Biological systems are often modeled as a system of ordinary differential equations (ODEs) with time-invariant parameters. However, cell signaling events or pharmacological interventions may alter the cellular state and induce multi-mode dynamics of the system. Such systems are naturally modeled as hybrid automata, which possess multiple operational modes with specific nonlinear dynamics in each mode. In this paper we introduce a model checking-enabled framework than can model and analyze both single- and multi-mode biological systems. We tackle the central problem in systems biology--identify parameter values such that a model satisfies desired behaviors--using bounded model checking. We resort to the delta-decision procedures to solve satisfiability modulo theories (SMT) problems and sidestep undecidability of reachability problems. Our framework enables several analysis tasks including model calibration and falsification, therapeutic strategy identification, and Lyapunov stability analysis. We demonstrate the applicablitliy of these methods using case studies of prostate cancer progression, cardiac cell action potential and radiation diseases.