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.

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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

Persistence of multispecies host-parasitoid interactions in spatially distributed models with local dispersal.

The growing importance of pathogens in biological control warrants an investigation into the likely effects of disease on the widely successful class of biological-control agents, the parasitoid. Our models consider a parasitoid that can exhibit a range of attack behaviors ranging from clumped to random. The pathogen is assumed to be transmitted by external stages that can span periods when the host is rare or absent. When these two natural enemies occur simultaneously in a given host, the result can be the elimination of one enemy by the competitor in all coinfected hosts or it can be some intermediate outcome. We show through an analysis of invasion that both natural enemies may coexist with constant, cyclic, or chaotic populations. Other outcomes are also possible. One enemy may exclude its competitor either unilaterally or on the basis of the initial densities of the host and the competing natural enemies. The particular outcome depends on four main factors: (1) the finite rate of increase of the host...

https://www.jstor.org/stable/info/2462139

The dynamics of host-parasitoid-pathogen interactions

This paper is concerned with the dynamical effects of spatial heterogeneity in host-parasitoid interactions with discrete generations. We show that the dynamical effects of any pattern of distribution of searching parasitoids in such systems can be assessed within a common, simple framework. In particular, we describe an approximate general rule that the populations of hosts and parasitoids will be regulated if the coefficient of variation squared (CV 2 ) of the distribution of searching parasitoids is greater than one. This criterion is shown to apply both generally and in several specific cases. We further show that CV 2 may be partitioned into a density-dependent component (direct or inverse) caused by the response of parasitoids to host density per patch, and a density independent component. Population regulation can be enhanced as much by density independent as by density-dependent heterogeneity. Thus the dynamical effects of any pattern of distribution of searching parasitoids can be assessed within the same common framework. The paradoxical impact of density-independent heterogeneity on dynamics is especially interesting: the greater the density independence, and thus the more scattered the data of percent parasitism against local host density, the more stable the populations are likely to be. Although a detailed analysis of host-parasitoid interactions in continuous time has yet to be done, evidence does not support the suggestion of Murdoch & Oaten (1989) that non-random parasitism may have quite different effects on the dynamics of continuous-time interactions. There appears to be no fundamental difference in the role of heterogeneity in comparable discrete- or continuous-time interactions. A total of 65 data sets from field studies have been analysed, in which percentage parasitism in relation to local host density have been recorded. In each case, estimated values of have been obtained by using a maximum likelihood procedure. The method also allows us to partition the CV 2 into the density dependent and density-independent components mentioned above. In 18 out of the 65 cases, total heterogeneity was at levels sufficient (if typical of the interactions) to stabilize the interacting populations (i.e. CV 2 > 1). Interestingly, in 14 of these it is the host-density-independent heterogeneity that contributes most to the total heterogeneity. Although heterogeneity has often been regarded as a complicating factor in population dynamics that rapidly leads to analytical intractability, this clearly need not necessarily be so. The CV 2 > 1 rule explains the consequences of heterogeneity for population dynamics in terms of a simple description of the heterogeneity itself, and provides a rough rule for predicting the effects of different kinds of heterogeneity on population regulation.

Michael P. Hassell

Papers

Persistence of multispecies host-parasitoid interactions in spatially distributed models with local dispersal.

The dynamics of host-parasitoid-pathogen interactions

Heterogeneity and the dynamics of host—parasitoid interactions

Metapopulations and equilibrium stability: The effects of spatial structure

The Dynamics of Arthropod Predator-Prey Systems