Mathematical Modeling of the Dynamics of Renewable Resources Used by the Population

In most situations where entities interact by sharing limited resources, controlling population density is crucial for maintaining ecosystem sustainability. Mathematical models have been widely used to explain systems arising from the interaction between populations and nature, particularly regarding resource utilization. The objective of our work is to develop a mathematical model that incorporates the effort of resource renewal and harvesting by populations, represented through a system of nonlinear differential equations. The mathematical analysis of the model developed in our study involved demonstrating that the model is well posed, admitting a unique solution within a defined domain, and identifying the equilibrium points and their stability in the nonlinear system. Numerical simulations were conducted to understand the dynamics of populations, resource utilization, and renewal by the population. The numerical simulation of global stability was also examined. Sensitivity analysis of the parameters was performed, revealing that the viability of the system depends on resource renewal. Specifically, the analysis identified three types of parameter variations: the first category influences the dynamics of natural resources in the form of sinusoidal and periodic changes, which may be continuous or discontinuous. The second category does not influence the dynamics. The third category includes parameters that affect the dynamics in the short term but not in the long term.