CS151 Fall 2012 Programming Project 3 Population Models In C

Cs151 Fall 2012programming Project 3population Modelsin Chapter 14

In this assignment, you will create a C++ program to model the logistic growth function for populations using recursion. Your program must include a recursive function to compute the population at a given generation, a function to gather user input for parameters, and output results to a tab-delimited file, including initial conditions. You will also analyze the behavior of the logistic model through multiple experiments, visualize results in Excel, and discuss the dynamics observed.

Paper For Above instruction

The logistic model, developed by Pierre Francois Verhulst, provides a more realistic depiction of population growth compared to the Fibonacci sequence by incorporating the concept of carrying capacity. The recursive formula for the logistic model is given as x_n+1 = r x_n (1 - x_n), where x_n is the population at year n scaled between 0 and 1, x₀ is the initial population, and r is the growth rate parameter. The goal of this project is to program this function in C++, analyze its behavior under different parameters, and generate outputs suitable for analysis and visualization.

The program will start by obtaining user input for the parameters r, initial population x₀, and the number of generations n. It must utilize a recursive function to compute population at each subsequent year. This function should include a termination condition: if the computed population exceeds 1, the recursion should stop, presuming the population has exceeded the environment's capacity. The recursive function will calculate and return the population for a given generation, with the base case being the initial population.

In addition, the program must include a function to gather user inputs, ensuring values are valid (for example, non-negative and within reasonable bounds). The results, including initial conditions and subsequent values, should then be written into a tab-delimited text file, facilitating easy importing into spreadsheet software like Excel for graphing and analysis.

For the analytical part, the program will be tested with various parameter combinations, and the outputs will be studied to observe different behaviors, such as convergence to equilibrium, oscillations, or chaos. By running at least ten experiments with different values of r and initial populations, students will identify how these parameters influence system dynamics, pinpoint where bifurcations occur, and interpret these behaviors in terms of nonlinear dynamics and chaos theory. Visualizations will help to see how small variations influence long-term behavior, illustrating concepts like sensitivity to initial conditions—a hallmark of chaotic systems.

The project aligns with the curriculum topics of recursion, nonlinear functions, and population modeling, emphasizing both programming competencies and analytical skills. Emphasis on commenting, code readability, and proper output formatting are essential for clarity and reproducibility. The assignment emphasizes understanding complex system behaviors through iterative simulation and encourages visualization to deepen insights into population dynamics and chaos theory.

References

• Verhulst, P. F. (1845). "Recherches mathématiques sur la loi d'accroissement de la population." Nouvelles Annales de Mathématiques.
• Fibonacci, Leonardo of Pisa (1202). Liber Abaci. The Fibonacci Numbers.
• May, R. M. (1976). "Simple mathematical models with very complicated dynamics." Nature, 261(5560), 459-467.
• Strogatz, S. H. (2015). Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering. Westview Press.
• Hastings, A., et al. (1993). "Chaos in ecology: Is it really happening?" Ecological Applications, 3(4), 547-554.
• Ott, E. (2002). Chaos in Dynamical Systems. Cambridge University Press.
• Alligood, K. T., Sauer, T. D., & Yorke, J. A. (1997). Chaos: An Introduction to Dynamical Systems. Springer-Verlag.
• Lorenz, E. N. (1963). "Deterministic nonperiodic flow." Journal of the Atmospheric Sciences, 20(2), 130-141.
• Guckenheimer, J., & Holmes, P. (1983). Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer-Verlag.