Mathematical Analysis of Logistic Population Growth under Environmental Carrying Capacity
Keywords:
Mathematical Biology, Logistic Growth Model, Population Dynamics, Carrying Capacity, Stability AnalysisAbstract
Population growth models are fundamental tools in mathematical biology for understanding how biological populations change over time under environmental constraints. Among these models, the logistic growth model is realistic as it takes into account environmental carrying capacity and is independent of the classical exponential growth model which assumes unlimited resources. In this paper we have thoroughly studied the logistic population growth model with a first-order nonlinear ordinary differential equation. Starting from the biological assumptions of density-dependent population regulation, we formulate and solve the logistic equation analytically using the method of separation of variables to get its explicit solution. We calculate the equilibrium points of the system and perform local stability analysis by linearizing the governing equation to understand the long-term behavior of the population. We explore the effects of the basic parameters of the logistic growth model (the growth rate, the carrying capacity and the initial population size) on the dynamics of the population and how it would eventually be at equilibrium. We compare exponential and logistic growth models, and show that the ecological impact of resource limitation in population models is important. Moreover, we discuss the applications of the logistic model in ecology, wildlife conservation, fisheries management, microbial population dynamics, agriculture and environmental management.
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