First-order difference equations arise in many contexts in the biological, economic and social sciences. Such equations, even though simple and deterministic, can exhibit a surprising array of dynamical behaviour, from stable points, to a bifurcating hierarchy of stable cycles, to apparently random fluctuations. There are consequently many fascinating problems, some concerned with delicate mathematical aspects of the fine structure of the trajectories, and some concerned with the practical implications and applications. This is an interpretive review of them.

Simple mathematical models with very complicated dynamics

Interactions between organisms are a major determinant of the distribution and abundance of species. Ecology textbooks (e.g., Ricklefs 1984, Krebs 1985, Begon et al. 1990) summarise these important interactions as intra- and interspecific competition for abiotic and biotic resources, predation, parasitism and mutualism. Conspicuously lacking from the list of key processes in most text books is the role that many organisms play in the creation, modification and maintenance of habitats. These activities do not involve direct trophic interactions between species, but they are nevertheless important and common. The ecological literature is rich in examples of habitat modification by organisms, some of which have been extensively studied (e.g. Thayer 1979, Naiman et al. 1988).

Organisms as ecosystem engineers

Cooperation is needed for evolution to construct new levels of organization. Genomes, cells, multicellular organisms, social insects, and human society are all based on cooperation. Cooperation means that selfish replicators forgo some of their reproductive potential to help one another. But natural selection implies competition and therefore opposes cooperation unless a specific mechanism is at work. Here I discuss five mechanisms for the evolution of cooperation: kin selection, direct reciprocity, indirect reciprocity, network reciprocity, and group selection. For each mechanism, a simple rule is derived that specifies whether natural selection can lead to cooperation.

/pdf/five-rules-for-the-evolution-of-cooperation-2bpocasc6i.pdf

Five Rules for the Evolution of Cooperation

http://osenberglab.ecology.uga.edu/wp-content/uploads/2015/09/Charnov-1976-TPB.pdf

Optimal foraging, the marginal value theorem.

The metacommunity concept is an important way to think about linkages between different spatial scales in ecology. Here we review current understanding about this concept. We first investigate issues related to its definition as a set of local communities that are linked by dispersal of multiple potentially interacting species. We then identify four paradigms for metacommunities: the patch-dynamic view, the species-sorting view, the mass effects view and the neutral view, that each emphasizes different processes of potential importance in metacommunities. These have somewhat distinct intellectual histories and we discuss elements related to their potential future synthesis. We then use this framework to discuss why the concept is useful in modifying existing ecological thinking and illustrate this with a number of both theoretical and empirical examples. As ecologists strive to understand increasingly complex mechanisms and strive to work across multiple scales of spatio-temporal organization, concepts like the metacommunity can provide important insights that frequently contrast with those that would be obtained with more conventional approaches based on local communities alone.

/pdf/the-metacommunity-concept-a-framework-for-multi-scale-4dcjhfvx5w.pdf

The metacommunity concept: a framework for multi-scale community ecology

Insect natural enemies as regulating factors

SUMMARY (1) Insect populations frequently suffer some density dependent effect in addition to mortality from insect parasitoids. This paper follows Wang & Gutierrez (1981) in examining the question: 'Does the ordering of density dependence and parasitism in the host life cycle have a significant effect on the dynamics of the interaction?' (2) The properties of three models in which density dependence either acts before or after parasitism are displayed in terms of the hosts' rate of increase and the extent to which parasitism depresses the host equilibrium below its carrying capacity. (3) From this analysis, we conclude that the sequence of these events in the host life cycle can indeed have a marked affect on the population dynamics. In particular, the model with parasitism acting before density dependence stands apart in permitting a stable host-parasitoid interaction with the host population above its parasite-free carrying capacity. (4) These conclusions have implications for the use of parasitoids as biological control agents.

Density dependence in host-parasitoid models

STUDIES of insect host–parasitoid interactions have contributed much to the consensus that spatial patchiness is important in the regulation of natural populations1–5. A variety of theoretical models predict that host and parasitoid populations, although unstable in the absence of environmental heterogeneity, may persist at roughly steady overall densities in a patchy environment owing to variation in levels of parasitism from patch to patch. Observed patterns of parasitism, however, have a variety of forms (with variation in attack rates among patches depending directly or indirectly on host density, or showing variation uncorrelated with host density). There is some confusion about the dynamical consequences of these different forms6,7. Here we first show how the dynamical effects of all these forms of environmental heterogeneity can be assessed by a common criterion. This 'CV2 > 1 rule' states that the overall population densities will remain roughly steady from generation to generation if the coefficient of variation squared (CV2) of the density of searching parasitoids in the vicinity of each host exceeds approximately unity. By partitioning CV2 into components, we show that both direct and inverse patterns of dependence on host density, and density-independent patterns, all contribute to population regulation in the same way. Second, we show how a maximum-likelihood method can be applied to the kind of field data that are usually available (that is, percentage parasitism versus local host density) to estimate the components of CV2. This analysis indicates that heterogeneity is large enough to stabilize dynamics in 9 of 34 published studies, and that density-independent heterogeneity is the main factor in most cases.

Host–parasitoid associations in patchy environments

This review begins with a description of the parasitoid life-style and the ecological and evolutionary factors which generate the remarkable diversity of insect parasitoids. We then describe the various ways that parasitoids have been used in the biological control of insect pests, and survey their success to date. The use of parasitoids remains largely an art, aided by past experience of success and failure. A more fundamental approach, involving basic research and theory, has not as yet contributed significantly to practical biological control. We explore the potential for such a science of parasitoid use and review basic research on parasitoid ecology and evolution which is of particular relevance to biological control. Mathematical models are used to identify and examine those parasitoid and host attributes which lead to successful biological control. Factors such as parasitoid foraging behaviour, fecundity, larval survival and sex ratio are shown to be important in influencing the depression of host populations and/or the stability of host–parasitoid interactions after depression. Multiple release is discussed and a model for inundative release of parasitoids is explored.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S003118200005366X

Parasitoids as biological control agents-a fundamental approach

SUMMARY (1) The dynamics of a predator-prey, or parasitoid-host, interaction are considered where the predator or parasitoid is a generalist whose population is buffered against changes in the particular prey being considered. (2) The interaction is then broadened to include, in addition, a specialist natural enemy, and three questions are examined within this framework. (i) Under what conditions can a specialist 'invade' and persist in an existing generalist-prey interaction? (ii) How does the addition of the specialist natural enemy alter the prey's population dynamics? (c) How does the relative timing of specialist and generalist in the prey's life cycle affect the dynamics of the interaction? (3) The following conclusions emerge. (i) A specialist can invade and co-exist more easily if acting before the generalists in the prey's life cycle. (ii) A three-species stable system can readily exist where the prey-generalist interaction alone would be unstable or have no equilibrium at all. (iii) In some cases the establishment of a specialist leads to higher prey populations than existed previously with only the generalist acting. (iv) In some cases, a variety of alternative stable states are possible, either alternating between two-species and three-species states, or between different three-species states.

http://max2.ese.u-psud.fr/epc/conservation/PDFs/HIPE/Hassell1986.pdf

Michael P. Hassell

Papers

Insect natural enemies as regulating factors

Density dependence in host-parasitoid models

Host–parasitoid associations in patchy environments

Parasitoids as biological control agents-a fundamental approach

Generalist and specialist natural enemies in insect predator-prey interactions