Although we have tried to determine consensus views in the literature, the transdisciplinary nature of this field makes the knowledge landscape rugged and fluid. Although we will not provide technical details, readers will be able to design their models and conduct analyses based on what is provided here to suit their own needs. Most of the models we cite have been elucidated in great detail elsewhere and can be implemented in any computational language. We keep the mathematical details to a minimum and provide intuitive explanation of their meaning and rationale we also discuss, using simple terminology wherever possible, key procedures that lead to deductive conclusions. To embrace the complexity of ecological networks we need to introduce a few simple mathematical models and associated concepts that are fundamental to network analyses, visualisation and the ideas we develop. We start unpacking these issues here and will dive deeper into each in subsequent chapters. We partition related issues into six topics (network interactions, structures, stability, dynamics, scaling and invasibility). This book deals with the roles and impacts of the entangled web of biotic interactions that an alien species partakes in as it infiltrates ecological networks.
Whereas local stability is favored only by specific types of alternative food, persistence of prey and predators is promoted by a much wider range of food types. These dynamics can well be understood from the occurrence of an abrupt (or at least steep) change in the prey isocline. Yet, even if switching to alternative food does not stabilize the equilibrium, it may prevent unbounded oscillations and thus promote persistence. This range is notably narrow in a fine-grained environment. For suboptimal predators, a more gradual change will occur, resulting in stable equilibria for a limited range of alternative food types. Analyzing the population dynamical consequences of such stepwise switches, we found that equilibria will not be stable at all. A well-known result from optimal foraging theory is that when prey density drops below a threshold density, optimally foraging predators will switch to alternative food, either by including the alternative food in their diet (in a fine-grained environment) or by moving to the alternative food source (in a coarse-grained environment). We studied a model based on the simplifying assumption that the alternative food source has a fixed density. A more complete understanding of the effect of predator switching would therefore require the analysis of one-predator/two-prey models, but these are difficult to analyze.
This poses the question of under what conditions such switching-mediated stability is likely to occur. It has long been known that sigmoid (Holling's Type III) functional responses may stabilize an otherwise unstable equilibrium of prey and predators in Lotka-Volterra models. Sigmoid functional responses may arise from a variety of mechanisms, one of which is switching to alternative food sources.
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