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

In this study of arthropod predador-prey systems Michael Hassell shows how many of the components of predation may be simply modeled in order to reveal their effects on the overall dynamics of the interacting populations Arthropods, particularly insects, make ideal subjects for such a study because their generation times are characteristically short and many have relatively discrete generations, inviting the use of difference equation models to describe population changes Using analytical models framed in difference equations, Dr Hassell is able to show how the detailed biological processes of insect predator-prey (including host-parasitoid) interactions may be understood Emphasizing the development and subsequent stability analysis of general models, the author considers in detail several crucial components of predator-prey models: the prey's rate of increase as a function of density, non-random search, mutual interference, and the predator's rate of increase as a function of predator survival and fecundity Drawing on the correspondence between the models and field and laboratory data, Dr Hassell then discusses the practical implications for biological pest control and suggests how such models may help to formulate a theoretical basis for biological control practices

The dynamics of arthropod predator-prey systems.

Mutual interference between searching insect parasites provides theoretical support for current biological control practices.

New Inductive Population Model for Insect Parasites and its Bearing on Biological Control

MOST environments are spatially subdivided, or patchy, and there has been much interest in the relationship between the dynamics of populations at the local and regional (metapopulation) scales1 Here we study mathematical models for host-parasitoid interactions, where in each generation specified fractions (µN and µp, respectively) of the host and parasitoid subpopulations in each patch move to adjacent patches; in most previous work, the movement is not localized but is to any other patch2 These simple and biologically sensible models with limited diffusive dispersal exhibit a remarkable range of dynamic behaviour: the density of the host and parasitoid subpopulations in a two-dimensional array of patches may exhibit complex patterns of spiral waves or spatially chaotic variation, they may show static 'crystal lattice' patterns, or they may become extinct This range of behaviour is obtained with the local dynamics being deterministically unstable, with a constant host reproductive rate and no density dependence in the movement patterns The dynamics depend on the host reproductive rate, and on the values of the parameters µN and µp The results are relatively insensitive to the details of the interactions; we get essentially the same results from the mathematically-explicit Nicholon–Bailey model of host-parasitoid interactions, and from a very general 'cellular automaton' model in which only qualitative rules are specified We conclude that local movement in a patchy environment can help otherwise unstable host and parasitoid populations to persist together, but that the deterministically generated spatial patterns in population density can be exceedingly complex (and sometimes indistinguishable from random environmental fluctuations)

Spatial structure and chaos in insect population dynamics

where NS represents the survivors after Pt parasites have searched for Nt hosts resulting in P+ 1 parasite progenyt. All assumptions about parasite searching behaviour are here contained in the functionf[Pt,Nt]. If we consider the simplest case where the parasite population is specific and synchronized temporally with its host population, we can write the following generalized model for a host-parasite interaction:

http://www.blackwellpublishing.com/pdf/42-3-pp-693-726.pdf

Stability in insect host-parasite models

/pdf/density-dependence-in-single-species-populations-54vpd2ihmq.pdf

Michael P. Hassell

Papers

The dynamics of arthropod predator-prey systems.

New Inductive Population Model for Insect Parasites and its Bearing on Biological Control

Spatial structure and chaos in insect population dynamics

Stability in insect host-parasite models

Density-dependence in single-species populations