## Mathematics in Population BiologyThe formulation, analysis, and re-evaluation of mathematical models in population biology has become a valuable source of insight to mathematicians and biologists alike. This book presents an overview and selected sample of these results and ideas, organized by biological theme rather than mathematical concept, with an emphasis on helping the reader develop appropriate modeling skills through use of well-chosen and varied examples. Part I starts with unstructured single species population models, particularly in the framework of continuous time models, then adding the most rudimentary stage structure with variable stage duration. The theme of stage structure in an age-dependent context is developed in Part II, covering demographic concepts, such as life expectation and variance of life length, and their dynamic consequences. In Part III, the author considers the dynamic interplay of host and parasite populations, i.e., the epidemics and endemics of infectious diseases. The theme of stage structure continues here in the analysis of different stages of infection and of age-structure that is instrumental in optimizing vaccination strategies. Each section concludes with exercises, some with solutions, and suggestions for further study. The level of mathematics is relatively modest; a "toolbox" provides a summary of required results in differential equations, integration, and integral equations. In addition, a selection of Maple worksheets is provided. The book provides an authoritative tour through a dazzling ensemble of topics and is both an ideal introduction to the subject and reference for researchers. |

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### Contents

Some General Remarks on Mathematical Modeling | 1 |

Birth Death and Migration | 7 |

Unconstrained Population Growth for Single Species | 13 |

Von Bertalanffy Growth of Body Size | 33 |

Sigmoid Growth | 51 |

The Allee Effect | 65 |

Asymptotic Equality | 75 |

DiscreteTime SingleSpecies Models | 81 |

Some Nonlinear Demographics | 273 |

Background | 283 |

The Simplified KermackMcKendrick Epidemic Model | 293 |

Generalization of the MassAction Law of Infection | 305 |

The KermackMcKendrick Epidemic Model with | 311 |

SEIR S Type Endemic Models for Childhood Diseases | 317 |

AgeStructured Models for Endemic Diseases | 341 |

Endemic Models with Multiple Groups or Populations | 383 |

Dynamics of an Aquatic Population Interacting with | 107 |

Population Growth Under Basic Stage Structure | 151 |

The Transition Through a Stage | 185 |

Stage Dynamics with Given Input | 211 |

Demographics in an Unlimiting Constant Environment | 239 |

Some Demographic Lessons from Balanced Exponential Growth | 255 |

Appendix A Ordinary Differential Equations | 421 |

Appendix B Integration Integral Equations and Some Convex Analysis | 453 |

Some MAPLE Worksheets with Comments for Part 1 | 493 |

519 | |