Mathematical modeling of biological systems using differential equations
*Satish V. KakadeCorresponding authorsatishvkakade@yahoo.co.inDepartment of Community MedicineKrishna Institute of Medical SciencesKrishna Vishwa Vidyapeeth (Deemed to be University)Karad, Maharashtra, 415539, IndiaView full profile → , Rahul Mishrarahul.mishra@rungta.orgDepartment of Computer Science and EngineeringRungta College of Engineering & TechnologyRungta International Skills UniversityBhilai, Chhattisgarh, 490024, IndiaView full profile → , Swapnil Chaudharyswapnilchaudhari68@gmail.comDepartment of MathematicsIndira College of Engineering and ManagementPune, Maharashtra, 410506, IndiaView full profile → , H. Savithrisavithri-maths@dayanandasagar.eduDepartment of MathematicsDayananda Sagar College of EngineeringBangalore, Karnataka, 560111, IndiaView full profile → , Madhur Tanejamadhur.taneja.orp@chitkara.edu.inCentre for Research Impact and Outcome (CRIO)Chitkara UniversityRajpura, Punjab, 140401, IndiaView full profile → , Makineedi Raja Baburajababu.makineedi@adityauniversity.inDepartment of Information TechnologyAditya UniversitySurampalem, Andhra Pradesh, 533437, IndiaView full profile → , Vinod Kumar Naidu Pamulurivknpamuluri@outlook.comDepartment of Mechanical EngineeringPragati Engineering CollegeSurampalem, Andhra Pradesh, 533437, IndiaView full profile →
* Corresponding author · click or hover a name for details
- Received:
- 01 Apr 2026
- Published Online:
- 27 Oct 2026
- Article type:
- Research Article
- Language:
- EN
- Article no.:
- JIM-2836
- Pages:
- 2945–2956
Abstract
Nonlinear interactions and temporal evolution in a biological system can be mathematically modeled by differential equations and can be analysed quantitatively. Combining biologically relevant growth, mortality, resource limitation, interaction, regulation, equilibrium, stability and numerical solution mechanisms, this study suggests an integrated framework. The novelty is to combine the three elements of nonlinear biological dynamics, Jacobian-based stability assessment, and comparative numerical evaluation, in a coherent modeling framework. The Euler, Heun, Midpoint, third-order Runge–Kutta (RK3) and fourth-order Runge–Kutta (RK4) methods were compared at a step size of 0.10. The results show that RK4 gives the highest accuracy of 98.8%, with the lowest RMSE of 0.012, and the lowest absolute error of 0.009 when compared to Euler with 91.6% accuracy and RMSE of 0.084. RK3 achieved 97.5% accuracy with 0.025 RMSE. These results confirm numerical reliability for studying nonlinear dynamics, equilibrium behavior, stability and responses of systems as a function of their parameters.
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References
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