Oxygen is a crucial element for cellular respiration. Based on oxygen concentration, tumor regions can be categorized as normoxic, hypoxic, and necrotic. Hypoxic tumor cells switch their metabolism from aerobic glycolysis to anaerobic glycolysis. As a result, lactate is produced in hypoxic regions and is used as an alternative metabolic fuel by normoxic tumor cells. The consumption of lactate and oxygen by tumor cells does not follow a linear pattern. Scientific studies suggest that oxygen consumption and lactate production are non-linear phenomena. In this study, we propose a two-dimensional mathematical model to investigate lactate dynamics in avascular tumors with various initial shapes, such as circular, elliptical, and petal, and to explore its growth patterns in the context of non-linear interactions between oxygen and lactate. In certain human tumors, particularly in kidney, skin, and liver, multiple tumors may emerge within a tissue domain simultaneously. We also examine how the growth patterns of multiple tumors evolve within a shared domain. Cyclic hypoxia, commonly observed in solid tumors, results from oxygen fluctuations over time at the tumor site. Additionally, we analyze lactate dynamics and tumor growth patterns in environments with cyclic hypoxia. In order to simulate the proposed model, we use finite element based COMSOL Multiphysics 6.0 interface. The simulated results show excellent agreement with experimental data. Our findings reveal that the initial tumor shape significantly influences the lactate distribution and the tumor's internal structure. Furthermore, the simulations indicate that multiple tumors eventually merge into a single tumor. We also observe that cyclic hypoxia with short periodicity increases tumor volume.