1}}=1.1\), \({k_{2}}=0.9\), \({k_{3}}=0.9\), \({k_{4}}=3.9\), \(\mu=0.8\), \(\rho =0.5\), \(T=20\), \(y(0)=0.13\), and \(z(0)=5\). (a) The solution trajectory projected on \((y,z)\)-plane. (b) The corresponding time course of the number of active osteoclasts \((y)\) tending towards zero. (c) The corresponding time course of the number of active osteoblasts \((z)\) exhibiting positive oscillation.