1.1 The Malthusian Model 1.2 Nonlinear Models 1.3 Analyzing Nonlinear Models 1.4 Variations on the Logistic Model 1.5 Comments on Discrete and Continuous Models Dynamic Modeling with Difference Equations Xiong You Department of Applied Mathematics Nanjing Agricultural University Feb. 2009 X. YouEND/START 1.1 The Malthusian Model 1.2 Nonlinear Models 1.3 Analyzing Nonlinear Models 1.4 Variations on the Logistic Model 1.5 Comments on Discrete and Continuous Models Topics in this chapter 1 1.1 The Malthusian Model 2 1.2 Nonlinear Models 3 1.3 Analyzing Nonlinear Models 4 1.4 Variations on the Logistic Model 5 1.5 Comments on Discrete and Continuous Models X. YouEND/START 1.1 The Malthusian Model 1.2 Nonlinear Models 1.3 Analyzing Nonlinear Models 1.4 Variations on the Logistic Model 1.5 Comments on Discrete and Continuous Models The Malthusian Model Biological phenomena to investigate: growth and interactions of an entire population evolution of DNA sequences inheritance of traits spread of disease, etc. Biological systems are marked by change and adaptation Diffi culty: A large number of interactions and competing tendencies can make it diffi cult to see the full picture at once X. YouEND/START 1.1 The Malthusian Model 1.2 Nonlinear Models 1.3 Analyzing Nonlinear Models 1.4 Variations on the Logistic Model 1.5 Comments on Discrete and Continuous Models The Malthusian Model Biological phenomena to investigate: growth and interactions of an entire population evolution of DNA sequences inheritance of traits spread of disease, etc. Biological systems are marked by change and adaptation Diffi culty: A large number of interactions and competing tendencies can make it diffi cult to see the full picture at once X. YouEND/START 1.1 The Malthusian Model 1.2 Nonlinear Models 1.3 Analyzing Nonlinear Models 1.4 Variations on the Logistic Model 1.5 Comments on Discrete and Continuous Models The Malthusian Model Biological phenomena to investigate: growth and interactions of an entire population evolution of DNA sequences inheritance of traits spread of disease, etc. Biological systems are marked by change and adaptation Diffi culty: A large number of interactions and competing tendencies can make it diffi cult to see the full picture at once X. YouEND/START 1.1 The Malthusian Model 1.2 Nonlinear Models 1.3 Analyzing Nonlinear Models 1.4 Variations on the Logistic Model 1.5 Comments on Discrete and Continuous Models The Malthusian Model Biological phenomena to investigate: growth and interactions of an entire population evolution of DNA sequences inheritance of traits spread of disease, etc. Biological systems are marked by change and adaptation Diffi culty: A large number of interactions and competing tendencies can make it diffi cult to see the full picture at once X. YouEND/START 1.1 The Malthusian Model 1.2 Nonlinear Models 1.3 Analyzing Nonlinear Models 1.4 Variations on the Logistic Model 1.5 Comments on Discrete and Continuous Models The Malthusian Model Biological phenomena to investigate: growth and interactions of an entire population evolution of DNA sequences inheritance of traits spread of disease, etc. Biological systems are marked by change and adaptation Diffi culty: A large number of interactions and competing tendencies can make it diffi cult to see the full picture at once X. YouEND/START 1.1 The Malthusian Model 1.2 Nonlinear Models 1.3 Analyzing Nonlinear Models 1.4 Variations on the Logistic Model 1.5 Comments on Discrete and Continuous Models The Malthusian Model Biological phenomena to investigate: growth and interactions of an entire population evolution of DNA sequences inheritance of traits spread of disease, etc. Biological systems are marked by change and adaptation Diffi culty: A large number of interactions and competing tendencies can make it diffi cult to see the full picture at once X. YouEND/START 1.1 The Malthusian Model 1.2 Nonlinear Models 1.3 Analyzing Nonlinear Models 1.4 Variations on the Logistic Model 1.5 Comments on Discrete and Continuous Models Mathematical language is designed for precise description Describing complicated systems often requires a mathematical model X. YouEND/START 1.1 The Malthusian Model 1.2 Nonlinear Models 1.3 Analyzing Nonlinear Models 1.4 Variations on the Logistic Model 1.5 Comments on Discrete and Continuous Models Mathematical language is designed for precise description Describing complicated systems often requires a mathematical model X. YouEND/START 1.1 The Malthusian Model 1.2 Nonlinear Models 1.3 Analyzing Nonlinear Models 1.4 Variations on the Logistic Model 1.5 Comments on Discrete and Continuous Models The Malthusian Model Focus on: Modeling the way populations grow or decline over time Questions to drive mathematical models: Why do populations sometimes grow and sometimes decline? Must populations grow to such a point that they are unsustainably large and then die out? If not, must a population reach some equilibrium? If an equilibrium exists,