## Elements of Mathematical EcologyElements of Mathematical Ecology provides an introduction to classical and modern mathematical models, methods, and issues in population ecology. The first part of the book is devoted to simple, unstructured population models that ignore much of the variability found in natural populations for the sake of tractability. Topics covered include density dependence, bifurcations, demographic stochasticity, time delays, population interactions (predation, competition, and mutualism), and the application of optimal control theory to the management of renewable resources. The second part of this book is devoted to structured population models, covering spatially-structured population models (with a focus on reaction-diffusion models), age-structured models, and two-sex models. Suitable for upper level students and beginning researchers in ecology, mathematical biology, and applied mathematics, the volume includes numerous clear line diagrams that clarify the mathematics, relevant problems throughout the text that aid understanding, and supplementary mathematical and historical material that enrich the main text. |

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### Contents

SINGLESPECIES MODELS | 3 |

Harvest models bifurcations and breakpoints | 13 |

Stochastic birth and death processes | 25 |

Discretetime models | 43 |

Delay models | 70 |

Branching processes | 93 |

INTERACTING POPULATIONS | 107 |

To cycle or not to cycle | 116 |

SPATIALLY STRUCTURED MODELS | 267 |

Spatial steady states linear problems | 276 |

Spatial steady states nonlinear problems | 294 |

Models of spread | 311 |

AGESTRUCTURED MODELS | 345 |

The Lotka integral equation | 353 |

The difference equation | 365 |

The Leslie matrix | 377 |

Global bifurcations in predatorprey models | 140 |

Chemostat models | 161 |

Discretetime predatorprey models | 181 |

Competition models | 198 |

Mutualism models | 220 |

DYNAMICS OF EXPLOITED POPULATIONS | 237 |

STRUCTURED POPULATION MODELS | 265 |

The McKendrickvon Foerster PDE | 391 |

Some simple nonlinear models | 401 |

SEXSTRUCTURED MODELS | 413 |

425 | |

443 | |

447 | |

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### Common terms and phrases

asymptotically stable attractor bifurcation diagram birth and death birth rate boundary conditions capita carrying capacity characteristic equation chemostat coefficients coexist competition Consider constant corresponding curve decreases density dependence difference equation diffusion discrete-time dynamics ecology eigenvalues equilibrium point Euler–Lotka equation example exponential extinction Figure Fisher equation fishery fishing functional response growth rate harvesting homoclinic Hopf bifurcation increase individuals initial condition integral equation Jacobian Leslie matrix limit cycle linear logistic difference equation Lotka Lotka–Volterra males Mathematical negative nonlinear nontrivial equilibrium number of females oscillations parameter parasitoids partial differential equation periodic orbit phase plane Phase portrait pn(t population positive predator predator—prey models prey zero-growth isoclines probability generating function problem real roots Recommended readings reduces reproduction saddle point satisfies simple ſº solve spatial species stable age distribution stable manifold stable node substrate theorem transcritical bifurcation traveling wave unstable zero